Trajectory planning in time varying environments

I consider the problem of planning collision-free trajectories for a robot in time-varying environments (among stationary or moving obstacles). I then propose a framework and show how it leads to fast and efficient algorithms for trajectory planning. The key idea behind this framework is a space-time representation that leads to a natural decomposition of the spatial and temporal aspects of the problem. The basic approach is to partition the whole problem into two sub-problems: (i) planning a path to avoid collisions with stationary obstacles, and (ii) planning the velocity along this path to avoid collisions with the moving obstacles. The first sub-problem is called the Path Planning Problem, and the second sub-problem is called the Velocity Planning Problem. I call this the Path-Velocity decomposition. Within this decomposition paradigm, standard algorithms may be used to solve the path planning problem. Fast and efficient algorithms are then presented to solve the velocity planning problem. The crux of the algorithms lies in an explicit representation of time as an additional dimension. Moving obstacles give rise to time-varying constraints on the path of the robot. These constraints are represented as forbidden regions in the 2-dimensional path-time space. Computational geometric techniques are then used to determine a collision-free velocity profile. Furthermore, in conjunction with these computational geometric techniques, curve approximation splines are used to incorporate smoothness constraints in trajectories, while ensuring that they remain collision-free. The use of these algorithms is illustrated for (i) a polygonal robot moving among polygonal obstacles, and (ii) motion coordination of multiple polygonal robots. The use of these geometric trajectory planning algorithms with low-level local avoidance strategies in a hierarchical robot control system is explored.