  In order to perform highly dynamic and agile maneuvers, legged robots
typically spend time in underactuated domains (e.g. with feet off the ground)
where the system has limited command of its acceleration and a constrained
amount of time before transitioning to a new domain (e.g. foot touchdown).
Meanwhile, these transitions can instantaneously change the system's state,
possibly causing perturbations to be mapped arbitrarily far away from the
target trajectory. These properties make it difficult for local feedback
controllers to effectively recover from disturbances as the system evolves
through underactuated domains and hybrid impact events. To address this, we
utilize the fundamental solution matrix that characterizes the evolution of
perturbations through a hybrid trajectory and its 2-norm, which represents the
worst-case growth of perturbations. In this paper, the worst-case perturbation
analysis is used to explicitly reason about the tracking performance of a
hybrid trajectory and is incorporated in an iLQR framework to optimize a
trajectory while taking into account the closed-loop convergence of the
trajectory under an LQR tracking controller. The generated convergent
trajectories recover more effectively from perturbations, are more robust to
large disturbances, and use less feedback control effort than trajectories
generated with traditional methods.
