Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo

We describe a full-featured simulation pipeline implemented in the MuJoCo physics engine. It includes multi-joint dynamics in generalized coordinates, holonomic constraints, dry joint friction, joint and tendon limits, frictionless and frictional contacts that can have sliding, torsional and rolling friction. The forward dynamics of a 27-dof humanoid with 10 contacts are evaluated in 0.1 msec. Since the simulation is stable at 10 msec timesteps, it can run 100 times faster than real-time on a single core of a desktop processor. Furthermore the entire simulation pipeline can be inverted analytically, an order-of-magnitude faster than the corresponding forward dynamics. We soften all constraints, in a way that avoids instabilities and unrealistic penetrations associated with earlier spring-damper methods and yet is sufficient to allow inversion. Constraints are imposed via impulses, using an extended version of the velocity-stepping approach. For holomonic constraints the extension involves a soft version of the Gauss principle. For all other constraints we extend our earlier work on complementarity-free contact dynamics - which were already known to be invertible via an iterative solver - and develop a new formulation allowing analytical inversion.