Trajectory Planning for Systems with Homotopy Class Constraints

There are various applications where homotopy constraints are useful in trajectory generation for mobile robots. In this paper, we present a method to generate an optimal trajectory restricted to a particular homotopy class, which is specified by a given representative trajectory. The optimality is achieved by formulating the trajectory generation problem as a Mixed-Integer Quadratic Program (MIQP). An additional inequality constraint is introduced for the mixed-integers to partition the configuration space. We associate with any sequence of integer variables a word, so that each trajectory can be mapped to a word. We then construct a set of all words that are homotopically equivalent to a given word. For each word, we fix the integer variables of MIQP to find optimal time distribution in each region, by solving QP for each iteration, to obtain the locally optimal trajectory in the specified homotopy class. We illustrate an example of minimum acceleration trajectory generation on a plane with different homotopy constraints.