Structure-Preserving Constrained Optimal Trajectory Planning of a Wheeled Inverted Pendulum

The wheeled inverted pendulum (WIP) is an underactuated, nonholonomic mechatronic system, and has been popularized commercially as the Segway. Designing a control law for motion planning, that incorporates the state and control constraints, while respecting the configuration manifold, is a challenging problem. In this article, we derive a discrete-time model of the WIP system using discrete mechanics and generate optimal trajectories for the WIP system by solving a discrete-time constrained optimal control problem. Furthermore, we describe a nonlinear continuous-time model with parameters for designing a closed-loop linear-quadratic regulator (LQR). A dual control architecture is implemented in which the designed optimal trajectory is, then, provided as a reference to the robot with the optimal control trajectory as a feedforward control action, and an LQR in the feedback mode is employed to mitigate noise and disturbances for ensuing stable motion of the WIP system. While performing experiments on the WIP system involving aggressive maneuvers with fairly sharp turns, we found a high degree of congruence in the designed optimal trajectories and the path traced by the robot while tracking these trajectories. This corroborates the validity of the nonlinear model and the control scheme. Finally, these experiments demonstrate the highly nonlinear nature of the WIP system and robustness of the control scheme.