Motion Planning of Uncertain Ordinary Differential Equation Systems

Abstract

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and underactuated 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 reallife systems are affected by it, and poor robustness and suboptimal performance result if it is not accounted for in a given design. In this work uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a leastsquare collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and underactuated systems, prescribes deterministic actuator inputs that yield uncertain state trajectories. The inverse dynamics formulation is the dual to that of forward dynamics, and is only applicable to fully actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to underactuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories that yield uncertain unactuated states and uncertain actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space.