This work presents a novel nonlinear programming based motion planning framework that treats uncertain under-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, and poor robustness and suboptimal performance result if it’s not accounted for in a given design. System uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiencies of this approach enable 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 a hybrid dynamics formulation for under-actuated systems where actuated state and unactuated input trajectories are prescribed and uncertain unactuated states and actuated inputs are quantified. The benefits of the ability to quantify the resulting uncertainties are illustrated in a power optimal motion planning case-study for an inverting double pendulum problem. The resulting design determines a motion plan that minimizes the required input power—subject to actuator and terminal condition variance constraints—for all possible systems within the probability space.

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