Convex optimization is crucial in controlling legged robots, where stability and optimal control are vital. Many control problems can be formulated as convex optimization problems, with a convex cost function and constraints capturing system dynamics. Our review focuses on active balancing problems and presents a general framework for formulating them as second-order cone programming (SOCP) for robustness and efficiency with existing interior point algorithms. We then discuss some prior work around the Zero Moment Point stability criterion, Linear Quadratic Regulator Control, and then the feedback model predictive control (MPC) approach to improve prediction accuracy and reduce computational costs. Finally, these techniques are applied to stabilize the robot for jumping and landing tasks. Further research in convex optimization of legged robots can have a significant societal impact. It can lead to improved gait planning and active balancing which enhances their ability to navigate complex environments, assist in search and rescue operations and perform tasks in hazardous environments. These advancements have the potential to revolutionize industries and help humans in daily life.

翻译：凸优化在足式机器人的控制中至关重要，其中稳定性和最优控制是关键因素。许多控制问题可以表述为凸优化问题，通过凸代价函数和描述系统动力学的约束条件来建模。本综述聚焦于主动平衡问题，提出一个通用框架，将其表述为二阶锥规划（SOCP），以实现现有内点算法的鲁棒性和高效性。随后，我们讨论了一些先前关于零力矩点稳定性准则、线性二次型调节器控制以及反馈模型预测控制（MPC）方法的工作，以提高预测精度并降低计算成本。最后，将这些技术应用于稳定机器人完成跳跃和着陆任务。足式机器人凸优化的进一步研究可产生显著的社会影响，有助于改进步态规划与主动平衡，从而增强其在复杂环境中导航、执行搜救任务以及在危险环境中操作的能力。这些进展有望革新相关行业，并在日常生活中协助人类。