Optimal trajectory generation under homology class constraints

There are many applications where topology 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 homology class. The optimality is achieved by formulating the trajectory generation problem as a Mixed-Integer Quadratic Program (MIQP). We introduce binary variables that not only encode information about the satisfaction of geometric constraints, but also incorporate information about the homology class. We define the h-signature, a complete homology class invariant, as a quadratic function of the binary variables, which we subsequently convert to a linear function by variable substitutions. As a result, the suggested trajectory generation problem under homology class constraints can still be formulated as a MIQP, which can be solved by an anytime solver like CPLEX. We illustrate the method with examples of minimum acceleration trajectory generation under different homology class constraints with potential application to differentially-flat systems with a two-dimensional flat output space.