A Lie Group Formulation of Robot Dynamics

Abstract

In this article we present a unified geometric treatment of robot dynamics. Using standard ideas from Lie groups and Rieman nian geometry, we formulate the equations of motion for an open chain manipulator both recursively and in closed form. The recursive formulation leads to an O(n) algorithm that ex presses the dynamics entirely in terms of coordinate-free Lie algebraic operations. The Lagrangian formulation also ex presses the dynamics in terms of these Lie algebraic operations and leads to a particularly simple set of closed-form equations, in which the kinematic and inertial parameters appear explic itly and independently of each other. The geometric approach permits a high-level, coordinate-free view of robot dynamics that shows explicitly some of the connections with the larger body of work in mathematics and physics. At the same time the resulting equations are shown to be computationally ef fective and easily differentiated and factored with respect to any of the robot parameters. This latter feature makes the ge ometric formulation attractive for applications such as robot design and calibration, motion optimization, and optimal control, where analytic gradients involving the dynamics are required.