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. By generalizing the elimination ordering and accounting for MUJOCO-type soft constraints, we derive two original 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) LTL algorithm of Featherstone. The derivation through the LQR-constrained dynamics connection can make our algorithm accessible to a wider audience and enable cross-fertilization of software and research results between the fields

翻译：我们通过利用高斯最小约束原理构建等效的线性二次型调节器（LQR）问题，并采用动态规划求解，推导出一系列高效的约束动力学算法。该方法基于Popov和Vereshchagin（PV）开创性但鲜为人知的O(n + m²d + m³)求解器，其中n、m和d分别表示关节数、约束数和运动学树深度。我们对该原始PV求解器进行了阐释性推导，并将其扩展至允许在任何连杆上施加约束的浮动基座运动学树。我们建立了LQR对偶海森矩阵与逆操作空间惯性矩阵（OSIM）之间的新联系，从而实现了高效的OSIM计算，并进一步利用矩阵求逆引理加速该过程。通过推广消元顺序并考虑MUJOCO型软约束，我们推导出两种原创的O(n + m)复杂度求解器。数值结果表明，对于四足机器人及人形机器人等高维机器人，我们的算法可实现显著的仿真加速，其扩展性优于广泛使用的Featherstone O(nd² + m²d + d²m) LTL算法。通过LQR约束动力学关联的推导，我们的算法可被更广泛的受众理解和应用，从而促进两个领域之间软件与研究成果的交叉融合。