The problem of motion planning for affine control systems consists of designing control inputs that drive a system from a well-defined initial to final states in a desired amount of time. For control systems with drift, however, understanding which final states are reachable in a given time, or reciprocally the amount of time needed to reach a final state, is often the most difficult part of the problem. We address this issue in this paper and introduce a new method to solve motion planning problems for affine control systems, where the motion desired can have indefinite boundary conditions and the time required to perform the motion is free. The method extends on our earlier work on motion planning for systems without drift. A canonical example of parallel parking of a unicycle with constant linear velocity is provided in this paper to demonstrate our algorithm.
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