Multistage Approach for Solving the Optimal Control Problem for a Wheeled Inverted Pendulum with Infeasible Initial Guess

A robotic system constructed as a wheeled inverted pendulum (WIP) serves as an impressive demonstrator, since this kind of system is inherently non-linear, unstable and non-minimum-phase. But the same properties which make it impressive also introduce serveral difficulties, when it comes to control and trajectory planning. Therefore this paper shows a method for deriving a highly dynamic trajectory compliant with the system dynamics. The constraints inherent to the definition of this trajectory are non-convex. The constraint functions have a local minimum in an infeasible region. This poses a problem when the initial guess is within this infeasible reagion. For a WIP this happens when it has to pass (in limbo) underneath an obstacle. To overcome this, a multistage approach is proposed for the optimal control problem to evade this infeasible local minimum. After solving 4 stages of related optimization problems the optimal trajectory is obtained. Additionally the center of gravity of the examined WIP can be influenced slightly by the mounting point of the battery pack. By incorporating this into the optimization problem, a task dependent optimal system configuration can be found.