Control Synergies for Rapid Stabilization and Enlarged Region of Attraction for a Model of Hopping

Abstract

Inspired by biological control synergies, wherein fixed groups of muscles are activated in a coordinated fashion to perform tasks in a stable way, we present an analogous control approach for the stabilization of legged robots and apply it to a model of running. Our approach is based on the step-to-step notion of stability, also known as orbital stability, using an orbital control Lyapunov function. We map both the robot state at a suitably chosen Poincaré section (an instant in the locomotion cycle such as the mid-flight phase) and control actions (e.g., foot placement angle, thrust force, braking force) at the current step, to the robot state at the Poincaré section at the next step. This map is used to find the control action that leads to a steady state (nominal) gait. Next, we define a quadratic Lyapunov function at the Poincaré section. For a range of initial conditions, we find control actions that would minimize an energy metric while ensuring that the Lyapunov function decays exponentially fast between successive steps. For the model of running, we find that the optimization reveals three distinct control synergies depending on the initial conditions: (1) foot placement angle is used when total energy is the same as that of the steady state (nominal) gait; (2) foot placement angle and thrust force are used when total energy is less than the nominal; and (3) foot placement angle and braking force are used when total energy is more than the nominal.

One-Step Deadbeat Control of a 5-Link Biped Using Data-Driven Nonlinear Approximation of the Step-to-Step Dynamics

“…Three, the time-scale for solving the step-to-step control problem is of the order of half step-time, thus one may use a slow computer processor for online computation. Finally, although the instantaneous dynamics is under-actuated, by a judicious choice of control actions, it is possible to have over-actuation in the step-to-step dynamics, which one may exploit to achieve stabilization of wider range of initial conditions [41].…”

Control Policies for a Large Region of Attraction for Dynamically Balancing Legged Robots: A Sampling-Based Approach

“…The three control actions perform three distinct roles: foot placement cannot change the total energy but can convert the potential energy to the kinetic energy and vice versa, the braking force can only decrease the total energy, and thrust force can only increase the total energy. A detailed analysis of how these separate control actions combine to increase the ROA is provided elsewhere 27 . Figure 6 shows the associated MCOT (see Eq.…”