Development and application of a gradient method for solving differential games
David A. Roberts, Raymond C. Montgomery, United States. National Aeronautics and Space Administration, Langley Research Center
National Aeronautics and Space Administration, 1971 - Education - 46 pages
A technique for solving n-dimensional games is developed and applied to two pursuit-evasion games. The first is a two-dimensional game similar to the homicidal chauffeur but modified to resemble an airplane-helicopter engagement. The second is a five-dimensional game of two airplanes at constant altitude and with thrust and turning controls. The performance function to be optimized by the pursuer and evader was the distance between the evader and a given target point in front of the pursuer. The analytic solution to the first game reveals that both unique and nonunique solutions exist. A comparison between the gradient results and the analytic solution shows a dependence on the nominal controls in regions where nonunique solutions exist. In the unique solution region, the results from the two methods agree closely. The results for the five-dimensional two-airplane game are also shown to be dependent on the nominal controls selected and indicate that initial conditions are in a region of nonunique solutions.
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airplanes at constant analytic solution reveals appendix Application arg max constant altitude control derivatives control variations control vector different nominal differential equations drag coefficient equation 9 equations of motion Figure 16 game theory gradient method induced drag initial conditions insure convergence Integrate equations Isaacs iteration scheme iteration sequence Langley Research Center m/sec maneuvers max-min solution maximize the payoff METHOD FOR SOLVING min-max minimization phase minimum thrust nominal control table nominal controls selected nominal engagement nominal path nonunique region nonunique solutions exist optimal controls optimal paths optimal solution payoff with respect player playing space problem considered pursuer and evader pursuer's velocity pursuit-evasion games retrograde saddle-point solution scalar set of initial shown in figure simplified game steepest descent strategy terminal line terminal manifold termination condition time-optimal game trajectory transition matrix turning control two-airplane game update the controls usable variable VxP(xf x^-axis zero zero-sum differential games