Closed Form Time Derivatives of the Equations of Motion of Rigid Body Systems

TL;DR

This paper derives closed-form expressions for the first and second time derivatives of the equations of motion of rigid body systems using Lie group formulation, aiding control and design in robotics.

Contribution

It introduces a direct, closed-form formulation for derivatives of EOM of rigid bodies, improving upon recursive algorithms with more insight and simplicity.

Findings

Provides compact, easily parameterized equations

Enables smoother control trajectories

Facilitates analysis of robotic system dynamics

Abstract

Derivatives of equations of motion(EOM) describing the dynamics of rigid body systems are becoming increasingly relevant for the robotics community and find many applications in design and control of robotic systems. Controlling robots, and multibody systems comprising elastic components in particular, not only requires smooth trajectories but also the time derivatives of the control forces/torques, hence of the EOM. This paper presents the time derivatives of the EOM in closed form up to second-order as an alternative formulation to the existing recursive algorithms for this purpose, which provides a direct insight into the structure of the derivatives. The Lie group formulation for rigid body systems is used giving rise to very compact and easily parameterized equations.