Singularity-free Twist Limit Constraints for the Ball Joint 47

Ball joints are commonly used in graphics and robotics in the simulation of articulated rigid body systems. The ball joint orientation can be decomposed, non-uniquely, into swing and twist components. In modeling with ball joints, it can be advantageous to include limit constraints on the swing, twist, or both. While many physics engines include such limit constraints, we have found that twist limit constraints, when present, are lacking in robustness, typically due to unhandled singularities in commonly employed swing-twist decompositions, where the twist state is undefined. In this paper, we introduce two novel, robust models of the ball joint twist limit constraint. The first utilizes the Euler angle decomposition of the orientation, which has two singularity points. To robustly handle the twist limit constraint, we propose a novel treatment at singularities based on the physical behavior of a three-hinge system. Thus, our model significantly reduces the dramatic motion that can occur around the singularities, eliminating the need to directly avoid them. Second, we introduce an alternative incremental model that avoids singularities altogether during simulations. In this model, the twist is quantified by integrating the instantaneous twist velocity in time, ensuring that the singularity shifts from one time step to the next and remains far from the current state. We investigate the properties of both proposed models. We demonstrate that both models offer robust and practical solutions to common issues encountered in constraining the motion of ball joints.