Open-loop stable limit cycles are foundational to legged robotics, providing inherent self-stabilization that minimizes the need for computationally intensive feedback-based gait correction. While previous methods have primarily targeted specific robotic models, this paper introduces a general framework for rapidly generating limit cycles across various dynamical systems, with the flexibility to impose arbitrarily tight stability bounds. We formulate the problem as a single-stage constrained optimization problem and use Direct Collocation to transcribe it into a nonlinear program with closed-form expressions for constraints, objectives, and their gradients. Our method supports multiple stability formulations. In particular, we tested two popular formulations for limit cycle stability in robotics: (1) based on the spectral radius of a discrete return map, and (2) based on the spectral radius of the monodromy matrix, and tested five different constraint-satisfaction formulations of the eigenvalue problem to bound the spectral radius. We compare the performance and solution quality of the various formulations on a robotic swing-leg model, highlighting the Schur decomposition of the monodromy matrix as a method with broader applicability due to weaker assumptions and stronger numerical convergence properties. As a case study, we apply our method on a hopping robot model, generating open-loop stable gaits in under 2 seconds on an Intel Core i7-6700K, while simultaneously minimizing energy consumption even under tight stability constraints.

翻译：开环稳定极限环是足式机器人技术的理论基础，其固有的自稳定特性极大降低了对计算密集型反馈步态校正的依赖。以往方法主要针对特定机器人模型，本文提出了一种通用框架，可快速生成多种动力学系统的极限环，并能灵活施加任意严格的稳定性边界。我们将该问题表述为单阶段约束优化问题，并采用直接配点法将其转录为非线性规划问题，其中约束、目标函数及其梯度均具有闭式表达式。本方法支持多种稳定性表述形式。具体而言，我们测试了机器人学中两种常用的极限环稳定性表述：(1) 基于离散回归映射的谱半径，(2) 基于单值矩阵的谱半径，并测试了五种不同的特征值问题约束满足表述来界定谱半径。我们在机器人摆腿模型上比较了各种表述的性能和解的质量，指出单值矩阵的舒尔分解法因假设条件较弱且数值收敛性更强而具有更广泛的适用性。作为案例研究，我们将该方法应用于弹跳机器人模型，在Intel Core i7-6700K处理器上于2秒内生成开环稳定步态，同时即使在严格稳定性约束下也能实现能耗最小化。