Optimal Control of Nonholonomic Mechanical Systems

Tuesday, October 24, 2017 - 1:25pm - 2:25pm
Lind 305
Stuart Rogers (University of Minnesota, Twin Cities)
This talk investigates the optimal control of two nonholonomic mechanical systems, Suslov's problem and the rolling ball. Suslov's problem is a nonholonomic variation of the classical rotating free rigid body problem, in which the body angular velocity $\boldsymbol{Omega}(t)$ must always be orthogonal to a prescribed, time-varying body frame vector $\boldsymbol{xi}(t)$. The motion of the rigid body in Suslov's problem is actuated via $\boldsymbol{xi}(t)$, while the motion of the rolling ball is actuated via internal point masses that move along rails fixed within the ball. First, by applying Lagrange-d'Alembert's principle with Euler-Poincarè's method, the uncontrolled equations of motion are derived. Then, by applying Pontryagin's minimum principle, the controlled equations of motion are derived, a solution of which obeys the uncontrolled equations of motion, satisfies prescribed initial and final conditions, and minimizes a prescribed performance index. Finally, the controlled equations of motion are solved numerically by a continuation method, starting from an initial solution obtained analytically (in the case of Suslov's problem) or via a direct method (in the case of the rolling ball).