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Abstract-Although the problem of dynamic locomotion in very rough terrain is critical to the advancement of various areas in robotics and health devices, little progress has been made on generalizing gait behavior with arbitrary paths. Here, we report that perturbation theory, a set of approximation schemes that has roots in celestial mechanics and nonlinear dynamical systems, can be adapted to predict the behavior of nonintegrable state-space trajectories of a robot's center of mass, given its arbitrary contact state and center of mass (CoM) kinematic path. Given an arbitrary kinematic path of the CoM and known step locations, we use perturbation theory to determine phase curves of CoM behavior. We determine step transitions as the points of intersection between adjacent phase curves. To discover intersection points, we fit polynomials to the phase curves of neighboring steps and solve their differential roots. The resulting multi-step phase diagram is the locomotion plan suited to drive the behavior of a robot or device maneuvering in the rough terrain. We provide two main contributions to legged locomotion: (1) predicting CoM state-space behavior for arbitrary paths by means of perturbation theory, and (2) finding step transitions by locating common intersection points between neighboring phase curves. Because these points are continuous in phase they correspond to the desired contact switching policy. We validate our results on a human-size avatar navigating in a very rough environment and compare its behavior to a human subject maneuvering through the same terrain. I. STATE OF THE ARTIn dynamic walking we can classify techniques in various categories: (1) trajectory-based techniques, (2) limit cycle-based techniques, (3) prediction of contact, and (4) hybrids of the previous three.Trajectory-based techniques are techniques that track a timebased joint or task space trajectory according to some locomotion model such as the Zero Moment Point (ZMP). The state of the art of these methods includes generalized multi-contact locomotion behaviors, developed in [1] and more recenlty, a time delay extension to the ZMP method for locomotin in moderately uneven terrain, developed by [2].Prediction of contact placement are techniques that use dynamics to estimate suitable contact transitions to produce locomotion or regain balance. In [3], simple dynamic models are used to predict the placement of next contacts to achieve desire gait patterns. Finding feasible CoM static placements given frictional constraints was tackled in [4], [5]. In [6], stable locomotion, in the wide sense of not falling down, is studied by providing velocity based stability margins. This work is used to regain stability when the robot's is pushed out, and lead to the concept of Capture Point.Limit cycle based techniques were pioneered by McGeer [7] through the field of passive dynamic walking. In [8] the authors study orbital stability, and the effect of feedback control to achieve asymptotic stability. Optimization of open-loop stability is in...

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