Recent strides in nonlinear model predictive control (NMPC) underscore a dependence on numerical advancements to efficiently and accurately solve large-scale problems. Given the substantial number of variables characterizing typical whole-body optimal control (OC) problems - often numbering in the thousands - exploiting the sparse structure of the numerical problem becomes crucial to meet computational demands, typically in the range of a few milliseconds. Addressing the linear-quadratic regulator (LQR) problem is a fundamental building block for computing Newton or Sequential Quadratic Programming (SQP) steps in direct optimal control methods. This paper concentrates on equality-constrained problems featuring implicit system dynamics and dual regularization, a characteristic of advanced interiorpoint or augmented Lagrangian solvers. Here, we introduce a parallel algorithm for solving an LQR problem with dual regularization. Leveraging a rewriting of the LQR recursion through block elimination, we first enhanced the efficiency of the serial algorithm and then subsequently generalized it to handle parametric problems. This extension enables us to split decision variables and solve multiple subproblems concurrently. Our algorithm is implemented in our nonlinear numerical optimal control library ALIGATOR. It showcases improved performance over previous serial formulations and we validate its efficacy by deploying it in the model predictive control of a real quadruped robot.

翻译：非线性模型预测控制（NMPC）的最新进展凸显了其依赖于数值方法的进步，以高效且精确地求解大规模问题。鉴于典型全身最优控制（OC）问题所涉及的变量数量巨大——通常达到数千个——利用数值问题的稀疏结构对于满足计算需求（通常在几毫秒内）变得至关重要。求解线性二次调节器（LQR）问题是计算直接最优控制方法中牛顿步或序列二次规划（SQP）步的基本构建模块。本文专注于具有隐式系统动力学和对偶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。在此，我们提出了一种用于求解带对偶正则化的LQR问题的并行算法。通过块消元法重写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分割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。其性能优于先前的串行公式，并且我们通过将其部署于一个真实四足机器人的模型预测控制中验证了其有效性。