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I CAT 



AIRCRAFT CONFLICT ANALYSIS AND 

REAL-TIME CONFLICT PROBING USING 

PROBABILISTIC TRAJECTORY MODELING 

Lee C. Yang and James K. Kuchar 



Report No. ICAT-2000-2 
May 2000 



MIT International Center for Air Transportation 

Department of Aeronautics & Astronautics 

Massachusetts Institute of Technoiogy 

Cambridge, MA 02139 USA 



AIRCRAFT CONFLICT ANALYSIS AND REAL-TIME CONFLICT PROBING 
USING PROBABILISTIC TRAJECTORY MODELING 



by 

Lee C. Yang 

James K. Kuchar 



Department of Aeronautics and Astronautics 

Massachusetts Institute of Technology 

Cambridge, MA 02139 USA 

May, 2000 

ICAT-2000-2 



AIRCRAFT CONFLICT ANALYSIS AND REAL-TIME CONFLICT PROBING 
USING PROBABILISTIC TRAJECTORY MODELING 

by 

Lee C. Yang 

James K. Kuchar 

ABSTRACT 

Methods for maintaining separation between aircraft in the current airspace 
system have been built from a foundation of structured routes and evolved procedures. 
However, as the airspace becomes more congested and the chance of failures or 
operational error become more problematic, automated conflict alerting systems have 
been proposed to help provide decision support and to serve as traffic monitoring aids. 

The problem of conflict detection and resolution has been tackled from a number 
of different ways, but in this thesis, it is recast as a problem of prediction in the presence 
of uncertainties. Much of the focus is concentrated on the errors and uncertainties from 
the working trajectory model used to estimate future aircraft positions. The more 
accurate the prediction, the more likely an ideal (no false alarms, no missed detections) 
alerting system can be designed. 

Additional insights into the problem were brought forth by a review of current 
operational and developmental approaches found in the literature. An iterative, trial and 
error approach to threshold design was identified. When examined from a probabilistic 
perspective, the threshold parameters were found to be a surrogate to probabilistic 
performance measures. To overcome the limitations in the current iterative design 
method, a new direct approach is presented where the performance measures are directly 
computed and used to perform the alerting decisions. 

The methodology is shown to handle complex encounter situations (3-D, multi- 
aircraft, multi-intent, with uncertainties) with relative ease. Utilizing a Monte Carlo 
approach, a method was devised to perform the probabilistic computations in near real- 
time. Not only does this greatly increase the method's potential as an analytical tool, but 
it also opens up the possibility for use as a real-time conflict alerting probe. A prototype 
alerting logic was developed and has been utilized in several NASA Ames Research 
Center experimental studies. 



This document is based on the thesis of Lee C. Yang submitted to the Department of 
Aeronautics and Astronautics at the Massachusetts Institute of Technology in partial 
fulfillment of the requirements for the Degree of Doctor of Philosophy. 



Acknowledgments 



This research was supported by the NASA Ames Research Center under grant 
NAG-2-1 111. The authors are especially appreciative of Paddy Cashion, Sandy Lozito, 
Walt Johnson, Diane Carpenter, Paul Soukup, Laura Colletti, Barry Sullivan, Joe King, 
Greg Pisanich, Dominic Wong, Tony Wang, and Kevin Corker at NASA Ames Research 
Center for their support and input during the development and implementation of the 
conflict alerting logic. 

The research documented in this report would not have been possible without the 
help and input from Professor R. John Hansman, Professor Walt Hollister, and Professor 
Eric Feron at the Massachusetts Institute of Technology. 



Table of Contents 



1 Introduction 11 

1.1 Objectives „..__. 12 

1.2 Overview of Thesis ,....., 13 

2 Review of Alerting System Performance 15 

2.1 Generic Alerting System 15 

2.2 State-Space Representation of Alerting Systems 16 

2.2.1 State Trajectory 17 

2.2.2 Hazard Space 17 

2.2.3 Alert Space ...„..„„......„... 18 

2.3 Alerting Decision Outcomes 19 

2.4 Tradeoff Between Missed Detection and False Alarms 22 

2.5 Role of Uncertainty in the Alerting Outcome 25 

2.6 System Operating Characteristic Analysis , 28 

2.6.1 Probabilistic Trajectories and Probability of Conflict 28 

2.6.2 Nominal and Avoidance Trajectories 29 

2.6.3 SOC Curves 30 

2.7 Summary 37 

3 Conflict Detection and Resolution Methods 39 

3.1 Conflict Detection and Resolution 39 

3.2 Survey of Algorithmic Designs 44 

3.2.1 Current Operational Systems 44 

3.2.1.1 Traffic Alert and Collision Avoidance System (TCAS) 44 

3.2.1.2 Traffic and Collision Alert Device (TCAD) 48 

3.2.1.3 Ground Proximity Warning System (GPWS) 49 

3.2.2 Additional Algorithmic Designs 51 

3.3 Insights from Survey 52 

3.3.1 Variety of Threshold Metrics 52 

3.3.2 Prevalence of Ad Hoc Approach to Alerting Design 53 

3.3.3 Three Trajectory Projection Methods 55 

3.3.3.1 Single Path „....,.,., 56 

3.3.3.2 Worst Case 57 

3.3.3.3 Probabilistic 59 

3.3.4 Accounting for Uncertainties 60 



3.4 Summary * 61 

4 A Unified Approach to Improving Alerting System Performance 63 

4.1 Errors in the Trajectory Model 63 

4.1.1 Working Model (W) vs. "Truth" Model (T) 63 

4.1.2 Errors and Uncertainties in the Trajectory Model 66 

4.1.3 Effects of Modeling Errors on Performance Estimates 71 

4.2 (W = T) Reducing Trajectory Modeling Errors (Increase Accuracy of Model). 73 

4.2.1 (W -> T) Drive W Toward T - Improve Trajectory Modeling 76 

4.2.1.1 Utilize a Probabilistic Trajectory Model 76 

4.2.1.2 Utilize Sufficient and Accurate Information of the State 
Trajectory 76 

4.2.1.3 Update the Working Trajectory Model 80 

4.2.2 (T -> M) Drive T Toward M 81 

4.2.2.1 Utilize Conformance Boundaries 81 

4.2.2.2 Limit Operation 84 

4.3 Reduce Inherent Uncertainties (Reduce Uncertainty of Future Trajectory) 85 

4.3.1 Restrict Flight Path 88 

4.3.2 Establish Protocol (Training, Rules of the Road, Convention) 89 

4.3.3 Introduce Better Equipment.. 90 

4.3.4 Delay Alert (Minimize Size of T at Alert Time) 91 

4.4 Investigate Other Avoidance Maneuvers 92 

4.5 Design Issues 94 

4.6 Summary 95 

5 A Probabilistic Perspective of the Alerting Design Process 97 

5.1 A Probabilistic Perspective to the Ad Hoc Approach 97 

5.2 A New Direct Approach 101 

5.3 Implications from a Probabilistic Perspective 102 

5.3.1 Global Design vs. Situation-Specific Design 103 

5.3.2 Relating Performance Measures to Alerting Thresholds 1 14 

5.3.3 Using Performance Measures as Alerting Thresholds 117 

5.3.4 Continuous Update of Trajectory Model W 119 

5.4 Summary 120 

6 Probabilistic Analysis of Conflict 123 

6.1 The Trajectory Model i24 

6.2 Calculating the Probability of Conflict 129 

6.2.1 Monte Carlo Simulations 129 



6.2.2 Propagation Method , 131 

6.2.3 Computational Accuracy 136 

6.3 Conflict Probability Maps ,., , ,. 138 

6.4 Summary 139 

7 Example Applications 141 

7.1 Horizontal Conflict Examples 141 

7.2 Vertical Conflict Examples 146 

7.3 Summary 152 

8 A Probabilistic Real-Time Alerting Probe 153 

8.1 Alerting Probe Concept 153 

8.2 Prototype Alerting System 156 

8.3 The Alerting Thresholds 158 

8.4 Evaluation of Prototype System 162 

8.5 Simulation Studies 167 

8.6 Discussion 169 

8.7 Summary ....... 170 

9 Summary and Conclusions 171 

9.1 Summary 171 

9.1.1 Review of Alerting Systems and Alerting Performance 171 

9.1.2 Survey of Alerting Approaches 171 

9.1.3 A Unified Approach for Improving Alerting Performance 172 

9.1.4 Probabilistic Influence in Alerting System Design 172 

9.1.5 Methodology for Computing Conflict Probabilities 172 

9.1.6 Application of Methodology 173 

9.1.6.1 Conflict Analysis Tool 173 

9.1.6.2 Real-Time Conflict Probe 173 

9.2 Conclusions , 174 

References 177 



Appendix A. A Review of Conflict Detection and Resolution Modeling Methods 187 

A.l State Propagation 188 

A. 2 State Dimensions 191 

A. 3 Conflict Detection 191 

A.4 Conflict Resolution 192 

A. 5 Resolution Maneuvers 194 

A.6 Multiple Conflicts 195 

A.7 Other Model Elements 196 

Appendix B. Statistical Analysis of Global Distributions 201 

B.l Statistics of Combining 2 Distributions 201 

B.2 Statistics of Combining More Than 2 Distributions 204 

Appendix C. Conflict Detection Using Line- Volume Intersection 207 

C.l Relative Frame 207 

C.2 Line-Volume Intersection 209 

C.2.1 Horizontal Intersection 209 

C.2.2 Vertical Intersection 210 



10 



Chapter 1 
Introduction 



As the sky above becomes more congested, new concepts of Air Traffic 
Management (ATM) are being proposed to handle the expected growth [RTCA, 1995; 
Wangermann, 1994; Phillips, 1996, Brudnicki and McFarland, 1997]. To enable more 
efficient ways and procedures of moving traffic about in the airspace, many methods will 
require the relaxation of the rigid airway structure and in-trail spacing currently being 
used to maintain traffic separation. The new concept is based on the idea of reducing 
restrictions on individual flight paths. Consequently, to handle the increased traffic 
volume or possible loss of airway structure, automated traffic conflict detection and 
resolution tools would be required to aid pilots and/or ground controllers in ensuring safe 
separation at all times. 

To predict traffic conflicts, it is necessary to project the future positions of aircraft 
over time. However, uncertainty is inherent in the prediction of any future event and the 
same is true in conflict prediction. Due to random processes, there is variability and 
uncertainty in the aircraft trajectory that make it difficult to precisely determine the 
aircraft's location at future times. It is this deficiency that produces errors in the 
determination of conflict and causes problems in the design of an effective alerting 
system. 

The use of a probabilistic approach can be helpful when uncertainty is expected or 
prevalent. It allows assessment of the likelihood of specific outcomes and provides the 
end-user with additional information that could be beneficial in the decision-making 

11 



process. Probability methods can also be placed in the role of assessing the overall 
hazard level of the encounter situation and the difficulty of resolving the conflict at hand. 
By utilizing these properties, the effects of various uncertainty elements in the aircraft 
trajectory on alerting system performance may be analyzed. Of special interest is the 
importance of including additional intent information into the alerting scheme. 

When aircraft intent is added into the aircraft trajectory prediction, it can 
significantly reduce the uncertainty in the estimated future path and hence lower errors in 
conflict determination. The effects can be analyzed using probability metrics such as the 
false alarm rate or successful alert probability to examine the benefits of the added intent 
information in specific situations. However, one should be cautioned that erroneous 
intent assumptions could also lead to missed detections of conflicts and inaccurate 
estimates of hazard. Mistakes such as these occur from incorrect modeling of the aircraft 
trajectories and can actually lead to additional sources of estimation errors. 

1.1 Objectives 

There are several major objectives contained within this thesis. One is to explain 
how uncertainties in trajectory estimation significantly impact and hinder conflict alerting 
systems. A unifying concept is proposed to explain various methods of improving 
prediction and alerting performance in the presence of aircraft trajectory uncertainties. 
The framework ties together such approaches as improved sensor accuracy, added intent 
information, path conformance checking, reduced false alerts, and also probabilistic 
trajectory estimation into a single underlying basis for improving conflict prediction and 
alerting. In addition, a foundation is laid to explain how probability concepts are already 
embodied within the general framework of alerting system design. 



12 



Finally, this work presents a method used to model, analyze, and compute the 
likelihood of conflicts in the presence of uncertainties. The approach uses probability 
density functions to model potential trajectories and utilizes Monte Carlo simulations to 
calculate the likelihood of violating separation minimums. A technique was developed 
and refined to perform the calculations in a manner efficient enough to be used as a 
possible real-time conflict detection and resolution probe. 

1.2 Overview of Thesis 

To begin, Chapter 2 introduces the relevant terms and definitions used to describe 
alerting system performance. The notion of uncertainty in the state trajectory estimation 
and its effects on prediction and alerting is given here in the context of state-space 
terminology. Also, the tradeoff between false alarms and missed detections is presented 
using System Operating Characteristic curves. 

In Chapter 3, the problem of conflict detection and resolution is formally 
introduced. A brief overview of current operational and developmental conflict alerting 
approaches are discussed. Initial motivation will be provided for the use of the 
probabilistic approach as compared to other previous modeling efforts, though the 
foundation will be further substantiated in subsequent chapters. 

Chapter 4 gives a general guideline for improving alerting performance in the 
presence of uncertainties. It ties together various approaches from two main goals of 
increasing prediction accuracy and reducing inherent uncertainty. The importance and 
concept of uncertainties and trajectory modeling errors are also explained in terms of 
their degrading effects on the alerting system performance. 



13 



A probabilistic connection is developed in Chapter 5 from looking at the typical 
alerting system design approach from a different perspective. The paradigm provides 
further motivation for utilizing probabilistic trajectory models for conflict alerting and 
prediction. 

Chapter 6 discusses the methodology and tools used in this thesis for the 
calculations of the conflict probabilities (Monte Carlo simulations). The basic trajectory 
model is developed and accuracy in its modeling is discussed. 

In Chapter 7, several example conflict encounters are studied utilizing the 
methodology developed in this thesis. The potential of the method to handle very 
complicated situations is shown by these examples. 

Chapter 8 describes an effort of transforming probabilistic conflict analysis into a 
real-time alerting system. Issues related to the use of probability and the implementation 
of real-time conflict probing are discussed. Experience and lessons learned are brought 
forth. 

Chapter 9 provides a final summary and major contributions introduced in this 
thesis. 



14 



Chapter 2 



Review of Alerting System Performance 



At its root, conflict detection and resolution is a process of determining the 
existence of a possible hazard and alerting the air traffic controller and/or flight crew. 
Accordingly, it is worthwhile to begin by examining the operation of the alerting system 
in general. 

2.1 Generic Alerting System 

A hazard alerting system is one of several safety components typically found in 
complex human-operated systems [Kuchar, 1995]. Its purpose is to monitor potential 
threats and issue warnings to human operators when undesirable events are predicted to 
occur. A simplified diagram of a generic alerting system within the context of the entire 
operating environment is shown below in Figure 2-1. 

Information about the situation is measured by sensors and presented to the 
human operator via various types of displays. The same information or a subset of it can 
also be fed to an alerting system to help determine the possibility of a hazardous 
situation. In many cases, a hazard can be detected by the operator from the displays 
themselves; however, in other instances, the operator may not be fully aware of the 
situation or may need additional confirmation to aid in decision making. An alert is 
supposedly a prediction that an unsafe state may occur, but reliance is usually still on the 
human operator to make the final decision. The degree of automation can vary, with 
some alerting systems providing a simple warning, while others give additional resolution 



15 



advisories. It is also possible to have an alert fully automated to initiate a resolution, and 
the operator is only informed of the action. 



Situation 



Environment 



Controlled 
System 



Informational 
Displays 



Alerting 
System 



Operating 
Procedures 



Human 
Operator 



Figure 2-1: Generic Alerting System in Operation with Human Operator 

[Kuchar, 1995] 

2.2 State-Space Representation of Alerting Systems 

The use of state-space representation is a way of introducing the concepts and 
issues associated with alerting system design. The method was developed by Kuchar 
[1995, 1996] and is based on multivariable control system theory. The following is a 
brief review from that previous work. 

In the approach, the variables .*,(/), x 2 (t), . . . , x n (t) describe the states of the 
encounter situation at time t . These states can be thought of as the set of parameters 
utilized by the alerting system logic to characterize the dynamics of the threat condition. 
The state vector, x(t), is then defined as: 



16 



x(f) = [*,(/) x 2 (t) 



c,(Of 



(2.1) 



where n is the number of elements chosen to describe the situation. At any given time, 
t, the current state of the system, as known to the alerting logic, is at some particular 
point identified by x(t) in the n -dimensional state-space X. 

2.2.1 State Trajectory 

The states of the system will typically change over time during the course of 
operation. These changes in the state vector occur in accordance with the system 
dynamics, the environment, and the inputs from the human controller. The set of values 
of x(?) over a given time interval is the state trajectory. Figure 2-2 shows an example of 
a state trajectory as viewed in a State-Space Diagram. 

X2 



State-Space 
X 




State Trajectory 



X\ 



x(0 = [x,(0 x 2 (t)f 

Figure 2-2: Example State-Space Diagram 

2.2.2 Hazard Space 

In certain regions of the state-space X , there are domains where undesirable 
events can occur. These regions are termed hazard space (as denoted by H). Whenever 
x(0 is allowed to enter a region of hazard space, a missed detection has occurred and the 



17 



alerting system has failed to provide the necessary protection to prevent an unwanted 
event. An example of hazard space as depicted in a State-Space Diagram is shown in 
Figure 2-3. 



State-Space 
X 




X\ 



Figure 2-3: Example Hazard Space in State-Space Diagram 
2.2.3 Alert Space 

The alert space is defined as the set of all state-vectors, x(/), in which the alerting 
system will warn the operator in order to prevent a possible intrusion into hazard space. 
By definition, no alerts are generated when \(t) is outside the alert space. The 
boundaries of alert space are considered the alerting thresholds and basically define when 
alerts are given and when they are not. In a given state-space, X , the regions of alert 
space will be denoted as X A . 

In Figure 2-4, an example alert space is shown in a State-Space Diagram. When 
the state trajectory first enters alert space (point 1), an alert will be given. At this point, 
the alerting logic has decided that an intrusion into hazard space is likely if nothing is 
done to wam the human operator (solid line). In other words, the alerting thresholds have 
been surpassed. By initiating the alert, it is expected that some action will be performed 
(depicted at point 2) to alter the course of the state trajectory (dashed line) in order to 
prevent a hazard from taking place. There is usually some delay from point 1 to point 2 

18 



as the human operator decides on the appropriate response to take. Because an alert is a 
precursor warning to avoid a hazard, the alert space should encompass all regions of 
hazard space. 



Trajectory 
,* After Alert 



State-Space 
X 




Initial Projected 
Trajectory 



Figure 2-4: Example Alert Space in State-Space Diagram 
2.3 Alerting Decision Outcomes 

Ideally, an alert correctly notifies when a hazard will occur if nothing is done to 
alter the current situation of the system. To be more complete, the alert should also allow 
for absolute resolution of the threat if it is to be considered a safety feature. If both these 
elements are satisfied, then the alert is termed a correct detection (CD). If, however, the 
hazard is not prevented (whether or not an alert is given), the outcome would be 
considered a missed detection (MD) because the system has failed to provide the intended 
safe avoidance of the hazard. An alert that is given but was not necessary (because a 
hazard would not have occurred in the first place) is usually termed a false alarm (FA). 
For sake of completeness, normal operation with no threat and correctly indicated by the 
alerting system will be considered a correct rejection (CR). The complete decision 
outcome is diagrammed in Figure 2-5. 



19 




Figure 2-5: Alerting Decision Outcomes 

The corresponding outcomes can be graphically depicted using State-Space 
Diagrams as shown in Figure 2-6. The points 1 and 2 refer to when the alert is given and 
when the response action is initiated, respectively. The solid lines are used to indicate the 
state trajectory which would have occurred had the alert not been given, and the dotted 
lines refer to the new state trajectory from the response to an alert. Figure 2-6a (correct 
detection) has been discussed already, and along with Figure 2-6d (correct rejection), 
represent the two cases of an ideally operating alerting system. 



20 



Xi 



1 

* 




x(r) l/ 2 <'>>*. 




/^ Alert Space W/V 

x A 





Xl 



a) Correct Detection 




X\ 



b) Missed Detection (No Alert/Late Alert) 



X\ 



c) False Alarm (Nuisance Alert/Induced Hazard) 



x 2 



x(t) 

• — 



Alert Space 

X A 




x\ 



d) Correct Rejection 



Figure 2-6: State-Space Diagrams of Alerting Outcomes 



X] 




Xl 



21 



Trouble occurs when either a missed detection or a false alarm is experienced. In 
the case of a missed detection, an alert is needed (state trajectory will enter hazard space) 
but the alerting system fails in preventing the hazard from occurring. Sometimes, a 
missed detection is further sub-divided into two categories as shown in Figure 2-6b: 
missed detection due to no alert and missed detection due to late alert [Winder and 
Kuchar, 1999; Haissig, et al. 1999]. The former is most likely due to lack of information 
of the states of the system (both x(t) or H) or from design errors in the alerting 
algorithm. In the late alert case, either the operator is not given enough time to decide 
and perform the appropriate action, or for some reason, the warning is not heeded or is 
just ignored. 

False alarms, as depicted in Figure 2-6c, occur whenever an alert is given but the 
state trajectory would not have entered into hazard space without it. A false alarm can 
also be parsed down further into two sub-categories: false alarm resulting in no hazard 
(nuisance alert) and false alarm causing induced hazard [Drumm, 1996; Winder and 
Kuchar, 1999; Haissig, et al. 1999]. Both cases decrease efficiency and increase 
workload for the human operators involved. At first glance, it might seem that the former 
would be of little concern to safety. However, the increased occurrence of such nuisance 
alerts can directly impact the response of the human operator in actual emergency 
situations. This is especially important when quick and decisive action is called for in 
order to prevent a catastrophic loss of the system (e.g. collision between two aircraft). 

2,4 Tradeoff Between Missed Detections and False Alarms 

When a state trajectory first enters the alert space (assuming x(t) is coincidental 
with point 1 in Figure 2-4), an alert is initiated. As stated earlier, the boundaries of this 
alert space define the alerting threshold of the alerting logic. Since x(/) is only an 



22 



estimate of the current state of the threat condition, an element of prediction is inherently 
involved in determining the path that x(t) will follow. The decision to alert is based on 
the logic's prediction that an intrusion into hazard space is likely given this x(t). If the 
prediction is wrong, then a false alarm (FA) has occurred. If the prediction is correct, 
then either the alert prevents the hazard from occurring (CD) or the alert is too late in 
avoiding a hazard space incursion (MD). Thus the event of an alert results in one of 3 
mutually exclusive outcomes: CD, MD, or FA. The likelihood of any of these events 
occurring (given an alert) can be expressed in statistical properties such as the probability 
of correct detection, P( CD), missed detection, P(MD), and false alarm, P(FA). 

If the alert space is made relatively large, alerts will occur more often during the 
operation of the system. This is the conservative approach. It can reduce the number of 
missed detections but at the expense of an increased rate of false alarms. If the alert 
space is made relatively small, less alerts will occur (fewer false alarms), but at the 
expense of increased missed detections from late alerts. Thus, here lies the fundamental 
tradeoff between MD and FA in alerting system design: reducing P(MD) will increase 
P(FA) while reducing P(FA) will increase P(MD). The result is similar to the problem 
found in signal detection theory as demonstrated by Kuchar [1995, 1996] in his work on 
System Operating Characteristic (SOC) curves. An overview of the SOC technique is 
presented in a later section. 

The reason a larger alert space will generally increase P(FA) lies in the mere fact 
that more states are included in the alert space. This allows for a higher probability that 
an alert will be induced whether or not it is needed. There is much more room for error 
in the prediction that hazard space will be reached if nothing is done. Take, for example, 
Figure 2-7a where the alert space is relatively large compared to the hazard space. 
Depending upon the dynamics of the system, there can be a higher probability that the 



23 



state trajectory will not enter into the region of hazard space as compared to the case in 
Figure 2-7b where the alert space is smaller. Thus, the larger alert space will usually 
result in a higher rate of false alarms. 




a) Larger Alert Space (More Alerts/Higher Rate of F A/Lower Rate of MD) 



X2 




Xl 



b) Smaller Alert Space (Fewer Alerts/Lower Rate of FA/Higher Rate of MD) 

Figure 2-7: Effect of Alert Space Size on False Alarms 

However, the smaller alert space may end up sacrificing the ability to escape from the 
hazard (i.e. provide insufficient warning time) and thus lead to a higher rate of missed 
detections. This tradeoff is the fundamental design challenge which alerting system 
designers are often faced with. 



24 



2.5 Role of Uncertainty in the Alerting Outcome 

Due to the nature of prediction, the estimate of future events is inherently 
uncertain to some extent. As alluded to in the previous section, the path of the state 
trajectory is usually not known exactly. This leads to a statistical description of the 
alerting outcomes (e.g. P(CD), P(MD), and P(FA)). Much of this can be attributed to 
uncertainties with predicting the state trajectory from only the current state, x(t). 

Take, for example, Figure 2-8 where three different aircraft encounter scenarios 
are shown. In each case, the range ( r) is 100 nautical miles and the range rate ( r) is 566 
knots. If the alerting decision is to be based on only these two parameters, then the 
current state vector \(t) = [r r] would be identical for each of these three cases - the 
alerting algorithm would be unable to tell them apart. However, the outcome from each 
scenario is decidedly different. In the case of Figure 2-8a, a direct collision would occur, 
while in the other 2 cases, no real threat is encountered. 

The comparison between Figure 2-8a and Figure 2-8b is especially important to 
note because it shows just how much the predictive path of the vehicle states can come 
into play even with the same apparent initial conditions. Though the current position and 
velocity of each aircraft is the same in these cases, the latter would result in a false alarm 
if an alert were to occur at the present time. 



25 



400 kts 




400 kts 




a) Direct Collision Outcome 



b) No Collision Outcome 



289 kts 



@- 



■J*-: 



98 nmi 



■® 



I 



289 kts 



c) No Collision Outcome 

Figure 2-8: Example Encounters with the Same State Vector, 

x(r) = [r rf = [lOOnmi 566ktsf 

In the case of Figure 2-8c, the situation appears quite different to the previous two 
scenarios, but would actually be transparent to an alerting algorithm based on only 
relative range and range rate at the current time. Unless some other provision is included 
to differentiate the scenes (e.g. relative bearing), the alerting algorithm would likely treat 
all three scenario encounters the same at this particular instant in time. The consequence 
of this is a higher degree of uncertainty in the alerting decision outcome. This effect can 
be seen in the state-space representation shown in Figure 2-9 where the state trajectory of 
the three cases from the previous figure are plotted. In the figure, a hazard is assumed to 



26 



be a loss of separation of some predefined distance, such as 5 nautical miles or less, 
between aircraft. 




Hazard 
Space 



100 r [nmi] 



-800 



Figure 2-9: State-Space Representation of Figure 2-8 

As shown in the figure, 2 of the 3 scenarios would have incurred a false alarm if 
the decision was made to alert at x(0- The result would be a high rate of false alarms 
due to the inability to predict the outcome of the action from current state vector. 

Even if the decision to alert is justified, such as in Figure 2-8a, there can be 
further uncertainties that affect the new state trajectory in response to the alert. These 
uncertainties (e.g. response time and avoidance action) would inevitably influence the 
likelihood that the hazard could be avoided. The outcome would be a direct impact of the 
correct detection and missed detection (by late alert) rates, P(CD) and P(MD), 
respectively. 

Further uncertainties, such as those due to stochastic randomness, can enter into 
the problem as well. In the case of aircraft, this could include fluctuations in speed, 



27 



heading, or altitude of each vehicle involved. There may also be course changes not 
known at the present time that could by initiated by the flight crew to avoid weather or 
meet performance goals (e.g. time of arrival, fuel savings, flight comfort). The changes 
may also be inadvertent due to pilot blunders to maintain an expected course of flight. 
All this leads to uncertainties in the state trajectory which can affect the outcome of each 
alerting decision. 

2.6 System Operating Characteristic Analysis 

The approach of the System Operation Characteristic (SOC) method has its roots 
in signal detection theory. It was developed by Kuchar [1995, 1996] to help analyze and 
design alerting thresholds by examining the tradeoff between successful alerts and false 
alarms. Much of the method is based on the use of probabilistic trajectory analysis in the 
propagation of the state vector. 

2.6.1 Probabilistic Trajectories and the Probability of Conflict 

As stated before, the prediction of future events inevitably involves uncertainties, 
and the same is true of the state trajectory. In general, the path of the true state trajectory 
is not known exactly, and it can be assumed to be probabilistic and include uncertainties 
to some degree. The concept is detailed in the following Figure 2-10 where the shaded 
area represents possible state positions, or uncertainties, in the future of the system. 
Usually, but not always, the uncertainty in the trajectory will tend to grow with time as it 
naturally becomes more difficult to predict further into the future. 



28 



Uncertainty in 

Current State 

Estimates 




& 



0) 



3 
In 



Figure 2-10: State Trajectory with Uncertainties 

In state-space, the term conflict will refer to the occurrence of an undesirable 
event (i.e. hazard space incursion of the state vector). In estimating the likelihood of its 
occurrence, the term Probability of a Conflict, P(C), will be used. 

2.6.2 Nominal and Avoidance Trajectories 

To determine if an alert is warranted in a given situation, it is necessary to 
examine the hypothetical outcomes of the alert / no alert decision. If no alert is issued, 
the state continues along what will be termed the projected nominal trajectory, denoted as 
N. Similarly, in response to an alert, there is a different projected path that is taken called 
the avoidance trajectory, denoted A. Both N and A are, in general, probabilistic due to 
the uncertainties in the current and projected future states. During an avoidance action, 
there are many variables which may make it difficult to predict the exact path taken, 
especially with human involvement (e.g. different response times and actions). 
Trajectory A may also include the possibility that no action is taken in response to the 
alert. 



29 



The probability of conflict along N and along A are denoted P N (Q and P A (C), 
respectively.. In Kuchar [1995], the Probability of a False Alarm, P(FA), and the 
Probability of a Successful Alert, P(SA) , are defined as: 

P(FA) = 1 - P N (C) (2.2) 

P(SA) = 1 - P A (C) (2.3) 

In the above two equations, both P(FA) and P(SA) are conditional upon an alert being 
given. Also, P(SA) is specific to a particular avoidance trajectory, A. 

2.6.3 SOC Curves 

In previous work, Kuchar [1995, 1996] explored the performance tradeoff 
between false alarms and successful alerts. This technique is based on the System 
Operating Characteristic (SOC) Curve which facilitated the visualization of the exchange 
between the two parameters. In any conflict detection decision, there is usually some 
probability that the alert is not needed. Additionally, there is some probability that the 
alert is successful in prevent a conflict. As one varies the time at which the alert is 
generated, these probabilities trade off against one another as described by an SOC curve. 

In order to determine if an alert is successful, it is necessary to consider what 
resolution action occurs when the alert is given. Some conflict resolution maneuver must 
be assumed so that it can be determined whether a conflict is ultimately averted by the 
alert. Thus, a SOC curve is specific to both the encounter geometry and the type of 
resolution action that is prescribed. In simple terms, a SOC curve is a plot of P(SA) 
versus P(FA) along a specific nominal path, N, and for a specific avoidance maneuver, 
A 



30 



The choice of avoidance trajectories for conflict analysis will depend on the 
performance criteria to be met. The criteria can be safety-based, in which the trajectory 
is to reflect an aggressive maneuver, or it can be more cost driven, in which the trajectory 
represents a more strategic maneuver. 

A sample SOC plot is shown in Figure 2-11 for a path on a direct collision course 
to a hazard. The points 1 and 2 correspond to different alerting times, with point 1 being 
earlier than point 2. If the conflict decision is made while the hazard is far away, (upper 
right corner of the plot), the probability of a successful alert is likely to be very high 
( P(SA) -» 1); but because action is taken so early, the probability of a false alarm is also 
high (P( FA) -> 1). With the hazard far in the distance, there is typically too much 
uncertainty in the nominal N trajectory to alert without knowing if it was really 
necessary. As the conflict alert decision is delayed and the hazard continues with 
increasing threat, the probability of successful alerts and false alarms both decrease as 
shown by the curve. If alerts are delayed too long, the alerts will not be successful 
(P(SA) -> 0) and there will be no false alarms as well (P(FA) -> 0). 



1.0 



P(SA) 




P(FA) 



1.0 



Figure 2-11: Typical SOC Plot 



31 



The SOC curve shows the tradeoff between P(FA) and P(SA) as a function of 
the alerting threshold location for a series of decision points along a chosen path. The 
curve in Figure 2-1 1 clearly shows the effect of delaying the alert on reducing the chance 
of a false alarm. The corresponding drop in a successful alert, P(SA), is also evident. 
The location of the threshold can be examined relative to the desired level of nuisance 
alerts and safety margin. It is also possible to utilize the SOC curve as part of a 
preliminary design evaluation for setting the alert threshold. 

In Figure 2-12, a pictorial perspective on the underlying principle behind the SOC 
is given. Far away, with the hazard at location 1, there is usually sufficient uncertainty to 
warrant delaying an alert so that false alarms are not too predominant. However, waiting 
too long may result in an unavoidable hazard (missed detection). The state-space analogy 
was explained back in Figure 2-7 with the discussion on the alert space size. 

As mentioned earlier, high rates of nuisance alerts can lead to mistrust of a 
system, and thus, also indirectly impact safety. Deciding when to alert (i.e. threshold 
placement) is one of the most crucial elements in alerting design. The choice becomes 
obvious in Figure 2-12 when the avoidance trajectory, A, is superimposed over N. To 
ensure the alert is successful, it must be issued prior to a high likelihood of conflict along 
the avoidance trajectory (location 2). In other words, the alert must be in time for the 
aircraft to avoid the hazard. This appealing concept is captured nicely in SOC plots. 



32 




Figure 2-12: Delaying the Alert 

The shape of the SOC plot can provide a lot of information regarding the possible 
performance of the alerting system. A curve that allows placement of the threshold at the 
upper left corner (P( FA) = , P(SA) - 1) is considered ideal since there would be no 
false alarms and only successful alerts (see Figure 2-13). Due to uncertainties in the 
conflict dynamics, however, the SOC curve will generally lie somewhere below this 
optimal point. The closer a system is able to operate near this optimal point, the more 
effective the system will be in terms of providing an acceptable level of safety while 
minimizing unnecessary alarms. 



33 



Ideal Alert 
Threshold 



P(SA) 




P(FA) 



1.0 



Figure 2-13: Various Shapes of the SOC Curve 

A curve that lies diagonally from the origin (P(FA) = , P(SA) = 0) to the 
upper right corner (P(FA) = 1, P(SA) = 1) represents either a poorly chosen avoidance 
maneuver for the particular encounter or an inherently difficult situation due to the 
uncertainties involved. In such circumstances, the alert basically has no effect in altering 
the outcome of a conflict. The alert is just as likely to produce a conflict as if no alert 
was given. Thus the more the SOC curve deviates upward from the diagonal, the more 
likely a better alerting decision can be determined. 

It is also possible that an alert can induce a negative consequence in the encounter 
situation. In this case, the avoidance trajectory incurs a higher likelihood of a conflict 
that along the nominal trajectory when no alert is given. The resultant SOC curve would 
deviate below the diagonal as shown in Figure 2-13. 



34 



The characteristic bend or drop in P(SA) that is sometimes found in an SOC plot 
can be largely attributed to the specific avoidance maneuver being examined, but is also 
influenced by the underlying uncertainties in both the nominal and avoidance trajectories, 
N and A, as well as that from the hazard, H. In general, if no uncertainties were present 
and the future trajectories could be predicted with utmost precision, then an ideal alerting 
system would most likely be possible. A point at the upper left corner would exist 
provided the alert is given early enough. Different maneuvers would affect the required 
alert time, though. A 30 degree bank turn maneuver performed by an aircraft may 
provide an ideal system for a specific encounter if the alert is given prior to 15 seconds 
prior to predict conflict, while a 1000 feet per minute climb, in the same situation, might 
need the alert to be given 23 seconds ahead of time. But assuming there are no 
uncertainties involved, both maneuvers could provide perfectly ideal alerting thresholds 
(P(FA) = 0, P(SA) = 1). 

Each of the 4 extreme corners in the SOC diagram represents an absolute 
certainty condition. Thus, if no uncertainty is present in the trajectory or position of the 
aircraft, the state must lie at one of these corner positions. Either a conflict will exist 
along the nominal trajectory, N, or it does not. Either the conflict can be avoided with the 
avoidance trajectory, A, or it can not. These 4 extreme conditions are shown in Figure 2- 
14. Although it is assumed here that the trajectories are known perfectly, it is still 
possible for the SOC curve to be at any of the corner positions even with some 
uncertainties present in the trajectories. Most likely, however, the locus of points will lie 
somewhere within the boundaries of the 4 corners. 



35 




P(SA) 



N 




P(FA) 



a) No Alert Needed but Alert Causes No Conflict (P( FA ) = 1, P(SA) = 1) 




P(SA) 




P(FA) 



b) Alert Need and Alert Successful (P( FA) = 0, P(SA) = 1) 




P(SA) 



N 



P(FA) 



c) Alert Needed but Alert Unsuccessful (P(SA) = 0, P(SA) = 0) 



±\ L 



^ 



^ 



P(SA) 



N 



P(FA) 



d) No Alert Needed and Alert Induces Conflict (P( FA ) = 1, PfSAJ = 0) 
Figure 2-14: Four Corners of the SOC Diagram 



36 



Thus the shape of the SOC curve can also serve as a visualization tool to gauge 
the effects of uncertainties in the encounter scenario. If a high level of uncertainty 
existed in a trajectory (either N or A), then the curve would tend to diverge from the 
corner positions. The result could be used to determine if a more severe avoidance option 
is necessary, or even to consider a different type of maneuver altogether - one that is 
more robust to the uncertainties involved in the scenario. 

2.7 Summary 

The use of the state-space representation was shown as a way of presenting the 
concepts associated with alerting system design. It was used to explain the ideas behind 
alerting performance and the parameters associated with the different alerting decision 
outcomes: correction detection, missed detection, false alarm, and correct rejection. 
Also, the System Operating Characteristic technique was highlighted as a method for 
examining different performance tradeoffs and provides a framework for analyzing 
alerting performance in later chapters. 



37 



38 



Chapter 3 

Conflict Detection and Resolution Methods 



3.1 Conflict Detection and Resolution 

Methods for maintaining separation between aircraft in the current airspace 
system have been built from a foundation of structured routes and evolved procedures. In 
this framework, humans have been an essential element in this process due to their ability 
to integrate information and make judgements. However, because failures and 
operational errors can occur, automated systems have begun to appear both in the cockpit 
and on the ground to provide decision support and to serve as traffic conflict alerting 
systems. These systems use sensor data to predict conflict between aircraft, alert humans 
to the conflict, and may also provide commands and guidance to resolve the conflict. 
Together, these automated systems provide a safety net should normal procedures and 
controller and pilot actions fail to keep aircraft separated beyond established minimums. 

Recently, interest has grown into developing more advanced automation tools to 
detect and resolve traffic conflicts. These tools could make use of more advanced 
technologies, such as datalink of current aircraft flight plan information, to enhance 
safety and enable new procedures to improve air traffic flow efficiency. 

To begin, it is necessary to have a clear definition of what constitutes a 
conflict. For the majority of this thesis work, a conflict will refer to a situation in which 
an aircraft experiences a loss of minimum separation with another aircraft. In other 
words, the distance between them violates a preset criterion that is considered 



39 



undesirable. One example might be a 5 nautical mile horizontal distance between aircraft 
and a 1000 feet vertical separation (current Air Traffic Control standards). The result is a 
protected zone or volume of airspace surrounding each aircraft that should not be 
infringed upon by another vehicle (see Figure 3-1). The protected zone could also be 
defined much smaller depending upon the goals of the alerting system (e.g. parallel 
runway incursions). It could also be specified in terms of parameters other than distance, 
such as time. In any case, the underlying conflict detection and resolution functions are 
similar although the specific models and alerting thresholds would likely be different. 



5 rani , 
_ .| 


c^T _^> 


r *- ~3 



L000' 
L000' 



Figure 3-1: Example Protected Zone Around Aircraft 

Any traffic management system in which vehicles are monitored and controlled to 
prevent collisions has certain basic functional requirements. The objective of a conflict 
avoidance system is to predict the occurrence of a conflict, communicate (alert) the 
detected conflict to the human operator, and then, in some cases, assist to resolve the 
conflict situation. These three fundamental processes can be organized into several 
phases or elements as shown in Figure 3-2. 

To begin, the traffic environment must first be monitored and the appropriate 
aircraft state information must be collected and disseminated using sensors and 
communication equipment. These states provide an estimate of the current traffic 
situation (e.g. aircraft positions and velocities). However, due to sensor limitations, the 
information may not be complete enough to describe the actual situation. For example, a 



40 



system may only have access to range and range rate information and unable to determine 
bearing (recall the state-space example of Figure 2-9). Additionally, there is generally 
some uncertainty within the values of the available states. 




Environment 



State 
Estimates 



1 



Intent 



li 



Dynamic 
Trajectory- 
Model 



Metric 
Definition 



Projected 
States 




Detection 
Phase 



Resolution 
Phase 



Figure 3-2: Conflict Detection and Resolution Framework 

Information regarding the future intent of aircraft may also be available to the 
alerting algorithm. Such data might include the waypoints in flight plans, level-off 
altitudes in vertical maneuvers, or commanded heading during turns. The information 



41 



can be used to provide additional prediction accuracy to the future trajectories of each 
aircraft. It is also possible that the intent of an action will not be followed so there is 
likely some potential uncertainties in the information as well. 

Continuing on, a dynamic trajectory model is usually required to project the states 
into the future in order to predict whether a conflict will occur. This projection may be 
based solely on the current state information (e.g. a straight-line extrapolation of the 
current velocity vector) and may include additional intent information (e.g. the flight 
plan). As shown before in the previous chapter, the importance of this model cannot be 
understated as it has a direct impact on the overall performance of the system. Any 
prediction of future events inherently involves some uncertainties, and the choice of 
dynamic model to estimate future states is no exception. 

The parameters used for the actual alerting decision will be referred to as the 
alerting metrics. Here, information from the current and predicted states are combined to 
provide an overall measure of threat to the occurrence of a conflict. Some example 
metrics include the relative range, closure rate, predicted miss distance, or the estimated 
time to closest point of approach. These metrics form the state space of the alerting 
algorithm as explained in Chapter 2. 

Given the conflict metrics, a discrete decision (Conflict Detection) is then made 
regarding whether or not to inform the human operator of a threat. Often, this decision is 
based upon a simple check against specific thesholds (e.g. take action if predicted miss 
distance is less than 5 nautical miles), but could involve a more complex set of rules. The 
thresholds may include corrective adjustments or safety buffers to account for 
uncertainties as well. 



42 



Note, however, that the prediction of a conflict need not always require a 
notification. A conflict may be predicted, but its occurrence may be too far into the 
future or too uncertain to be considered a threat at the current time. The decision to alert 
could also hinge upon user preference, experience, or operational factors. For the 
purpose of this thesis work, a conflict is detected once it is both predicted to occur and it 
has been determined to be appropriate to alert the operator. 

In some cases, notification of a conflict is all that is required of the alerting 
system (the human operator is expected to resolve the conflict independently). In other 
cases, a Conflict Resolution phase may be initiated. This involves determining an 
appropriate course of action and transmitting that information to the operator. For 
example, the system might present to the pilot of an aircraft the target rate of climb or 
descent necessary to avoid a potential collision with another aircraft. Although conflict 
resolution is shown as a single block in Figure 3-2, it requires its own set of current state 
estimates, a resolution maneuver trajectory model, and decision criteria which may be 
different from those used in the Conflict Detection phase. This simplification in the 
figure was intended to make the schematic less cluttered without diminishing the 
meaning of the concepts. 

The response of the human to the alert is also critical to the design and efficacy of 
the alerting system as well. In many instances, the human's response can be included to 
some extent within the determination of the resolution maneuver, such as a 5 second 
delay. However, the human response is inevitably variable and needs to be considered as 
another source of uncertainty in the overall scheme of the conflict alerting process. 

In the framework of Figure 3-2, conflict detection can be thought of as deciding 
when action should be taken while conflict resolution can be looked upon as determining 



43 



how or what action should be performed. In practice, there may not always be a clear 
separation between alerting and resolution, however. Deciding when action is required 
may depend on the type of action to be performed; and similarly, the type of action that is 
required may depend on how early that action begins. 

The multitude of various metrics and thresholds and also the interdependence 
between conflict detection and resolution are factors which make alerting system 
development challenging and interesting because there are many feasible design 
solutions. As will be shown in the next section, there are a number of ways to tackle the 
problem and develop a feasible solution. A more difficult task is determining the best 
one. 

3.2 Survey of Algorithmic Designs 

To provide better insight into different methods of conflict detection and 
resolution, a literature review of previous research approaches and current operational 
and developmental systems was performed. A total of over 60 different papers were used 
and a more detailed discussion can be found in Appendix A. These methods do not 
represent an exhaustive list by any means, but are believed to encompass a majority of 
the recent approaches to the conflict detection and resolution problems. 

3.2.1 Current Operational Systems 

3.2. 1. 1 Traffic Alert and Collision Avoidance System (TCAS) 

The Traffic Alert and Collision Avoidance System (TCAS) has been the standard 
that many approaches to the conflict detection and resolution problem have been 
compared to. The system has been implemented on U.S. jet transports since the early 
1990's as concern over the potential of future mid-air collisions grew. The algorithm is 



44 



more complicated than will be explained here, so the reader is asked to refer to Ford 
[1986, 1987], Kuchar [1995], Miller et al. [1994], RTCA [1983], or Williamson [1989] 
for a more detailed description. 

In abbreviated terms, the TCAS logic calculates threat in the horizontal and 
vertical dimensions separately and alerts if both criteria are met. The algorithm is based 
on the relative range (r) and range rate (r), and also the relative altitude (h) and altitude 
rate (h). TCAS uses a two-stage process with a cautionary alert called the Traffic 
Advisory (TA) and a warning alert called the Resolution Advisory (RA). RAs provide 
vertical avoidance commands, but TAs are merely cautions and lack any resolution. The 
following discussion will focus on RA alerts only. 

The TCAS thresholds are actually more complex, but for the most part, can be 
summarized by what is commonly referred to as the Tau Criterion shown in Equation 3.1. 

r - DMOD ._ „ 
< T (3.1) 

-r 

T is a threshold parameter with units of time, and DMOD is a buffer distance used to 
account for slow closure rates, ensuring that aircraft will not drift closer than the DMOD 
distance without receiving an alert [Williamson, 1989]. Within the alerting logic, these 
two parameters are varied depending on the altitude and whether or not the aircraft are 
maneuvering vertically. These values are summarized in Figure 3-3. 

In the notation used in the figure, the first value listed for x is the alerting 
threshold for the TCAS equipped aircraft if it is level, or is climbing or descending in the 
same direction as the threat but at a lower rate; else the second value is used. 



45 



30,000' MSL 



20,000' MSL 



"g 10,000' MSL 



5,000' MSL 



2350' AGL 



1000' AGL 



1 

X = 30/35 


sec 


ZTHR 




800 


ft 


DMOD = 1. 


1 nmi 


ALIM 


= 


700 


ft 


X = 30/35 


sec 


ZTHR 


= 


700 


ft 


DMOD = 1. 


1 nmi 


ALIM 


= 


600 


ft 


X = 22/30 


sec 


ZTHR 


_ 


600 


ft 


DMOD = 0. 


8 nmi 


ALIM 


= 


400 


ft 


T = 20/25 


sec 


ZTHR 


_ 


600 


ft 


DMOD = 0. 


55 nmi 


ALIM 


~ 


350 


ft 






ZTHR 


= 


600 


ft 


X = 18/20 


sec 


ys. ALIM 


= 


W 1 


ft 


DMOD = 0. 


35 nmi> 














ZTHR 


= 


600 


ft 


X = 15/15 


sec 


_ ALIM 


= 


300 


ft 


DMOD = 0. 


2 nmi > 














Ground Alti 


tude 





MSL = Mean Sea Level 
AGL = Above Ground Level 



Figure 3-3: TCAS Version 6.04A RA Logic Parameters 

The vertical criterion is a little more complicated, but in essence, also utilizes a 
Tau Criterion to estimate the time to co-altitude. It includes various buffers and 
parameters (ZTHR, ALIM) that are variable depending on the flight altitude and relative 
vertical separation. 

The left side of the Equation 3.1 can also be thought of as an estimate of the time 
it will take for the range to decrease to a distance, DMOD, between aircraft [Miller et al., 
1994]. From this point of view, the TCAS logic is assuming a straight line projection 
model and DMOD is acting as a buffer to account for possible deviations or sources of 
error. 

A possible state-space representation of the alerting logic at work is shown in 
Figure 3-4. Here, the aircraft are assumed to be in level flight (30,000 feet) and traveling 



46 



in opposite directions, each with a velocity of 400 knots. For case A, the opposing 
aircraft are on course for a direct collision; while in case B, the aircraft will miss by 5 
nautical miles. 



V 



A 



5 nmi 



« 



B 



r-DMOD 



(sec) 




30 sec 



Figure 3-4: TCAS Example 

Notice that for a range greater than about 10 nautical miles, it becomes 
increasingly more difficult for the TCAS logic to differentiate between the two cases. 
The r threshold is 30 seconds for this particular scenario, which is just below the lowest 
point for which case B would trigger an alert. Trying to extend the warning time of 
TCAS in its present form would only introduce an increase in false alarms as shown from 



47 



this simple example. For instance, raising r beyond 40 seconds would inevitably cause 
an RA alert for case B even though the aircraft would be expected to miss each other by 5 
nautical miles or more. 

The parameters DMOD, ZTHR, ALIM, and r effectively determine the 
frequency with which RAs are given. Reducing these values will reduce the alert rate 
and number of disruptions caused by false alarms [Miller et al., 1994]. However, the 
tradeoff is the risk of missed alerts due to insufficient warning time. The desire is a 
balance between false alarms and collision protection that TCAS is intended to provide. 

To achieve this balance, the various design parameters (e.g. DMOD, r) were set 
using an iterative, trial-and-error approach run through literally millions of simulation 
scenarios involving many hypothetical encounter geometries [Miller et al., 1994]. 
Modifications were also made from data and user comments during actual in-flight 
operations. 

3.2.1.2 Traffic and Collision Alert Device (TCAD) 

The Traffic and Collision Alert Device (TCAD) is a low cost, low complexity 
conflict alerting system directed at the general aviation industry [Ryan and Brodegard, 
1997]. Its algorithm logic for conflict detection is based simply on range and altitude 
deviations alone, pilot selectable in one of three sensitivity levels: en route, standard, and 
terminal (see Figure 3-5). The basic function of TCAD is provided in an audible alarm 
and a numerical display of the range and relative altitude whenever another aircraft 
penetrates the alert space set by the pilot. If multiple aircraft are encountered, the data 
from the most prominent threat (based first on altitude, then on range) is shown. 



48 



v^ 



En route 



Standard Terminal 



Figure 3-5: TCAD Alert Zones 

The threshold parameters (range and relative altitude) defining each alert zone are 
actually left to the discretion of the pilot. If the threshold is set too high, extraneous false 
alarms will result; if it is set too low, warning time is compromised. The result is a 
tradeoff between false alarms and missed detections which the pilot must optimize to his 
or her own preference. 

3.2.1.3 Ground Proximity Warning System (GPWS) 

Although the Ground Proximity Warning System (GPWS) is not designed to 
prevent traffic collisions, many of the problems encountered in alerting design can be 
seen its development (e.g. prediction of future hazards in presence of uncertainty, and 
tradeoff between false alarms and missed detections). In the case of GPWS, the hazard is 
terrain and the system is intended to prevent crashes while in controlled flight (no 
mechanical failures). The system has been mandated in jet transport aircraft in the U.S., 
and since its introduction in 1975, the number of Controlled Flight Into Terrain (CFIT) 
accidents has been reduced considerably [Kuchar, 1995]. However, CFTT accidents still 
occur worldwide and remain the leading cause of aircraft fatalities. Poor pilot response, 
either from delayed reaction or inadequate avoidance maneuver, was found to have 
contributed to many of these accidents. Previous experience with nuisance alerts was 
suspected to have played a role in a number of these poor responses [DeCelles, 1991]. 



49 



At first glance, overly sensitive alerts appear to be conservative and on the safe 
side. The effects from false alarms seem to be primarily on efficiency. However, when 
alerts are judged to be erroneous or unnecessary, trust in the system validity becomes an 
issue. If the fallacy is excessive, then complacency may set in resulting in delayed pilot 
response times, inappropriate actions, or even no action at all. Thus safety becomes a 
direct fallout of false alarms when human operators are involved. 

For GPWS, the dilemma comes mainly from the limited amount of terrain hazard 
information available to the system. Only the altitude from the current position, both 
Mean Sea Level (MSL) and Above Ground (AGL) is utilized in the calculations. It is 
based on only one dimension of the terrain - the altitude directly below the aircraft. No 
information is available to the GPWS regarding the terrain ahead or to the side. The 
GPWS must perform a derivative calculation from altitude separation alone and 
extrapolate the expected time to impact from this closure rate estimate. The prediction of 
the upcoming terrain hazard can be highly inaccurate from this information as shown 
below in Figure 3-6. 



£=» 



Extrapolated 
Terrain Profile 



True Terrain 
Profile 



Extrapolated 
Terrain Profile 




True Terrain 
Profile 



a) Accurate Extrapolation 



b) Inaccurate Extrapolation 



Figure 3-6: GPWS Measurement and Prediction of Terrain [Kuchar, 1995] 



50 



The lack of knowledge of the terrain features prominently shows the difficulty in 
the design and optimization of alerting systems when uncertainty is involved. In the case 
of the GPWS, the uncertainties come from the lack of information about the upcoming 
terrain features as well as the large variability that can occur in pilot response times and 
avoidance actions. These factors led to an iterative, evolutionary design, both from 
simulated scenarios and actual case studies, which proposes to balance unnecessary alerts 
and sufficient warning times [Bateman, 1994]. 

3.2.2 Additional Algorithmic Designs 

There are many other methods which have been proposed to handle the conflict 
detection and resolution problem (see Appendix A). The problem is not limited to 
aerospace applications, but spans other areas such as automobile, naval, and robotic fields 
as well. Some are based on range, estimated miss distance, expected time to conflict, 
optimal escape paths, or force/potential fields. In all these cases, the element of 
prediction is inherently involved in determining the future behavior of the vehicles. The 
prediction occurs both in detecting a conflict and in resolving it. In many instances, the 
resolution requires the vehicles to comply exactly to the computed avoidance route in 
order to obtain full benefit from the calculated solution. However, as stated previously, 
there are uncertainties involved that will likely influence the final result. 

Often, the solution is optimized and applied to only a few example scenarios. 
Most examples are given for 2-D horizontal conflicts, though it is noted sometimes that 
the methodology could be extended to the vertical dimension as well. Much of this is due 
to the relative ease of the solution in the planar case, especially for pairwise vehicle 
encounters where analytical solutions exist. Also, the system's overall performance can 
only be judged in real-life operation, or estimated in simulations over a large number of 



51 



encounter situations. If done in simulations, the performance result is often stated in 
probabilistic terms such as false alarm and missed detection rates. Parameters in the 
algorithm are typically set to balance these performance measures. For example, the 
maximum look-ahead time of a conflict probe might be set to 20 minutes to minimize 
nuisance alerts (or in the case of TCAS, around 30 to 45 seconds), but still provide 
reasonable warning time. 

3.3 Insights from Survey 

Based on the review of the different methods of conflict detection and resolution, 
there were several important insights gained as described below. 

3.3.1 Variety of Threshold Metrics 

There did not appear to be a clear winner or single solution to the problem of 
conflict alerting. There were several different combinations of metrics that were 
mentioned, yet no analytical proof to determine the optimal set. Mostly, the variables 
utilized in the operational systems were the ones which were obtainable with the limited 
type of sensors available to the designers at the time of implementation. Some of the 
more commonly mentioned metrics used for conflict detection were range, range rate, 
altitude, altitude rate, expected time to closest point of approach, estimated miss distance, 
and probability of a conflict. Notice that all these variables have a natural correlation to 
the existence of a future conflict. Take range, for example. The likelihood that two 
vehicles would ever interfere with one another is intuitively higher if their separation is 
10 miles, as opposed to being 100 miles apart (there is simply more volume of airspace 
possible for the future trajectory). 



52 



In general, the more information available for use in the alerting algorithm, the 
better the chances for improving prediction. For example, TCAS (which currently uses 
range and range rate in its horizontal threshold criterion) could benefit from additional 
data to predict expected miss distance [Burgess et al., 1994]. Also, one of the more 
talked about items in recent literature is the concept of intent information. This type of 
predictive information would allow for a better estimate of the future state trajectories, 
and thus, reduce the uncertainty in the entire conflict alerting process. 

3.3.2 Prevalence of Ad Hoc Approach to Alerting Design 

Though there were many possible metrics utilized in the different conflict alerting 
approaches, it was often mentioned that the appropriate setting of the thresholds was a 
tradeoff between overly sensitive nuisance alerts and adequate warning times. Where the 
process was actually described [Drumm, 1996; Bradley, 1992; Warren, 1997, Miller et 
al., 1994; Williamson, 1989], the settings were determined from an iterative, ad hoc 
approach using scenario simulations. The concept is shown in Figure 3-7. 




Alert 
Threshold 
Criteria 

X A 



Validation 
Simulation 



I 



Performance 
Measures 

P(SA) 
P(FA) 



Figure 3-7: Ad Hoc Approach 



53 



Initially, the thresholds are set at predetermined values and put through a series of 
test scenarios. Often, these are done through Monte Carlo simulations and the thresholds 
are adjusted accordingly, depending on its performance. It is a cyclic process where the 
newly adjusted thresholds are repeatedly tested in simulation and the results tabulated 
with the alerting performance commonly judged by use of probabilistic means. The 
design must usually pass specified probabilistic parameters very similar to those already 
discussed in this work (e.g. false alarm rate, missed detection rate). If they are not met, 
the thresholds must be readjusted and the cycle repeats itself several times before it 
becomes satisfactory to the designers. In the progression of the alerting system through 
actual implementation, the thresholds will likely be altered again to compensate for 
problems encountered in the field. This type of threshold adjustment is considered an ad 
hoc process and can be expected to some degree from any design. Some references to 
this type of approach can be found in the TCAS design [Miller et al., 1994; Drumm, 
1996; Bradley, 1992; Williamson, 1989] and the Airborne Information for Lateral 
Spacing (AILS) system [Winder and Kuchar, 1999]. 

3.3.3 Three Trajectory Projection Methods 

The importance of the dynamic trajectory model in the conflict alerting system 
design cannot be understated, especially in the presence of uncertainty. As defined by 
Neelamkavil [1987], 

"A model is a simplified representation of a system (or process or theory) 
intended to enhance our ability to understand, predict, and possibly control the 
behaviour of the system. " 

The use of a trajectory model, either in the design of the thresholds or directly in the 
alerting logic itself, is simply to predict the occurrence of a possible conflict situation and 



54 



the likelihood of avoiding it. In Chapter 4, the effects and consequences due to modeling 
errors will be discussed in detail, but for now, a quick look will be given to some of the 
more common dynamic modeling methods used in conflict detection systems. 

Past approaches to conflict analysis have usually relied on one of two propagation 
methods in the modeling of future aircraft trajectories: 1) single path and 2) worst case. 
These two approaches can be considered deterministic models in that a definitive output 
or conclusion is produced for a given set of state inputs. A conflict either will occur, or it 
will not, resulting in a binomial output of a hit (1) or miss (0), respectively. 

On the other hand, the outcome of a single event cannot usually be determined 
perfectly from a stochastic type model. For a specific set of state inputs, the result is not 
a precise outcome (the same inputs can produce a different output). There is some 
random variability involved and thus the output is usually expressed in terms of statistical 
properties or probabilities. Incidentally, a stochastic system will approach a deterministic 
one when the probability of the outcome is either or 1 for each specific set of input 
states. 

Take, for example, Figure 3-8, where two aircraft are in parallel, level flight with 
some fluctuations or variability in their flight paths. In one instance (Figure 3-8a), the 
aircraft never violate each other's protected zone. However, in Figure 3-8b, given the 
same set of initial aircraft state conditions, minimum separation is now violated at some 
future time. This is the concept of stochastic or probabilistic trajectories and is the main 
focus behind the methodology developed within this thesis. 



55 



■:*:■ 



© © 



a) No Conflict Occurs 




b) Conflict Occurs 



Figure 3-8: Example of Random, Stochastic Trajectory 

3.3.3. 1 Single Path Approach 

In the single path approach, future aircraft positions are assumed to follow one 
specific trajectory (in many cases, usually along straight line projections from the current 
estimated velocity vector). This is the simplest approach and conflicts can easily be 
determined if the trajectory is not too complicated; for example, when heading and 
altitude changes are simply modeled as consecutive straight line segments. Figure 3-9 
shows an example with two aircraft traveling in opposite directions. Assuming the 
current velocity vectors are held constant, the event of a conflict can be readily 
determined from analytical geometry. For the particular case shown in Figure 3-9, no 
conflict would be found. Many examples exist in the literature which utilize the single 



56 



path trajectory method. A small sample set can be found in Andrews, 1978; Kosecka et 
al., 1997; Krozel et al., 1996; Sridhar and Chatterji, 1997; Bilimoria et al., 1996; Durand 
et al., 1995; Eby, 1994; Ford 1986, RTCA, 1983; Havel and Husarcik, 1989; Love, 1988; 
and Zeghal, 1994. 



<§ 




Figure 3-9: Example Single Path Approach for Conflict Prediction 

It is obvious that the single path model does not compensate for uncertainties in 
the dynamic trajectory. The outcome is deterministic and results in either a hit (1) or a 
miss (0). In this type of modeling scheme, uncertainty and possible variability in flight 
trajectories are typically considered in the following stages of the metric definition or the 
threshold criteria using an iterative ad hoc approach to set the various threshold 
parameters, sometimes with appropriate safety buffers. 

3.3.3.2 Worst Case Approach 

In the worst case approach, every possible path (within expert reasoning) is 
considered and limited usually only by the aircraft's aerodynamic capabilities. The 
method is actually compensating for trajectory uncertainty within the dynamic model, but 
accuracy in estimating the occurrence of conflict is somewhat compromised (the concept 
of prediction accuracy is covered in the Chapter 4). Figure 3-10 is an example of this 



57 



approach for the same situation as in Figure 3-9. Unlike the single path case, a conflict 
would be predicted here. The worst case assumption takes the conservative approach of 
underlining safety and it would seem most useful in short, time critical situations. In long 
term conflicts, it may be less practical since the swath of volume that could be 
encompassed by all the possible trajectories would be enormous. For instance, an aircraft 
that can be expected to climb or descend at 2000 ft/min would engulf an airspace 20,000 
ft above and below it in after just 10 minutes. Previous work employing worst case 
analysis can be found in Ford and Powell, 1990; RTCA, 1995; Ratcliffe, 1988; Shepard 
et al., 1991; Shewchun and Feron, 1997. 




Figure 3-10: Example Worst Case Approach for Conflict Prediction 

In order to confine airspace coverage, the prediction envelope is typically range or 
time limited at the metric or alerting threshold stages (e.g. look ahead time is bounded to 



58 



less than 5 minutes or range to less than 50 miles). Again, this tends to lead to an ad hoc 
approach of maintaining a desirable coverage without being overly conservative. For 
example, would 5 minutes look ahead time still be too large, or would 3 minutes be more 
appropriate? Again, the answer may require simulations to establish a suitable medium. 

3.3.3.3 Probabilistic Approach 

In between the previous two modeling methods, however, is a middle ground 
where the various possible trajectories are weighted by their probability of occurrence. 
Uncertainties are modeled directly within the state estimates and the dynamic model. 
This approach has the distinct advantage of forcing an explicit, quantifiable measure of 
the uncertainty and accuracy affecting the conflict estimate. 

Both the single path and worst case approaches provide either a hit (1) or a miss 
(0) in their evaluation of a conflict directly from the dynamic model. In the probabilistic 
approach, the prediction is in terms of the likelihood of a conflict, a value that can be 
explicitly attributed to the effects of uncertainties in the situation. A weighted value 
between and 1 is determined referring to the estimated probability of a conflict, P(C) 
in the future. It was shown in Chapter 2 that P(C) is a direct indicator of (or link to) 
alerting performance. Thus this method provides a means for direct examination of the 
various levels and sources of uncertainty in the aircraft trajectory on the alerting system 
performance. Individual parameters can then be analyzed for their impact on the design. 

To some degree, both the single path and worst case approaches can be 
considered subsets or special cases of the probabilistic approach (much the same way that 
a stochastic system can approach a deterministic one). In the single path method, it is 
assumed that the aircraft will follow a particular course with an absolute probability of 
1.0 with no possibility of deviation, so the distribution is a single discrete point. For the 



59 



worst case approach, any possibility of conflict in the desired region of interest is 
rounded conservatively upward to a P(C) of 1.0. It does not really matter what 
trajectory distribution is inferred here, since any likelihood of intrusion would be 
considered off limits. 

Though the probabilistic approach has been utilized sparingly in the past [Kuchar, 
1996; Paielli and Erzberger, 1997; Heuvelink, 1988; Rome and Kalafus, 1988; Taylor, 
1990; Bakker and Blom, 1993; Williams, 1993; Warren, 1997, Prandini et al., 1999; 
Innocenti et al., 1999], this thesis work expands the technique to a host of more complex 
encounter scenarios and adds a more theoretical basis underlying the purpose of the 
methodology. With the advent of high speed computing, the feasibility of the sometimes 
arduous or complex probability calculations is shown to be easily realized. This notion 
will be explained further in a later chapter. 

3.3.4 Accounting for Uncertainties 

In some approaches to the problem, the aircraft trajectories are assumed to be 
known exactly in 4-D. The determination of a conflict in this manner is relatively 
straightforward. Either a conflict will occur or it will not. Either a conflict can be 
avoided or it cannot. Approaches such as these are mainly concerned with obtaining a 
optimal solution based on some monetary or workload cost function. 

However, in many cases, there is usually some leeway in the design approach to 
account for uncertainties that may occur in the prediction of a conflict and the ability to 
avoid it. Using TCAS as an example, the parameters such as DMOD, ZTHR, and ALIM 
act as buffers to account for uncertainties in the prediction process. This is a common 
approach with the single path projection method. The actual values of the buffers are set 
using an iterative, trial and error process (ad hoc) as discussed in Section 3.3.2. The final 



60 



parameters are set from a balance between nuisance alerts (due to false alarms) and late 
alerts (due to missed detections). 

In the worst case aproach, the uncertainties in the future path are accounted for 
somewhat in the path prediction, but not explicitly. It does not take into account the 
likelihood for which each of the paths would occur. All paths within bounds are 
considered equally likely. Also, a look ahead time limit is usually employed so as not to 
envelop too much open airspace. The actual boundaries are again commonly set using an 
iterative, trial and error process to set a balance between false alarms and missed 
detections. 

In the probabilistic approach, the uncertainties are modeled directly within the 
trajectory estimation. This provides direct accountability of the sources of the errors in 
the prediction process. It is the most direct method of including uncertainties in the 
conflict estimation. As will be explained in a later chapter, the ad hoc approach is merely 
an indirect method of injecting uncertainties in the alerting system design 

3.4 Summary 

In this chapter, the problem of conflict detection and resolution was introduced. 
The resulting framework was the building block upon which subsequent discussions 
could be made. A survey of current operational and developmental approaches to the 
conflict alerting problem was also performed to provide insight into the design process 
and help determine underlying themes. 



61 



62 



Chapter 4 

A Unified Approach to Improving Alerting System 
Performance 



As mentioned in the previous two chapters, the performance of an alerting system 
is often measured in terms of probabilities and prediction outcomes. For example, the 
false alarm and missed detection rates are indications of a system's ability to correctly 
determine the likelihood of an undesirable event (e.g. violating minimum separation). In 
this sense, the entire alerting problem can be perceived as a prediction problem in the 
presence of uncertainties. To improve performance is thus to increase the prediction 
accuracy of the alerting system. One way of achieving this is to reduce errors in the 
trajectory model. Another way is to make the future trajectory more easily predictable. 

4.1 Errors in the Trajectory Model 

4.1.1 Working Model (W) vs. "Truth" Model (T) 

To be able to estimate P(C) for either the nominal (N) or avoidance (A) 
trajectories, it is necessary for the alerting system logic to develop an approximate 
working trajectory model, W, for each aircraft as part of the estimation process. The 
detection of conflict is basically determined by where the modeling scheme predicts the 
position of the aircraft to be later in time. The importance of an appropriate model 
should not be understated since it is the defining source of the trajectory prediction. 
Errors in the model used by the alerting logic can increase the chances of missed 
detection of hazard and also add to false alarms. The choice of modeling schemes is 



63 



already complicated by the existence of uncertainties in the aircraft trajectories as 
explained earlier. Common approaches for modeling were discussed in Section 3.3.3, but 
the following will explain in further detail the impact of the models on conflict 
prediction. 

Let T be defined as the probabilistic state trajectory in the state-space of the 
alerting system. Thus, for a given state vector, x(t), the future state trajectory can be 
described by T. In this thesis, T will be referred to as the "truth " model and represents 
the best estimate of the uncertainties in the future states of the conflict situation. As 
shown in the example back in Figure 2-9, the state trajectory can have multiple outcomes 
from a single initial state, x(r). Thenmpredictability of a trajectory can be caused by 
many factors that are random such as wind, autopilot behavior, human actions, etc. and is 
thus a stochastic process. T can then be thought of as the trajectory defining the true 
probability of conflict, P T (C), at any given time. 

Now let W represent the working trajectory model that is actually being utilized 
by the alerting system to estimate the future states of the system. The two models, W and 
T, are depicted in Figure 4.1. It goes without saying that ideally one would like W to be 
an exact copy of the true probabilistic trajectory, T, for all time t; thus, the probability of 
conflict predicted from the working model, P W (C), would be the same as the true P T (Q . 
Unfortunately, uncertainties in the true trajectory make this all but unlikely except for a 
short time step into the future. In addition, as shown in Figure 2-8, multiple encounter 
situations with the same apparent state can lead to additional uncertainties in W. 

Note that for now, the subscript T will be used to explicitly differentiate the true 
conflict probability P T (Q as opposed to that obtained from the working model W. 
However, in general and for subsequent chapters, it is impossible to actually utilize 



64 



P T (Q since the true trajectory distribution is unknown. The alerting decision is based on 
W, but the actual situation is really dependent on T. In the discussions on the SOC 
curves from the previous chapters, it was assumed the true stochastic trajectory, T, was 
known and being used in the plots. 




Figure 4-1: Working Trajectory Model (W) and "Truth" Model (T) 

The error in the model, W, can be a result of many factors. It may be due to the 
limited amount of information available, oversimplification of the true trajectory, T, or 
even mistaken assumptions about the flight path. Sufficient and accurate state 
information about each aircraft is vital in the modeling process, but sometimes it is not 
available due to constraints on the equipment and technology currently available. For 
example, the Traffic Alert and Collision and Avoidance System (TCAS) is unable to 
obtain accurate relative bearing estimates of the surrounding aircraft because of 
limitations in the onboard equipment. It must rely on relative range and altitude data to 
estimate the conflict situation. Closure rates in the horizontal and vertical dimensions are 
estimated from derivative calculations and are used in the alerting decision. However, 
the relative bearing, which can be used to estimate the closest miss distance assuming a 



65 



straight line projection, can not be obtained with sufficient accuracy using analytical 
means [Burgess et al., 1994]. 

As mentioned earlier, the single path and worst case propagation methods are 
common approaches taken in modeling the trajectories. Both can be considered 
simplifications of the true path of the aircraft. In the case of the single path trajectory 
method, W would be narrowed to a single track with probability 1.0. The aircraft would 
not be predicted to veer off this path. In the worst case method, W would encompass the 
realm of all likely paths with the addendum that any possibility of a conflict 
{ Pw(C) > 0.0 } would be considered a violation. The single path and worst case 
approaches represent the two extremes of the modeling spectrum. Both have advantages 
and disadvantages which can depend on the situation. The single path approach may be 
preferred if the true trajectory is known with a high degree of confidence. The worst case 
approach lends itself to dealing with many of the uncertainties by setting a more sensitive 
alerting criteria. 

However, whenever the working model W does not correctly correspond to the 
"truth" model, T, errors in the prediction of conflict will ultimately result. The more 
accurate the working model, the more likely P V (Q will approach P T (C), and the better 
the prediction of conflict. Ultimately, this will influence the potential performance of any 
conflict alerting system. 

4.1.2 Errors and Uncertainties in the Trajectory Model 

To obtain a better understanding of the effects from modeling deficiencies, an 
examination can be made by considering the position distributions of the aircraft from 
both the "truth" and working modeled trajectories at a given time t into the future. A 
representative depiction of these position distributions is shown in Figure 4-2. The area 



66 



of T intersecting the hazard, H, is the true probability of conflict P T (C \ t) at the particular 
time, / . The value predicted by the working model is P w (C | f) . In general, however, the 
values need to be computed along the entire path of the probabilistic trajectory and not at 
just one particular instant in time. 

PT(CJt) 



P^(CJt) 



Figure 4-2: Prediction of Conflict at Time t 



The effectiveness of the working model, W, depends on its ability to accurately 
represent the "truth" trajectory, T, and predict the value of the true conflict probability, 
P T (C\t). There are several ways in which the model will differ from the true 
distribution. In the same context as in the previous figure, Figure 4-3 shows the position 
predicted by the model to be in error by a general displacement from the true 
probabilistic distribution at time t. Because of uncertainties, it is natural to expect 
displacement error especially if the prediction time is long. Also, information such as a 
heading change, if not included in the model, will show up as a displacement. Most 
likely, the causes are a result of insufficient information either from sensor equipment or 
misunderstanding of the pilot's expected course of action. 



67 



©* 



*s>i. 



-% 



'** 



^ 



■*-. 



°^ 




w 



Figure 4-3: General Displacement Error in Working Model 

Displacements can easily result in either a missed detection of a true hazard 
(Figure 4-4a) or a false indication of hazard (Figure 4-4b). Certainly, the ability to detect 
all forthcoming conflicts is of paramount importance. An alerting system would be 
basically useless otherwise. The effects of false alarms on the alerting paradigm was 
mentioned earlier in Chapters 2 and 3. 



p T rc/t; 




Pwfc/t; 




a) Missed Detection by Model 

^w(Q = o,p T (C) > o 



b) False Detection by Model 

P*(Q > o, p t (C) = o 



Figure 4-4: Incorrect Predictions from Displacement Error 



68 



Genera] displacement errors are more likely to occur when a single path 
projection method is being used. The reason lies in the fact that the approach does not 
allow for variability in the predicted path. Even small inaccuracies in sensor readings 
(e.g. velocity, bearing) or wind variations can render significant displacement errors on 
the order of a mile in just 10 minutes. Generally, the approach used to compensate for 
the uncertainties and variations is to define a safety buffer zone about the predicted 
positions along the path. The net effect is much like a single tubular trajectory of 
constant width as shown in Figure 4-5. A conflict would then be declared if the hazard is 
predicted to pass within the specified distance from the modeled path. The main idea is 
contain some or most of T with the buffer region to compensate for the uncertainties of 
the future path. 



Increased 
Buffer Size 




Conflict 
Predicted 



Figure 4-5: Safety Buffer Solution to Single Path Projection Approach 

Differences in the shape and size of the distributions can also be expected due to 
uncertainties and incorrect modeling assumptions. In Figure 4-6, the working model's 
position distribution is shown much smaller than the true distribution, indicating not 
enough of the uncertainties were accounted for. Again, this situation would appear to 



69 



occur often in the single path propagation approach. This type of occurrence can easily 
result in missed detection of the true hazard unless the conflict is very nearby and with 
high probability of conflict. 

p T (c/t; 




Figure 4-6: Modeled Distribution Too Small 

Missed Detection by Model {P^(Q = 0, P T (C) > 0} 

In a similar fashion, the modeled distribution could end up being much larger than 
the true one (Figure 4-7) leading to excessive predictions of conflict when none exists - a 
condition leading to needless nuisance alarms. The worst case conservative approach 
discussed earlier is a possible example of this happening. 




P*(CJt) 



Figure 4-7: Modeled Distribution Too Large 

False Alarm {P V (C) > 0, P T (C) = 0} 



70 



Errors from the working model just add to the difficulty already present from the 
uncertainties within the "truth" model, T, and will only lead to higher rates of erroneous 
alerts or missed detection of conflict. The performance of the trajectory model can be 
defined simply as the difference between its predicted probability of conflict and the true 
probability of conflict. 

AP(C) - %(C) - P T (C)\ (4.1) 

In actual application, the working trajectory model is what is used in the conflict 
analysis. Whenever AF(C) * 0, additional errors in the hazard assessment are incurred 
and only increase the difficulty in deciding the appropriate action to take. Ideally, one 
would prefer the working model to match the uncertainties in T, but usually this will not 
be the case. Methods which only set P^(C) to be either or 1 will cause some important 
information to be lost in the analysis, especially effects of the inherent uncertainties in the 
future trajectory on alerting performance { P(FA) and P(SA)}. The significance of 
calculating these parameters is not only to have the ability to estimate the performance of 
the alerting design, but also to examine the benefits from reducing the various levels of 
uncertainty in the trajectory prediction. 

4.1.3 Effects of Modeling Errors on Performance Estimates 

The effect of errors in the model can severely hamper and alter the design and 
analysis of the alerting system. A poor model does not represent the true situation at 
hand and can lead to uninformed decisions based on inaccurate data. It becomes much 
more difficult to fully assess the current conflict state and set an appropriate threshold. 
The SOC curve would be deceiving and may show the conflict situation looking better or 
worse than it really is. Without decent trajectory models, setting good alerting thresholds 
then becomes an achievement merely by chance. 



71 



In cases where uncertainty in W is modeled as smaller than in T, the faulty SOC 
curve will show up as being better (moving more toward the corners). Recall that when 
there is no uncertainty, the operating point must lie in one of the 4 corners. This gives a 
false sense of security that does not really exist. The single path modeling approach has 
basically this net effect. It assumes very little, if any, variability in its path prediction; 
thus it is very forthright in its estimate of conflicts - it either exists or it does not (binary). 
It is a mere simplification of the problem that if that situation (i.e. no uncertainty in the 
trajectory) was truly reflected in the conflict, then alerting would be greatly simplified. 

In an alerting system, the threshold is based on the working model, W, being 
employed by the logic, but the actual performance is due to the probabilistic trajectory, T. 
Thus, two SOC curves really exist - one based on the designed tradeoff (due to W) and 
the other on the true tradeoff (actual T). A point on the W curve maps to some point on 
the T curve as shown by the dashed line in Figure 4-8 (the mapping need not be one-to- 
one, however). An inaccurate model may then induce an alert with results not expected 
by the analysis using the modeled W trajectory. In the particular case of Figure 4-8, the 
actual performance of the system will have a higher false alarm rate and lower successful 
alert rate than the intended design. 



1.0 



P(SA) 




P(FA) 



1.0 



Figure 4-8: Effect of Underestimating Uncertainty 



72 



Overestimating the uncertainty of the "truth" trajectory, T, can also lead to 
problems. When the uncertainty in W is taken to be larger than in T, the SOC analysis 
will have the appearance of being in a more difficult conflict situation than it really is. 
When more uncertainty exists, the curve will tend more towards the diagonal line from 
the lower left corner to the upper right (Figure 4-9). The effect may cause one to alert 
earlier than necessary believing successful avoidance would be jeopardized otherwise; 
when in fact, it would be the false alert rate that is really compromised (alerting too early 
increases risk of false alarms). 



1.0 



P(SA) 




P(FA) 



1.0 



Figure 4-9: Effect of Overestimating Uncertainty 

4.2 (W = T) Reducing Trajectory Modeling Errors (Increase Accuracy of Model) 

The general effects of modeling errors (which were just explained) can lead to 
misrepresentation of the actual hazard at hand. In order to effectively make well 
informed alerting decisions, one must have an appropriate assessment of the current 
threat situation . The objective is to increase the prediction accuracy of the trajectory 
model, W, used by the alerting logic. This is achieved by increasing the degree to which 
W = T. 



73 



It is extremely important to keep in mind that the "truth" model of the trajectory. 
T, is still considered a stochastic process with a random outcome for a single initial 
realization . If this were not true, then T would not have uncertainties, and the above 
relation would imply that an ideal alerting system could almost always be designed (no 
false alarms, always safe avoidance). The statement W = T simply denotes that the two 
trajectory distributions should exhibit the same parametric characteristics of a random 
process such as the mean, standard deviation, and dispersion form. It indicates the desire 
to properly match the alerting logic's estimate of a hazard to the actual, and allows the 
accurate assessment of the true probability of conflict, P T (C), and thus P{FA) and 
P(SA) as well. The SOC method requires a curve to match as best it can to the true 
conflict situation, else the curve would be wrong and misguided as is shown in Figure 4- 
10. This is true of any performance analysis method. 

The natural randomness in the problem still has to be dealt with, however. For 
any single event, such as an alert, the outcome is still probabilistic even if W = T. Take, 
for example, a flip of an unbiased coin. Even if the model of the outcome is perfect (50% 
chance of heads, 50% chance of tails), for any given toss, there is a 50% chance of being 
wrong - akin to a P(FA) = 0. 50. For any given alert, the outcome of succeeding or 
giving a false alarm is still probabilistic (the trajectories are random processes with an 
associated variance). This is the notion of inherent uncertainties and will be dealt with in 
the next section. 



74 



1.0 



P(SA) 




1.0 



P(FA) 



1.0 



P(SA) 




P(FA) 



1.0 



a) Poor Model 



b) Better Model 



Figure 4-10: SOC Comparison 

For now, the concentration will remain on increasing the trajectory modeling 
accuracy, W = T. In Figure 4-1 1, a schematic for two approaches to this theme is given: 
driving W toward T, and driving T toward W. 




Wh>T 



Th-»W 



Probabilistic Trajectory Model 
Accurate Information of State 

Trajectory 
Continuous Update of W 



• Conformance Boundaries 

• Limit Operation 



Figure 4-11: Approaches to Reducing Modeling Errors 



75 



4.2.1 (W -> T) Drive W Toward T - Improve Trajectory Modeling 

4.2.1.1 Utilize a Probabilistic Trajectory Model 

The notion that W should equal T would imply that W should also exhibit 
characteristics of uncertainties, thus leading to a probabilistic modeling approach for the 
dynamic model of the alerting system. It is just one way of improving the accuracy of the 
conflict prediction process. This is the direct approach of dealing with the uncertainties - 
by independent modeling of the various sources and causes, quantitatively and explicitly. 
This includes uncertainties in flight path due to human factors and possible blunders. 

The single path and worst case approaches, in general, are more likely to give 
inaccurate predictions of conflict (modeling errors), and must therefore account for the 
inaccuracies with other methods of instilling uncertainties into the analysis as explained 
in Chapter 3 (e.g. iterative modification of the threshold parameters and criteria). The 
goal of using a probabilistic trajectory for W is thus to directly reduce the chance of 
modeling errors in the dynamic modeling phase of Figure 3-2. 

4.2.1.2 Utilize Sufficient and Accurate Information of the State Trajectory 

In developing a probabilistic trajectory model, W, one would prefer to have it 
coincide as much as possible with the random characteristics of the true trajectory, T. In 
order to do so and thus reduce the amount of error in the conflict prediction, sufficient 
and accurate information regarding T is necessary. In other words, it would be preferable 
to have more and better prediction information, assuming it is not in error. Else, the 
overreliance on a false assumption of the trajectory model can lead to problems. 

Take for example, the current velocity vector of an aircraft. Using it to predict the 
position 5 minutes in advance would lead to a large displacement error if the aircraft was 



76 



currently banked in a turn. Now adding the bank angle to the prediction would improve 
matters some, but only for a short extrapolation time because it is unlikely the plane 
would continue to bank in full circles. The addition of a commanded heading set by the 
pilot (intent) would be very helpful in this case. 

Though it is highly unlikely that all the parametric estimates of T are available to 
an alerting system for modeling W, a sufficient amount of information should be utilized 
in order to give a reasonable estimate of the hazard situation. Sufficient is subjective and 
really depends on the goal of the system and the specific situation at hand - as it is with 
any type of modeling scheme. 

The amount and type of aircraft state information necessary for the trajectory 
model W is somewhat arbitrary. Past conflict avoidance methods (see Kuchar and Yang, 
1997) have shown the use of a variety of different combinations of state variables. The 
only state variable usually required is with regard to positional data since the conflict 
criteria are based on separation standards. All other state variables would certainly help 
improve the prediction of the future trajectory if utilized appropriately. However, 
sometimes the additional information may not be necessary as the impact on performance 
might only be minimal. It really depends on the situation in which the alerting system is 
operating. 

Take the two examples shown in Figure 4-12. Figure 4-12a shows two aircraft in 
a simultaneous parallel approach, and Figure 4- 12b has two aircraft converging on an 
intersecting waypoint along two separate airways angled at a certain f3 degrees apart. As 
long as the alerting system is designed and limited to a very specific type of encounter 
situation, it is possible to have a reduced number of variables to define the working 
model W. This is because of the implicit information already embedded within the 



77 



specific scenario - the relative bearing in the case of Figure 4-12. It is conceivable that a 
dynamic model may not even be necessary as the relative position between aircraft may 
be sufficient enough to differentiate threat and non-threat situations. This is analogous to 
mapping or regression modeling where the number of variables required to sufficiently 
model a problem increases with the complexity and the uncertainty of the environment. 
If the alerting system is needed to handle many types of possible encounters (such as 
aircraft coming from all directions), then it becomes more necessary to be able to 
differentiate between the situations and more defining variables are required. A more 
thorough discussion on this topic will be given in Chapter 5. 





<* 




a) Parallel Approach b) Intersecting Airways at Waypoint 

Figure 4-12: Example Encounter Situations 



78 



In general, the more sensors and information available about the aircraft and 
hazard (such as from data-link), the better the ability of the system logic to match W to 
T. As previously discussed, the lack of certain information has been a recurrent problem 
in several aviation related alerting systems such as the GPWS and TCAS. However, 
caution should be used in assuring the information is utilized properly so as to truly 
match W = T. Else false reliance on the data will only lead to misguided decisions. The 
single path approach to modeling, for example, usually assumes a constant velocity even 
in vertical maneuvers, yet the velocities have been observed to fluctuate significantly 
especially in climbs and descents. Also, projecting a vertical maneuver over a long 
period of time is usually meaningless since the aircraft would be expected to level off 
sooner or later. This leads to the following sub-section of utilizing intent information to 
further improve prediction accuracy. 

The use of additional intent information has been a recent topic in conflict 
avoidance lately, though an exact definition as to its meaning has not been thoroughly 
presented. For this thesis, intent will be taken as any information that will support the 
prediction accuracy of future aircraft positional states . Commanded heading, level-off 
altitude, or next waypoint could be considered intent information, though they must 
accurately depict the current situation in order to satisfy W — > T. 

Intent can also have uncertainties and must be adequately accounted for or else its 
limitations must somehow be expressed in the analysis. Take, for example, an aircraft 
(B) with the expected intention of leveling off 1000 ft above another aircraft (A) as 
shown in Figure 4-13. If aircraft B only has a 75% chance of following through with the 
level-off, the alerting system must acknowledge this information and decide when to alert 
to supply sufficient clearance to avoid the hazard and still hold false alarms at bay. 



79 



T 



A 



1000 

L 



/ 



Figure 4-13: Uncertainty in Intent Information 

Another example is the parallel approach scenario where aircraft are landing 
simultaneously on two different but closely spaced runways. The intent is obvious - fly 
straight, at a low rate of descent, down to the respective runway. However, if this intent 
were 100% foolproof, then aircraft could be allowed to be spaced extremely close 
laterally (subject to wake vortex constraints) if sufficient guidance and sensor systems 
were onboard (i.e. use of Differential GPS). The problem occurs when human pilots 
blunder or when weather, wind, or equipment failure become an issue. Spaced too close, 
the parallel runways may not provide the aircraft with adequate time to avoid a conflict if 
the intent is not followed. Overconfidence in the use of intent information can lead to 
trouble if it does not accurately depict the true uncertainty of the trajectory, T. 

4.2.1.3 Update the Working Trajectory Model 

In order to consistently obtain accurate predictions of the current threat situation, 
the dynamic model, W, should ideally be continuously adjusted to match changing 
conditions. It should really be adaptive to different state or intent information that 
becomes available. Recall that the TCAS Tau Criteria thresholds actually change 



80 



depending upon altitude, climbing or descending, and direction of flight (see Figure 3-3). 
Though these values were designed using hand-picked simulation scenarios, they prove 
the necessity of adjusting for varying flight conditions. 

Take an example where an aircraft begins to enter airspace where inclement 
weather is abundant. It becomes more likely now that the trajectory of the aircraft is 
more variable and may change course to avoid certain areas. The dynamic working 
model, W, should then allow for higher possibilities of aircraft varying off their present 
course. It should be updated accordingly so that W matches T as much as possible to 
reduce prediction errors. Assuming the aircraft will maintain a current heading, steady 
level flight when that behavior is unlikely will only lead to additional modeling errors. 
At the other extreme, over-modeling of uncertainties can also lead to incorrect 
predictions. 

It is likely that the intent of an aircraft will change many times over the course of 
a flight. To assure proper conflict detection, the dynamic model should be coupled to the 
updated information and change accordingly. If possible, it would be best if a pre-check 
of the new intent trajectory was examined for immediate conflicts before allowing the 
intent to be carried out. However, this might be out of the scope of an alerting probe 
concept and more toward a centralized Air Traffic Management effort. 

4.2.2 (T -> W) Drive T Toward W 

4.2.2.1 Utilize Conformance Boundaries 

One way to improve algorithm prediction accuracy is to enforce conformance of 
the aircraft trajectory through the use of secondary alerts. The princ i ple is to alert not 
because of a conflict, per se. but because the aircraft is deviating from the model that 



81 



would allow the conflict lo gic to more easily predict the outcome of a future conflict . 
This method can be especially useful when aircraft are closely spaced together and 
expected to fly a certain path or pattern. It makes it much easier for the conflict 
avoidance algorithm to work as expected with less error. A very simple working model, 
W, could be utilized (e.g. a single path projection) and the associated aircraft could be 
forced to conform to a specific route. However, it can be somewhat constraining to the 
flight of the aircraft since it is bound to restrictions on its path. 

An example of conformance monitoring is shown in Figure 4-14 where an aircraft 
is expected to follow a specific path marked by conformance bounds. The boundaries 
usually provide some leeway for fluctuations from both environment, aircraft, and human 
variations. Both horizontal and vertical margins may be delineated so as to contain the 
path of the aircraft in 3-D. With future advancements, 4-D conformance monitoring 
(adding time) may be possible. 

If the boundaries are exceeded by the aircraft, then an alert can be given to warn 
the pilot to return back on track. It is also possible to have a new path recomputed once 
the aircraft deviates from the original routing. 

The usefulness and simplicity of the conformance approach is marked by the 
number of prototype conflict systems which utilize this method. Examples include the 
Center-TRACON Automation System (CTAS) [Isaacson and Erzberger, 1997], 
EUROCONTROL's Medium Term Conflict Detection (MTCD) system [Vink et al., 
1997], the User Request Evaluation Tool (URET) [Wanke, 1997; Brudnicki et al., 1977], 
and the No Transgression Zone (NTZ) for parallel runway landings [Shank and Hollister, 
1994; Carpenter, 1996]. Even motor vehicles on the inter-state highway system utilize a 
form of conformance check with the lane bumps indicating deviation outside of the 



82 



current lane. Notice that a worst case approach to highway design would probably 
require individual lanes to be over a hundred feet in width, yet they are usually only 
about 12 feet wide. 




Planview 



Profile View 



Figure 4-14: Example of Conformance Boundaries 

To some extent, the current ATC system with its rigid airway structure can be 
thought of as conformance monitoring by the human controllers. Aircraft are maintained 
in their correct flight paths and place heavy trust in the controller to alert and clear them 
out of danger. 



83 



4.2,2.2 Limit Operation 

Sometimes there are instances in which an alerting system may not work well 
and the thus the operation of the system will be limited to use for certain types of 
encounters only. For example, the TCAS algorithm is intended for near term conflicts on 
the order of less than 1 minute. Extending the original algorithm to long term encounters 
would likely incur deteriorating performance as shown back in the simple example of 
Figure 3-4. Without accurate bearing information, the TCAS logic simply cannot obtain 
satisfactory conflict prediction results in long term encounter situations [Burgess et al., 
1994]. Thus the alerting system is limited to those operations in which it is capable of 
handling or predicting. 

In another example, certain flight procedures are sometimes tailored to meet the 
limitations of the alerting system so that it may obtain adequate performance (e.g. 
minimize false alarms). For instance, in a situation often referred to as the "Seattle 
Encounter" [Drumm, 1996], an intruder aircraft is descending toward a TCAS equipped 
aircraft but levels off at an altitude above it. TCAS initially predicts a collision and 
issues an alert for the TCAS aircraft to climb. However, the intruder levels off above the 
TCAS aircraft resulting in a false alarm (the alert was not necessary). Not only has a 
false alarm occurred, but the situation can actually induce a hazard with TCAS aircraft 
climbing into the intruder. Because of the tendency to produce false alarms and a 
possible hazard situation, it has been recommended that aircraft slow their rates of 
descent when approaching their final altitude [Mellone and Frank, 1993]. The effect is a 
modification of T in order for the alerting logic to correctly anticipate (or predict) the 
leveling off of the intruding aircraft. 



84 



4.3 Reducing Inherent Uncertainties (Reduce Uncertainty in Future Trajectory) 



Once the working dynamic model, W, is presumed to be a good representation of 
the "truth" trajectory, T, the decision to alert becomes a tradeoff between false alarms 
and successful avoidance. Assume the SOC plot of the current alerting design looks like 
that shown in Figure 4- 15a. The curve designates the performance expected for various 
possible alerting points along a particular nominal path. 



1.0 



P(SA) 




1.0 



P(SA) 



P(FA) 



1.0 




P(FA) 



1.0 



a) High Uncertainties 



b) Reduced Uncertainties 



Figure 4-15: Improved Performance Depicted in SOC Plot 

The plot in Figure 4- 15a shows a system that would not have great performance 
because of the difficulty in setting a threshold without sacrificing either P( FA) or 
P(SA). The effect is likely due to the amount of uncertainty in the true trajectories of 
the nominal (N) and/or avoidance (A) paths. It is an inherent characteristic of this 
alerting system given the random processes involved. This will be termed the inherent 
uncertainty of the system. In Figure 4- 15b, there is less inherent uncertainty in the 
system. 



85 



To be able to improve the performance of this system, the uncertainties in the 
trajectories must be reduced so as to increase the prediction outcome for a single event in 
a random process environment. The situation is analogous to reducing the variance (cr 2 ) 
in a prediction. Take the coin flip example again. Inherent uncertainty is highest with an 
unbiased coin (the variance a 2 = p(\ - p) is maximum at p = 0.5). If the coin was 
biased toward 90% chance of heads and only 10% chance of tails, then the chances of 
predicting a head or tail with a single flip would be much better (variance is lower). The 
inherent uncertainty of this biased coin would be less than the unbiased one. The effect is 
basically to make the outcome more deterministic . 

In terms of the SOC plot, reducing inherent uncertainty would drive the points to 
the outer perimeters making it easier to determine more suitable threshold locations. In 
the limit of no uncertainties, all possible threshold locations must exist at one of the 
corner positions as explained back in Chapter 2. For example, Figure 4-16 shows two 
aircraft on a nominal direct collision course in a 90° encounter situation. The SOC plot 
shows the deviation of the curve from the ideal position as the bearing accuracy of both 
aircraft is varied from a = 0° to a = 5° (normal distribution) in 1° increments. This 
simulation is based on a point-mass model of the aircraft flying along a straight path with 
the different random heading errors. The standard 5 nautical mile separation was used as 
the definition of conflict in this example, and the avoidance maneuver was a 20° turn 
after 10 seconds. 



86 



a= 1 ( 



P(SA) 





Figure 4-16: Example of Increasing Uncertainty (Heading) on SOC Curve 

Inherent uncertainty is a function of the distribution and size of the "truth" 
trajectory relative to the hazard at the time of the prediction . The "truth" model can refer 
to either the nominal (N) or avoidance (A) paths. Analogous to the coin flip example, 
high uncertainty in the future path makes it more difficult to accurately predict the 
outcome for any single event. This can be seen in Figures 4-15 and 4-16 where higher 



87 



levels of uncertainty make it more difficult to determine whether or not a conflict will 
occur. In the limit of no uncertainty, the conclusion is deterministic and binomial. 

In the following sections, various ways of reducing the inherent uncertainty and 
thus improving alerting performance are given. It should be realized that these methods 
are not newly proposed ideas, but simply brought together to show that they all really fall 
under the category of reducing the inherent uncertainty of the underlying random process. 
The purpose, of course, is to increase the chances of correctly predicting the outcome of a 
conflict and the ability of avoiding it (i.e. to make results more deterministic). 

Much of the effort is in reducing the uncertainty in the future track. Sensor 
inaccuracies, autopilot control behavior, weather changes, and variable winds are all 
contributing factors. However, much of the uncertainty involving the future path of an 
aircraft is a result of the human pilot in control and the decisions he/she makes; basically, 
not knowing what the pilot is going to do. Because of the high variability between 
humans, it is unlikely the course of action followed by each pilot would be the same in 
any given situation. Several methods are possible to help decrease these variabilities and 
are discussed below. 

4.3.1 Restrict Flight Path 

The position of an aircraft traveling at 450 knots with the ability to bank 30 
degrees would, in just 6 minutes, encompass a nearly circular region of 45 nautical mile 
radius about its current position [Andrews and Hollister, 1997]. With a possible climb or 
descent rate of 2000 ft/min, the volume engulfed would reach 12,000 feet above and 
below the aircraft. Thus, in order to reduce the number of possible trajectories, some 
form of restrictions need to be placed on the flight path. Constraints on the trajectory 
would effectively narrow the region of uncertainty (see Figure 3-5b) and produce a more 



88 



definitive outcome on whether or not a conflict would occur. In other words, P(C) , as 
well as P(FA), would shift more toward 1 or 0. 

This appears to be the simplest method and is currently implemented for enroute 
traffic today. Aircraft follow along in pre-assigned airways and at defined altitudes 
making it easier for pilots and controllers to predict potential conflicts. The system is not 
without shortcomings however. A tremendous amount of airspace is left under-utilized 
leaving many to believe a more efficient means of operation is necessary to handle the 
current congestion today and the increased air traffic demand in the future. 

There is a movement under way to relax the current system of rigid airways and 
in-trail spacings to increase flexibility for more efficient operation. This notion of a Free 
Flight environment with less restrictions in course adjustments has been a source of much 
research and discussion lately [RTCA, 1995; Phillips, 1996]. Of course, the increase in 
flexibility will nonetheless increase the potential for more difficult conflict encounters 
and added uncertainty. The final report on Free Flight implementation by the Radio 
Technical Committee on Aeronautics [RTCA, 1995] suggests conformance to maneuver 
limits as an interim solution. For example, aircraft might be restricted to a 20 degree 
heading change within a 15 minute time period. Constraints placed on such maneuvers 
have been shown to significantly reduce the rate of conflict encounters [Andrews and 
Hollister, 1997]. 

4.3.2 Establish Protocol (Training, Rules of the Road, Convention) 

The use of established rules-of-the-road with training can help increase the 
likelihood that pilots will follow specific patterns of flight behavior. The uncertainty in a 
pilot's action, and ultimately the aircraft's flight path, can thus be reduced. For example, 
rules-of-the-road regarding the right-of-way (depending on relative aircraft positioning) 



89 



may help determine the expected trajectories each aircraft may fly. The importance of 
taking humans into account as a large contributor of uncertainties cannot be overlooked. 
A study of commercial jet accidents resulting in a complete hull loss has placed the flight 
crew as the primary cause in 70% of the accidents from 1988-1997 [Boeing, 1998]. 

Also, requiring communication between pilots or the controller prior to making 
any course correction would inevitably help reduce uncertainty in the expected flight 
trajectory. This has the notion of intent information but is explicitly forcing the pilots to 
decide and communicate immediate changes in intent before allowing the action to be 
taken. The idea certainly has merit and some have even proposed a new ATM 
environment where principled negotiation between pilots and controllers is the basis of an 
established protocol [Wangermann and Stengel, 1994]. The explicit communication of 
aircraft intent is a possibility for reducing the set of likely trajectories one can expect 
when deciding on potential conflict situations. 

4.3.3 Introduce Better Equipment 

Even with all the restrictions placed on the pilot to maintain a specific course, 
inevitably there will still be some random uncertainties involved. As mentioned 
previously, sensor inaccuracies, variations in wind, autopilot capabilities, and inherent 
aircraft dynamics all play a role in introducing variability in the aircraft trajectory. 
Empirical data from observing current aircraft maintaining a given track have shown 
deviation perpendicular to the nominal track (cross-track) that is approximately Gaussian 
on the order of 1 nautical mile standard deviation [Paielli and Erzberger, 1997]. 
Fluctuation in speed, due primarily to wind effects, was also observed to be upwards of 
15 knots (one standard deviation) normally distributed [Paielli and Erzberger, 1997; 
Wanke, 1997]. Current technological advances in sensors (most notably the Global 



90 



Positioning System, GPS) and in computer hardware and software may be able to provide 
improved accuracy on future air transports. Future developments may also allow for 4-D 
path following capabilities. 

The introduction of highly automated equipment onboard the aircraft is not 
without its critics, however, especially if it involves automation in the cockpit and new 
allocation of piloting tasks. The addition of new technology may likely alter the way 
aircraft are flown and may introduce new modes of human error. A more thorough 
examination of the underlying human-machine process would thus be necessary and is an 
area of continuing research. 

4.3.4 Delay Alert (Minimize Size of T at Alert Time) 

Though reducing the uncertainty in the trajectory would be ideal, sometimes it is 
not always possible. It may be impossible to alter the operating environment or add 
sophisticated instruments and sensors to reduce the variability in the flight path. To the 
alerting system designer, the only remaining alternative for decreasing the unnecessary 
alert rate is to delay the alert as long as possible. In effect, the delay is used to wait for a 
more definitive determination of a conflict before proceeding to warn the pilot and/or 
controller. This has the effect of minimizing the uncertainty in the trajectory at the time 
of the alert (the position distribution of T is smaller). Usually, the uncertainty in the 
position of an aircraft will grow with time (the exception might be 4-D trajectories). This 
effect is depicted in Figure 4-17. The shorter the prediction time, the smaller the 
uncertainty. Examination of Equation 2.2 shows that as P(C) tends towards 1, P(FA) 
will approach 0. Delaying the alert also has economic and reduced workload benefits as 
well, since nuisance alerts incur costs from maneuvering unnecessarily. 



91 



$- = = ::<£>___ (^ 




I ^ Prediction time into the future 

Figure 4-17: Increase in Uncertainty Due to Prediction Time 

Of course there is a tradeoff to all this; namely that the delayed alert may be 
placing the aircraft at a higher risk of danger. The pilots would have less time and 
options to undertake avoidance action when the alert is given. It may also be possible 
that the human pilot will disagree with the delayed timing of the alert, thus resulting in 
mistrust of the system as well [Pritchett, 1999]. 

4.4 Investigate Other Avoidance Maneuvers 

If the performance of an alerting system still proves insufficient to meet design 
goals given the methods discussed above, then another option is to utilize a different 
conflict resolution. The determination of P(SA) discussed in the last chapter is specific 
to a given avoidance trajectory, A, which, as a reminder, still includes uncertainties and is 
a stochastic process. Various horizontal or vertical maneuvers could be tried including 
the addition of speed control or cooperative maneuvering between aircraft, or a more 
severe or drastic maneuver could be employed. Figure 4-18 shows how a larger climb 
rate could be used to increase the chances of avoiding another aircraft descending into a 
conflict situation. 



92 



*"«>«i^egj^;^> 




a) Original Climb Maneuver 



*— ■M^piy^ 




b) Increase Climb Rate 

Figure 4-18: Utilizing a Different Maneuver to Avoid Hazard 

However because of human involvement, there may be higher uncertainties 
associated with more complex maneuvers. The result may be worse performance than 
expected because of larger uncertainties in the avoidance trajectory, A. The pilot may 



93 



also be less willing to perform complicated or severe maneuvers if for some reason 
he/she did not deem the threat to be real. In any case, it is always good to have viable, 
multiple avoidance options available to the pilot during emergency situations for added 
safety. 

4.5 Design Issues 

Reducing modeling errors and reducing inherent uncertainties results in one effect 
- it increases the ability to predict the outcome of a single stochastic event. The idea is, in 
effect, making the process more deterministic and thus increasing the performance of the 
alerting design. Any of the methods discussed above can be used individually or in 
combination. 

To improve performance, one can either reduce the uncertainties in T, and then 
design an alerting system to match (reduce T and W -> T), or one can reduce the 
uncertainties in W and then enforce trajectory conformance (reduce W and T — > W). 
Either way, the optimal performance (in terms of conflict alerting) will occur when the 
true trajectory is deterministic (no uncertainty, 4-D path). The current ATM environment 
with its heavily structured airway system and ATC monitoring appears to allow for easy 
predictions of localized conflicts. There are restrictions and constraints in the system 
which make the trajectory more deterministic in many cases. 

The notion of Free Flight seems to be contrary to this idea, however. It is 
probably unlikely that aircraft would be allowed to fly randomly about in the airspace. 
As explained in this chapter, the larger the uncertainties in the trajectories, the more 
difficult it is to determine and prevent possible conflicts. In Andrews and Hollister 
[1997], an analytical model was used to determine that a significant increase in conflict 
rate would result if aircraft maneuvering was left unconstrained. An increase in the 



94 



number of conflicts coupled with a drop in alerting performance could only lead to 
problems. The end result would be a loss of efficiency and increase in workload rather 
than the more efficient system originally sought after. 

One way around this is to have aircraft provide and confirm intent information 
prior to any changes in the current intended course. The information must be accurate, 
and some form of conformance monitoring would be helpful, else a modeling error would 
occur in the alerting system. Of course, the system should check for possible conflicts 
with the new path prior to allowing the changes to be made. 

4.6 Summary 

In this chapter, the problem of collision avoidance was recast as a problem of 
prediction in the presence of uncertainties. The importance of trajectory modeling was 
examined as a major source of errors in the outcome of conflict alerting. Without 
uncertainties, the problem would be greatly simplified. The issue of improving 
performance then becomes one of increasing prediction accuracy of the conflict situation 
and its resolution. 



95 



96 



Chapter 5 

A Probabilistic Perspective to the Alerting Design 
Process 



In previous chapters, the use of iterative, ad hoc adjustments was discussed as a 
common method for setting threshold parameters. This approach to alerting design can 
be thought of as an implicit method of dealing with uncertainties. It is implicit because, 
as will be shown, the simulations are indirectly accounting for the uncertainties in the 
encounter situation. 

In this chapter, a probabilistic perspective to the alerting design process will be 
examined and discussed in detail. It provides a different view to the current practice of 
locating suitable thresholds based on iterative searches using trial and error. Probabilistic 
elements will be shown to be embedded within the ad hoc approach, and thus the design 
is, in essence, influenced by probabilistic or stochastic concepts which may at first not 
appear to be present. A new, direct method will also be proposed to overcome some of 
the limitations in the ad hoc approach. 

5.1 A Probabilistic Perspective to the Ad Hoc Approach 

During the operation of an alerting system a discrete decision is made to either 
remain silent or issue an alert to warn the human operator. Typically, this decision is 
based on metrics and whether or not they exceed critical values defining the alerting 
thresholds. The manner in which these threshold parameters are set often follow an 
iterative, ad hoc approach as explained back in Section 3.3.2. 



97 



To reiterate, Figure 5-1 diagrams the general, iterative design process often seen 
in setting alerting threshold parameters. It usually begins with some working dynamic 
trajectory model (W) upon which the alert metrics are used to describe the encounter 
situation. The metrics can be thought of as forming the state-space of the alerting system. 
In some instances, the choice of metrics may be constrained by the type of information 
available, in which case the choice of working models may also be limited. For example, 
limited sensor information in the TCAS system allows the use of only range and range 
rate in the formulation of its horizontal alerting criterion, and a single path working 
model is used. Attempts to include information on the expected miss distance to improve 
prediction accuracy could not be achieved with the current equipment because of 
difficulties in estimating relative bearing angles between aircraft [Burgess et al., 1994]. 





/ 


Ak 










/ 






Dynamic 
Trajectory- 
Model 

W 




I Alert;: 
Me t r i c s j Thre sho Id 
1 Criteria 


^ 


Simulation 
J^Modefe"^ 






^ 






■ Igapg-gijgj^ 


mR*?- 










/ 






v 




Modify Threshold 
Until Performance 




Performance 
Measures , 


Performance 
Reqmts 


Meets or Exceeds 
Requirements 






P(l 




ZR 



Figure 5-1: Current Design Process (Iterative Ad Hoc Approach) 



In the common ad hoc approach, the thresholds would likely be initial settings 
(from some combination of analysis and user expertise), but usually require some fine 
tuning from test scenarios through simulations as shown in Figure 5-1. As a reminder, 



98 



X A represents the alert space of the threshold criteria (Section 2.2.3). The number of test 
scenarios could run in the tens or hundreds of thousands , and would be used to evaluate 
the performance {e.g. P(FA) and P(SA) } over the various simulation runs. For 
example, changes to the original TCAS design were tested using over 1 million 
hypothetical encounter geometries [Miller et al., 1994]. 

Adjustments and modifications to the thresholds or even the metric variables (the 
feedback path in Figure 5-1) would then proceed until a satisfactory setting is achieved. 
The values used to determine performance can themselves form a state-space which will 
be denoted as Z . The symbology, Z*. in the figure is meant to represent the 
performance requirements that need to be satisfied by the alerting system. For example, 
Z R could be the region of performance state-space where P(FA) < 0.10 and 
P(SA) > 0.95. If these requirements are not met, then the parameters in the threshold 
are iteratively adjusted until satisfactory performance is achieved. 

There are some very important insights when the process is portrayed in the 
manner shown in Figure 5-1. The depiction looks quite similar to a neural network 
scheme where the simulations define a "truth" model from which the thresholds are 
adjusted to meet or optimize performance parameters. 

From previous discussions, W is the working model being utilized by the alerting 
logic to predict conflicts; and T represents the "truth" model of the trajectory which 
determines the actual P(C). T is still a random process and can be considered as an 
ensemble of individual trajectories. Thus, the simulations, being a collection of 
scenarios, can be interpreted as representing T. Taken together, the simulation scenarios 
are a probabilistic distribution (though discrete, it could be inferred to be a sample from a 
continuous distribution). The scenarios represent the variability or uncertainty in the true 



99 



aircraft trajectories. If the distribution of the scenarios were changed, the threshold 
values would likely change also. 

The performance measures, as it turns out, are commonly error and success rates 
(such as false alarms, successful alerts, or missed detections) as described earlier. Notice 
that even if the simulation scenarios are changed, the specification for the minimum level 
of performance will likely remain unaltered. If a design must achieve 99% success with 
less than 5% false alerts, then those requirements would not change with different sets of 
simulation runs. If they cannot be met, then changes in the metrics or threshold settings 
must be amended to obtain satisfactory results against the chosen scenarios. Thus the 
alerting thresholds can really only be considered to be indirect measures of the alerting 
performance. In essence, the design p rocedure is a mapping of the threshold metrics to 
the performance measures . 

For example, if the thresholds were based on range ( r), range rate ( r), and 
predicted miss distance (m), then the probability of a false alarm and a successful alert 
would be some function of these variables, P(FA) = f(r, r, m) and P(SA) = g(r, r, m), 
respectively. The functions, f() and g(), would be specific to the scenarios used. In 
general, P(FA) and P(SA) can be expressed as a mapping from the threshold settings, 
X A , to the performance metrics as denoted in Equations 5.1 and 5.2. 

P(FA) = f(X A ) (5.1) 

P(SA) = g(X A ) (5.2) 

where: X A = threshold metric settings (alert space region) 

/() = false alarm mapping function (scenario specific) 
g() = successful alert mapping function (scenario specific) 



100 



The governing functions, f() and g(), are typically not explicitly expressed or 
defined during the design process. Thus, it can become nearly impossible to predict the 
outcome or even make informed comparisons between different sets of simulation runs. 
This approach can lead to ambiguities and is really an indirect method of including 
uncertainties missed by W in the dynamic modeling stage. 

5.2 A New Direct Approach 

In Figure 5-2, a new approach to the alerting process is presented. As opposed to 
Figure 5-1 , the intermediate block of metrics is removed in favor of directly estimating 
the probabilistic measures in which to make the alerting decision. The idea is to bypass 
the middle step since the results of the scenario simulations are being utilized to adjust 
the parameters in the threshold metrics in the first place . In Figure 5-1, the notion is that 
the probabilities { P(FA) , P(SA) } are functions of the set threshold metrics; thus in terms 
of the cause-effect relationship, the set thresholds determine the probabilistic 
performance. The concept is somewhat reversed in Figure 5-2 where the probability 
values determine when and where to alert. Because of the feedback in Figure 5-1 to meet 
pre-determined probabilistic requirements, it is really the probabilistic parameters that 
drive the threshold placement. As mentioned before, in effect, the threshold placement is 
really just a function of the probabilistic performance measures and the probabilistic 
distribution of simulation scenarios. 

In the concept of Figure 5-2, the working trajectory model is made to match as 
closely as possible to the "truth" model (W = T). In doing so, the alerting algorithm is 
obtaining a direct prediction of the likelihood of conflicts and the ability to avoid them. 
These values can then be utilized as the threshold metrics in the state-space of Z with the 
alerting criteria denoted by Z* (performance requirements). 



101 



Dynamic 

Tra j e.ct.ory 

Model 



W =1T 



Performance; 
Metrics 

P(SA) 
P(FA) 



Alert 

Threshold 

Criteria 



Optional 



Metrics 



o: 



I Alert 
^Threshold 
f Criteria 



Figure 5-2: New Direct Approach 

As indicated by the dashed line in Figure 5-2, it may be possible to map the 
probabilistic values to other metric variables in another state-space, X, with alert space 
X A . In doing so, this may allow for easier interpretation of the threshold logic since 
probabilistic values may not always be clearly understood by the human operator. 
However, this may not always be possible unless a one-to-one mapping of variables 
exists. The problem is akin to the same type of dilemma associated with inverse 
kinematics. 

5.3 Implications from a Probabilistic Perspective 

The approach shown in Figure 5-2 requires a direct modeling of the uncertainties 
in the trajectories of the aircraft, which in turn can help determine the impact and 
influence of each source of uncertainty on the alerting performance. This direct link 
gives rise to some very important implications that can have a significant impact in the 
design and analysis of alerting systems. To fully understand the consequences requires a 
detailed explanation of the differences as well as the association between Figures 5-1 and 



102 



5-2. For the remainder of this work, the former method will be referred to as the ad hoc 
ap proach , and the latter as the direct approach . 

There are some very important ramifications to notice here. First, the ad hoc 
method tends to develop a global threshold setting as opposed to a situation-specific 
threshold, one that is individually tailored to the current encounter situation . As will be 
shown, a global threshold tends to exhibit a higher level of uncertainty and reduced 
overall level of performance! Second, the approach is also heavily influenced by the 
distribution of test scenarios used for the simulations . Using an alternative sample of test 
cases could change the performance outcome resulting in a different set of threshold 
values. In effect, the thresholds could be highly biased toward certain conflict conditions 
while ignoring or discounting other possible encounters. Also, a complete rerun of the 
iterative process would be needed again to formulate a new set of threshold parameters. 
Finally, the ad hoc approach can be considered a functional mapping of the performance 
state-space. Z. to a different domain of state-space. X . Though this can be advantageous 
under certain conditions or applications, it can also be a disadvantage when the 
complexity of the mapping is considered. 

5.3.1 Global Design vs. Situation-Specific Design 

A global design refers to a process in which the simulations used to set the 
thresholds are based on an aggregate mixture of different encounter scenarios; while a 
situation- specific design only considers the current situation at hand. To illustrate this 
concept, consider the following example. Take an automobile company designing a 
"world" car to be sold globally under one baseline model. There are some obvious 
advantages to such a tactic, of course, since it may be minimizing labor, parts, design, 
and advertising costs. Suppose this company gathered the following data shown in 



103 



Figure 5-3a on the height of drivers in country A, B, and C. One of the design (or 
performance) requirements is to be able to seat 95% of the drivers comfortably. 



Hi, ox, n\ 




V2, <*2, n 2 



VG, Og, N 




M3- a 3> n 3 



a) Individual Distributions 



b) Global Distribution 



Figure 5-3: Example of Global Design Distribution 

Assume the data show that 95% of drivers sampled were between 5'0" and 5'6" in 
country A, 5*2" and 5'8" in country B, and 5'4 and 5' 11" in country C. These distributions 
will be associated with the random variables T v T 2 , and T 3 , respectively; with 
corresponding means, variances, and sample sizes of /*,, a x , n,; // 2 , cr 2 , n 2 ;and /z 3 , a 3 , 
n i . If the company were to design different cars for each of these markets, the size and 
dimensions of each car would more than likely be tailored to meet the requirements of 
each country separately. However, if restricted to a one car design in which a combined 
global distribution, T G (see Figure 5-3b), is utilized as the test data, then some 
compromises and added difficulties would be encountered. The combined distribution 
would have the following mean, fi G , and variance, a G (see Appendix B for derivations): 



104 



^ G yv ' yv 2 N 3 



"l _2 , »2 _2 , "3 „2 



N "' N 2 N V 



"l ,,2 _L "2 M 2 ■ "3 „2l -2 



^= ^^ + ^^ 2 + ^^ a + M + ^; + w-« < 5 - 2 > 



/v ' iv yv 



where: 



TV = n x + n 2 + n 3 



It very important to note here the following characteristics: 

minfju,, ]x x , &) < H G < max(jU,, ^ 2 , j/ 3 ) ( 5 - 3 ) 

ct* > minfof, cr 2 2 , cr 3 2 ) (5.4) 

There are three significant consequences that come out of these equations and also 
graphically from Figure 5-3. First (from Equations 5.1 and 5.3), the mean of the global 
distribution will not be the same as the mean of the individual distributions (modeling 
error); unless, of course, ^, = n 2 = ^ 3 . In fact, it is probably unlikely that n c would 
be equal to even one of the individual means. Second (from Equations 5.2 and 5.4), the 
variance, or spread, of the global distribution is larger than at least one of the other 
individual distributions (increase in overall uncertainty). The increase is due to the 
additional dispersion caused by the conglomeration of the different distributions located 
at different positions (the effect of the individual means on the global variance, a 2 G , can 
be seen in Equation 5.2). These additional terms will be called the across-sample 
variance as opposed to the in-sample variance, a 2 , of each separate distribution. And 
finally (from Equations 5.1 and 5.2), the global distribution can be heavily biased toward 
individual distributions by having uneven sample sizes . One consequence is that 
performance in certain situations may be compromised more than others. The global 



105 



performance may appear to be adequate, but in individual situations, the outcome could 
be completely unsatisfactory even though it was included in the simulations. 

Sometimes, there is not a clear line defining a global distribution. For instance, 
the different countries could be broken down further into male and female drivers, each 
with its own separate distribution. These could even be taken down more by, say, age 
group or household income. The appropriate amount to divide out depends on the 
problem itself. There may not be any justification to utilize the knowledge. For example, 
it may not make any sense to design a car specifically for women only. How specific the 
distributions need to be will likely depend on costs or what is the appropriate level of 
uncertainty that can be afforded. Also, the information or data may not exist to allow for 
more specific categorizations. 

The graph in Figure 5-4 diagrams the level of uncertainty in a stochastic process. 
A deterministic system would fall on the farleft of the bar while a completely chaotic and 
unpredictable system would be to the far right. A stochastic system would have some 
amount of inherent uncertainty built in (i.e. no matter how much information is available, 
the realization of a single event is still not completely predictable). This is indicated by 
the location of the vertical line in the figure and represents the uncertainty inherent in T . 
Errors due to modeling will increase the overall uncertainty of the system by inducing 
additional components to the right of this line (e.g. Equation 5.2). Predictability and 
performance of the system is thus degraded if there are sufficient errors in the models 
used in the design process. 



106 



Overall 
Uncertainty 
1 



Inherent Modeling 

Uncertainty Errorg 

' \ I ► 



Perfectly 



Absolutelv 



Deterministic Chaotic 

' ^ Level of Uncertainty- 



Predictability 



Figure 5-4: Level of Uncertainty 

The examples above allow a better understanding of the importance of utilizing 
appropriate simulation models. Returning back to conflict avoidance and the ad hoc 
approach of Figure 5-1, the consequences can be re-worded in the following manner with 
reference to the terms defined in Chapter 4: 

i) Use of a global distribution would likely result in modeling errors and 

therefore increase the overall uncertainty of the system since the design is not 
individually tailored to the current encounter situation. The threshold would 
instead be based on a weighted average of thousands of sample scenarios 
which may not even be applicable to the current situation at hand (W ■* T). 

ii) Virtually any threshold setting can be "shown" to be optimal by merely 
weighting certain encounters to take place more often. This could occur 
inadvertently, of course, but may easily lead to inappropriate results and 
conclusions. 

To further clarify these concepts, a simplified conflict simulation example will be 
given. Figure 5-5 shows a possible subset of sample simulation scenarios that might be 



107 



used as test cases for the ad hoc approach. The aircraft in these six test cases each have 
stochastic trajectories with some random distribution so that repeated runs would result in 
different paths being taken. For the purpose of this discussion, these scenarios will be 
labeled T„ T 2 , . . . , T 6 with the indices to designate the different encounter conditions. 
Notice that the simulations represent the "truth" model of trajectories for which the 
alerting logic is to be tested against to determine the system's performance. For 
clarification, the six cases shown in Figure 5-5 are just a minute subset of thousands of 
simulation scenes to be used in the ad hoc process. For the sake of simplicity, assume 
there are a total of q different encounter situations in the simulations so that T„ T 2 , . . . , 
T q span the entire distribution of scenarios. In the testing of TCAS, there were literally 
hundreds of thousands of sample simulations used to help determine the appropriate 
threshold parameters [Drurnm, 1996; Williamson and Spencer, 1989; Miller et al., 1994]. 

The performance of the threshold logic could be determined by use of the 
definitions of P(FA) and P(SA) given back in Sections 3.2, or more precisely in the 
following form, 

_ number of alerts that were unnecessary ,- $\ 

G total number of alerts 

number of times an alert is successful in avoiding a conflict ._ ,. 

P (SA) = — P.oj 

G total number of alerts 

In the ad hoc approach, these values are actually global or overall performance 
metrics which will be denoted with the subscript G. The reason for the differentiation is 
because the metrics are computed over the entire spectrum of simulation scenarios using 
the same threshold setting. They are a compilation, or weighted average, of the system 
threshold's performance over all situations, T,, T 2 , . . . , T q . 



108 





Ti 



T 2 





T 3 



T 4 





T 5 



T 6 



Figure 5-5: Sample Simulation Scenarios 



109 



A threshold designed in this manner would suffice if the particular threshold 
setting works well in each of the individual T { cases (where the subscript, i, denotes 
some particular scenario from 1 to q). However, such a setting is really a compromise 
between the various test cases and is not optimized to deal with each of the individual 
situations separately. In fact, the particular threshold could be detrimental in certain 
cases, and yet, in the global metric, still appear to perform adequately. It has already 
been shown that this type of design approach, which is based on using a distribution of 
different scenes in the simulations, will lead to an overall drop in system performance 
because of the increase in modeling error and also an increase in the overall uncertainty 
of the process. 

To elaborate a little more on the above statement, assume an example where the 
current situation is T 3 of Figure 5-5 (two aircraft crossing at near right angles). Keeping 
in mind the situation is still stochastic, the most accurate way of determining an 
appropriate alert threshold would be to utilize a dynamic model W = T 3 to predict the 
likelihood of a conflict. This is the method of the direct approach. The threshold should 
not be determined with any influence, whatsoever, from any of the other test cases, T,, 
T 2 , . . . , T q (except for T 3 ). They have no bearing on the current encounter and their 
influence could actually be detrimental to the decision to alert for the current situation T 3 . 

Take the case of T 5 . The two middle aircraft in that scenario appear to be in a 
similar conflict encounter as T 3 ; however, the additional surrounding aircraft would more 
than likely require a different set of alerting criteria to account for the loss of lateral 
maneuvering available to resolve the original conflict { P(SA) will be affected}. A 
threshold optimally set to handle T 3 may, on the other hand, be ineffective, or even 
hazardous, in the case of T 5 . 



110 



The ad hoc approach which uses an aggregate of the individual scenarios would 
simply be obtaining a weighted, global threshold. The alerting performance on average 
would be less than optimal because of modeling errors and increased uncertainties 
induced into the design. Utilizing only one threshold criterion to handle both T 3 and T 5 
would result in a system that would not be best suited for either case individually, but 
instead would be a compromise between the two. 

Take the case of the Ground Proximity Warning System. If the thresholds were to 
be designed based on an equal distribution of flat, medium, and high sloping terrain 
cases, then one might expect to find compromises in performance in each of the 
individual circumstances. The threshold would be averaged out to be optimized globally, 
but the overall uncertainties (from lack of information about the terrain features) would 
be large. The result is high rates of false alarms over relatively flat terrain, but 
inadequate warning time in mountainous terrain. Changing the distribution of the 
simulations would only result in biasing the thresholds toward certain types of terrain 
conditions and would not be a good solution to deal with the problem since jet transports 
are expected to be flown almost anywhere in the world. 

It is, however, possible to break the thresholds down into multiple scenario- 
specific groupings using "if-then" statements or include additional metrics, provided, of 
course, the information is known. For example, if the aircraft is currently over flat terrain 
or the ocean, then one set of thresholds could be utilized; if it is in a region of high 
mountains, then another set of thresholds would be invoked. 

In the case of the traffic example of Figure 5-5, the breakdown can then be 
expressed as Equation 5.7. 



Ill 



P(FA) 



P(SA) 



'/>(*?). if T, 

/ 2 (X^, if T 2 

ml). ^ t 3 

' : (5.7) 

SxCtih if T, 

&rX^, if T 2 

8,(^1), if T 3 



where: 

X A = threshold metric settings (alert space) 

f() = false alarm mapping function (scenario specific) 
g() = successful alert mapping function (scenario specific) 

This design process can be very time consuming and complex if many different 
situations are to be addressed separately and the iterative procedure of Figure 5-1 must be 
repeated for each one. This basically leads to the ad hoc approach of alerting design. As 
explained earlier, TCAS, for instance, utilizes different Tau Criteron values and DMOD 
buffers for different altitudes and encounter situations (see Figure 3-3). There are a fair 
amount of changes in threshold parameters just to account for climbing/descending 
aircraft and altitude, even for a seemingly simple design in which aircraft are assumed to 
fly in straight, constant velocity paths and only use vertical evasive maneuvers. 

In the evaluation of TCAS, the MITRE Corporation generated a large database of 
pairwise aircraft encounters from actual recorded tracks in the United States airspace 
[McLaughlin and Zeitlin, 1992]. Using this database, MITRE defined 10 types of 
vertical encounter geometries (Figure 5-6) which were considered to encompass all 
aircraft maneuvers observed. In evaluating the performance of the system, a large 
number of simulations were used to cover each of these 10 encounter classes [Drumm, 



112 



1996]. Changes to the threshold parameters were then suggested due to the results of 
these simulations. 



Class 



Class 5 



Class 1 



Class 6 



/ 



Class 2 



Class 7 



Class 3 



Class 8 



Class 4 



Class 9 



Arrows represent aircraft vertical profiles 

Figure 5-6: TCAS Encounter Types Defined by MITRE 

[Drumm, 1996] 

In another example, Zeghal [1994] divides horizontal planar conflicts into 3 
separate classes of encounters when using a force potential method. Three different sets 
of equations are defined to best express the threat of collision for: 1) head-on, 2) 
overtaking, and 3) tangent encounters. The need to derive a separate threshold metric for 
different encounter situations illustrates the desire to veer toward a more situation- 
specific design in order to maintain performance. 

This method is one way of dealing with this problem, but it be can be a tedious 
process of breaking up and grouping the scenarios to cover all possible encounter 
geometries and flight conditions. In a more general conflict alerting environment (such 
as Free Flight) when waypoints, intent information, multiple aircraft, and 3-D encounters 



113 



are all fused together, it becomes extremely difficult to utilize such a scheme to 
amalgamate all the individual situations and develop separate thresholds for each one. In 
the ad hoc approach shown in Figure 5-1, it would seem necessary to perform the task 
iteratively for each scenario T„ T 2 , . . . , T q in order to map out a different alert space, 
Xf , for individual encounters. This would be true unless, of course, one could pick a set 
of threshold variables which would allow settings to be virtually invariant of the 
encounter situation. In fact, this is the approach shown in Figure 5-2 and the topic of 
discussion in the following sections. 

5.3.2 Relating Performance Measures to Alerting Thresholds 

In using the ad hoc approach, the choice of state-space variables for the threshold 
can be quite variable as was shown in the survey of alerting methods (Appendix A). If 
so, then what constitutes a viable or sufficient set of variables? Is the use of only the 
range variable ( r) adequate? Or is the time to closest point of approach (r CPA ) or 
expected miss distance (m) also needed? The answer actually depends on two factors: 
1) the type of situations to be encountered, and 2) the performance requirements. 

It was mentioned earlier in this chapter that the ad hoc approach of Figure 5-1 can 
be thought of as a mapping of performance measures back to another set of threshold 
variables due to the iterative feedback adjustments of threshold parameters. Referring 
back to Equations 5.1 and 5.2, the mapping equations, f() and g(), are governed by the 
scenarios used in the simulations; and the performance measures, P(FA) and P(SA), are 
used to judge the efficacy of the threshold setting, X A , Thus, the encounter scenarios and 
the performance requirements are the only defining factors which can determine whether 
the choice of threshold variables will be adequate. 



114 



Figure 5-7 shows a conceptual illustration of mapping thresholds in the state- 
space of X to the state-space of performance measures, Z. An example of performance 
state-space, Z, might be the variables of the SOC diagram, P(FA) and P(SA). The alert 
space in X is denoted by the region, X* , and the required performance region to be met 
in Z will designated Z R . When X A is mapped into Z, it actually becomes a single state 
vector z A . If z A is outside the region of Z R , as depicted in the leftmost illustration of 
Figure 5-7, then the performance requirement is not met and the threshold parameters 
need to be adjusted until a suitable performance, z A , is obtained. This is shown in the 
series of drawings going from left to right, and represents the iterative search and fine 
tuning of the feedback loop back in Figure 5-1. Notice that X A is changed in each step. 









N 


I 


z 


z R 


z A 


P(SA) 







P(SA) 



z R 


\ 
z A 





P(SA) 



z R 









P(FA) 



P(FA) 



P(FA) 



Figure 5-7: Mapping to Performance State-Space 

If an acceptable X A cannot be found, then there are four possible options. The 
first is to change to a different set of threshold variables (i.e. change the state-space, X). 
The threshold metrics may not have been appropriate for the encounters, or else there 
may have been an insufficient number of variables to handle the complexity of situations. 



115 



The second option, which was explained in the last section, is to partition out the 
thresholds to handle more situation-specific groupings. Basically, different sets of 
threshold criteria are used for different encounter scenarios. 

Take the example shown in Figure 5-8a where an alerting threshold, X A , is used 
for three specific types of encounters with the corresponding mapping functions f x (), 
f 2 (), and f 3 (). In this case, only f x () maps adequately to the required performance 
specifications. If X A were to be utilized for all three encounter situations, the overall 
performance of the system would be a weighted average of each of the individual 
outcomes. In Figure 5-8b, a second state-space, X', with different metric variables is 
used to derive adequate thresholds for f 3 (); while the original state-space but different 
parameter settings sufficed for f 2 (). The result is again an increased number of threshold 
metrics designed and tailored specifically for different types of encounters. 




P(SA) 



Yv'/\ 




hQ' 



•\ 









P(SA) 



Z R ' /* 



P(FA) P(FA) 

a) Global Threshold b) Situation-Specific Thresholds 

Figure 5-8: Use of Situation-Specific Thresholds (State-Space Explanation) 



116 



The third option is to simply limit the use of the alerting logic to specific types of 
encounters, basically what is done with TCAS. It can only be used for near term, last 
minute conflicts due to the lack of accurate bearing information in the logic. In the case 
of GPWS, it might be conceivable to have two separate threshold designs, one for 
mountainous terrain and one for flat terrain. The switch could be made manually by the 
pilot or better yet, automatically with some onboard database coupled to navigational 
data. 

The method of limiting the alerting logic to certain types of encounters is 
somewhat analogous to T -» W as discussed back in Chapter 4. The idea is to maintain 
good results; albeit in restricted circumstances. Sometimes it is out of necessity to cope 
with the limitations of the design, such as with the lack of available information to the 
system. Other times, the functional requirements may not warrant or call for the 
additional capabilities (e.g. initial requirements for TCAS were for short term conflicts 
only). 

The fourth and final option is to utilize a different resolution strategy. Given that 
the performance is partly based on the ability to avoid a conflict, it is natural to assume 
some metric such as P(SA) , which is based on a specific avoidance maneuver, is 
included in the performance state-space. Since this was already discussed in Chapter 4, 
not much more on this topic will be mentioned here other than to say a different or a 
more drastic avoidance maneuver might be examined. 

5.3.3 Using Performance Measures as Alerting Thresholds 

In the previous section, the relationship between the performance metrics and 
alerting thresholds was examined. Now, one might ask why go through all the trouble 
testing and re-testing, adjusting and re-adjusting all the threshold parameters, when the 



117 



performance metrics, themselves, could be used as the alerting thresholds? It was already 
explained in Section 5.1 that the performance values were really driving the threshold 
settings in the ad hoc approach. If this is the case, then it appears that if the performance 
measures could be obtained directly in real-time, there would be no need to implement 
the additional iterative steps to map to what would essentially be a set of redundant 
metrics. 

The mapping procedure shown in Figure 5-1 leaves open many different possible 
variables for use as metrics without real analytical computations of conflict in the 
presence of uncertainty. In essence, it is bypassing the dynamic modeling stage of Figure 
3-2, either completely or partially while leaving the fine tuning to pattern matching. The 
reason for the required mapping is because of the disparity between the working 
trajectory model, W, and the "truth" model, T. Without the ability to obtain an accurate 
prediction of conflict directly from its own trajectory model (since W * T), the alerting 
logic is forced to trial and error methods. 

The result is akin to obtaining a simplified model of a probabilistic model, such as 
through correlation or regression modeling to find a simplified set of metric parameters to 
best fit probabilistic data. However, there is really no need for this since a prediction of 
alerting performance can be obtained di rectly by using probabilistic trajectory modelin g 
assuming W = T (the direct method). In the direct approach, there would be no modeling 
error provided W is a good depiction of T. The working trajectory model, W, is either a 
representation of the simulation scenarios in Figure 5-1 or a subset of them. In order to 
do so, W must be allowed to exhibit any trait that would have been charac terized in the 
simulations, including the likelihood of human errors and blunders. 



118 



5.3.4 Continuous Update of Trajectory Model W 

In the direct method as shown in Figure 5-2, there is a need to continuously 
update the dynamic model, W, utilized by the alerting system to keep up with the current 
situation. As long as the uncertainties in the trajectories can be modeled, the update 
process is a natural progression as new aircraft states and other data such as intent 
information is brought in to modify W. At any instant in time, the current aircraft states 
are projected into the future using W and the probabilistic values, P(FA) and P(SA) , are 
computed. The decision to alert is then made directly from these performance estimates. 

The direct approach which utilizes W = T is as situation-specific as one can get 
since the alert decision is based solely on any current information specific to the 
encounter. All knowledge of the current situation, including the effects of uncertainties, 
is contained in T . Take the example back in Figure 2-9 where only range and range rate 
were used to define an alerting threshold. The two variables are simply not sufficient to 
completely define a specific encounter situation. There is no information differentiating 
encounters at different bearings or predicted miss distances. Nor is there information 
regarding the effects of uncertainties or what type of intent information was involved. It 
is, however, conceivable to develop an infinite number of thresholds for every possible 
type of encounter scenario and store them in the alerting logic (much like the if-then 
statements of Equation 5.7). But this is not very practical if the alerting system were to 
be designed to handle multiple aircraft in 3-D flight and various types of flight 
conditions. 

The idea behind the direct approach and W = T is to allow the computation of the 
threat condition on the run as the situation occurs. It is analogous to many current 
computer chess programs which wait for a move to be made; then based on the current 



119 



configuration, propagate the probable moves of each chess piece (out to a finite number 
of moves ahead) and make a decision based on the results. This was the approach used 
by IBM's Deep Blue supercomputer in its highly touted and successful match against 
chess Grand Master Kasparov [Krauthammer, 1996]. Even in the limited confines of the 
chess board and the incredible processing power of today's supercomputers, it is nearly 
impossible to determine what all the moves should be prior to the start of the game 
(except for the first few moves the opening). There are just too many possible 
configurations even on the discrete space of a chess board. Instead, the simulations are 
performed by the computers on the run as the situation unfolds and the decisions are 
situation -specific based on the current configuration. 

In Chapter 8, this similar tactic is used to develop a real-time conflict alerting 
probe. By keeping W = T, the alerting decision is tailored specifically to the current 
conditions of the encounter. Any changes to aircraft state or intent information are 
accounted for directly and done as the situation occurs. This resolves the problem of pre- 
determining separate threshold metrics for every possible encounter situation. 

5.4 Summary 

In this chapter, the common ad hoc approach to alerting system design was re- 
examined from a different perspective. It was shown that probabilistic concepts of 
performance and uncertainties were embedded within the design process. As discussed 
with reference to the iterative method of Figure 5-1, most threshold metrics can be 
thought of as a set of simplified variables mapped to satisfy probabilistic performance 
criteria. The notion that probabilistic analysis and uncertainty drive the alerting system 
design is clearly seen in the feedback loop of Figure 5-1. 



120 



A new direct approach to alerting design was presented and shown be a more 
compact method of estimating performance directly without the unnecessary step of 
mapping back to a redundant set of threshold metrics. Since all information with regard 
to the current situation is contained in the characteristics of the probabilistic aircraft 
trajectories of T, properly modeling these trajectories in the conflict logic (W = T) allows 
for the most accurate prediction of the current encounter in a stochastic environment. 
This ensures the alerting decision is based on situation-specific information rather than a 
global set of data which was shown to have a degrading effect on performance. 



121 



122 



Chapter 6 



Probabilistic Analysis of Conflict 



As mentioned in Section 3.3.3.3, the use of probability estimation has been 
explored in conflict analysis before [Kuchar, 1996; Paielli and Erzberger, 1997; 
Heuvelink, 1988; Rome and Kalafus, 1988; Taylor, 1990; Bakker and Blom, 1993; 
Williams, 1993; Warren, 1997, Prandini et al., 1999; Innocenti et al., 1999]. In previous 
work, Paielli and Erzberger [1997], developed a viable analytical solution to determine 
the probability of a conflict for two aircraft maintaining a straight ahead course. Their 
approach used Gaussian uncertainties to model along- and cross-track error and can be 
rapidly solved and implemented in real-time. If more complex uncertainties (e.g. non- 
Gaussian, 3-D trajectories, aircraft changing course, pilot reaction times) are modeled, it 
becomes increasingly difficult to obtain an explicit analytical solution. 

In this thesis, a Monte Carlo based methodology is employed. The approach can 
intake a large and complex assortment of probabilistic distributions without added 
difficulty. The complications of estimating the future trajectory were explained in the 
previous chapters and it was shown how modeling errors could adversely affect the 
conflict prediction and alerting process. Because the approach is based on Monte Carlo 
simulations, there is a great deal of flexibility built into handling difficult trajectory 
models. However, since the simulations are iterative, significantly more processing 
power is required for the computations than, say, the method employed by Paielli and 
Erzberger [1997]. Also, concerns arise on the stochastic nature of the process to achieve 



123 



repeatability of the results. Nevertheless, a systematic approach can be devised to obtain 
fairly fast results with sufficient bounds on the accuracy of the values. 

6.1 The Trajectory Model 

To calculate the probability of a conflict, P(C), the positions of the involved 
aircraft must be projected into the future using some form of trajectory model as was 
discussed in Chapter 2. Essentially, the subscript W has been dropped off P(C) with the 
understanding that only a working model estimate of the actual truth trajectory can be 
used in simulation. The implications of this were discussed back in Chapter 4, and it has 
further consequences when intent information is included in the model as will be 
explained later in another chapter. 

Figure 6-1 is a pictorial representation of an aircraft in flight showing some of the 
possible parameters which may affect the uncertainty in the future trajectory. The 
modeled parameters might include uncertainty in the current position estimate, future 
along- and cross-track position variability, and the potential for and magnitude of course 
changes. 



124 




Figure 6-1 : Potential Sources of Uncertainty in Trajectory 

The approach of this thesis assumes that the uncertainty of the future path can be 
approximated by an ensemble of possible trajectories weighted by the likelihood of their 
occurrence. The work involves modeling each parameter that could influence the flight 
of the aircraft as a probabilistic distribution, and then using random sampling to generate 
variations in flight path during successive Monte Carlo iterative runs. Figure 6-2 shows 
the baseline model that was used. In this thesis, the aircraft with the alerting system will 
be termed the host aircraft while other vehicles involved in the encounter will be denoted 
as intruder aircraft. 

Uncertainty in the current position is modeled after the accuracy of combined 
Global Positioning System (GPS) and Inertial Navigation System (INS) estimates, and is 
shown as a normally distributed random variable with standard deviation of 50 meters 
laterally and 30 meters vertically. For level flight, course drift in the future trajectory is 
modeled as a 15 knot standard deviation speed fluctuation (along-track error) and a 1 



125 



nautical mile standard deviation cross-track error. These tracking error values were 
based on data obtained empirically from observations of current traffic by Paielli and 
Erzberger[1997]. 



Host 
Aircraft 



Intruder 
Aircraft 



Uncertainty Parameter 



Lateral Position Error 



Vertical Position Error 



Speed Fluctuation 
(Along-Track Variability) 



Cross-Track Variability 



Modeled Distribution 



Gaussian 




Gaussian 



Gaussian 



Gaussian 



a = 50 meters 




a = 30 meters 




a = 1 5 knots 




a = 1 nautical mile 



Figure 6-2: Baseline Trajectory Model 

Additional model parameters can easily be included into the Monte Carlo 
simulations without much added difficulty or loss in computational speed. The more 
common ones utilized within the scope of this thesis work are displayed in Figure 6-3. 
They include provisions for the likelihood of random course changes in heading and 
altitude; plus pilot response latency during avoidance maneuvers used in conflict 
resolution analysis. The specific distributions chosen serve only as one possible model 
and undoubtedly other distributions can be used. The modifications are relatively simple 
with the Monte Carlo approach, and usually only involve sampling from a different 
distribution and possibly making some appropriate changes in the program algorithm to 
reflect the nature of the adjustments in trajectory path. For example, in descending flight, 



126 



fluctuation in speed has been observed to increase slightly to 20 knots standard deviation 
and the vertical rate can vary with a standard deviation of 300 ft/min for a 1500 ft/rnin 
descent, both normally distributed [Erzberger et al., 1998]. These additional parameters 
can be added easily to a Monte Carlo sampling algorithm without much effort and with 
little loss in processing speed. 



Intruder 

Aircraft 

Only 



Host 

Aircraft 

Only 



Uncertainty Parameter 



Likelihood of 
Heading Change 



Likelihood of 
Altitude Change 



Avoidance Response 
Latency 



Modeled Distribution 




t(hrs) -20° 0° 20° 

time magnitude 




time 



t (hrs) 



0' 10,000' 

magnitude 



Gammc 




mean = 1 min. 



12 3 
minutes 



Figure 6-3: Additional Model Parameters 

The task of modeling human behavior is extremely difficult to begin with, and 
any attempt to quantify the likelihood of the pilot in altering the aircraft's present course 
should be handled with caution. As Figure 6-3 shows, heading and altitude changes were 
modeled as Poisson processes with an average rate of occurrence defined by the 
parameters A, and A 2 , respectively. The distributions are formally known as exponential 
distributions in probability theory [Drake, 1967] and show the likelihood of the first- 
order interarrival time (i.e. first occurrence of course change). The units of these 
parameters are arbitrary, but were taken as A, turns per hour and A 2 altitude changes per 
hour. Another possibility could have been a distance based unit such as altitude changes 
per mile. The magnitude of a random course maneuver was modeled to be uniformly 



127 



distributed to a specified limit such as ± 20 degree in heading or within 10,000 ft in 
altitude. 

The purpose of including random course changes into the trajectory is simply that 
they do occur when considering trajectories on a statistical basis and are most likely the 
largest source of uncertainty in the prediction process. When not included, the outcome 
becomes very similar to the single path model shown back in Figure 2-3, and conflict 
determination and resolution can often become overly simplistic. Simply choosing to 
ignore the possibility of pilot actions because of the complexity undermines the true 
difficulty involved in conflict prediction and analysis. The mere fact that researchers 
examine worst case methods indicates the concern over this problem. Also the situation 
is exasperated in light of the current push for less restrictions and more flexibility for 
rerouting in the newly termed Free Flight environment [RTCA, 1995; Phillips, 1996]. 

The modeling of the course adjustments into the trajectory serves to better 
understand the impact of their occurrence on the entire conflict prediction process. 
Again, the distributions used in the model are only estimates and cannot be expected to 
perfectly match the exact outcome. The purpose is to capture the essence of the 
uncertainties which may lead to possible conflict encounters in the desired time frame. 
Even if the values of the parameters are unknown, the impact of changes in the 
parameters can be evaluated to determine their relative importance in the conflict 
assessment or help determine trends. This in turn will help focus future efforts on 
improving trajectory estimation. 



128 



6.2 Calculating the Probability of Conflict 

6.2.1 Monte Carlo Simulations 

Once the distributions of the trajectory model are developed, the probability of 
conflict, P(C), between aircraft can be obtained by extrapolating their positions out into 
the future. The goal is to determine the likelihood that one or more intruder aircraft will 
violate the protected zone of the host aircraft of interest, thus determining the level of 
threat to the host. For the discussions in this thesis, unless specifically stated otherwise, 
the protected zone is defined to be a cylinder 5 nautical miles in radius and extending 
1000 feet above and below the host aircraft. 

Figure 6-4 shows an example of the predicted position distributions for a single 
aircraft traveling with a nominal speed of 400 knots. Intent information of a 45° right 
turn at a waypoint 100 nautical miles ahead was assumed to be known. At each time 
shown in the figure, the aircraft is predicted to lie within the corresponding region with a 
probability of 0.9999. 

Figure 6-4 was generated from Monte Carlo simulations using some of the 
baseline trajectory distributions shown back in Figure 6-2. It included along-track 
fluctuations (Gaussian with standard deviation a = 15 knots) and cross-track variability 
(Gaussian with standard deviation a = 1 nautical mile at steady-state). At t = minutes, 
the position of the aircraft is known exactly since no sensor errors were included in this 
example for simplicity. As shown, the predicted position error grows both along-track 
and cross-track in time, but generally follows the intended path. 

If for some reason there is uncertainty that the aircraft will make the intended turn 
at the waypoint, an additional confidence probability can be included. In such a case, the 



129 



position distribution would split into two separate regions: one for the case in which the 
turn is followed, and one for the case in which the turn is not followed. A situation where 
this type of modeling might prove especially useful is in vertical conflict analysis where 
an intruding aircraft may not be entirely trusted to level off at the expected altitude. 

150 



t = 20 min 



t = 1 5 min 



t = 10mln 



f 



nin j& 



I - 5 min § 



t = 2 min | 



-50 



4* 



100 



50 






(0 
O 



z 



fo 



50 



Nautical Miles 



Figure 6-4: Example Projected Position Uncertainty 

The probability of a conflict, P(C), can be obtained by extrapolating each 
aircraft's position in a similar manner. Given the initial locations, speeds, and headings 
of the aircraft, the P(C) can be estimated through Monte Carlo simulation. Each Monte 
Carlo run consists of propagating the trajectories over time (using point-mass dynamics) 
and determining whether separation minimums of the protected zone are violated. The 
trajectories vary randomly with each run according to the uncertainty distributions chosen 
to define the trajectory model (e.g. Figures 6-2 and 6-3). In each iteration, a random 



130 



sampling from each distribution is chosen and used for the trajectory propagation. For 
instance, one run might have the intruder make a 14 degree heading change 1 minute into 
the flight; while another run may have the intruder follow a straight line path for over 30 
minutes. After a certain number, N, of Monte Carlo runs, a count of the number of 
protected zone intrusions, x, is totaled. Dividing x by N is then an unbiased estimator 
of P(C). A schematic of the Monte Carlo iterative process is shown in Figure 6-5. 



Aircraft State 
Estimates 



Intent 
Information 



Probabilistic 
Trajectory Model 



Aircraft 
Dynamics 

V 

4* 



N iterations 



Monte Carlo 
Simulations 





1=1 


^=- 



X 

conflict 



P(C) = x I N 



Figure 6-5: Monte Carlo Simulation 

6.2.2 Propagation Method 

When propagating the aircraft into the future, one possible approach is to check 
for a protected zone violation at the end of incremental time intervals as depicted in 
Figure 6-6a. For each time interval, Ar, the position of each aircraft is calculated and 
horizontal and vertical ranges are checked against minimum separation requirements. 
This method requires that the intervals be small enough so that intrusions which might 



131 



occur in between each end point are not missed. However, reducing Ar can greatly 
increase the computational time. The problem becomes a tradeoff between the maximum 
projection time and the time required to calculate P(C). 

A more computationally efficient approach can be devised by assuming the 
trajectories to be comprised of a series of straight line segments with instantaneous 
trajectory modifications. This simplified assumption is represented in Figure 6-6b, where 
change points approximate key course changes in the trajectories previously depicted in 
Figure 6-6a. In between change points, the velocity vector of each aircraft is constant. 
Separate change points reflect a new heading, altitude rate, or speed change in the 
trajectory. 

The simplification can lead to some inaccuracies from the trajectory model due to 
general displacement errors as discussed in Section 4.1.2. This is due mainly to the step 
changes around the transition regions induced by the simplified model. If deemed 
necessary, an added lag time can be included prior to the step changes to account for 
aircraft dynamics during the maneuver. Krozel et al. [1997] found the approximation of 
step maneuvers (bank angle and vertical rate) to adequately match simulated Boeing 737 
dynamics provided a 2 to 5 second lag was included prior to initiation of turn and altitude 
changes. For speed maneuvers, they conceded to using an acceleration or deceleration 
component to better model the relatively slow dynamics of aircraft speed changes. 

The approach is further simplified by transformation into a relative coordinate 
frame such as one with respect to the initial host aircraft position (shown in Figure 6-6c). 
The protected zone is placed around the origin representing the position of the host, and 
the relative trajectory of the intruder aircraft is propagated. Because of the assumptions 
made, the trajectory is comprised of straight line segments with each endpoint 



132 



corresponding to a course or speed deviation by either the host or the intruder aircraft. 
The task is then to determine if any individual line segment passes through the protected 
zone around the host aircraft at the origin. Analytic geometry can be used to derive the 
solution for the intersection between the equation of lines (either finite or infinite) and a 
3-D volume (the protected zone cylinder). The equations can be found in Appendix C. 




Instantaneous 
Maneuvei 




a) Incremental Time Steps 



b) Straight Line Approximation 



Change Points 




Host Maneuver 
Intruder Maneuver 



Line-Volume 
Intersect 



c) Line- Volume Intersection in Relative Frame 
Figure 6-6: Aircraft Trajectory Propagation 



133 



Not only does this method detect conflicts along the entire path, rather than at 
discrete points; but the computational time is decreased by orders of magnitude compared 
to the incremental time approach of Figure 6-6a. Also, the method is insensitive to the 
time scale of the projection (the equations for a line-volume intersection are applicable to 
an infinite line); it only depends on the number of course or speed changes that occur 
between both aircraft. 

In some instances, it may be desirable to model the course transitions more 
accurately, as would be the case if an encounter is expected to be in the vicinity of the 
maneuver transition. If the maneuver is far ahead into the future, an instantaneous 
maneuver is less likely to be a factor as the uncertainty in the path grows over time. 
When the maneuver is expected in the near future, a more accurate representation of the 
maneuver may be in order. This might be the case if an intruding aircraft is relatively 
close and the crucial conflict point is somewhere near the region of the course change. 

Take, for instance, the example shown in Figure 6-7a where the host aircraft 
(white) is currently in a turn toward a target waypoint. A trajectory modeled with an 
instantaneous turn (dashed line with sudden path change of Ay/) may be overly 
simplistic since the actual turn radius can be on the order of 10 nautical miles or so 
depending on the speed and bank angle. This could lead to a missed detection of the 
conflict with the intruder aircraft (black) shown in the picture. Thus, it is more accurate 
to include additional line segments to better represent the actual change in the heading 
over time. Figure 6-7b shows one additional change point, A, added to better 
approximate the path of the host aircraft during the heading transition. 

The turn radius ( R) can be estimated from the intended bank angle (0), speed 
(u), and gravitational acceleration (g) using Equation 6.1. 



134 



R 



v 



gjtan0| 



(6.1) 



The center of the turn circle can be approximated to be in the direction perpendicular to 
the current aircraft heading and at a distance R away. Using geometry, the position of 
point A can then be determined as a tangent line from the turn circle to the target 
waypoint position. The result is a two-segment path (shown as a solid line in Figure 6- 
7b). 




a) Instantaneous Heading Change 




b) New Segmented Turn Maneuver 



Figure 6-7: Heading Change Model with Bank Angle 



135 



If more accuracy is desired, the turn can be further sub-divided into additional 
straight line segments, though at a cost to computational time. The choice depends on the 
goals of the conflict analysis. For very critical, short scenario analysis such as the case 
for parallel approach studies, the need for a more accurate path model would be 
important. For long term conflict probing, the desire for further look-ahead time may 
take precedence over near term conflict prediction accuracy. 

6.2.3 Computational Accuracy 

Because Monte Carlo simulations are inherently stochastic, a discussion on the 
computational accuracy and performance is warranted. The problem posed in calculating 
P(C) is basically that of estimating a value of proportion, p. Each iterative run is a 
binomial process in which a conflict (minimum separation criteria violated) occurs or it 
does not. The number of conflicts, x, divided by the total number of iterative runs, N, 
provides an unbiased estimator of p with variance a 2 given by 

p = - (6-2) 

N 

2 = pa - p) {63) 

N 

From the Central Limit Theory in probability, the sum of N independent, 
identically distributed random variables will approach that of a normal distribution as the 
number N -> °° [Drake, 1967]. Thus when the number of iterations, N, is sufficiently 
large (Johnson [1994] suggests N > 200 for the range of 0.75 > p > 0.925), the 
normal approximation to the binomial distribution can be used to construct an 
approximate confidence interval for the binomial parameter, p [Johnson, 1994]. For 3a 
standard deviation accuracy (99.7%), the error in using x/N (Equation 6.1) to estimate 



136 



the true P(C) can be computed from Equation 6.4. Noting that Equation 6.4 is a 
maximum at p = 0.5, the upper bound of the 99.9% confidence error can be found by 
using Equation 6.5. 



3(7 



pa - P) 



N 



(99.7%) 



(6.4) 



3 °™ 2^N 



(99.7%) 



(6.5) 



The tradeoff between the number of iterative Monte Carlo runs, N, and accuracy 
in the estimate of P(C) is evident from Equations 6.4 and 6.5. The effect can be 
visualized in Figure 6-8 where upper error bound (Equation 6.5) is plotted versus N. 



3a 



max 




10000 20000 30000 

N Iterations 



40000 



50000 



Figure 6-8: Monte Carlo Accuracy as a Function of Iterations 

For the examples in this paper, N = 10, 000 was used as a compromise between 
speed and accuracy, providing a 3cr error in P(C) of at most ±0.015. There are 
diminishing returns on improved accuracy as the number of iterations increases beyond 



137 



this point at the expense computational processing. Originally developed on a Silicon 
Graphics, Inc. Indigo Elan 4000 Workstation (purchased in 1993), computational time to 
obtain P(C) from 10,000 iterations was on the order of 1 second for one pair of aircraft. 
Use of newer workstations (Silicon Graphics Indigo R 10000 and Octane MXE) have 
shown a 2 to 3-fold increase in speed; bringing the up possibility of using the Monte 
Carlo simulations as part of a real-time conflict alerting probe. There are some 
advantages to such a system, and this concept will be brought up in the next chapter. A 
test bed system has already been incorporated into part-task simulators at Massachusetts 
Institute of Technology (MIT) and NASA Ames Research Center. 

6.3 Conflict Probability Maps 

Given the relative speed, heading, and altitude between a host and intruder 
aircraft, a conflict probability map can be constructed to display the locations where the 
intruder aircraft currently must be in order to result in a conflict at some later time. As an 
example, assume two aircraft are co-altitude and both flying with a velocity 400 knots 
with offsetting headings of 30 degrees. For simplicity, assume the host is flying directly 
North at a heading of 360° and the intruder's heading is 330°. Figure 6-9 shows a conflict 
map of the likelihood of conflict for this specific encounter scenario. The host aircraft is 
shown in white at the lower left. 



138 



40 



30 " 



10 

® 20 



« 10 



- 



n 

L 




Protected Zone = 5 nmi radius 
1 1 1 -T— 

10 20 30 

Nautical Miles 



40 50 



Figure 6-9: Example Conflict Probability Map 

The plot shows the conflict probabilities for an intruder aircraft in the surrounding 
airspace relative to the host aircraft. For example, an intruder in the position shown in 
the figure (black aircraft) will cause a conflict in the future with probability 
P(C) = 0.45 . If the intruder were farther north or east of the host aircraft, the 
probability of conflict would be lower. The plot shows actual data based on 10,000 
Monte Carlo simulations at each 1 nautical mile spacing. The trajectory model was based 
on the uncertainties presented in Figure 6.2 and the random heading change of Figure 6.3 
with A, turns/hr. The magnitude of heading change was limited to 20° in either direction 
and uniformly distributed (see Figure 6.3). 

6.4 Summary 

In this chapter, a method was developed to calculated the probability of conflict, 
P(C), based on Monte Carlo simulations. This approach allows a large number of 
complex variables to be handled easily and efficiently in what would otherwise be 
problems without tractable analytical solutions. Error bounds on the accuracy of the 



139 



calculations can also be determined. In addition, a method to visualize the conflict 
situation through the use of probability contours was presented. 



140 



Chapter 7 



Example Applications 



In this chapter, several examples will be given to illustrate the utility of the 
probabilistic methodology in different encounter scenarios. The examples demonstrate 
the relative ease in which the methodology can be applied to both simple and complex 
situations. The method can handle encounters in 3-D, with or without intent information, 
and probabilistic trajectories. Since the approach is based on Monte Carlo simulations, 
the derivations are not hindered by the complexity of the encounters. 

7.1 Horizontal Conflict Examples 

As a simple example, assume two aircraft (host and intruder) are co-altitude and 
both flying with a velocity of 400 knots in opposite directions. If the intent of each 
aircraft is known, then potential conflict situations can be predicted. Assume that both 
aircraft have declared their intentions to maintain their current speed, heading, and 
altitude. This might be inferred, for example, through datalink of autopilot mode control 
settings. 

The potential conflict map as obtained through Monte Carlo simulation is shown 
in Figure 7-1 (using the baseline trajectory uncertainty model from Figure 6.2). As a 
reminder, in this example the intruder aircraft is traveling in the opposite direction as the 
host. The chart is shown relative to the host aircraft located at the origin (0, 0) with its 
track pointing up. The top of the chart is 200 nautical miles ahead of the host aircraft and 
represents a 15 minute time frame. Contours of constant conflict probability are shown 



141 



starting at P(C) = 1.0 around the host aircraft and decreasing in increments of 0.1. For 
example, the intruder aircraft shown in the figure 100 nautical miles ahead of the host 
aircraft will produce a probability of nearly 1.0. Variability and coarseness of the 
contours are a result of the accuracy of the Monte Carlo simulations. In this case, 
because the trajectory uncertainties are small, the corridor where aircraft must be located 
to generate conflicts is relatively narrow. Although the example shown is for a specific 
relative geometry and speed, similar maps can be generated for any situation. 



zuu 
180 






160 




- 


140 




- 


co 120 




- 


Nautical M 

00 o 

o o 


- 


Y 

| ■ 


60 


* 


40 


- 


- 


20 


_ 


4 





{ 





-50 

Nautical Miles 



50 



Figure 7-1: Intruder and Host Maintain Course 

A more interesting case to observe is when aircraft may change course at some 
time within the foreseeable future. In many cases, the intentions of each aircraft are not 



142 



known for certain, but information regarding rules-of-the-road, past experience, or flight 
restrictions can be helpful in establishing the likelihood of various trajectories. In Figure 
7-2, the intruder aircraft is still headed in the opposite direction as the host, but now no 
explicit intention to maintain a straight course is assumed. 





180 




160 




140 





120 


1 

"cfl 

o 

"■4— » 

■3 
CO 

z 


100 
80 




60 




40 




20 








-50 



Nautical Miles 



Figure 7-2: Potential for Intruder Course Change 

For this particular case, the likelihood that the intruder would make a heading 
change is modeled as Poisson distribution with an average rate of A, =4 turns per hour 
(see Figure 6-3). Also, the hypothetical flight rules in the airspace are assumed to require 
aircraft to restrict heading changes to less than 20° within a 15 minute period. Thus, 
potential changes in heading were modeled with a uniform distribution between ±20°. 



143 



The resultant conflict map is shown in Figure 7-2, again using contour spacing of 
probabilities of 0. 1 . Note that the probability of conflict decreases more rapidly as one 
moves farther from the host aircraft The same intruder 100 nautical miles ahead of the 
host will now cause a conflict with a probability of approximately P(C) - 0.83 because 
there is some chance that the intruder will perform a turn. 

In the next example shown in Figure 7-3, additional intent information regarding 
knowledge of waypoints is added. For this case, the intent is supplied by the host aircraft 
in terms of 3 future waypoint locations in which the host will shift its flight path laterally. 
Again, the conflict map is shown with contour spacings of 0.1 . Here, the intruder aircraft 
100 nautical miles ahead of the host will not create a conflict as long as the intended path 
is followed. 

Comparing Figures 7-2 and 7-3 provides some insight into the potential benefit of 
intent information. Consider for example the flight path shown in Figure 7-3. If the 
host's waypoint information was not used in the conflict detection, the situation would 
likely be modeled as shown in Figure 7-2, resulting in a conflict alert. Such a conflict 
would be unnecessary, however, since as Figure 7-3 shows, there would not be a conflict 
with the intruder aircraft. The intent information has improved the modeling accuracy of 
the trajectory (W -» T) and thus increases the chance of making an better informed 
decision about alerting. 



144 




Nautical Miles 

Figure 7-3: Host Aircraft Following Waypoints 

Conflict maps can also be utilized in the examination of avoidance maneuver 
options for conflict resolution. Figure 7-4 shows an example 30° right turn avoidance 
maneuver made by the host aircraft in response to a conflict alert in the example from 
Figure 7-2. An additional uncertainty was included to represent variability in pilot 
response time in initiating the turn maneuver. The latency time was modeled as a 
Gamma distribution with an average of 1 minute and skewed with a 95% probability of 
the maneuver occurring within 2 minutes. 

Comparison of Figure 7-2 and Figure 7-4 shows the effect of the avoidance 
maneuver on the probability on conflict. Similar analysis can be performed to determine 



145 



what other avoidance options (e.g. heading, speed, or altitude changes) can be used for 
the resolution. For multiple aircraft in the airspace, the maneuver can be checked to see 
if it induces addition conflicts which would not have occurred without it. 





200 




180 




160 




140 


CO 

1 

TO 
O 

(0 

Z 


120 

100 

80 




60 




40 




20 







-50 




Nautical Miles 



Figure 7-4: Host Aircraft Turns 30° 

7.2 Vertical Conflict Examples 

To more fully illustrate the utility of the probabilistic methodology (especially in 
more complex situations), several examples will be shown here to analyze the effects of 
intent information on conflicts in the vertical plane. Rather than depict conflict maps, 



146 



however, the discussion will revolve around false alert and missed detection rates using 
the SOC methodology. In these examples, two aircraft are flying in opposing directions 
with the intruder currently above the host aircraft by 5000 feet. Suddenly the intruder 
descends directly toward the host aircraft at 1000 feet per minute. The uncertainties are 
modeled using the baseline model shown back in Figure 6-2, and a conflict is defined as a 
loss of minimum separation of 5 nautical miles in the horizontal plane and 1000 feet in 
the vertical plane. 

Two cases will be considered here. In the baseline case, it is not known whether 
the intruder will level off at some point or continue its descent beyond the host aircraft's 
altitude. The vertical profile of the intruder is modeled such that it is equally likely that 
the intruder will level off at any altitude within 10,000 feet of its initial descent point. 
Thus, a conflict may exist (the intruder continues to descend into the host) or a conflict 
may not exist (the intruder levels off safely above the host). The situation is depicted in 
Figure 7-5a. 

In the intent case, datalinked information from the intruder indicates that it will 
continue its descent at 1000 feet per minute through the host aircraft's altitude (Figure 7- 
5b). For simplicity, it is assumed here that the aircraft maintains this descent rate 
perfectly. However, there may likely be variability in the descent rate as pointed out by 
Paielli and Erzberger [1999]. Fortunately, the Monte Carlo method presented in this 
thesis could very easily handle this additional parameter without much additional 
complexity or loss of computational performance. 



147 



Initial Descent 
Altitude Known 



Avoidance 
Maneuver 
1000ft/min 

4 




a) Baseline Case 



E 



5 sec 
Delay 

— * 



Avoidance 
Maneuver 
1000ft/min 

4 



"^^■ESE^^™ 



k 




5NMI 



b) Intent Case 



Figure 7-5: Vertical Conflict Examples 



SOC curves for both cases are plotted in Figure 7-6. The assumed resolution 
maneuver is a 5 second delay when the conflict alert occurs, followed by a 1000 feet per 
minute climb. Variability in these response parameters could be added to the analysis as 



148 



well, but are not included here for simplicity. The intent case SOC curve is shown by 
the solid line which happens to follow along the y-axis; the SOC curve for the baseline 
case is shown by the dashed line. Operating points for each case are shown in terms of 
the time at which the conflict alert occurs in increments of 10 seconds relative to the time 
to Closest Point of Approach (CPA) (assuming a straight line projection of the current 
velocity vector). 



1.0 
0.9 




0.4 
0.3 
0.2 
0.1 




140 -300 s 



130 s 



120 s 




300 s 



130 s 



120 s 



110s 



.100 s 



Baseline Case 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

P(FA) 
Figure 7-6: SOC Comparison of Baseline vs. Intent Cases 

The SOC curve in Figure 7-6 shows that essentially an ideal alerting decision 
could be made in the intent case provided that the intended path was indeed followed by 
the intruder aircraft. By alerting at any time prior to 140 seconds before CPA, the host 
aircraft could avoid a protected zone violation with approximately 100% confidence 



149 



{ P(SA) ~ 1.0 }. Simultaneously, because the uncertainties in this case are relatively 
limited, the probability of false alarms is approximately 0. Alerting with less than 140 
seconds to CPA reduces the probability of a successful alert as shown. 

In the baseline case, there is a possibility of the intruder aircraft leveling off above 
the host aircraft. Thus, a climbing resolution maneuver may actually induce a conflict 
that would not otherwise have occurred. The net effect is an increase in the probability of 
a false alarm (a conflict would not have occurred) and a decrease in the probability of a 
successful alert (the avoidance maneuver induces a conflict). As a result, the 
performance of an alerting system in the baseline case would be lower than in the intent 
case. 

Notice that the baseline SOC curve shows that the successful alert probability, 
P(SA), cannot be increased beyond approximately 0.8 without greatly increasing the 
false alarm probability. Assuming that the intentions of the intruder aircraft were not 
known, this example shows the difficulty placed on the alerting system designer to 
develop a suitable threshold for this type of encounter. A viable option would be to 
examine a more aggressive climb maneuver, or else look into a different avoidance 
option all together. Figure 7-7 shows the SOC curves for a 1000 feet per minute descent 
and a 30° turn avoidance maneuvers. The climb maneuver from the previous figure is 
redrawn for comparison. 

The advantage of utilizing a descent or turn maneuver in this particular encounter 
situation is obvious from the SOC diagram. Both these maneuvers allow for a threshold 
setting that is more ideal than the original climb maneuver (curves are closer to the ideal 
position of P(FA) = 0, P(SA) = 1). In addition, they also allow for the alert to be 
delayed longer before action is required. For example, the plots indicate that a descent 



150 



maneuver can be delayed until 80 seconds to CPA before a large drop off in P(SA) is 
experienced. For the turn maneuver the delay can be as long as 100 seconds prior to 
CPA. In comparison, the climb option requires the alert to be given much earlier (140 
seconds to CPA) if P(SA) is to be kept above 0.8. In this situation, the longer the delay, 
the better since the intruder aircraft may level off in altitude and an alert would never be 
needed. 

80s 




Figure 7-7: SOC Comparison of Climb, Descent, and Turn Maneuvers 

The results shown in Figure 7-7 are very interesting when examined from the 
perspective of uncertainties. Maneuvers which deviate away from regions of high 
uncertainty will allow for increased alerting performance. In essence, the outcome of the 



151 



conflict will become more certain. For example, the descending option allows the host 
aircraft to move away from the possible regions of airspace that might be occupied in the 
future by the intruder aircraft. As a result, the outcome of a possible conflict is more 
certain. The same is true with regard to the turning maneuver. By utilizing a horizontal 
escape maneuver, the host aircraft is removing itself from the major source of uncertainty 
(i.e. the intruder leveling off) that is involved in the encounter. The results help 
substantiate the desire to include horizontal resolution advisories in new versions of 
conflict avoidance systems. 

7.3 Summary 

The case studies presented in this chapter demonstrate the utility of the 
probabilistic methodology to handle both simple and complex encounter situations. The 
examples showed how intent information could be used to improve the quality of the 
conflict detection problem by increasing the prediction accuracy. Both probability 
contour maps and SOC curves were used in presenting the results. 



152 



Chapter 8 

A Probabilistic Real-Time Alerting Probe 



The previous chapters of this thesis have detailed most of the theoretical and 
computational issues of analyzing conflicts using the probabilistic approach. Its uses as 
an evaluative tool to investigate conflict scenarios have been shown with examples from 
the last chapter. Now this chapter will describe taking probability analysis one step 
further to the development of a real-time alerting probe. The first section will explain the 
rationale behind the endeavor and the next section will describe the most recent update of 
the logic to run Monte Carlo simulations concurrently with the real-time updates of 
aircraft state and intent information. Finally, the advantages and disadvantages behind 
this work will be discussed. 

8.1 Alerting Probe Concept 

In order to better understand the potential advantages and disadvantages of 
probabilistic threshold criteria, a prototype alerting logic was developed based on the 
concept of probabilistic conflict calculations discussed in the previous chapters. The 
basis for the logic follows very much in line with the concepts developed for the SOC 
curves (explained in Section 2.6) using P(FA) and P(SA) and the direct approach of 
Chapter 5. 

The design of the prototype alerting system was guided in part by NASA 
requirements for their experiments in their Advanced 747-400 Full Motion Simulator as 
well as their less complicated part-task simulators. The logic was tailored to an airborne 



153 



system where conflicts were expected to be resolved primarily on the flight deck, though 
the concept could be extended to a ground-based system to aid ATC. The first prototype 
assumed only that current state information (position, speed, and heading) was available 
through inter-aircraft datalink such as from Automatic Dependent Surveillance-Broadcast 
(ADS-B). The alerting system was specifically designed to provide ample warning time 
so that strategic maneuvers could be examined and coordination between flight crews 
could be carried out. 

A multi-staged threshold approach was utilized to provide a series of alerts to 
indicate trends in conflict hazard. This approach allowed the means of implementing the 
alert to be tailored to the level of threat. Low probability threats resulted in relatively 
passive alerts such as changing the color of a traffic symbol. High probability, urgent 
threats produced aural warnings to actively inform the pilots or controllers of the conflict. 

The multi-staged approach is shown in the schematic diagram of Figure 8-1. 
Three stages (marked 1, 2, and 3) produced changes in the traffic display symbology in 
the cockpit of the host aircraft. As implemented, the outermost threshold provided an 
initial indication of potential threat more than 10 minutes into the future and up to 200 
nautical miles away. In the NASA 747-400 simulator, a hollow traffic symbol on the 
map display would change color when the first threshold was exceeded and the flight 
crew was expected to initiate verbal communication with the intruding aircraft in an 
effort to coordinate a resolution. If the encounter continued, an additional stage would 
inform the crew of the heightening threat by filling in the traffic symbology on the map 
display. At stage 3, an aural Alert Zone Transgression (AZT) was provided to the flight 
crew indicating that action must be taken to resolve the conflict. At this point, there was 
still ample time to coordinate resolution with other aircraft. If the conflict continued 



154 



without resolution, a fourth level of alert called the Authority Transition (AT) would 
inform ATC to take control over the conflict situation. 



TCAS 



Authority 
Transition 




Protected Zone 
(5 nmi radius) 



Other Aircraft 
In Vicinity 



Figure 8-1: Multi-Stage Alerting Probe Concept 

The prototype alerting logic was overlaid on top of the current TCAS logic which 
was not modified and kept in the simulation setup as an independent, final warning 
system. But because the alerting thresholds on the present TCAS 6.04A version are 
based on limited variables (range and closure rate), TCAS cannot accurately predict 
whether a conflict will occur beyond a few minutes. TCAS can track traffic within a 
range of 40 nautical miles and its earliest alert can be triggered approximately 1 minute 
prior to the projected closest point of approach [RTCA, 1983; Nordwall, 1997]. 

The newest version of TCAS (v7.0) became available in 1999, and increased the 
range to 100 nautical miles using ADS-B via mode-S to transmit additional position, 
heading, and vertical speed information [Klass, 1998]. The prototype alerting system 



155 



could also be expected to utilize this same aircraft state data to estimate future 
trajectories. 

Additional requests by NASA required the alerting probe be able to handle 
various types intent information (i.e. next waypoints, commanded headings, commanded 
altitudes) if made available. The difference in alerting thresholds with and without intent 
can be significant as was shown in the example applications of Chapter 7. The 
availability of intent allows for the more accurate prediction of future aircraft states and 
thus improve the outcome of the alerting process. 

8.2 Prototype Alerting System 

The ability to change the parameters of the trajectory model becomes important 
when intent information is to be considered in the probability computations. Depending 
on the type of intent information available, the parameters need to be reflected in the 
trajectory model to reduce errors in the estimates of conflict as explained in Chapters 4 
and 5 (W = T). Also, if the intent of an aircraft changes, the trajectory model needs to 
adapt to the new information in order to determine possible conflicts along the new path. 

In response to new directives within NASA to explore the use of intent 
information in conflict detection and resolution, a new alerting logic was necessary to 
accommodate their experimental requirements. A new setup was needed to handle the 
various levels of intent information, if available, and also adjust dynamically to changes 
in that information. 

Utilizing the propagation method discussed in Section 6.2, specifically that shown 
in Figure 6-6c, a new alerting system was devised based on running Monte Carlo 
simulations in near real-time flight. Aircraft state and intent information are passed to the 



156 



Monte Carlo simulation engine to be processed, then the hazard level based on the 
conflict probabilities { P(FA) and P(SA) } is returned after computation is completed. 
Additional information regarding possible avoidance options and the location of the 
closest point of approach (assuming straight line extrapolation of the current velocity 
vectors) were also computed by the alerting logic at the request of NASA. The structure 
is shown in Figure 8-2. 



AIRCRAFT 




Cockpit Display 




A 



Aircraft States 



Intent Information 



ALERTING LOGIC 



Probabilistic Monte Carlo 
Simulations of Future Trajectories 



ProbaMIiSe 
Traj*ck»y Modal 



r^=b 



1 



a 



a 



Output to Cockpit 

• Hazard Level 

• Avoidance Options 

• Closest Point of Approach 



Alerting Thresholds 

• P(FA) 

• P(SA) 



Figure 8-2: Prototype Alerting System Based on Real-Time Monte Carlo 
Calculations 

Currently, the system handles intent information of 4 types: only current state and 
derivative information, future 3-D positional targets (waypoints, top and bottom of climb 
and descent), target headings, and target altitudes. These forms were chosen to satisfy 
requirements for experiments to be conducted by NASA. 



157 



This setup is flexible such that the parameters of the trajectory models may be 
adjusted according to the intent information provided (W = T). If intent is available, the 
alerting logic develops the state trajectory based on that information. If there is no 
information on intent, then the logic assumes a possibility of the surrounding aircraft 
deviating from their current track. The premise is similar to the results shown back in 
Figures 7-1 and 7-2. In the former case, the intent of intruding aircraft is to maintain its 
current heading resulting in a very long, narrow region of high conflict probability. In 
the latter case, the intruder aircraft is modeled with the possibility of deviating from its 
current track which results in a wider spread of the probability contours. 

8.3 The Alerting Thresholds 

In order to satisfy the requirements set forth by NASA, the prototype alerting 
system uses four stages of alert as shown in Figure 8-1. The first three stages produce 
alerts in the cockpit that are intended to aid the flight crew in resolving the conflict before 
tactical maneuvering is required. At the fourth stage, ATC is notified to issue commands 
to provide traffic separation. To set the conditions at which these stages are triggered, it 
is necessary to examine the tradeoffs between P(FA) and P(SA) . This requires 
balancing the likelihood of a conflict against the ability of the host aircraft to avoid a 
conflict. Since P(SA) is specific to different avoidance options, five standard conflict 
resolution maneuvers were considered: 

1) Left Heading Change of 30° 

2) Right Heading Change of 30° 

3) Climb or Descent of 2000 ft/min 

4) Speed Increase of 50 kts 

5) Speed Decrease of 50 kts 



158 



These maneuvers serve as benchmarks for estimating the ability of the host 
aircraft to avoid a conflict. When the intruder is far from the host aircraft, any of these 
five maneuvers could be used to resolve the conflict. As the intruder nears the host 
aircraft, some of these maneuvers may no longer provide the required separation between 
aircraft. The premise behind the alerting logic is that if a sufficient number of these 
maneuvers are still available to the pilot, the alert can be delayed. When the pilot's 
options begin to disappear, an alert should be issued. The concept is illustrated in Figure 
8-3. 

As shown in the figure, initially the hazard (e.g. intruder aircraft) is sufficiently 
far away that left and right turns and climb and descent maneuvers can easily avoid the 
hazard (Figure 8-3a). As the hazard closes in on the host aircraft (Figure 8-3b), the 
options to resolve the conflict will diminish as different avoidance maneuvers can no 
longer safely able to avoid the conflict. As shown in Figure 8-3b, the right turn maneuver 
is depict as an ineffective option to safely avoid the incoming intruder. 



159 




^^ 



a) Avoidance Maneuvers Still Available 




b) Avoidance Maneuvers Begin to Diminish 

Figure 8-3: Loss of Available Avoidance Options 

A maneuver was defined to be available to the host aircraft if, by performing the 
maneuver, the probability of a conflict was reduced to less than 0.05, i.e. P(SA) > 0.95. 
The five maneuver options listed above included the probabilistic response time depicted 
earlier in Figure 6.3 (with a mean latency of 1 minute). Thus, when a maneuver was 



160 



deemed to be not available, safe separation could still be achieved if the pilot reacted 
more quickly or more aggressively than assumed in the model. 

During actual simulator runs, the logic would determine the number of avoidance 
maneuvers remaining or available, N, to resolve a conflict with the intruder. The latest 
version of the alert logic computed these values in near real-time using Monte Carlo 
simulation during runtime. The logic would compute P(FA) from P(C) 
{ P(FA) = 1 - P(C) } and also the various values of P(SA) for the five standard 
avoidance options. By comparing N with P(FA), the appropriate alert stage was defined 
as shown in Table 8-1. 

The leftmost column of Table 8-1 shows the probability of a conflict if the host 
aircraft continues along its current trajectory. This assumes that the intruder's trajectory 
can be represented by the model discussed earlier. The rightmost column shows P(FA), 
which as discussed earlier is related to P N (C) by Equation 2-2. The other columns 
indicate the defined alert stages as a function of N. Generally, the more options 
available to the pilot, the lower the alert stage. For example, if P(FA) is 0.35 and there 
are two avoidance maneuvers available, then the alert stage is 2. If P(FA) drops below 
0.3 or if N is reduced to one, then the alert stage increases to 3. If P(FA) drops below 
0. 1 , then the AT stage is triggered. 



161 



Table 8-1: Alert Level Classification 





Number of Avoidance Maneuvers Remaining, iV 




P N (C) 


None 


One 


Two 


Three + 


P(FA) 


0.0-0.1 


- 


- 


- 


- 


0.9-1.0 


0.1-0.2 


1 


1 


- 


- 


0.8-0.9 


0.2-0.3 


1 


1 


- 


- 


0.7-0.8 


0.3-0.4 


2 


1 


1 


- 


0.6 - 0.7 


0.4-0.5 


2 


2 


1 


1 


0.5-0.6 


0.5-0.6 


3 


2 


2 


1 


0.4 - 0.5 


0.6-0.7 


3 


3 


2 


2 


0.3-0.4 


0.7-0.8 


AT 


3 


3 


2 


0.2-0.3 


0.8-0.9 


AT 


3 


3 


3 


0.1-0.2 


0.9-1.0 


AT 


AT 


AT 


AT 


0.0-0.1 



8.4 Evaluation of Prototype System 

To better understand the underlying design process, the thresholds from Table 8-1 
can be mapped into SOC curves. Figure 8-4 shows SOC curves for two co-altitude 
aircraft on a collision course along flight paths at right angles to one another. SOC 
curves corresponding to each of the five resolution maneuver options are shown. 



162 







Alert Stage 




1.0 


AT 


3 2 


1 


0.95 


- 5***> 






0.9 




^ climb/descend 




0.8 




2000 ft/min 




0.7 




N right 30° 




0.6 




' left 30° J / 




v> 0.5 

0. 








0.4 






slower 50 kts 


0.3 








0.2 




// faster 50 kts 



0.1 



0.0 



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 
P(FA) 

Figure 8-4: SOC Curve for Aircraft on Perpendicular Tracks 

When the intruder is far from the host aircraft, the situation maps into the upper 
right of the plot: it is likely that a conflict will not actually occur { P(FA ) -> 1 } and it is 
likely that any avoidance action would resolve the situation { P(SA) for each of the five 
avoidance maneuvers is 1 }. Data for Figure 8-4 were not obtained beyond 200 nautical, 
and thus the SOC curves in the figure do not extend all the way to the upper right corner. 

As the intruder continues on a collision course, it becomes more clear that a 
conflict will occur: P( FA) decreases and the situation moves from right to left along the 
curves. Thus, P(FA) is related to the distance between aircraft and to the time before 
closest point of approach. As P(FA) decreases, P(SA) also decreases in differing 
amounts according to the different SOC curves. The effectiveness of a given maneuver 



163 



depends on how slowly its P(SA) decreases. When a curve's value of P(SA) drops 
below 0.95, the corresponding avoidance maneuver is no longer available. Thus, as the 
situation progresses to the left in Figure 8-4, the different avoidance maneuvers become 
unavailable, in order, from speed changes to turns and finally to climb or descent. Thus, 
the SOC curves show that for this case, vertical maneuvers are the most effective. 

The first maneuvers to become unavailable are the speed change maneuvers, at 
P(FA) of approximately 0.9. This is because large speed changes are generally required 
to resolve conflicts in the time scales under consideration. 

Until P(FA) drops below approximately 0.25, turns and climb/descent avoidance 
maneuvers will still provide the required separation. At approximately P(FA) = 0.25, 
however, a 30° left turn maneuver is no longer an option. At approximately 
P(FA) = 0.2, the 30° right turn is also no longer an option. When P(FA) reaches 
approximately 0.1, the climb/descend options become unavailable. 

At a given value of P(FA), N corresponds to the number of SOC curves that 
have values above P(SA) = 0.95. Figure 8-4 also shows when the four alert stages are 
triggered as a function of P(FA). Cross-referencing with Table 8-1, stage 1 is triggered 
when N is three or more and P(FA) drops to 0.6. Stage 2 is triggered when P(FA) 
drops to 0.4 and Stage 3 is triggered when N drops to two. Finally, the AT stage is 
triggered when N drops to zero. Although Figure 8-4 shows SOC curves for a direct 
collision between two aircraft on perpendicular flight paths, other geometries produced 
similar patterns. 

The five avoidance maneuvers used here are intended to represent strategic 
maneuver limits. It should be reiterated that a large response time (mean = 1 min.) is 
modeled in the avoidance maneuvers (see Figure 6-3) and that when N is zero, the host 

164 



aircraft can still maneuver out of the conflict. A more aggressive, tactical maneuver such 
as a 45° heading turn or a combined climbing turn may still be available when the five 
assumed strategic maneuvers are not. 

Further examination of the SOC curves show that speed changes make only a 
limited contribution to the prototype logic. In many cases, a speed change of greater than 
50 knots is required for adequate separation with 95% confidence. As can be seen from 
Figure 8-4, the SOC curves for the speed maneuvers deviate only slightly from the 
diagonal. Thus, it is difficult to provide successful, necessary alerts with speed control 
alone. Similar difficulties with relying on speed control are mentioned by Krozel, et al. 
[1996] using a much different conflict analysis method based on optimal control theory. 

Figure 8-5 shows the observed times in which the alert stages were triggered for 
the perpendicular crossing case of two aircraft on a direct collision course. This is the 
same situation described by the SOC curves in Figure 8-4. Alert stage 1 is triggered 12.3 
minutes prior to the time of Closest Point of Approach (CPA). Stages 2 and 3 are 
triggered at approximately 8. And .8 minutes to CPA, respectively. If the conflict is not 
resolved, ATC is notified to take over authority (at the AT stage) at 3.3 minutes to CPA. 
Finally, TCAS produces a traffic advisory (TA) at approximately 45 seconds and a 
resolution advisory (RA) at 35 seconds to CPA. 



165 



4 






^ 



Direct Collision 



TCAS RA 
TCAS TA 



CPA 



12.3 min. 



8.5 



5.8 




0.6 min. 



Time to Closest Point of Approach (CPA) 
Figure 8-5: Alert Time Line: Direct Collision (90° Crossing Angle) 

Figure 8-6 shows a slight different encounter in which two aircraft are not on a 
direct collision course but will pass within 6 nautical miles of one another. Stage 1 is 
triggered 6.5 minutes before CPA, and stage 2 is triggered 2.2 minutes before CPA. A 
TCAS TA is also generated at approximately 30 seconds before CPA. When the traffic 
passes the host aircraft, the alert stages gradually decrease. Thus, the logic increases the 
alert stage as the potential for a conflict rises and reduces the alert stage as it becomes 
less likely that the intruder could turn and cause a conflict. 



166 



^ 



A 



$ 



6 nmi Miss 



ih 



CPA 



TCASTA 



6.5 min. 



2 

2.2 0.5 
0.2 



+0.5 min. 



0.0 



Time to Closest Point of Approach (CPA) 
Figure 8-6: Alert Time Line: 6 nmi Minimum Separation (90° Crossing Angle) 
8.5 Simulation Studies 

At the NASA Ames Research Center, the prototype alerting logic was 
incorporated in several aircraft- ATC simulator experiments as part of a study on pilot 
decision-making aids for new Air Traffic Management environments [Johnson et al., 
1997; Cashion et al. 1997; Battiste and Johnson, 1998; Cashion and Lozito, 1999; Dunbar 
et al., 1999; Johnson et al., 1999]. In these studies, enroute conflicts were scripted to 
examine pilot response and to exercise the alerting logic. 

In operation, the alerting logic was used to trigger the four stages of alerts 
discussed earlier. Additionally, the probability data were used in one study to determine 
the magnitude of maneuvering required to resolve conflicts at a specified level of 
confidence. The pilots in the study were given an interactive tool to explore different 
maneuvering possibilities. These maneuvers were compared against the probability data 
to determine whether the conflict would be resolved with 95% confidence. The cockpit 



167 



display then indicated to the pilots whether the proposed maneuver was likely to be 
successful. 

Preliminary results from the NASA studies show that the pilots could successfully 
resolve conflict without ATC guidance in most cases. AT alert stages were only 
observed in scenarios where the intruding aircraft was purposely diverted toward the host 
aircraft at close proximity. However, a more complete analysis is required to more fully 
evaluate the alerting logic and to determine the potential impact of the airborne conflict 
resolution in air traffic management. Additional studies are now under way to exercise 
the logic utilizing the various intent information that might be available to aircraft in the 
future. 

Early test runs from the intent studies showed some interesting results. With 
intent, aircraft could be allowed to be spaced in relatively close proximity of one other as 
shown in the example of Figure 8-7. Here the two aircraft are on parallel tracks and 
indicating an intent to maintain the same heading. Such intent allows the aircraft to be 
spaced a short distance apart since the alerting logic is expecting the intent to be 
followed. However, blunders or sudden course changes by one of the aircraft would 
result in an immediate hazard situation with little time for appropriate action on the part 
of the other aircraft. The original intent information has suddenly become a detriment to 
the overall safety of the system. Thus in order to utilize intent effectively and safely, 
some use of conformance check to maintain the intent is necessary so that W = T. If the 
intent is to be changed, it should also be cleared first with the alerting logic so that there 
is no disparity between W and T, or a conflict in the new course. 



168 




Figure 8-7: Error in Intent Information 



8.6 Discussion 



The Monte Carlo approach used in the alerting logic has the advantage that it can 
handle complex, 3-D encounters with complicated error distributions specific to each 
conflict situation. It uses whatever information is available to describe the conflict and 
makes a direct prediction of the alerting outcome {i.e. P(FA), P(SA) Jbased on that 
information. If a change in the working model (W) is required, such as a change in flight 
plans, the Monte Carlo simulations should reflect the new updates (W = T). 

However, there are some limitations that should be considered when utilizing the 
Monte Carlo approach presented here. A sufficient amount of processing power is 
required to perform the vast number of Monte Carlo simulations at any instant in time. 
The methodology presented in Section 6.2 has allowed a relatively efficient way of 
computing these probabilities in near real-time (on the order of a quarter of second for 
each pair of aircraft). As the number of aircraft increases in the vicinity, the longer it will 
take to compute the desired probabilistic values. 

Another consideration involves the scope of the conflict. The resolution 
maneuvers used to develop the alerting logic are based on the immediate problem of 
avoiding a conflict and do not consider the additional maneuvering required to return to 



169 



the original flight path. Thus, the logic does not incorporate issues such as increased fuel 
burn or flight time in the decision to alert. It may be necessary to incorporate cost-based 
metrics into the alerting logic as efficiency becomes an increased priority. This might be 
achieved, for example, by weighting avoidance maneuver options by the additional cost 
of deviation each option would incur. 

Finally, centralized traffic management issues have been ignored. Because, as 
assumed in the NASA experimental studies, pilots have initial responsibility for traffic 
separation, ground controllers could have difficulty when suddenly presented with a 
conflict that was not resolved by the flight crews. Additional conflict detection and 
resolution aids must be provided for ground controllers to enable them to return to the 
traffic management loop and handle traffic once they are alerted to the conflict. 
Alternatively, it may be more appropriate for all conflict detection and resolution 
activities to be performed on the ground. In either case, the design approach presented in 
this thesis could be applied in an air, ground, or mixed mode of operation to develop 
future alerting systems. 

8.7 Summary 

In this chapter, a conflict alerting probe was developed from the concepts 
explained in the previous chapters. It is a novel approach based on the utility of near 
real-time Monte Carlo simulations to predict conflicts and alerting performance. The 
thresholds are based on the concepts of the SOC methodology first presented by Kuchar 
[1995, 1996] and entails modeling uncertainties directly into the aircraft trajectory model. 
Some preliminary results for some of the NASA studies are discussed as well as some the 
advantages and disadvantages of the approach. 



170 



Chapter 9 



Summary and Conclusions 



9.1 Summary 

9.1.1 Review of Alerting Systems and Alerting Performance 

A brief overview of the state-space approach to describing alerting systems was 
given to provide a foundation for further discussion into the problems associated with 
alerting system design and performance. Terms such as false alarms, missed detections, 
and correct detections were presented in the realm of state-space. The problem of 
conflict detection and resolution was formally introduced, and the importance of the 
tradeoff between the different performance parameters (e.g. false alarms, missed 
detections) was discussed. A review of System Operating Characteristic (SOC) analysis 
was also given. 

9.1.2 Survey of Alerting Approaches 

A survey of different conflict alerting approaches was made to gain insight into 
the problem of designing a conflict probe to handle complex encounters (3-D, multi- 
aircraft, intent information, uncertainties). It was found that a large variety of methods 
existed in literature with no single, apparent underlying theme to drive designs. 
However, there was a prevalence of an iterative, ad hoc approach using test scenarios to 
set the threshold parameters. Also, three major trajectory propagation methods for 
predicting and resolving conflicts were identified: single path, worst case, and 
probabilistic. 

171 



9.1.3 A Unified Approach for Improving Alerting Performance 

The alerting problem was recast as a prediction problem in the presence of 
uncertainties. The performance measures often used to gauge and set threshold 
parameters rely on the accuracy of the prediction. The importance of trajectory modeling 
in the prediction process was emphasized and errors in the model were shown to reduce 
prediction accuracy. A unified approach to improving alerting performance was stated 
which evolved around increasing prediction accuracy through better trajectory modeling 
(W — > T, T — » W) and reducing inherent uncertainties. 

9.1.4 Probabilistic Influence in Alerting System Design 

The iterative, ad hoc approach to alerting threshold design was revisited from a 
probabilistic standpoint. It was shown that probabilistic concepts of performance and 
uncertainties were embedded within this design process of setting threshold metric 
parameters. Also, a new direct approach to alerting design was presented and shown be a 
more compact method of estimating performance directly without the unnecessary step of 
mapping back to a redundant set of threshold metrics. This new approach is based on 
modeling all known information (including uncertainties and intent information) about 
the encounter directly into the working trajectory model used by the alerting logic to 
predict future aircraft positions. The result is a situation-specific design that is tailored to 
each individual encounter and not compromised from a globally averaged threshold. 

9.1.5 Methodology for Computing Conflict Probabilities 

A method of computing the probabilistic parameters was presented using a Monte 
Carlo simulation method. Uncertainties and intent information were modeled into the 
aircraft trajectories to predict the likelihood of conflicts and avoidance options. A 



172 



relatively simple idea of utilizing a relative frame coordinate system allowed the use of 
the Monte Carlo simulations to be performed in near real-time. The accuracy of the 
computations was discussed and appears to be within the necessary scope for use in a 
real-time conflict alerting probe. In addition, a method to visualize the conflict situation 
through the use of probability contours was presented. 

9.1.6 Application of Methodology 

The methodology developed in this thesis was utilized both as an analytical tool 
and as the basis for a real-time conflict alerting probe. 

9.1.6.1 Conflict Analysis Tool 

Through the use of probability contour maps and SOC curves, the benefits of the 
method was shown to be able to handled both simple and complex types of encounters 
with relative ease. Examples were presented using intent information, various types of 
uncertainties, and different resolution options. In one example, the merits of utilizing 
horizontal maneuvering proved to be more effective than a climb maneuver in a certain 
type of vertical encounter situation. 

9.1.6.2 Real-Time Conflict Probe 

Several versions of a real-time conflict alerting probe based on probabilistic 
thresholds were developed for use in NASA Ames Research Center simulator facilities. 
The latest version employs direct Monte Carlo simulations run in real-time and can 
include information on certain types of intent if available. 



173 



9.2 Conclusions 

The contributions from this thesis work are as follows: 

1. The importance of uncertainties in the alerting design process was identified. Much 
of the work discussed in this thesis evolved from the notion that uncertainties must be 
dealt with at some point within the alerting system. Without uncertainties in a 
conflict situation, a perfect alerting threshold could be designed. It is because of 
uncertainties and errors in prediction that make the conflict detection and resolution 
problem difficult. 

2. A unifying concept for improving alerting system performance was developed based 
on insight from recasting the problem as decision-making in the presence of 
uncertainties. The foundation is based on the realization that performance metrics 
such as false alarm rate and missed detections are measures of prediction accuracy. 
Thus to improve performance requires increasing prediction accuracy since the 
performance metrics are based on obtaining a correct prediction of a future hazard or 
the ability to avoid it. The result led to the concept of reducing modeling errors (W = 
T) and reducing inherent uncertainties (make future outcomes more deterministic). 
Common methods used to improve alerting performance (e.g. delaying alerts, 
including intent information, enforcing flight restrictions, conformance monitoring) 
were shown to belong in one of these two categories. 

3. A probabilistic connection to alerting design was shown to exist in the common ad 
hoc approach of setting threshold parameters. Uncertainties in the aircraft trajectories 
are injected into the design by use of the test simulation scenarios. The outcome of 
the simulations are probabilistic measures of performance which end up driving the 
final threshold settings. The result is that the thresholds are really a simplified 



174 



mapping to probabilistic performance measures such as P(FA) and P(SA). The 
iterative feedback of the ad hoc approach explains the ability to utilize various 
combinations of metric parameters by different alerting logics. 

4. Global versus situation-specific designs were identified. Using statistical 
computations, it was shown how global designs were a compromise between more 
situation-specific designs. The analysis provided an explanation to past problems 
encountered by current operational systems such as GPWS and TCAS. It is also the 
rationale behind tailoring thresholds to individual encounter situations especially as 
the complexity of the operating environment increases (multi-aircraft, 3-D 
encounters). 

5. A new direct approach was extended from previous work by Kuchar [1995, 1996] 
where uncertainties and intent are modeled explicitly in the aircraft trajectory model 
of the alerting logic. The method allows for a direct prediction of the performance 
measures and is inherently situation-specific to each encounter scenario. Thresholds 
can then be designed in the state-space of the performance measures {e.g. P(FA) and 
P(SA) } with the help of SOC plots. 

6. A methodology was developed to compute the probabilistic values in near real-time 
setting up the possibility for rapid analysis and also for conflict probing. 

7. Example problems were shown utilizing probability contour plots and SOC analysis. 
These examples showed the usefulness and capability of the method to handle both 
simple and complex encounter situations with relative ease. For instance, the benefit 
of intent information was shown in one example and the benefit of horizontal 
avoidance maneuvering in another. 



175 



8. The actual implementation of a prototype conflict alerting probe for flight simulation 
studies was developed for experimental use at MIT and NASA Ames Research 
Center facilities. The logic is based on near real-time Monte Carlo predictions of 
conflict and conflict avoidance. The logic can handle complex multi-aircraft, multi- 
intent, 3-D encounters. 



176 



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Appendix A 

A Review of Conflict Detection and Resolution 
Modeling Methods 



A number of methods have been proposed to automate air traffic conflict 
detection and resolution, but there have been little cohesive discussion or comparative 
evaluation of differing approaches. This appendix presents a survey of 62 recent 
methods, several of which are currently in use or under operational evaluation. This is by 
no means an exhaustive list, but it is believed to encompass a majority of the recent 
approaches to the problem. The taxonomy includes: method of dynamic state 
propagation, dimensions of state information, conflict detection threshold, conflict 
resolution method, maneuvering dimensions, and management of multiple aircraft 
conflicts. 

Nine of the models that were examined are existing operational systems in use or 
which have been evaluated in the field: Airborne Information for Lateral Spacing (AILS) 
[Waller and Scanlon, 1996], Center / TRACON Automation System (CTAS) [Isaacson 
and Erzberger, 1997], Ground Proximity Warning System (GPWS) [RTCA, 1976] and 
the recent Enhanced version (EGPWS) [Bateman 1999], Precision Runway Monitor 
(PRM) [FAA, 1991], Traffic Alert and Collision Avoidance System (TCAS) [RTCA, 
1983], Traffic and Collision Alert Device (TCAD) [Ryan and Brodegard, 1997], User 
Request Evaluation Tool (URET) [Brudnicki et al., 1997], and a prototype conflict 
detection system for the Cargo Airline Association [Kelly, 1999]. The remaining 
approaches range from abstract concepts to prototype conflict warning systems being 



187 



evaluated or used in laboratories. Five of the models were developed for robotic, 
automobile, or naval applications [Coenen et al., 1989; Iijima et al., 1991; Taylor, 1990; 
Chakravarthy and Ghose, 1998; Lachner, 1997], but are still applicable to aviation. 

To provide a consistent basis upon which to describe the approaches, each one is 
classified by the manner in which it is explicitly described in its reference. An approach 
defined here to address only horizontal conflicts, for instance, could potentially be 
extended to work in 3-D (and the need for such an extension may be mentioned in the 
reference), but such an addition was not specifically described in the reference. As 
another example, if a model computes aircraft missed distance but does not define an 
explicit conflict detection threshold, the model is not classified as providing conflict 
detection even though the model could be adapted to perform such a task. 

A.l State Propagation 

Because conflict detection and resolution can only be as reliable as the ability of 
the model to predict the future, the most important difference between modeling 
approaches involves the method by which the current states are projected into the future 
Three fundamental extrapolation methods have been identified: single path, worst case, 
and probabilistic. 

In the single path approach, the current states are projected into the future along a 
single trajectory without direct consideration of uncertainties. An example would be 
extrapolating the aircraft's position based on its current velocity vector (Figure A- la). 
The single path projection method is straightforward and provides a best estimate of 
where the aircraft will be based on the current state information. In situations where 
aircraft trajectories are very predictable (such as when projecting only a few seconds into 
the future), a single path model may be quite accurate. Single path projections, however, 



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do not directly account for the possibility that an aircraft may not behave as expected - a 
factor that is especially important in longer term conflict prediction. Generally, this 
uncertainty is managed by introducing a safety buffer, minimum miss distance, or time to 
closest point of approach threshold at which point a conflict will be detected. 



Single Path 
(a) 




Worst case 
(b) 




Probabilistic 
(c) 



Figure A- 1: Propagation Methods 

The other extreme of dynamic modeling is to examine a worst case projection. 
Here, it is assumed that an aircraft will perform any of a range of maneuvers. If any one 
of these maneuvers could result in a loss of minimum separation, then a conflict is 
declared. The result is a swath of potential trajectories which is monitored to detect 
conflicts with other aircraft (Figure A- lb). Worst case approaches are conservative in 
that they can trigger alerts whenever there is any possibility of a conflict within the 
definition of the worst case trajectory model. If such conflict inducing maneuvers are 
unlikely, protecting against them may severely reduce overall traffic capacity due to a 
high false alert rate. Accordingly, the worst case approach may be appropriate when it is 
desirable to determine if a conflict is possible, or for air traffic concepts in which aircraft 
are procedurally constrained to remain within a given maneuvering corridor. Each 



189 



corridor then becomes the boundary of the worst-case aircraft trajectories, and conflicts 
can be predicted based simply on whether corridors intersect at the same point in time. 

In the probabilistic method, uncertainties in the model are used to develop a set of 
possible future trajectories, each weighted by its probability of occurrence. Fore 
example, a distribution of future aircraft positions could be obtained by modeling 
uncertainty in along-track and cross-track guidance (Figure A-lc). A probabilistic 
approach provides an opportunity for a balance between relying too heavily on a aircraft 
adhering to a single trajectory versus relying too heavily that an aircraft performs worst 
case maneuvers. The advantage of a probabilistic approach is that decisions can be made 
on the fundamental likelihood of a conflict - safety and false alarm rates can be assessed 
and considered directly. The probabilistic method is also the most general - the single 
path and worst case models can be considered subsets of probabilistic trajectories. The 
single path trajectory corresponds to a case in which the aircraft will follow a given (e.g. 
maximum likelihood) trajectory with probability 1.0; the worst case model is one in 
which the aircraft will follow any trajectory with equal likelihood. However, the logic 
behind a probability-based system may be difficult to convey to operators, possibly 
reducing confidence in their usage [Pritchett, 1996]. There may also be difficulties in 
modeling the probabilities of the future trajectories with which aircraft may follow. 

Tables A-l, A-2, and A-3 provide an organized listing of the 62 approach 
methods. To conserve space, only the first author is listed in cases where multiple 
authors are listed on a publication. The three tables are segregated by the propagation 
method taken: single path, worst case, and probabilistic. Five columns are used to 
organize the models: State Dimensions, Conflict Detection, Conflict Resolution, 
Resolution Maneuvers, and Multiple Conflicts. 



190 



A.2 State Dimensions 

The Dimensions column shows whether the state information used in the 
approaches involves purely horizontal plan (H), vertical plane (V), or both (HV). The 
majority of approaches cover either 3-D or the horizontal plane; only GPWS focuses 
solely on the vertical plane. Some models may be easily extended to cover additional 
dimensions than are shown here, but such extension is not explicitly described in the 
reference. 

A.3 Conflict Detection 

The Detection column indicates (with a check mark) whether a modeling 
approach explicitly defines when a. conflict alert should be issued. Approaches that do 
indicate an explicit threshold may provide valuable tools and metrics upon which conflict 
detection decisions can be made, but do not precisely draw the line between predicted 
conflict and non-conflict. Additionally, models shown to not provide conflict detection 
may be primarily concerned with the resolution of the conflict rather than the in 
determining when action should begin. Although developing conflict resolution methods 
are important, at some point it will be necessary to define conflict detection thresholds 
and examine the false alarm / missed detection tradeoff. Approaches that are shown to 
provide conflict detection may use an extremely simple criterion (e.g. current range 
between aircraft) to determine when a conflict exists or may use a more complex set of 
threshold logic. 



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A.4 Conflict Resolution 

The Resolution column shows the method by which a solution to a conflict is 
generated. Five categories are included here: Prescribed (P), Optimized (O), Force Field 
(F), Manual (M), and no resolution (— ). 

Prescribed resolution maneuvers are fixed during system design based on a set of 
predefined procedures. For example, GPWS issues a standard "PULL UP" warning 
when a conflict with terrain exists. GPWS does not perform additional computation to 
determine an optimal escape maneuver. AILS [Waller and Scanlon, 1996] and Carpenter 
and Kuchar [1997] assume that a fixed climbing-turn maneuver is always performed to 
avoid traffic on a parallel approach. Prescribed maneuvers may have the benefit that 
operators can be trained to perform them reflexively. This may decrease response time 
when a conflict is issued. However, prescribed maneuvers are, in general, less effective 
than maneuvers that are computed in real-time since there is no opportunity to modify the 
resolution maneuver (the maneuver is performed open-loop to some extent). In many 
conflicts, it will be necessary to adapt the resolution maneuver to account for unexpected 
events in the environment, or to reduce the aggressiveness of the maneuver should the 
conflict be resolved more easily than first predicted. 

Optimization approaches typically combine a kinematic model with a set of cost 
metrics. An optimal resolution strategy is then determined by solving for the trajectories 
with the lowest cost. TCAS, for example, searches through a set of potential climb or 
descent maneuvers and selects the least-aggressive maneuver that provides adequate 
protection [RTCA, 1983]. This requires the definition of appropriate cost functions - 
typically projected separation, or fuel or time, but costs could also cover workload. 
Developing cost functions may be fairly straightforward for economic values, but 



192 



difficult when modeling human utilities. Because current interest in this field is generally 
centered on strategic resolution of conflicts before immediate tactical evasion is required, 
economic costs and operator workload will be important to the system design. 

Some of the models denoted as using optimized conflict resolution apply 
techniques such as game theory, genetic algorithms, expert systems, or fuzzy control to 
the problem. Expert systems use rule-based methods to categorize conflicts and decide 
whether to alert and/or resolve a conflict. These models can be complex and require a 
large number of rules to completely cover all possible encounter situations. Additionally, 
it may be difficult to certify that the system will always operate as intended, and the 
"experts" used to develop or train the system may in fact not use the best strategy in 
resolving conflicts. However, the rule base, by design, may be easier for a human to 
understand or explain than an abstract mathematical algorithm. 

Force field approaches treat each aircraft as a charged particle and use modified 
electrostatic equations to generate resolution maneuvers. The repulsive forces between 
aircraft are used to define the maneuver each performs to avoid a collision. A force field 
method, while attractive in the sense that a conflict resolution is continuously available 
using relatively simple equations, may have some pathologies that require additional 
consideration before they can be used in operation. For example, force field methods 
may assume that aircraft continuously maneuver in response to the changing force field, 
or that aircraft can vary their speed over a wide range. This requires a high level of 
guidance on the flight deck and increases complexity beyond issuing simple heading 
vectors, for example. Several human-in-the-loop implementations of the force field 
approach, however, have shown that the method can be effective if properly applied 
[Duong and Hoffman, 1997; Hoekstra et al., 1998; Zeghal and Hoffman, 1999]. 



193 



Some models allow the user to generate potential conflict resolution solutions and 
obtain feedback as to whether the trial solution is acceptable. These models are denoted 
as handling a Manual solution in the table. The benefit of a manual solution is that it is 
generally more flexible in the sense that it is based on human intuition (using information 
that may not be available to the automation). For example, weather information that is 
not available to the conflict detection and resolution system may be important when 
considering a conflict resolution maneuver. Automated solutions that do not take 
relevant environmental information into account will likely produce nuisance solutions 
that the human finds unacceptable. 

A " — " in the Resolution column indicates that the model does not provide an 
explicit output of an avoidance action or feedback on a user-defined trial solution. These 
models perform conflict detection but are not designed to explicitly consider conflict 
resolution. In some cases, successful conflict resolution is presumed (the focus of the 
approach is only on detecting or counting conflicts). 

A. 5 Resolution Maneuvers 

The Maneuvers column indicates what dimensions of resolution maneuvers are 
allowed. Possible maneuver dimensions include Turns (T), Vertical maneuvers (V), and 
Speed changes (S). The notation TV, for example, means that either turns or vertical 
maneuvers may be performed (but not both simultaneously). In some cases, combined 
maneuvers may be commanded or performed, indicated by CO. Thus C(TV), for 
example, indicates that a simultaneous climbing or descending turn may be performed. 

Generally, providing more maneuvering dimensions allows for a more efficient 
solution to a conflict. However, it does place additional responsibility on the operator in 



194 



the sense that a more complex maneuver must be controlled and monitored, possibly 
increasing response time and workload. 

A.6 Multiple Conflicts 

Finally, the Multiple column describes how the model handles more than two 
traffic conflict simultaneously. This can take two forms: Pairwise (P), in which multiple 
conflicts are addressed sequentially in pairs; and Global (G) in which the entire situations 
is examined simultaneously. 

In a realistic traffic environment, it will be necessary that a conflict detection and 
resolution system be able to manage more than one conflict at a time. In a pairwise 
approach, if one conflict solution induces a new conflict, the original solution may need 
to be modified until a conflict-free solution is found. This is the approach taken by 
TCAS, for example, and is effective but also could potentially fail in certain situations. A 
global solution, while potentially more complex, may be more robust. For example, 
consider the situation shown in Figure A-2. On the left, a pairwise solution is shown. 
The aircraft on the left detects a conflict with a co-altitude threat at a certain preset time 
before collision, and attempts to climb or descend. Neither solution is acceptable since it 
results in a conflict with another aircraft. On the right, a global solution considers all 
three threat aircraft simultaneously and determines that the climb or descent maneuver 
must begin earlier than the baseline threshold time in order to safely resolve the conflict. 
At the least, models should be examined in multi-aircraft situations to determine their 
robustness to this type of problem. 



195 



baseline 
alerting threshold 



alert time increased 
due to other hazards 





Pairwise Solution 



Global Solution 



Figure A-2: Pairwise vs. Global Solution 

A.7 Other Model Elements 

In addition to the six factors used to distinguish between modeling approaches in 
Tables A-l, A-2, and A-3, there are several other issues to be considered but are not fully 
described here. These issues include specifically which current states and metrics are 
used to make conflict detection and resolution decisions, how uncertainty is managed in 
the model, and the degree to which the model assumes coordination between aircraft 
involved in a conflict. 

Consideration of the states that are used in conflict detection and resolution is 
important because these states represent the means by which the system observes the 
environment. Some approaches use a simplified set of states which reduces sensor 
requirements, but increases the uncertainty in which the conflict detection and resolution 
decisions can be made. An additional set of data that will be valuable in strategic conflict 
detection is aircraft intent information such as a programmed flight plan. This 
information can be used to better model the future trajectory of the aircraft, and thereby 
be better able to make correct alerting decisions. 



196 



The manner in which uncertainties are managed in the design of a conflict 
detection and resolution system varies widely. Most approaches to the problem combine 
the uncertainties into a spatial safety buffer to reduce missed detection probability and 
also incorporate a look-ahead time boundary to limit false alarms. This provides for a 
reasonable accommodation of uncertainties, but it may not be as effective or accurate as 
more complete, probabilistic trajectory models. 

Coordinate conflict resolution between aircraft has two primary benefits. First, 
the required magnitude of maneuvering can be reduced when both aircraft maneuver 
cooperatively as opposed to the case when only one aircraft maneuvers. Second, 
coordination helps ensure that aircraft do not maneuver in a direction that could prolong 
or intensify the conflict. However, a system designed assuming that coordination will 
occur should also be evaluated in cases in which coordination is not carried out as 
planned. This would provide some measure of the robustness of the system to a datalink 
failure or pilot error. For example, TCAS was found to perform poorly in situations in 
which one aircraft did not respond to the recommended advisory [Drumm, 1996]. In fact, 
it was deemed unproductive to analyze such encounters in depth since it was felt to 
completely overshadow any other factors contributing to poor performance. However, if 
such situations do occur in actual operation, the problem must be resolved else it can only 
lead to devastating consequences. 



197 



Model 



Table A-l: Single Path Trajectory Propagation 



Dimensions Detection Resolution Maneuvers Multiple 



Andrews 


H 







T 


P 


Chakravarthy 


H 


— 





C(ST) 


P 


Tomlin 


H 


— 


o 


T 


G 


Irvine 


HV 


— 





C(STV) 


P 


Ota 


HV 


— 





C(TV) 


G 


Kosecka 


H 


— 


F 


C(ST) 


G 


Zeghal (1998) 


H 


— 


F 


C(ST) 


G 


Eby 


HV 


— 


F 


C(STV) 


G 


Sridhar 


H 


V 


— 


— 


P 


Bateman (EGPWS) 


HV 


V 


— 


— 


— 


Havel 


HV 


V 


— 


— 


P 


Kelly 


HV 


V 


— 


— - 


P 


Ryan (TCAD) 


HV 


V 


— 


— 


P 


RTCA (GPWS) 


V 


V 


P 


V 


— 


FAA (PRM) 


H 


V 


P 


C(TV) 


P 


Bilimoria 


HV 


V 


P 


STV 


P 


Burgess 


H 


V 





TV 


P 


Coenen 


H 


V 


o 


ST 


P 


Gazit 


H 


V 





VT 


P 


Harper 


H 


V 





C(ST) 


G 


Iijima 


H 


V 





ST 


P 


Burdun 


HV 


V 


o 


C(STV) 


P 


Durand 


HV 


V 





T 


G 


Ford 


HV 


V 


o 


V 


P 


Krozel 


HV 


V 


o 


STV 


P 


Love 


HV 


V 


o 


TV 


P 


Schild 


HV 


V 


o 


C(TV) 


P 


RTCA (TCAS) 


HV 


V 





V 


P 


Hoekstra 


HV 


V 


F 


C(STV) 


P 


Zeghal (1994) 


HV 


V 


F 


C(STV) 


G 


Duong 


HV 


V 


M/F 


C(STV) 


P 



198 



Table A -2: Worst Case Trajectory Propagation Methods 



Model 



Dimensions Detection Resolution Maneuvers Multiple 



Lachner 


H 


— 





C(ST) 


P 


Ratcliffe 


HV 


V 


— 


— 


P 


Shepard 


HV 


V 


— 


— 


P 


Shewchun 


HV 


V 


— 


— 


P 


Waller (AILS) 


HV 


V 


p 


C(TV) 


P 


Vink 


HV 


V 


M 


C(STV) 


P 



Table A-3: Probabilistic Trajectory Propagation Methods 



Model 



Dimensions Detection Resolution Maneuvers Multiple 



Heuvelink 


H 


— 


— 


— 


P 


Paielli 


H 


— 


— 


— 


P 


Taylor 


H 


— 


— 


— 


P 


Bakker 


HV 


— 


— 


— 


P 


Innocenti 


H 


— 


F 


C(ST) 


G 


Rome 


H 


V 


— 


— 


P 


Warren 


H 


V 


— 


— 


P 


Williams 


HV 


V 


— 


— 


P 


Carpenter 


H 


V 


p 


C(TV) 


P 


Prandini 


H 


V 





T 


P 


Krozel 


HV 


V 





STV 


P 


von Viebahn 


HV 


V 


o 


TV 


P 


Isaacson (CTAS) 


HV 


V 


M 


C(STV) 


P 


McNally (CTAS) 


HV 


V 


M 


C(STV) 


P 


Brudnicki (URET) 


HV 


V 


M 


C(STV) 


P 


Yang 


HV 


V 


M 


C(STV) 


P 



199 



200 



Appendix B 



Statistical Analysis of Global Distributions 



The following will show the computations used to determine the descriptive 
statistics of combining two probability distributions into one global distribution. The 
results can then be extended to a combination of more than two distributions. 

B.l Statistics of Combining 2 Distributions 

Let f(x) and f 2 (x) be two separate distributions from which to sample from. 
The mean and variance of each function will be //,, of and jU 2 , a\, respectively. Also, 
the fraction of times each distribution is sampled will be denoted as a, and a 2 , and thus 
the combined probability density function, f(x), will have a distribution of: 

f(x) = a,f(x) + a 2 f 2 (x) (B.l) 

a, + a 2 = 1 (B.2) 

The expected value or mean, pL G , of this global distribution can be computed as follows: 

= f xf(x)dx 

= £^W + o 2 / 2 W)i C (B.3) 

= aj xf(x)dx + a 2 J xf 2 (x)dx 



201 



(B.4) 



The mean of the combined, global distribution is just a weighted average of the 
individual distribution means. Thus min(^,, /i 2 ) < f.i G < max(^,, /i 2 ). 

The variance can be derived from the second central moment: 

a 2 c = E[(x - E[x}f] 

= £ (* " E[x]f[aJ } (x) + a 2 f 2 (x)]dx 

= £(* 2 - 2x ^o + Vlt^/if*) + <hki x ^ 
= a, £ x 2 f i (x)dx + a 2 j^x 2 f 2 (x)dx - 
2[i G a x fxf ] (x)dx - 2fl G a 2 \ xf 2 (x)dx + 

= a,E,[x 2 ] + a 2 E 2 [x 2 ] - 2fi G a 2 E 1 [x] - 2n G a 2 E 2 [x] + /4(l) 
- a,E\x 2 ] + a 2 E 2 [x 2 ] - 2n G (a 2 E,[x] + a 2 E 2 [x]) + fi 2 G (l) 
= ^[x 2 ] + a 2 E 2 [x 2 ] - 2^ G (a 2 ^ + a 2 }X 2 ) + £{1) 
= a,£,[x 2 ] + a 2 E 2 [x 2 ] - 2/^ G (fl G ) + fi 2 a (l) 
= a,(a 2 x + tf) + a 2 {a\ + n\) - fi 2 G 
= (atf + a 2 a 2 2 ) + (a,^ 2 + a 2 \i 2 2 ) - f£ 

The term in the left-most parenthesis is just the weighted average of the individual 
variances. It can be shown that the other terms in the equation must be greater than or 
equal to so that the overall result will be larger than the weighted average of the 
variances. The proof is through the use of the Lagrange multiplier method with the 
constraint of Equation B. 2. The idea is to show that (a s ^l 2 + a 2 n\) - /i 2 > by 
proving the minimum value cannot be negative. Thus, find the minimum of: 

A = (o^, 2 + a 2 n 2 2 ) - a4 (B ' 5) 



202 



with the constraint of a, + a 2 = 1. 



L = A + A(a, + a 2 - l) 

= (a^, 2 + a 2 H 2 2 ) - H 2 G + A(a, + a 2 - l) 

= (a,/!, 2 + a 2 n 2 2 ) - (a x fx x + a 2 n 2 ) 2 + A(a, + a 2 - l) (B.6) 

= (a,^ 2 + a 2 ^ 2 ) - (afo 2 + 2a x a 2 n x \i 2 + a 2 2 fl 2 2 ) + k(a x + a 2 - l) 

= (a x - a 2 )^ 2 + (a 2 - a 2 )^ 2 - 2a x a 2 n x fi 2 + l(a x + a 2 - l) 

Now minimize L: 

^. = = (1 - 2a> 2 - 2a 2 ^/i 2 + A (B.7) 

-^1 = o = (1 - 2a 2 )[i\ - 2a x \i x \i 2 + A (B.8) 

da 2 

^ = = a, + a 2 - 1 (B.9) 

(7/1 

Solving Equations B.7 - B.9 results in the following: 

A = [i x ii 2 (B.10) 



1 
a, = — 

1 2 

1 

a, = — 

2 2 



(B.ll) 



(B.12) 



203 



= (i - (i)>. 2 + (i - u)> 2 - mm^ 

= (i - iK + (t " iK - (7)^2 (B.13) 

= 1 (fif + \i\ - 2^fi 2 ) 

= Hh - fh) 1 

> 

The contribution of A to the new, combined variance, a 2 G , is thus to increase the 
spread of the distribution due to the separation of the individual distribution means, //, 
and jj, 2 . 

B.2 Statistics of Combining More Than 2 Distributions 

The statistical parameters of combining more than 2 distributions can be obtained 
in a similar manner as in the previous section. For example, the probability density 
function, f(x), for sampling from 3 distributions is: 

f(x) = ajjx) + a 2 f 2 (x) + a 3 f 3 (x) (B.14) 



a, + a 2 + a 3 = 1 (B.15) 



And the mean and variance can be computed as follows: 

= j xf(x)dx 

= £ x^ix) + aj 2 (x) + a 3 f 3 (x)}ix (B.16) 

= a, £ xf l (x)dx + a 2,Q x f 2 {x)dx + a 3.L W*^ 
= a,^, + a 2i u 2 + 03^3 



204 



(B.17) 



al = E[(x - E[x]f] 

= £(* ~ E[x]f[a x f x (x) + aj 2 (x) + a 3 f 3 (x)]dx 

= j°^(x 2 - 2x\i G + nl)[a x f x (x) + a 2 f 2 (x) + a 3 f 3 (xj\dx 

= a x \ x 2 f x (x)dx + a 2 \ x 2 f 2 (x)dx + a 3 j x 2 f 3 (x)dx - 

2\i G a x f xf l (x)dx - 2\i G a 2 \ xf 2 (x)dx - 2pL G a 3 [ xf 3 (x)dx + 

H 2 G \j[aJ x (x) + a 2 f 2 (x) + a 3 f 3 (x)]dx 

= a x E x [x 2 ] + a 2 E 2 [x 2 ] + a 3 E 3 [x 2 ] - 
2n G a 2 E x [x] - 2/l G a 2 E 2 [x] - 2^ G a 3 E 3 [x] + p 2 G (l) 

= OjEJx 2 ] + a^J* 2 ] + OjEjj* 2 ] - 
2/" G («2^[^] + a 2 E 2 [x] + a 3 E 3 [x]) + H 2 G {1) 

= a x E x \x 2 ] + a 2 E 2 \x 2 \ + a 3 E 3 [x 2 } - 
2n G (a 2 v x + a 2 li 2 + a 3 n 3 ) + fi 2 G {l) 

= ^[x 2 ] + a 2 E 2 [x 2 ] + a 3 E 3 [x 2 ] - 2fl G (n c ) + ^ 2 (1) 
= ^(a 2 + rf) + a 2 (a 2 2 + tf) + a 3 (a 2 3 + ^ 3 2 ) - ,u 2 
= (a,a 2 + a 2 a\ + a 3 a]) + (a^f + a 2 n\ + a 3 ji 2 ) - \i 2 G 

Again, the left-most terms in parenthesis are just the weighted average of the individual 
distribution variances; while the remaining terms are due to the difference in the means of 
the distributions and increase the overall variance of the combined probability density 
function. 



205 



206 



Appendix C 



Conflict Detection Using Line- Volume Intersection 



The determination of conflict between 2 moving objects is relatively simple if the 
path of the objects can be approximated as straight line segments. When placed in the 
relative frame of one of the objects, the solution becomes one of calculating the 
intersection of a line (relative velocity vector) with a protected volume surrounding the 
origin. If an intersection occurs, then minimum separation is violated. 

C.l Relative Frame 

Working in the relative frame of one of the aircraft can greatly simplify the 
computational complexity of the conflict determination problem. For this discussion, the 
aircraft at the origin will be termed the ownship, and the problem is to determine if the 
other aircraft (intruder) will pass through the protected volume surrounding the origin 
position. 

Figure C-la shows the encounter in absolute frame with individual trajectories 
depicted as straight line segments. A key change in the velocity vector constitutes a new 
segment. For Figure C-lb, the situation is shown in the relative frame of the ownship 
aircraft. Each change point is thus a change the relative velocity vector between the two 
aircraft. The task is then to determine if any of the line segments intersect the protected 
volume at the origin. If there is an intersection, the protected volume will be violated and 
a conflict is declared. 



207 



Instantaneous 
Maneuver 




a) Absolute Frame Encounter 




Change Points jr Host Maneuver 

Intruder Maneuver 



Line-Volume 
Intersect 



b) Relative Frame Encounter 



Figure C-l: Absolute vs. Relative Frame 



208 



C.2 Line- Volume Intersection 

The solution to solving the intersection between a line (relative velocity vector) 
and a cylinder (protected volume) can be split up into horizontal and vertical dimensions. 
The idea is to determine first if there is an intersection in the horizontal plane. If there is, 
then the line connecting the 2 points of horizontal intersection is cross-checked in the 
vertical domain to see if it lies within or passes through the cylinder. 

For the discussion in this appendix, DMOD will refer to the horizontal radius of 
the protected zone and ZMOD will denote the minimum vertical separation (see Figure 
C-2). 

DMOD . 
_ .J 



^^ 



31 



ZMOD 
ZMOD 



Figure C-2: Cylindrical Protected Zone Parameters 

C.2.1 Horizontal Intersection 

Let (d x , d y , d) be the starting position of a line segment which begins at time 
t and ends at t e . The direction of the line can just the relative velocity vector 
v = [vj v y vj . Any other point, p, on the line can be found from: 

P = {d x + vjt - tj, d y + v y (t - tj, d z + v z (t - t j) (C.l) 

In the horizontal plane, the time when the line intersects DMOD can be found from the 
equation of a circle: 



209 



K + V J< - O) 2 + (d, + v,(t - tjf = DMOD 2 (C.2) 

(v, 1 + vj)(r - t f + 2(vA + v,J y )(r - ,„) + (d 2 + < - DMOD 2 ) = (C.3) 

which has the form of quadratic equation: 

a(t - t f + b (t-t o ) + c = (C.4) 



(' " '•) = Ta iC) 



Note that is the solution assumes an infinite line. 

There are 3 possible solutions depending on the value of the radicand, b 2 - 4ac . 
If b 2 - Aac < 0, then no real solution exists and the line will never intersect the circle. 
If b 2 - Aac = 0, then only one solution exists and the line intersects tangent to the 
circle of radius DMOD. If b 2 - Aac > 0, then two solutions exist as the line passing 
through the circle. 

C.2.2 Vertical Intersection 

C.2.2.1 b 2 - Aac < 
A conflict does not exist so no there is no need to check the vertical intersection. 
C.2.2.2 b 2 - Aac = 

The line intersects tangent to the circle edge. Thus, if a conflict exists, the 
intersection point must lie within ZMOD in the vertical dimension. Using the solution 
from Equation C.5, the position of the intersection point can be found from Equation C.l. 
The z-component must be checked to see if it lies within ±ZMOD. If it does, then one 



210 



final check needs to be made to make sure t < t < t e , If this is satisfied, then a conflict 
exists along this line segment. 

C.2.2.3 b 2 - 4ac > 

In this case, the line intersects the circle at 2 distinct points, p x and p 2 , at times, t { 
and t 2 , respectively. It will be assumed that t x < t 2 . The following set of C code can be 
used to determine if the vertical domain intersection is also satisfied: 



#define AND && 

#define OR | | 

#define MAX(a,b) (((a) > (b) ) 

#define MIN(a ; b) (((a) < (b) ) 

#define HIT 1 

#define MISS 

#define ZMOD 1000.0 /* (ft) vertical threshold "*/ 






(a) 
(a) 



(b)) 
<b)) 



short conflict; 
double tO; 
double te; 
double tl; 
double t2; 
double vz ; 
double dz ; 
double zl; 

double z2; 



/* 
/* 
/* 
/* 
/* 
/* 
/* 
/* 
/* 
/* 
/* 



0=miss, l=hit 

(sec) start of line segment 
(sec) end of line segment te>t0 
(sec) 1 st pt. of DMOD hit 
(sec) 2 nd pt. of DMOD hit t2>tl 
(fps) vertical velocity comp. 
(ft) start of line segment 
(ft) 1 st pt. of DMOD hit or 

start of line segment 
(ft) 2 nd pt. of DMOD hit or 

end of line segment 



*/ 
*/ 
*/ 
*/ 
*/ 
*/ 
*/ 
*/ 
*/ 
*/ 
*/ 



if ( (t2 < 0.0) OR (t < tl) ) 

conflict = MISS; 
else { 

zl = vz*(MAX(t0, tl) - tO) + dz; 

z2 = vz*(MIN(te, t2) - tO) + dz; 

if ((zl <= ZMOD) AND (z2 >= -ZMOD)) 

conflict = HIT; 
else if ((zl > ZMOD) AND (z2 <= ZMOD)) 



211 



conflict = HIT; 
else if ((zl < -ZMOD) and (z2 >= -ZMOD) ) 

conflict = HIT; 
else 

conflict = MISS; 
} 

The first if-statement checks to see if the intersection occurs outside of the two 
endpoints of the line segment. If it does, then no conflict will exist within the time t to 

The first nested if-statement covers an intruder aircraft that is either currently 
within the ownship's protected volume (Figure C-3a) or enters it from the side (Figure C- 
3b). The next if-statement handles an intruder coming from above the protected zone 
(Figure C-3c), while the last if-statement takes into account the intruder entering from 
below (Figure C-3d). 



212 



fe =5 ^ 



ZMOD 



-ZMOD 



a) Intruder Inside Protected Zone 



£^ 



ZMOD 



-ZMOD 



b) Intruder Entering from Side 




ZMOD 



-ZMOD 



c) Intruder Entering from Top 




d) Intruder Entering from Bottom 



Figure C-3: Vertical Conflict Dimension 



213