揭示足式机器人多接触运动中摆动结构的自然涌现机制
On the Emergence of Pendular Structure in Multi-Contact Locomotion
- 从角动量变化率最小化出发，推导出最优运动模式趋向摆动力分布
- 双足支撑时摩擦锥限制角动量变化率下界，临界水平加速度可解析表达
- 理论与仿真验证一致，适用于对称步态设计与控制器优化参考
在足式机器人控制中，线性倒立摆模型（LIPM）常作为建模选择而非控制器成本的自然偏好。本文通过一个小型质心最优控制问题（OCP），惩罚角动量变化率，分析其最优解的形态。结果表明：在满秩支撑时，最优解趋向于由动量雅可比奇异值分解决定的摆动力模式，常数由足距几何决定，与实验吻合度达16%；在双足支撑（如四足小跑）时，摩擦锥引入角动量变化率下界，权重调节无法克服；同时发现临界水平加速度处存在非光滑可行性拐点，且可闭式表达。若增加要求非零角动量变化率的任务项，最优解将偏离摆动集，且偏移可预测。所有结论均与经典ZMP/DCM框架一致。在质点四足模型和Unitree Go1机器人上进行仿真测试（开环QP与扭矩级闭环控制器），并指出渐近理论在闭环系统中的适用边界。
原文摘要 · Abstract (English)
LIPM is everywhere in legged-locomotion control, but almost always as a modeling choice rather than as something the controller's cost actually prefers. This note tries to make that link more explicit. Working from a small centroidal OCP that penalizes the rate of angular momentum, we look at what its optimum tends to look like. Three things come out. With full-rank stance, the optimum drifts toward a pendular force pattern at a rate determined by the SVD of the moment Jacobian; the constant is set by foot-span geometry and matches the experiments to within 16%. With N=2 stance, as in trot, the friction cone introduces a lower bound on $\|\dot{H}_G\|$ that no amount of weight tuning fixes; we also see a non-smooth feasibility kink at a critical horizontal acceleration that we can write in closed form. Adding a task term that asks for a nonzero $\dot{H}_G$ moves the optimum off the pendular set in a predictable way. None of this is far from the classical ZMP/DCM picture. We test these claims on a point-mass quadruped and on the Unitree Go1 in MuJoCo (open-loop QP and a torque-level closed-loop controller), and we note where the asymptotic story stops being a good description of what the closed loop actually does.
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