Angular Divergent Component of Motion: A step towards planning Spatial DCM Objectives for Legged Robots

TL;DR

This work introduces the idea of spatial DCM, which adds an angular objective to the existing linear DCM theory, and combines it into a 3D linear and 1D angular DCM framework.

Abstract

In this work, the Divergent Component of Motion (DCM) method is expanded to include angular coordinates for the first time. This work introduces the idea of spatial DCM, which adds an angular objective to the existing linear DCM theory. To incorporate the angular component into the framework, a discussion is provided on extending beyond the linear motion of the Linear Inverted Pendulum model (LIPM) towards the Single Rigid Body model (SRBM) for DCM. This work presents the angular DCM theory for a 1D rotation, simplifying the SRBM rotational dynamics to a flywheel to satisfy necessary linearity constraints. The 1D angular DCM is mathematically identical to the linear DCM and defined as an angle which is ahead of the current body rotation based on the angular velocity. This theory is combined into a 3D linear and 1D angular DCM framework, with discussion on the feasibility of simultaneously achieving both sets of objectives. A simulation in MATLAB and hardware results on the TORO humanoid are presented to validate the framework's performance.

Figures (6)

• Figure 1: Spatial DCM is a framework allowing for linear and angular rotation planning for complex behaviors such as traversing rubble, climbing through a window, and compensating a linear and angular load.
• Figure 2: The VRO set-points, $\phi_{\rm vro}$, define the desired angular DCM, $\xi_{\rm a}^d$, which drives the motion of the angular DCM, $\xi_{\rm a}$, and body orientation, $\theta$.
• Figure 3: The angular DCM, $\xi_{\rm a}$, is ahead of orientation, $\theta$, which is repelled by the VRO, $\phi_{\rm vro}$.
• Figure 4: Image of SRBM, angular DCM, $\xi_{\rm a}$, linear DCM, $\xi_{\rm l}$, force vector, $\mathbf{f}_{\rm ext}$, which gets projected from the CoP, $\mathbf{r}_{\rm cop}$, on the physical robot.
• Figure 5: Simulation results of the 3D linear and 1D angular DCM framework simultaneously completing walking and rotating behaviors. The 3D linear motion has been projected to the ground plane.