  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.
