Many problems in robotics involve multiple decision making agents. To operate efficiently in such settings, a robot must reason about the impact of its decisions on the behavior of other agents. Differential games offer an expressive theoretical framework for formulating these types of multi-agent problems. Unfortunately, most numerical solution techniques scale poorly with state dimension and are rarely used in real-time applications. For this reason, it is common to predict the future decisions of other agents and solve the resulting decoupled, i.e., single-agent, optimal control problem. This decoupling neglects the underlying interactive nature of the problem; however, efficient solution techniques do exist for broad classes of optimal control problems. We take inspiration from one such technique, the iterative linear-quadratic regulator (ILQR), which solves repeated approximations with linear dynamics and quadratic costs. Similarly, our proposed algorithm solves repeated linear-quadratic games. We experimentally benchmark our algorithm in several examples with a variety of initial conditions and show that the resulting strategies exhibit complex interactive behavior. Our results indicate that our algorithm converges reliably and runs in real-time. In a three-player, 14-state simulated intersection problem, our algorithm initially converges in < 0.25s. Receding horizon invocations converge in < 50 ms in a hardware collision-avoidance test.
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