A hybrid dynamical extension of averaging and its application to the analysis of legged gait stability

Abstract

We extend a smooth dynamical systems averaging technique to a class of hybrid systems with a limit cycle that is particularly relevant to the synthesis of stable legged gaits. After introducing a definition of hybrid averageability sufficient to recover the classical result, we illustrate its applicability by analysis of first a one-legged and then a two-legged hopping model. These abstract systems prepare the ground for the analysis of a significantly more complicated two legged model: a new template for quadrupedal running to be analyzed and implemented on a physical robot in a companion paper. We conclude with some rather more speculative remarks concerning the prospects for further extension and generalization of these ideas.

Symmetries and periodic orbits in simple hybrid Routhian systems

“…It would be interesting to consider perturbed simple hybrid Routhian systems, similarly to the framework given in [43], while the perturbation preserves the symmetry, in order that we can turn the neutrally stable periodic orbit into a stable limit cycle. This will be explored in a future work by considering an adaptation of the averaging method and approximate for hybrid systems given in [17] and [50], respectively.…”

Analytically-Guided Design of a Tailed Bipedal Hopping Robot

“…Specifically, in this paper, we take initial steps toward a formal morphological reduction [23] of the 4 actuator, 12 DoF Jerboa robot onto the 1 actuator, 2 DoF TVH template by using hybrid averaging [7,24] to project the large and unintuitive 1 parameter space of the Jerboa onto its equilibrium hopping height, our candidate metric of hopping performance (13). In so doing, we (a) enable selection of control gains for metrically accurate hopping height control ( Fig.…”

“…In [7], we introduced an analytical tool for ε-perturbation stability analysis of hybrid limit cycles. 3 A key insight is to leverage the perturbation parameter ε > 0 in order to analytically study the effects of feedback control parameters as well as mechanical design parameters, on the fixed point (as we do in this paper) or the stability properties of the hybrid system [3].…”

“…with the goal of obtaining a pair of slow (a v ) and fast (ψ v ) coordinates useful for averaging. 7 In this instance, a v is the square root of the mechanical energy of the oscillating mass (with slow dynamics), while ψ v is a phase coordinate increasing through stance (with fast dynamics).…”

“…1) whose pronounced kinematic asymmetries simultaneously confer a diverse array of steady state and transitional behaviors [5] while at the same time challenging the intuitive leap from kinematic to dynamical symmetry that has guided legged robotics since its pioneering articulation by Raibert [6]. By conjecturing (but not yet proving) the hybrid averageability [7] of a (kinematically highly asymmetric) 2 DoF template-the "Tailed Vertical Hopper" (TVH) depicted in Fig. 2(a)-we obtain an analytically exact representation of the putative manner in which its physical and control parameters determine its steady state behavior.…”