Efficient Constrained Dynamics Algorithms Based on an Equivalent LQR Formulation Using Gauss' Principle of Least Constraint

We derive a family of efficient constrained dynamics algorithms by formulating an equivalent linear quadratic regulator (LQR) problem using Gauss' principle of least constraint and solving it using dynamic programming. Our approach builds upon the pioneering (but largely unknown) O(n+m^2d+m^3) solver by Popov and Vereshchagin (PV), where n,m and d are the number of joints, number of constraints and the kinematic tree depth respectively. We provide an expository derivation for the original PV solver and extend it to floating-base kinematic trees with constraints allowed on any link. We make new connections between the LQR's dual Hessian and the inverse operational space inertia matrix (OSIM), permitting efficient OSIM computation, which we further accelerate using matrix inversion lemma. We generalize the elimination ordering and support MuJoCo-type soft constraint models to obtain O(n+m) complexity solvers. Our numerical results indicate that significant simulation speed-up can be achieved for high dimensional robots like quadrupeds and humanoids using our algorithms as they scale better than the widely used O(nd^2+m^2d+d^2m+m^3) LTL algorithm of Featherstone.