Overview
Decomposing trajectory optimization for gaits on a quadrupedal robot to that of two bipedal robots.
Abstract
A robotic system can be viewed as a collection of lower-dimensional systems that are coupled via reaction forces (Lagrange multipliers) enforcing holonomic constraints. Inspired by this viewpoint, this paper presents a novel formula- tion for nonlinear control systems that are subject to coupling constraints via virtual “coupling” inputs that abstractly play the role of Lagrange multipliers. The main contribution of this paper is a process—mirroring solving for Lagrange multipliers in robotic systems—wherein we isolate subsystems free of cou- pling constraints that provably encode the full-order dynamics of the coupled control system from which it was derived. This dimension reduction is leveraged in the formulation of a nonlinear optimization problem for the isolated subsystem that yields periodic orbits for the full-order coupled system. We consider the application of these ideas to robotic systems, which can be decomposed into subsystems. Specifically, we view a quadruped as a coupled control system consisting of two bipedal robots, wherein applying the framework developed allows for gaits (periodic orbits) to be generated for the individual biped yielding a gait for the full-order quadruped. This is demonstrated through walking experiments of a quadrupedal robot in simulation and on rough terrains.