Optimal Control of a 5-Link Biped Using Quadratic Polynomial Model of Two-Point Boundary Value Problem

Abstract

To walk over constrained environments, bipedal robots must meet concise control objectives of speed and foot placement. The decisions made at the current step need to factor in their effects over a time horizon. Such step-to-step control is formulated as a two-point boundary value problem (2-BVP). As the dimensionality of the biped increases, it becomes increasingly difficult to solve this 2-BVP in real-time. The common method to use a simple linearized model for real-time planning followed by mapping on the high dimensional model cannot capture the nonlinearities and leads to potentially poor performance for fast walking speeds. In this paper, we present a framework for real-time control based on using partial feedback linearization (PFL) for model reduction, followed by a data-driven approach to find a quadratic polynomial model for the 2-BVP. This simple step-to-step model along with constraints is then used to formulate and solve a quadratically constrained quadratic program to generate real-time control commands. We demonstrate the efficacy of the approach in simulation on a 5-link biped following a reference velocity profile and on a terrain with ditches. A video is here: https://youtu.be/-UL-wkv4XF8.

Data-driven Step-to-step Dynamics based Adaptive Control for Robust and Versatile Underactuated Bipedal Robotic Walking

“…For example, [15], [16] uses data to optimize existing controller parameters. Data-driven reduced order models are considered in [17], [18], where a reduced-order model of the hybrid dynamics or the step-to-step (S2S) dynamics for a specific robot is obtained from simulation data. Using a similar concept, a linear discrete reduced-order S2S dynamics model is learned in [19], [20] by extending a Hybrid-LIP based S2S approximation in [21]: both of which have a particular inputstate dynamics structure, where they consider the step size as the input to control horizontal center of mass (COM) states of bipedal walking.…”