In this thesis we present a novel approach to sensor-based motion planning developed using the mathematical tools provided by the field of nonsmooth analysis. The work is based on a broad body of background material developed using the tools of differential topology (smooth analysis), that is limited to simple cases like a point or circular robot. Nonsmooth analysis is required to extend this background work to the case of a polygonal robot moving amidst polygonal obstacles. We present a detailed nonsmooth analysis of the distance function for arbitrary configuration spaces and use this analysis to develop a planner for a rotating and translating polygonal mobile robot. Using the tools of nonsmooth analysis, we then describe a one-dimensional nonsmooth roadmap of the robot's freespace called the Nonsmooth Critical Set + Nonsmooth Generalised Voronoi Graph (NC_RIT+NGVG) where the robot is equidistant to a number of obstacles, in a critical configuration or passing between two obstacles. We then use the related field of nonsmooth control theory to develop several provably stable control laws for following and exploring the nonsmooth roadmap. Finally, we implement a motion planner in simulation and for a real polygonal mobile robot, thus verifying the utility and practicality of the nonsmooth roadmap.

"This thesis investigates several non-linear analogues of Lagrange functions in the hope of answering the question 'Is it possible to generalise Lagrange functions such that they may be applied to a range of nonconvex objective problems?' The answer to this question is found to be yes for a particular class of optimization problems. Furthermore the thesis asserts that in derivative free optimization the general schema which is most theoretically and practically appealing involves the reformulation of both objective and constraint functions, whilst the least practically successful approach for everything but the most simple convex case is the augmented Lagrangian approach." / Doctor of Philosophy

"This thesis investigates several non-linear analogues of Lagrange functions in the hope of answering the question 'Is it possible to generalise Lagrange functions such that they may be applied to a range of nonconvex objective problems?' The answer to this question is found to be yes for a particular class of optimization problems. Furthermore the thesis asserts that in derivative free optimization the general schema which is most theoretically and practically appealing involves the reformulation of both objective and constraint functions, whilst the least practically successful approach for everything but the most simple convex case is the augmented Lagrangian approach." / Doctor of Philosophy