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 Riemannian 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 expresses the dynamics entirely in terms of coordinate-free Lie algebraic operations. The Lagrangian formulation also expresses 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 explicitly 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 effective and easily differentiated and factored with respect to any of the robot parameters. This latter feature makes the geometric formulation attractive for applications such as robot design and calibration, motion optimization, and optimal control, where analytic gradients involving the dynamics are required.

Geometric methods and formulations in computational multibody system dynamics

“…The first version of (54) was reported in [132], and both forms in [125,126]. Common to both formulations (54) is the exclusive use of body-fixed reference frames, while no joint frame is necessary.…”

“…The body-fixed version of the recursive relations (58) and (62) was reported in [125,126,132]. The spatial version was presented in [115].…”

“…The expressions for the body-fixed acceleration were presented in [125,126,132]. The role of the Jacobian was, however, not discussed.…”

“…The only information needed are the instantaneous joint screw coordinates in body-fixed representation as it was discussed in [125,126,132]. It is common to rephrase the Lagrange equations as the equations of geodesic in the configuration space V n equipped with the mass matrix as Riemannian metric…”

Recent Advances on the Algorithmic Optimization of Robot Motion

“…To represent robot systems and their dynamics, we use a set of analytical tools for multibody systems analysis based on the mathematics of Lie groups and Lie algebras [13,11]. In the traditional formulation, a rigid motion can be represented with the Denavit-Hartenberg parameters as a 4x4 homogeneous transformation T (θ, d) ∈ SE(3), where θ is the rotation about the z-axis and d is the translation along it.…”