Motion Planning of Uncertain Ordinary Differential Equation Systems

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and 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. In this work 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, 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 under-actuated systems, prescribes deterministic actuator inputs which yield uncertain state trajectories. The inverse dynamics formulation is the dual to the forward dynamic, 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 under-actuated 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 which yield uncertain unactuated states and 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

Applications of the local state-space form of constrained mechanical systems in multibody dynamics and robotics

This thesis explores several areas in dynamics which can be viewed as applications of the local state-space form of a mechanical system. The simulation of mechanical systems often involves the solution of differential algebraic equations (DAEs). DAEs occur in every mechanism containing kinematic loops. Such systems can be found in a wide range of areas including the aerospace, automotive, construction, and farm equipment industries. The numerical treatment of DAEs is a topic which is relatively recent and continues to be studied. One can regard DAEs as ordinary differential equations (ODEs) on certain invariant manifolds after index reduction. Thus, the numerical solutions of the DAEs can be obtained through integration of their underlying ODEs. In certain circumstances, difficulties may occur since the numerical solutions of the underlying ODE can drift away from the invariant manifold. In this thesis, the underlying ODEs are locally transformed into ODEs of minimal dimension via local parameterizations of the invariant manifold. By their nature, such ODEs are local and implicit, but their solutions do not suffer from the drift phenomenon. Since the states of these minimal ODEs are independent, they are known as a local state-space form of the equations of motion. This work focuses on generalizing the application of the local state-space form and applying it towards problem areas in multibody dynamics and robotics. The first application of the local state-space form is in deriving a formulation of dynamics called the Singularity Robust Null Space Formulation. This formulation utilizes several aspects of the singular value decomposition for an approach which is efficient, does not fail at singularities, and is better suited than most near singularities. The second application area in this work is the study of the linearized mechanical system. Since the linearized model is also useful in optimization and implicit integration problems, an efficient recursive algorithm for its construction is derived. The algorithm appeals to a formulation of the dynamics found in robotics to ease in a coherent derivation