Christoffel-Consistent Coriolis Factorization and Its Effect on the Control of a Robot

Among the many choices in the matrix-vector factorization of the Coriolis and centripetal terms satisfying the skew-symmetry condition in system dynamics, the unique factorization, called Christoffel-consistent (CC) factorization, has been proposed. We derived the unique CC factorization in the Lie group context and examined the impact of Christoffel inconsistency in Coriolis matrix factorization on the dynamic behavior of robot systems during both free motion and interaction with humans, particularly in the context of passivity-based controllers and augmented PD controllers. Specifically, the question is: What are the advantages of using the CC factorization, and what is the effect of non-CC factorization on the robot's dynamic behavior, which has been rarely explored? We showed that Christoffel inconsistency generates unwanted torsion, causing the system to deviate from the desired trajectory, and this results in undesirable dynamic behavior when controlling the system, especially when the dynamics of the robot is described by twist and wrench. Through simulation and a real-world robot experiment, this phenomenon is verified for the first time.