Towards a systematic and unified method to solving nonholonomic systems.

Abstract

Understanding locomotion is one of the most ongoing quests in the world of mechanics and robotics; ranging widely from legged locomotion to any biomechanically inspired system. While traditional frameworks, such as the Lagrangian and general Hamiltonian formulations, have served to provide a base platform for having an idea about the equations of motion of a system, it has equipped the user with no knowledge on the geometry of the mechanical systems with the nonholonomic constraints and no clarification on its implications. In addition, the resulting equations are of high index Differential-Algebraic equations (DAE's) with the constraints added at the force level as Lagrangian multipliers. In this work, the formulation of reduced constraint Hamiltonian is utilized to model the dynamics of mechanical systems with nonholonomic constraints. Rather than enforcing the constraints by introducing Lagrange multipliers -- extra variables and dimensions -- to the equations of motion, the constraints are integrated at the geometry level. This allows the definition of a reduced constrained Hamiltonian and a Poisson structure which in turn are utilized to express the full dynamics of the system in a series of first order differential equations. A major contribution of this work it to make the formulation of such equations of motion accessible. A 20-Step method is designed which requires the knowledge only of: the coordinates of the system , the Lagrangian , the constraint equations, the generalized forces, and the parameters of the variables. As a first step, the method finds the full symmetry group of the system on which a map is built transforming the dynamics from a general manifold to a reduced constraint submanifold.