Dynamic Constraint-based Optimal Shape Trajectory Planner for Shape-Accelerated Underactuated Balancing Systems

This paper presents an optimal shape trajectory planner for shape-accelerated underactuated balancing systems, which are destabilized by gravitational forces. These systems have unactuated shape variables and fully actuated external variables. They also have the same number of actuated and unactuated degrees of freedom. Their equations of motion result in nonholonomic acceleration/dynamic constraints, which relate the acceleration of external variables to the position, velocity and acceleration of shape variables. This paper describes a procedure to use the dynamic constraints for planning shape trajectories, which when tracked will result in optimal tracking of desired external configuration trajectories. Examples of planned optimal shape trajectories for the 3D ballbot system, which is a 3D omnidirectional wheeled inverted pendulum, are also presented.