变量阻抗MPC中虚假可行性问题研究
False Feasibility in Variable Impedance MPC for Legged Locomotion
AI总结：
本文研究了变量阻抗MPC中因误将关节刚度视为瞬时决策变量导致的可行性问题，通过分析参数可行集与真实可行集的差异，提出α参数用于描述不匹配区域，并通过一维跳跃单足和平面弹簧-loaded倒立摆验证了该问题。
摘要：
变量阻抗模型预测控制（MPC）公式通常将关节刚度视为瞬时决策变量。由此产生的可行集严格包含在第一阶执行器动力学下可物理实现的集合。本文将其识别为公式错误而非建模近似，正式区分基于参数的可行集F_param与真实可行集F_real，并通过无量纲参数α=ωsT（执行器带宽乘以任务时间尺度）表征不匹配区域。对于一维跳跃单足，证明在α_crit阈值以下，没有可行的刚度命令能实现参数预测。数值验证显示随着α减小，偏差增长单调，预测的标度在十种参数组合中保持（对数-对数R²=0.986）。机制转移至平面弹簧-loaded倒立摆动力学证实了质心和支撑时间偏差为主要后果，随区域变化的摩擦效应为次要可观测量。第二个阈值α_infeas<α_crit确立了一个地板，低于该阈值限制可接受刚度范围无法修复可实现性，关闭了保守调优的反对意见。在预测状态中增加刚度可构造性地关闭不匹配。
英文摘要：
Variable impedance model predictive control (MPC) formulations often treat joint stiffness as an instantaneous decision variable. The resulting feasible set strictly contains the physically realizable set under first-order actuator dynamics. We identify this as a formulation error rather than a modeling approximation, formalize the distinction between the parameter-based feasible set F_param and the realizable set F_real, and characterize the regime of mismatch via the dimensionless parameter α = ωsT (actuator bandwidth times task timescale). For the 1D hopping monoped, we prove that below an analytical threshold α_crit derived in closed form from task physics, no admissible stiffness command realizes the parameter-based prediction. Numerical validation in 1D shows monotonic deviation growth as α decreases, with the predicted scaling holding across ten parameter combinations (log-log R2 = 0.986). Mechanism transfer to planar spring-loaded inverted pendulum dynamics confirms center-of-mass and stance-timing deviation as the primary consequence, with regime-dependent friction effects as a tertiary observable. A second threshold α_infeas < α_crit establishes a floor below which restricting the admissible stiffness range cannot repair realizability, closing the conservative-tuning objection. Augmenting the prediction state with stiffness closes the mismatch by construction.
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