To the best of our knowledge, an exact solution to the visibility-based pursuit–evasion problem with point agents and polygonal obstacles addressed in this work is not known. Given the above, in this work, we present approximate algorithms, but feasible and with other desirable properties, for such a pursuit–evasion game. Our new method combines asymptotically optimal motion planning based on sampling (more specifically, optimal probabilistic roadmaps) and the value iteration of dynamic programming. In addition, our formulation finds solutions for the evader when there are singular surfaces, which previous work could not find. In this work, we obtain two main theoretical results. We first prove that the proposed discrete formulation is correct (that the method obtains the solution for the discretization of the given configuration space). Subsequently, combining random graph results, Bellman's optimality principle, and limits, it is proved that, as the number of samples tends to infinity, our approximate discrete formulation becomes the continuous formulation corresponding to the Hamilton–Jacobi–Isaacs equation. This results in a feasible method that improves its approximation as the number of samples increases. Simulation experiments in 2-D and 3-D environments with obstacles that are simply and multiplicattively connected exemplify the results of the new method and show the advantages over previous work.

In this article, we study properties of local planners for nonholonomic dynamical systems to achieve asymptotic global optimality in a sampling-based planner, namely, the RRT*. Formal results are presented concerning length optimal trajectories under a nonholonomic metric, more specifically, an analysis about necessary or sufficient conditions for the local planners is provided. The analysis makes use of the case of finding time-optimal trajectories for differential drive robots. Later, relevant concepts for complex robotic systems such as detachability and time dominance are introduced. Detachability is a property over the cost functional such that, if respected, it allows the reusability of previously designed local planners for simpler systems into more complex ones. The introduced concepts are further illustrated through a robotic system comprised of a mobile manipulator robot equipped with a camera in the arm’s end-effector, whose task is to move between states maintaining a desired object within its field of view. Experiments in a physical mobile manipulator robot are shown, which validate the proposed theoretical modeling.