Motion Planning of Uncertain Fully-Actuated Dynamical Systems: An Inverse Dynamics Formulation

This work presents a novel nonlinear programming based motion planning framework that treats uncertain fully-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it; ignoring uncertainty during design may lead to poor robustness and suboptimal performance. System uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, new design questions related to uncertain dynamical systems can now be answered through the new framework. Specifically, this work presents the new framework through an inverse dynamics formulation where deterministic state trajectories are prescribed and uncertain actuator inputs are quantified. The benefits of the ability to quantify the resulting actuator uncertainty are illustrated in a time optimal motion planning case-study of a serial manipulator pick-and-place application. The resulting design determines a feasible time optimal motion plan—subject to actuator and obstacle avoidance constraints—for all possible systems within the probability space. The forward dynamics formulation (using deterministic actuator inputs and uncertain state trajectories) is presented in a companion paper.