Exponentially stabilizing continuous-time controllers for periodic orbits of hybrid systems: Application to bipedal locomotion with ground height variations

Abstract

This paper presents a systematic approach for the design of continuous-time controllers to robustly and exponentially stabilize periodic orbits of hybrid dynamical systems arising from bipedal walking. A parameterized family of continuous-time controllers is assumed so that (1) a periodic orbit is induced for the hybrid system, and (2) the orbit is invariant under the choice of controller parameters. Properties of the Poincaré map and its first- and second-order derivatives are used to translate the problem of exponential stabilization of the periodic orbit into a set of bilinear matrix inequalities (BMIs). A BMI optimization problem is then set up to tune the parameters of the continuous-time controller so that the Jacobian of the Poincaré map has its eigenvalues in the unit circle. It is also shown how robustness against uncertainty in the switching condition of the hybrid system can be incorporated into the design problem. The power of this approach is illustrated by finding robust and stabilizing continuous-time feedback laws for walking gaits of two underactuated 3D bipedal robots.

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

“…The stability criterion serves as a nonlinear constraint on the human and prosthesis output functions used to define a gait [43] because a usable gait must be stable. If the zero dynamics are stable, the stability of the full system can be easily proven if the output feedback controller v ij in Eq.…”

Prosthetic leg control in the nullspace of human interaction

“…The vast majority of virtual constraints used in bipedal robots are holonomic [9]–[17], so we consider virtual holonomic constraints $$h\left(q\right)=0,$$ where $h:\mathcal{Q}\to {\mathrm{double-struckR}}^2$ for an actuated knee and ankle, i.e., one virtual constraint per actuated degree-of-freedom (DOF).…”

“…In previous work on bipedal robots (e.g., [9]–[17]), virtual constraints are used to control the actuated joints specified by h 0 ( q ) = ( q 2 , q 3 ) T to a desired trajectory $h_{\mathrm{normald}}\left(ϴ\right(q\left)\right)\in {\mathrm{double-struckT}}^2$ as a function of a monotonic quantity $ϴ\left(q\right)\in \mathbb{R}$. This quantity, known as the phase variable (or timing variable ), provides a unique representation of the gait cycle phase to drive forward the desired joint patterns in a time-invariant manner.…”

“…The desired constraints are enforced by torque control laws that linearize and stabilize the input-output dynamics associated with the constraint outputs. This method has been widely successful in controlling autonomous walking robots without human interaction [9]–[17]. In the application to powered prosthetic legs, the input-output dynamics generally depend on the human interaction forces, through which the human can influence the behavior of the leg.…”

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

“…This paper also presents sufficient conditions on the Poincaré map to guarantee the convergence of the iterative BMI algorithm at a finite number of iterations. We previously applied a BMI algorithm for the systematic design of centralized feedback controllers in [25]–[27], whereas this paper presents a BMI framework for designing decentralized controllers. A class of novel decentralized controllers is first developed and then the BMI algorithm is improved for tuning the local controllers.…”