The controlled motion of a rolling ball actuated by internal point masses that move along arbitrarily-shaped rails fixed within the ball is considered. Application of the variational Pontryagin's minimum principle yields the ball's controlled equations of motion, a solution of which obeys the ball's uncontrolled equations of motion, satisfies prescribed initial and final conditions, and minimizes a prescribed performance index. The controlled equations of motion are solved numerically using a predictor-corrector continuation method, starting from an initial solution obtained via a direct method, to realize trajectory tracking and obstacle avoidance maneuvers.