September 11, 2014 International Journal of Control PaperIJC˙final To appear in the International Journal of Control Vol. 00, No. 00, Month 20XX, 1–25 A Visual Feedback-based Time-Optimal Motion Policy for Capturing an Unpredictable Evader David Jacobo a , Ubaldo Ruiz a , Rafael Murrieta-Cid a , Hector M. Becerra a* and Jose Luis Marroquin a a Centro de Investigaci´on en Matem´aticas, Guanajuato M´ exico (e-mails:{jguillen; ubaldo; murrieta; hector.becerra; jlm}@cimat.mx). (Received May 2014) In this paper, we address the pursuit/evasion problem of capturing an unpredictable omnidirectional evader using a Differential Drive Robot (DDR) in an obstacle-free environment. We present three main contributions: i) We provide a state feedback-based time-optimal motion policy for a differential drive robot. The motion policy is based on a partition of the state space. One main contribution of this paper is to provide algebraic equations of the regions’ boundaries of this partition in terms of the state space coordinates. ii) We estimate the state of the evader based on images using the 1D trifocal tensor. We propose a new formulation of the estimation of the evader’s state relative to the pursuer. iii) We present a bound, for conventional cameras, over the pursuer’s field of view that guarantees that, if the evader is initially visible, it will remain visible (inside the camera’s view) regardless of its motion strategy, until the capture condition is achieved. We also present an implementation of the pursuer’s motion policy, the estimation of the evader’s state and also present simulation results of the pursuit/evasion game. Keywords: Visual Feedback; Unpredictable Evader; Visual Servo-control; State Estimation; Nonholonomic Constraints 1. Introduction In Ruiz et al. (2013), we have considered the kinematic problem of capturing an omnidirectional evader using a Differential Drive Robot (DDR) in an obstacle-free environment. The DDR is faster than the evader, but it can only change its direction of motion at a maximum rate that is inversely proportional to its maximal translational speed (Balkcom & Mason (2002)). The game is over when the distance between the DDR and the evader is smaller than a critical value l. The DDR wants to minimize the capture time while the evader wants to maximize it. In that work, we presented closed-form representations of the motion primitives and time-optimal strategies for each player in open-loop. We proposed a partition of the playing space into mutually disjoint regions where the strategies of the players are well established. This partition is represented as a graph which exhibits properties that guarantee global optimality. We also analyzed the decision problem of the game and presented the conditions defining the winner. In this paper, we also consider the problem of capturing an omnidirectional evader using a DDR, but the players have different objectives. The DDR also wants to capture the evader in minimal time but now we assume that the evader moves unpredictably. The main motivation of this work is as following. Let us assume that the partition of the playing space is found. Let us also assume that the pursuer executes its time-optimal policy 1 but the evader moves unpredictably instead (i.e. its motion policy is unknown by the pursuer). If the pursuer executes its optimal policy in open-loop, * Corresponding author. E-mail: hector.becerra@cimat.mx 1 A policy is a rule that tells each player the control it has to apply at each time instant. 1