Image feedback based optimal control and the value of information in a differential game

In this paper, we address pursuit-evasion problems in which the pursuer is a Differential Drive Robot (DDR) that attempts to capture an omnidirectional evader. From the Nash property it follows that if the evader deviates from its maximum potential speed then the capture time shall not increase for a pursuer that does not deviate from its Nash equilibrium motion strategy. However, it is not immediately clear how the pursuer could exploit that evader's deviation from its maximum potential speed, which might correspond to situations where the evader's capabilities may degrade with time, for example, battery depletion in an autonomous vehicle, or fatigue in an animal evader. This can be considered as a scenario of an evader in which the set of admissible controls varies with time. In the present paper we consider such scenario. In our first result, we propose an alternative strategy for the pursuer, which, for certain scenarios, further reduces the capture time compared to the strategy based on the maximum potential evader's speed. In our second result, we show that, under non-anticipative strategies, a pursuer strategy that uses the instantaneous evader speed alone, does not always guarantee to improve the payoff for the pursuer, nor the capture of the evader. Hence, we conclude that the evader's location is the relevant information for the pursuer to know. Later, we present vision-based control laws that implement the optimal pursuer strategy. The optimal pursuer strategy is characterized by a partition of the reduced space (a representation of the game in the pursuer's body-attached coordinate system) in which each region maps to an optimal pursuer action. We consider the case for which the pursuer is equipped with an omnidirectional catadioptric camera. Finally, in our third result we show that the location of the evader on the image can be directly used by the pursuer to define its motion strategy, in spite of the distortion of the state space suffered on the image. That is, the pursuer is able to apply its motion strategy using the image without explicitly reconstructing the evader's position. This approach is computationally efficient, and robust to occlusions and noise in the image.