Quadratic programming (QP) underpins real-time robotics by enabling efficient, constrained optimization in state estimation, motion planning, and control. In legged locomotion and manipulation, essential modules like inverse dynamics, Model Predictive Control (MPC), and Whole-Body Control (WBC) are inherently QP-based, demanding reliable solutions amid tight timing, energy, and computational resources on embedded platforms. This paper presents a comprehensive analysis and benchmarking study of QP solvers for legged robotics. We begin by formulating the standard convex QP and classify solvers into principal algorithmic approaches: interior-point methods, active-set strategies, operator-splitting schemes, and augmented Lagrangian/proximal approaches, while also discussing solver code generation for fixed-structure QPs. Each solver is examined in terms of algorithmic structure, computational characteristics, and its ability to exploit problem structure and warm-starting. Performance is reviewed using publicly available benchmarks, with a focus on metrics such as computation time, constraint satisfaction, and robustness under perturbations. Unified comparison tables yield practical guidance for solver selection, underscoring trade-offs in speed, accuracy, and energy efficiency. Our findings emphasize the synergy between solvers, tasks, and hardware -- e.g., sparse structured IPMs for long-horizon MPC and dense active-set for high-frequency WBC to advance agile, autonomous legged systems, with emerging trends toward ill-conditioned, conic, and code-generated deployments.

翻译：二次规划（QP）通过实现状态估计、运动规划与控制中的高效约束优化，构成了实时机器人技术的基石。在足式运动与操作中，逆动力学、模型预测控制（MPC）和全身控制（WBC）等核心模块本质上均基于QP，需要在嵌入式平台上严格的时间、能量与计算资源限制下获得可靠解。本文对足式机器人领域的QP求解器进行了全面分析与基准测试研究。我们首先构建标准凸二次规划问题，并将求解器划分为主要算法类别：内点法、有效集策略、算子分裂方案以及增广拉格朗日/邻近方法，同时讨论了固定结构QP的求解器代码生成技术。针对每种求解器，我们从算法结构、计算特性、问题结构利用能力及热启动支持等方面进行剖析。通过公开基准测试评估其性能，重点关注计算时间、约束满足度及扰动下的鲁棒性等指标。统一的对比表格为求解器选择提供实践指导，揭示了速度、精度与能效之间的权衡关系。研究结果强调了求解器、任务与硬件之间的协同效应——例如长时域MPC适用稀疏结构化内点法，高频WBC适用稠密有效集法，以推动敏捷自主足式系统的发展，并指出当前研究正朝着病态问题、锥规划及代码生成部署等新兴趋势演进。