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We derive a family of efficient constrained dynamics algorithms byformulating an equivalent linear quadratic regulator (LQR) problem using Gaussprinciple of least constraint and solving it using dynamic programming. Ourapproach 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 ofjoints, number of constraints and the kinematic tree depth respectively. Weprovide an expository derivation for the original PV solver and extend it tofloating-base kinematic trees with constraints allowed on any link. We make newconnections between the LQR's dual Hessian and the inverse operational spaceinertia matrix (OSIM), permitting efficient OSIM computation, which we furtheraccelerate using matrix inversion lemma. By generalizing the eliminationordering and accounting for MUJOCO-type soft constraints, we derive twooriginal O(n + m) complexity solvers. Our numerical results indicate thatsignificant simulation speed-up can be achieved for high dimensional robotslike quadrupeds and humanoids using our algorithms as they scale better thanthe widely used O(nd^2 + m^2d + d^2m) LTL algorithm of Featherstone. Thederivation through the LQR-constrained dynamics connection can make ouralgorithm accessible to a wider audience and enable cross-fertilization ofsoftware and research results between the fields

Efficient Constrained Dynamics Algorithms based on an Equivalent LQR Formulation using Gauss' Principle of Least Constraint