Lagrangian Dynamics ∗

The motion of a mechanical system is related via a set of dynamic equations to the forces and torques it is subject to. In this work we will be primarily interested in robots consisting of a collection of rigid links connected through joints that constrain the relative motion between the links. There are two main formalisms for deriving the dynamic equations for such mechanical systems: (1) Newton-Euler equations that are directly based on Newton’s laws, and (2) Euler-Lagrange equations that have their root in the classical work of d’Alembert and Lagrange on analytical mechanics and in the work of Euler and Hamilton on variational calculus. The main difference between the two approaches is in dealing with constraints. While Newton’s equations treat each rigid body separately and explicitly model the constraints through the forces required to enforce them, Lagrange and d’Alembert provided systematic procedures for eliminating the constraints from the dynamic equations, typically yielding a simpler system of equations. Constraints imposed by joints and by other mechanical components are one of the defining features of robots so that it is not surprising that the Lagrange’s formalism is often the method of choice in the robotics literature.