Design and experimental implementation of a hybrid zero dynamics-based controller for planar bipeds with curved feet

Abstract

This paper extends the use of virtual constraints and hybrid zero dynamics (HZD), a successful control strategy for pointfoot bipeds, to the design of controllers for planar curved foot bipeds. Although the rolling contact constraint at the foot-ground interface increases complexity somewhat, the measure of local stability remains a function of configuration only, and a closed-form solution still determines the existence of a periodic orbit. The formulation is validated in experiment using the planar five-link biped ERNIE. While gaits designed for point feet yielded stable walking when ERNIE was equipped with curved feet, errors in both desired speed and joint tracking were significantly larger than for gaits designed for the correct radius curved feet. Thus, HZD-based control of this biped is shown to be robust to some modeling error in the foot radius, but at the same time, to require consideration of foot radius to achieve predictably reliable walking gaits. Additionally, under HZD-based control, this biped walked with lower specific energetic cost of transport and joint tracking errors for matched curved foot gait design and hardware compared to matched point-foot gait design and hardware.

Stable, Robust Hybrid Zero Dynamics Control of Powered Lower-Limb Prostheses

“…where ν F =[ q̇ x q̇ y ] T is the velocity of the fixed point on the biped and ν rel is the velocity of the foot relative to the velocity of the fixed point [22]. The relative velocity does not depend on q̇ x or q̇ y .…”

“…HZD-based control has successfully generated stable walking for 2D-point- [20], 3D-point- [21], and 2D-curved-foot [22] bipedal robots. In addition, HZD-based models can predict healthy human walking [23] and robustly enforce simplified model dynamics [24].…”

Velocity disturbance rejection for planar bipeds walking with HZD-based control

“…Walking gaits with average speeds of 0.60 (Gait I) and 0.75 m s (Gait II) represent ERNIE's middle and high experimental walking speeds [18]. K T and K L were tuned for Gait I by exploring the range 0-1.0 s m in 0.25 increments and unchanged for Gait II.…”

“…This paper designs and experimentally validates such a heuristic-based controller for the five-link biped ERNIE [17] in Fig. 1, whose physical parameters with curved feet installed [18] are listed in Table I. II. HZD-BASED & ORBITAL STABILIZATION CONTROL Both HZD-based and orbital stabilization control use the zero dynamics [19] of an underactuated system, which for planar bipeds are found using feedback linearization [15].…”

“…Lastly, the experimental and simulated gait velocities must match since the I coordinate measures errors from the planned orbit. In experiments with ERNIE, a torso offset is commonly used to compensate for small inaccuracies in the simulation model [18], shifting the desired zero dynamic orbit, thus causing the experimental velocity to deviate from the simulated value. The orbital stabilization controller attempts to correct this deviation, which can lead to gait failure.…”