Thesis Defense:  Real-Time Trajectory Generation
for Constrained Nonlinear Dynamical Systems Using
Non-Uniform Rational B-spline Basis Functions

Melvin Flores

Abstract:  The thesis describes a new method for obtaining minimizers for optimal control problems whose minima serve as control policies for guiding nonlinear dynamical systems to achieve prescribed goals under imposed trajectory and actuator constraints. The major contribution of the present work resides in the approximation of such minimizers by piecewise polynomials functions expressed in terms of a linear combination of non-uniform rational B-spline (NURBS) basis functions and the judicious exploitation of the properties of the resulting NURBS curves for use in trajectory generation.

In particular, by exploiting the two structures combine in a NURBS curve | NURBS basis functions and a union of overlapping polytopes constructed from the coecients of the linear combination, we are able to separate an optimal control problem into two subproblems | guidance and obstacleavoidance, making the original problem tractable. This is accomplished by laying out the union of overlapping polytopes in such a way that they delineate a section of space that avoids all obstacles and then manipulating the NURBS basis functions to obtain trajectories that are guaranteed to remain bounded by this section of space without explicitly including the conjunction of disjunctions naturally induced from obstacles into the guidance problem.