Path Optimization Through A Bounded Region

Abstract

The shortest path between two points is a straight line. However, finding the optimum path from one point to another with the introduction of a given boundary condition on the domain between the two points does not possess such a trivial solution. The boundary condition is expressed in the form of a twice-differentiable function and serves as an upper or lower bound for the desired path depending on characteristics of the boundary function such as concavity and extrema behaviors on the domain of interest. The obtained path that maximizes arc length without violating the boundary function is deemed the Rubber Band Solution. This solution type is named due to its similarities to the path that a rubber band created when it is stretched around an obstacle by minimizing the potential energy from the elastic forces in the band. Since we seek an analytic solution, we first consider using boundary arcs that are circles. We then generalize to boundary arcs that can be described by an differentiable function by finding the circle of curvature at the local max/min of the function on the desired domain. The case where the boundary arcs are circles serves as an approximation for the optimized path around any of the more general boundary functions.

National Conference on Undergraduate Research (NCUR) at University of North Carolina Asheville, Asheville, NC, April 7-9, 2016.