CARTAN'S CONNECTIONS APPLIED TO KINEMATIC AND DYNAMIC CALCULATION IN ROBOTICS: QUATERNIONS APPROACH

Diego Colón¹

¹ Escola Politécnica da USP

doi:10.20906/CPS/COB-2015-0491

Resumo

Cartan's 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 system with multiple reference frames, like a robot or a complex mechanism. The concept also can be used in different representations of rotation and twists, like in the Lie groups of unit quaternions and unit dual quaternions. In this work, we apply Cartan's connection to the kinematic and dynamic calculation of a system of multiple reference frames, that could represent a robotic chain or mechanism. Differently from previous works, we represent the transformation between frames as quaternions / unit quaternions, which are known to be better mathematical representations from the numerical point of view. We generalize, to this new quaternion representation, previous results like extended Newton's equation, covariant derivative and Cartan's connection.

Palavras-chave: dual quaternions; robotics; Lie groups; Screw Theory