Virtual Holonomic Constraints for Euler-Lagrange Control Systems

Virtual Holonomic Constraints for Euler-Lagrange Control Systems Alireza Mohammadi Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2016 In this thesis we investigate virtual holonomic constraints (VHCs) for mechanical control systems. A VHC is a relation among the configuration variables of a mechanical system that does not physically exist, but can be emulated via feedback in a precise sense that is defined within this thesis. An example of VHC is the requirement that the end effector of a robot should only move along a plane. Over the past decade, VHCs have raised to prominence in robotics as they have been successfully employed to induce stable walking motions in biped robots. There is a growing body of evidence that, for complex motion control problems, the VHC paradigm might be more appropriate than the traditional reference tracking approach. Motivated by the hope to establish a new paradigm for complex motion control problems in robotics, this thesis makes contributions in two main directions. First, the thesis presents a complete theory employing VHCs for the stabilization of repetitive “behaviors” in underactuated mechanical control systems with degree of underactuation one. The theory in question has a number of components, the development of which takes us along a journey that includes, among other things, the solution of an inverse Lagrangian problem, the development of an algorithm to implicitize parametric constraints, and the stabilization of closed orbits by means of VHCs parametrized by states of dynamic compensators. The second direction of this thesis is the exploration of the hypothesis that the design