Cartan connection applied to dynamic calculation in robotics

Abstract

A Cartan connection is an important mathematical object in differential geometry that generalizes, to an arbitrary Riemannian space, the concept of angular velocity (and twists) of a non-inertial reference frame. In fact, this is an easy way to calculate the angular velocities (and twists) in systems with multiple reference frames, like a robot or a complex mechanism. The concept can also be used in different representations of rotations and twists, like in the Lie groups of unit quaternions and unit dual quaternions. In this work, the Cartan connection is applied to the kinematic and dynamic calculations of systems with multiple reference frames that could represent a robotic serial chain or mechanism. Differently from previous works, the transformation between frames is represented as unit quaternions, which are known to be better mathematical representations from the numerical point of view. It is also generalized to this new quaternion representation, previous results like the extended Newton’s equation, the covariant derivative and the Cartan connection. An example of application is provided.