  In motion planning problems for autonomous robots, such as self-driving cars,
the robot must ensure that its planned path is not in close proximity to
obstacles in the environment. However, the problem of evaluating the proximity
is generally non-convex and serves as a significant computational bottleneck
for motion planning algorithms. In this paper, we present methods for a general
class of absolutely continuous parametric curves to compute: (i) the minimum
separating distance, (ii) tolerance verification, and (iii) collision
detection. Our methods efficiently compute bounds on obstacle proximity by
bounding the curve in a convex region. This bound is based on an upper bound on
the curve arc length that can be expressed in closed form for a useful class of
parametric curves including curves with trigonometric or polynomial bases. We
demonstrate the computational efficiency and accuracy of our approach through
numerical simulations of several proximity problems.
