4.7 DISCUSSION
ATC-360 showed that dynamic programming results in logic that provides the lowest possible alert rate for a speci.ed safety level under the assumption that the model used for optimization is correct. The results in this section demonstrate that even when the model used for optimization is inaccurate, the logic can still perform signi.cantly better than TCAS in terms of preventing near mid-air collisions and reducing the alert rate. This section also explored the use of robust dynamic programming to further improve the robustness of the optimized logic to modeling errors, and experiments showed reduced sensitivity to both parametric and structural modeling errors. A state-based robustness analysis demonstrated logic robustness across the entire state space.
The robustness analysis in this section used encounter models. Although the encounter models were useful in generating large collections of encounters to stress test the system, further analysis will involve evaluating the logic robustness on recorded radar data. One complication with using recorded radar data is that there are relatively few near mid-air collisions in the airspace, in part due to the e.ects of TCAS. However, evaluation of the optimized logic on recorded encounters can aid in tuning the model and choosing an appropriate cost function. Even if the model is not perfectly tuned due to the sparseness of the recorded radar dataset, the optimized logic should still perform well as shown by the experiments in this section.
0 10203040
τ (s)
(a) 0ft/s2 environment noise.
0 10203040
τ (s)
(b) 8ft/s2 environment noise.
0 10203040
τ (s)
(c) 15ft/s2 environment noise.
Figure 13. Probability of NMAC across the discrete state space evaluated at .xed environment noise values.
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5. COLLISION AVOIDANCE IN THREE SPATIAL DIMENSIONS
The
previous
section
applied
the
logic
optimized
for
a
simple
two-dimensional
model
(Section
3)
to three-dimensional encounters. Although the optimized logic performed better than TCAS, it was limited by its inability to adequately model the horizontal dynamics. One way to better model encounters in three spatial dimensions is to add additional state variables. However, naively discretizing this (greatly expanded) state space would signi.cantly increase the computational and storage requirements.
This section presents a general approach for approximating solutions to problems where only some of the state variables are controllable. In the collision avoidance problem, only the state variables representing the vertical motion of the aircraft are controllable. It is assumed that the state variables governing the horizontal motion of the aircraft evolve independently of the decisions made by the collision avoidance system. The approximation method involves decomposing the problem into two separate subproblems, one controlled and one uncontrolled, that can be solved independently o.ine using dynamic programming. During execution, the results from the o.ine computation are combined to determine the approximately optimal action from the current state. After explaining the general approximation method, which may be applied to a variety of di.erent problems, this section applies it to collision avoidance.
5.1 PARTIAL CONTROL ASSUMPTIONS
It is assumed that the state is represented by a set of variables, some controlled and some un-controlled. The state space of the controlled variables is denoted Sc, and the state space of the uncontrolled variables, Su. The state of the controlled variables at time t is denoted sc(t), and the state of the uncontrolled variables at time t, su(t). The solution technique may be applied when the following three assumptions hold:
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