  In autonomous systems, a motion planner generates reference trajectories
which are tracked by a low-level controller. For safe operation, the motion
planner should account for inevitable controller tracking error when generating
avoidance trajectories. In this article we present a method for generating
provably safe tracking error bounds, while reducing over-conservatism that
exists in existing methods. We achieve this goal by restricting possible
behaviors for the motion planner. We provide an algebraic method based on
sum-of-squares programming to define restrictions on the motion planner and
find small bounds on the tracking error. We demonstrate our method on two case
studies and show how we can integrate the method into already developed motion
planning techniques. Results suggest that our method can provide acceptable
tracking error wherein previous work were not applicable.
