  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. Further,
we describe a nonlinear continuous-time model with parameters for designing a
closed loop LQ-controller. 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
LQ-controller 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.
