Hybrid Zero Dynamics

Using ‘virtual constraints’ to compress bipedal walking into a low-dimensional system, then designing and proving a gait stable.

Hybrid zero dynamics was proposed by Westervelt, Grizzle, and Koditschek in 2003 for underactuated bipedal robots — point-foot robots, for instance, can't apply ankle torque to the ground at all. Walking is a ‘hybrid’ system: continuous dynamics while a leg swings, and a discrete jump at footstrike, with impact and a leg switch. The method chooses a phase variable θ that advances monotonically with the gait, and uses feedback to make every actuated joint angle q_a track a specified function h(θ), i.e., enforce y = q_a − h(θ) = 0 — these are the ‘virtual constraints.’ The robot is thereby compressed onto a low-dimensional surface, and whatever underactuated part remains is the zero dynamics. A Poincaré map is then used to check whether the state returns to the same periodic orbit (limit cycle) after each step's impact, which is how gait stability is proven. It doesn't require the foot to stay flat on the ground — the main way it differs from ZMP-based methods.

ExampleUniversity of Michigan's Grizzle group used virtual constraints plus a gait library to control Cassie in 2018: about six weeks after receiving the robot, they had it walking on sidewalks, grass, snow, and sand, and even balancing while standing on a Segway.

Also called

HZD, Virtual Constraints, HZD Control

Related

Underactuation · Bipedal Locomotion · Limit Cycle · Zero Moment Point · Agility Robotics Cassie · Gait Phase (Phase Clock)

Sources

Grizzle & Chevallereau: Virtual Constraints and Hybrid Zero Dynamics for Realizing Underactuated Bipedal Locomotion (arXiv:1706.01127)
Gong et al.: Feedback Control of a Cassie Bipedal Robot: Walking, Standing, and Riding a Segway (arXiv:1809.07279)