Open-loop stable limit cycles are foundational to the dynamics of legged robots. They impart a self-stabilizing character to the robot's gait, thus alleviating the need for compute-heavy feedback-based gait correction. This paper proposes a general approach to rapidly generate limit cycles with explicit stability constraints for a given dynamical system. In particular, we pose the problem of open-loop limit cycle stability as a single-stage constrained-optimization problem (COP), and use Direct Collocation to transcribe it into a nonlinear program (NLP) with closed-form expressions for constraints, objectives, and their gradients. The COP formulations of stability are developed based (1) on the spectral radius of a discrete return map, and (2) on the spectral radius of the system's monodromy matrix, where the spectral radius is bounded using different constraint-satisfaction formulations of the eigenvalue problem. We compare the performance and solution qualities of each approach, but specifically highlight the Schur decomposition of the monodromy matrix as a formulation which boasts wider applicability through weaker assumptions and attractive numerical convergence properties. Moreover, we present results from our experiments on a spring-loaded inverted pendulum model of a robot, where our method generated actuation trajectories for open-loop stable hopping in under 2 seconds (on the Intel Core i7-6700K), and produced energy-minimizing actuation trajectories even under tight stability constraints.

翻译：摘要：开环稳定极限环是腿式机器人动力学的基础，它为机器人步态赋予自稳定特性，从而减轻了对基于反馈的高计算量步态校正的需求。本文提出一种通用方法，可在给定动力系统中快速生成具有显式稳定性约束的极限环。具体而言，我们将开环极限环稳定性问题构建为单阶段约束优化问题（COP），并利用直接配点法将其转化为一个非线性规划问题（NLP），其中约束、目标函数及其梯度均具有闭式表达式。稳定性问题的COP公式基于两种方法发展：（1）基于离散返回映射的谱半径；（2）基于系统单值矩阵的谱半径，其中谱半径通过特征值问题的不同约束满足形式进行界定。我们比较了每种方法的性能和解质量，并特别强调了基于单值矩阵的舒尔分解方案——该方案因假设条件更弱和数值收敛性更优而具有更广泛的适用性。此外，我们展示了在弹簧负载倒立摆机器人模型上的实验结果：该方法在2秒内（基于Intel Core i7-6700K处理器）生成了开环稳定跳跃的驱动轨迹，即使在严格稳定性约束下也能产生能量最优的驱动轨迹。