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

For bipedal robots to walk over complex and constrained environments (e.g., narrow walkways, stepping stones), they have to meet precise control objectives of speed and foot placement at every single step. This control that achieves the objectives precisely at every step is known as one-step deadbeat control. The high dimensionality of bipedal systems and the under-actuation (number of joint exceeds the actuators) presents a formidable computational challenge to achieve real-time control. In this paper, we present a computationally efficient method for one-step deadbeat control and demonstrate it on a 5-link planar bipedal model with 1 degree of under-actuation. Our method uses computed torque control using the 4 actuated degrees of freedom to decouple and reduce the dimensionality of the stance phase dynamics to a single degree of freedom. This simplification ensures that the step-to-step dynamics are a single equation. Then using Monte Carlo sampling, we generate data for approximating the step-to-step dynamics followed by curve fitting using a control affine model and a Gaussian process error model. We use the control affine model to compute control inputs using feedback linearization and fine tune these using iterative learning control using the Gaussian process error enabling one-step deadbeat control. We demonstrate the approach in simulation in scenarios involving stabilization against perturbations, following a changing velocity reference, and precise foot placement. We conclude that computed torque control-based model reduction and sampling-based approximation of the step-to-step dynamics provides a computationally efficient approach for real-time one-step deadbeat control of complex bipedal systems.