The multi-contact nonlinear complementarity problem (NCP) is a naturally arising challenge in robotic simulations. Achieving high performance in terms of both accuracy and efficiency remains a significant challenge, particularly in scenarios involving intensive contacts and stiff interactions. In this article, we introduce a new class of multi-contact NCP solvers based on the theory of the Augmented Lagrangian (AL). We detail how the standard derivation of AL in convex optimization can be adapted to handle multi-contact NCP through the iteration of surrogate problem solutions and the subsequent update of primal-dual variables. Specifically, we present two tailored variations of AL for robotic simulations: the Cascaded Newton-based Augmented Lagrangian (CANAL) and the Subsystem-based Alternating Direction Method of Multipliers (SubADMM). We demonstrate how CANAL can manage multi-contact NCP in an accurate and robust manner, while SubADMM offers superior computational speed, scalability, and parallelizability for high degrees-of-freedom multibody systems with numerous contacts. Our results showcase the effectiveness of the proposed solver framework, illustrating its advantages in various robotic manipulation scenarios.

翻译：多接触非线性互补问题（NCP）是机器人仿真中自然出现的挑战。在精度与效率两方面均实现高性能仍然是一个重大难题，尤其是在涉及密集接触和刚性交互的场景中。本文基于增广拉格朗日法（AL）理论，提出了一类新的多接触NCP求解器。我们详细阐述了如何将凸优化中AL的标准推导过程，通过迭代求解代理问题并随后更新原-对偶变量，以适应多接触NCP的处理。具体而言，我们提出了两种针对机器人仿真量身定制的AL变体：基于级联牛顿的增广拉格朗日法（CANAL）和基于子系统的交替方向乘子法（SubADMM）。我们证明了CANAL能够以精确且鲁棒的方式处理多接触NCP，而SubADMM则为具有大量接触的高自由度多体系统提供了卓越的计算速度、可扩展性和并行化能力。我们的结果展示了所提出求解器框架的有效性，并阐明了其在各种机器人操作场景中的优势。