-Recent strides in model predictive control (MPC)underscore a dependence on numerical advancements to efficientlyand accurately solve large-scale problems. Given the substantialnumber of variables characterizing typical whole-body optimalcontrol (OC) problems -often numbering in the thousands-exploiting the sparse structure of the numerical problem becomescrucial to meet computational demands, typically in the range ofa few milliseconds. A fundamental building block for computingNewton or Sequential Quadratic Programming (SQP) steps indirect optimal control methods involves addressing the linearquadratic regulator (LQR) problem. This paper concentrateson equality-constrained problems featuring implicit systemdynamics and dual regularization, a characteristic found inadvanced interior-point or augmented Lagrangian solvers. Here,we introduce a parallel algorithm designed for solving an LQRproblem with dual regularization. Leveraging a rewriting of theLQR recursion through block elimination, we first enhanced theefficiency of the serial algorithm, then subsequently generalized itto handle parametric problems. This extension enables us to splitdecision variables and solve multiple subproblems concurrently.Our algorithm is implemented in our nonlinear numerical optimalcontrol library ALIGATOR. It showcases improved performanceover previous serial formulations and we validate its efficacy bydeploying it in the model predictive control of a real quadrupedrobot. This paper follows up from our prior work on augmentedLagrangian methods for numerical optimal control with implicitdynamics and constraints.

翻译：模型预测控制（MPC）的最新进展表明，其依赖于数值方法的进步来高效精确地求解大规模问题。鉴于典型全身最优控制（OC）问题涉及大量变量——通常达到数千个——利用数值问题的稀疏结构以满足计算需求（通常为数毫秒级别）变得至关重要。在直接最优控制方法中，计算牛顿步或序列二次规划（SQP）步的基础模块涉及求解线性二次型调节器（LQR）问题。本文聚焦于具有隐式系统动力学和双重正则化的等式约束问题，这是先进内点法或增广拉格朗日求解器的典型特征。我们提出了一种用于求解含双重正则化LQR问题的并行算法。通过利用块消去法重写LQR递推关系，我们首先提升了串行算法的效率，随后将其推广以处理参数化问题。这一扩展使我们能够拆分决策变量并同时求解多个子问题。我们的算法已在非线性数值最优控制库ALIGATOR中实现。相较于先前的串行形式，该算法展现出更优的性能，我们通过将其应用于真实四足机器人的模型预测控制中验证了其有效性。本文是我们先前关于含隐式动力学与约束的数值最优控制增广拉格朗日方法研究的后续工作。