Variational Integrators and Graph-Based Solvers for Multibody Dynamics in Maximal Coordinates

Simulators for mechanical systems are widely used, for example in testing and verification [1, 2], model-based control strategies [3, 4], or learning-based methods [5, 6]. However, many common simulators have numerical difficulties with more complex mechanical systems involving constraints [7]. Such constraints can represent joints connecting rigid bodies, which may form kinematic loops, for example, in exoskeletons. Constraints can also be used to confine the movement of bodies, for example, to model joint limits in robotic arms, or to describe contact with other bodies or the environment in walking and grasping. Exactly enforcing such constraints can cause numerical issues, for example, due to the stiff nature of contact interactions. To alleviate these numerical issues, simulators often allow small constraint violations by representing all constraints as spring-damper elements as in MuJoCo [8] and Brax [9], or by accepting interpenetration of bodies as in Drake [10] and Bullet [11]. Small violations can sometimes be acceptable, for example, contact interpenetration in the order of micrometers for meter-scale walking robots. But millimeter or centimeter violations, for example in MuJoCo, can be considered too large. Employing these methods and accepting larger constraint violations for stable simulations contributes to the sim-to-real gap, a major issue in robotics [12].