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.

It has been hypothesized by many that foot design can influence gait. This idea was investigated in both simulation and hardware for the five-link, planar biped ERNIE controlled under the Hybrid Zero Dynamics paradigm. The effects of walking speed, foot radius, and foot center of curvature location on gait efficiency and kinematics were investigated in a full factorial study of gaits optimized using a work-based objective function. In most cases, the simulation correctly predicted the trends observed in hardware, indicating that simulation can be used for foot design. As expected, increasing walking speed decreased the energetic efficiency. The dominant effect of speed on joint kinematics was to alter the timing of the peak hip flexion. Increasing foot radius up to the length of the shank improved the energetic efficiency and increased the range of motion of the hip and knee joints. Shifting the foot center of curvature location forward altered the energetic efficiency in a manner that interacted with changes in foot radius. The energetically optimal foot center of curvature location was coincident with the shank for a large foot radius and shifted far in front of the shank for a small foot radius. In all cases, the forward shift increased the range of motion of the hip and knee joints. Therefore, a robot designer can achieve similar energetic benefits across a range of speeds with either a larger radius foot or a smaller radius foot whose center of curvature is located forward of the shank.

Robots walking under hybrid zero dynamics (HZD)-based control are susceptible to velocity disturbances because the controller is typically designed for one speed. LQRbased orbital stabilization control is one means to address this issue using feedback on the unactuated velocity. The approach, though, is difficult to implement in real time experimentally even on planar bipeds with relatively few links. This work extracts simple heuristics from simulated planar bipeds rejecting velocity disturbances under orbital stabilization control to approximate that functionality. The heuristics are layered on top of traditional HZD-based control of a five-link planar biped robot for experimental validation. Results show that the heuristically modified controller yields more efficient and more stable walking for the biped than does HZD-based control alone. It also enables rejection of larger decelerating disturbances and more rapid return to the desired walking cycle.

The locomotion of legged robots is inherently underactuated, which creates control challenges in terms of rejecting large disturbances. A detailed understanding of how the control authority of a robot evolves over a gait trajectory has the potential to inform the design of controllers that offer superior disturbance rejection capabilities without compromising the efficiency benefits that typically accompany underactuated legged robots. Previous work has shown how the system velocities of an underactuated mechanical system can be decomposed into directions aligned with the inputs, or controlled directions, and directions orthogonal to the inputs, or uncontrolled directions, and applied that decomposition to drive wheeled robots to rest. This decomposition fundamentally provides a measure of the instantaneous control authority of the robot. This paper examines how the same techniques can be applied to inform the control of biped robots walking with periodic gaits. This problem differs from those previously studied in that it necessarily involves ground impacts and non-zero desired velocities. A representative example of a two-link planar biped walking on flat ground shows how a simple open loop controller that implements heuristics identified through the velocity decomposition to make use of the available control authority can improve disturbance rejection when added to a hybrid zero dynamics-based controller.