Modeling and Estimation of Motion Over Manifolds with Motion Capture Data

Abstract

Modeling the dynamics of complex multibody systems, such as those representing the motion of animals, can be accomplished through well-established geometric methods. In these formulations, motions take values in certain types of smooth manifolds which are coordinate-free and intrinsic. However, the dimension of the full configuration manifold can be large. The first study in this dissertation aims to build low-dimensional models models from motion capture data. This study also expands on the so-called learning problem from statistical learning theory over Euclidean spaces to estimating functions over manifolds. Experimental results are presented for estimating reptilian motion using motion capture data. The second study in this dissertation utilizes reproducing kernel Hilbert space (RKHS) formulations and Koopman theory, to achieve some of the advantages of learning theory for IID discrete systems to estimates generated over dynamical systems. Specifically, rates of convergence are determined for estimates generated via extended dynamic mode decomposition (EDMD) by relating them to estimates generated by distribution-free learning theory. Some analytical examples illustrate the qualitative behavior of the estimates. Additionally, a examination of the numerical stability of the estimates is also provided in this study. The approximation methods are then implemented to estimate forward kinematics using motion capture data of a human running along a treadmill. The final study of this dissertation contains an examination of the continuous time regression problem over subsets and manifolds. Rates of convergence are determined using a new notion of Persistency of Excitation over flows of manifolds. For practical considerations, two approximation methods of the exact solution to the continuous regression problem are introduced. Characteristics of these approximation methods are analyzed using numerical simulations. Implementations of the approximation schemes are also performed on experimentally collected motion capture data.