为双足机器人设计可计算的稳定轨迹不变集，提升运动控制鲁棒性。
Differentiable Invariant Sets for Hybrid Limit Cycles with Application to Legged Robots
- 用参数化嵌入法扩展连续系统可达集计算，构建周期轨道附近不变集。
- 在简化双足模型上验证：一步后过近似集严格包含于初值集，证明不变性。
- 适用于需要高鲁棒性的腿式机器人控制器设计，适合控制与系统领域研究者。
对于呈现周期行为的混合系统，分析包含极限环的不变集是研究闭环系统鲁棒性的自然方式。然而，在接触频繁的网络物理系统（如腿式机器人）中，此类集合的计算可能非常耗时。本文将现有基于参数嵌入的连续系统可达集过近似方法扩展至简化双足机器人模型的名义轨迹，提出三步法：(i) 计算名义连续流周围的过近似可达集，(ii) 识别与守卫面的交点，(iii) 将交点通过重置映射传递。若一步后的过近似可达集严格包含于初始集，则可形式化验证该混合周期轨道的前向不变集。我们使用 immrax（一个基于 JAX 的参数化可达集计算库）在双足步行者模型上数值验证该条件，并将其嵌入双层优化框架，用于设计最大化不变集尺寸的跟踪控制器。
原文摘要 · Abstract (English)
For hybrid systems exhibiting periodic behavior, analyzing the invariant set containing the limit cycle is a natural way to study the robustness of the closed-loop system. However, computing these sets can be computationally expensive, especially when applied to contact-rich cyber-physical systems such as legged robots. In this work, we extend existing methods for overapproximating reachable sets of continuous systems using parametric embeddings to compute a forward-invariant set around the nominal trajectory of a simplified model of a bipedal robot. Our three-step approach (i) computes an overapproximating reachable set around the nominal continuous flow, (ii) catalogs intersections with the guard surface, and (iii) passes these intersections through the reset map. If the overapproximated reachable set after one step is a strict subset of the initial set, we formally verify a forward invariant set for this hybrid periodic orbit. We verify this condition on the bipedal walker model numerically using immrax, a JAX-based library for parametric reachable set computation, and use it within a bi-level optimization framework to design a tracking controller that maximizes the size of the invariant set.
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