The effects of foot geometric properties on the gait of planar bipeds walking under HZD-based control

Abstract

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.

Dynamic Output Controllers for Exponential Stabilization of Periodic Orbits for Multidomain Hybrid Models of Robotic Locomotion

“…HZD-based controllers have been validated numerically and experimentally for (i) 2D and 3D bipedal robots, including RABBIT [17,18], MA-BEL [19][20][21], ERNIE [22], AMBER [23], ATRIAS [24][25][26][27], and DURUS [28,29] prototypes, (ii) powered prosthetic legs [30][31][32][33], (iii) exoskeletons [34], (iv) monopedal robots [35,36], and (v) quadruped robots [37]. In the HZD approach, a set of output functions, referred to as virtual constraints, is defined for the continuous-time dynamics of the system and asymptotically driven to zero by partial linearizing feedback controllers [38].…”

“…The power of the analytical approach is ultimately demonstrated through designing a nonlinear dynamic output feedback controller for walking of a 3D humanoid robot with 18 state variables and 325 controller parameters. 22 Variables for the controller synthesis, variables for the observer synthesis W ,γ,δ Decision variables for the BMI optimization ∆ξ,∆η Increment in controller and observer parameterŝ A,A Approximate Jacobian matrix, convex hull of Jacobian matrices…”

Decentralized Feedback Controllers for Robust Stabilization of Periodic Orbits of Hybrid Systems: Application to Bipedal Walking

“…In particular, virtual constraints are defined as holonomic output functions y ( x ) for continuous-time portions of hybrid models of walking and they are typically enforced (i.e., y ≡ 0) by centralized I-O linearizing feedback laws [51]. Virtual constraint controllers have been numerically and experimentally validated for stable 2D and 3D underactuated bipedal robots [14], [53]–[55], [58] as well as 2D powered prosthetic legs [28]–[30], [47]. For mechanical systems with more than one degree of underactuation, the stability of the periodic gait depends on the choice of the virtual constraints [13], [41].…”

Decentralized feedback controllers for exponential stabilization of hybrid periodic orbits: Application to robotic walking

“…In this approach, holonomic output functions y ( x ) define desired virtual constraints, i.e., y ≡ 0, that are typically enforced by centralized input-output (I/O) linearizing feedback laws during the continuous-time portion of the hybrid system. Virtual constraint controllers have been validated numerically and experimentally for stable 2D and 3D underactuated bipedal robots [8], [29], [30], [34], [35] as well as 2D powered prosthetic legs [16]–[18]. The stability of walking gaits in the virtual constraints approach depends on the choice of the output functions [25].…”