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An alternative to developing collision avoidance logic directly is to model the problem to be solved and then apply computational techniques to automatically derive the optimal logic with respect to a set of performance metrics. One advantage of this approach is that it is much easier to specify performance metrics and develop models of aircraft behavior than to develop robust pseudocode. The di.cult task of ensuring that the logic performs well in all possible encounter situations is left to computers, which are better equipped than human designers to reason about the low-probability events that can occur during the course of an encounter that can result in collision.
The computational technique that enables the e.cient optimization of the collision avoidance logic is called dynamic programming (DP). Although DP was originally developed over .fty years ago and has been applied to a wide variety of di.erent problems, it has not been applied to collision avoidance until recently. In FY09, the TCAS Program O.ce sponsored an initial investigation into this method, and the research was summarized in Project Report ATC-360. While the results were promising, the performance was limited by assumptions made by the model.
This report discusses work in FY10 on relaxing previous assumptions. The new model allows advisories to be strengthened and reversed, as with the existing version of TCAS. Because it not possible to know exactly the future trajectories of aircraft, the provision for changing the initial advisory signi.cantly reduces collision risk. Motion in three spatial dimensions is now captured by the model, allowing the logic to better determine when an alert is necessary. This report introduces probabilistic pilot response models, which can result in logic that is more robust to variability in response. The logic now uses explicit models to account for sensor noise, resulting in improved performance with realistic surveillance systems. This report shows how to coordinate maneuvers between aircraft and how to resolve threats from multiple aircraft.
The logic optimized by DP signi.cantly outperforms TCAS in simulation according to the standard safety and operational performance metrics. Both the DP logic and the TCAS logic were evaluated on the same collection of encounters generated by a high-.delity encounter model. Millions of simulated encounters show that the DP logic provides a lower probability of mid-air collision while reducing the alert rate. Although some further development is required, the DP approach appears viable for developing robust collision avoidance logic.

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ACKNOWLEDGMENTS

This report is the result of research and development sponsored by the TCAS Program O.ce at the FAA. The authors appreciate the assistance provided by the TCAS Program Manager, Neal Suchy, who has been supportive of pursuing new ideas and innovation for advancing aviation safety.
This work has greatly bene.ted from discussions with Thomas B. Billingsley, Barbara J. Chludzinski, Ann C. Drumm, Matthew W. M. Edwards, Leo P. Espindle, J. Daniel Gri.th, William
H. Harman, Tomás Lozano-Pérez, Leslie P. Kaelbling, James K. Kuchar, Donald E. Maki, Frans
A. Oliehoek, Wesley A. Olson, David A. Spencer, Selim Temizer, Panayiotis C. Tzanos, Roland E. Weibel, Richard E. Williams, and M. Loren Wood.

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TABLE OF CONTENTS

Page

Executive
Summary
iii Acknowledgments
v List
of
Illustrations
xi List
of
Tables
xiii
1. INTRODUCTION
1
1.1 Design
Considerations
1
1.2 Background
3
1.3 Proposed
Approach
5
1.4 Recent
Advances
6
1.5 Overview
7
2. PROBLEM
FORMULATION
9
2.1 Markov
Decision
Processes
9
2.2 Partially
　
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