Deformable Multibody Modeling for Model Predictive Control in Legged Locomotion with Embodied Compliance

TL;DR

This work addresses stabilizing dynamic gaits for legged robots with embodied spine compliance. It introduces a unified deformable multibody model and predictive inertia constructs (PDI and CCPDI) to capture deformation within a model predictive controller. Simulation experiments on a quadruped demonstrate that CCPDI-enabled MPC stabilizes trot for both rigid and compliant spines, with more balanced ground reaction forces and substantially improved inertia prediction accuracy, and shows robustness across a range of spine parameters. The approach offers a practical pathway to leverage spine compliance for enhanced agility and energy efficiency in legged locomotion, with future work toward real hardware verification.

Abstract

The paper presents a method to stabilize dynamic gait for a legged robot with embodied compliance. Our approach introduces a unified description for rigid and compliant bodies to approximate their deformation and a formulation for deformable multibody systems. We develop the centroidal composite predictive deformed inertia (CCPDI) tensor of a deformable multibody system and show how to integrate it with the standard-of-practice model predictive controller (MPC). Simulation shows that the resultant control framework can stabilize trot stepping on a quadrupedal robot with both rigid and compliant spines under the same MPC configurations. Compared to standard MPC, the developed CCPDI-enabled MPC distributes the ground reactive forces closer to the heuristics for body balance, and it is thus more likely to stabilize the gaits of the compliant robot. A parametric study shows that our method preserves some level of robustness within a suitable envelope of key parameter values.

Figures (8)

• Figure 1: Simulation Models and Robot Prototypes. (a) The simulation model with a rigid spine. (b) The robot prototype with a rigid spine. (c) The simulation model with a compliant prismatic spine. (d) The robot prototype with a compliant prismatic spine.
• Figure 2: Deformable Body Approximation. (a) The initial configuration of a body (grey): each sub-body $\mathcal{B}_i$ (blue) has a local frame $\prescript{}{j}{f}_i$, a CoM (purple cross-circle) and a MoI (purple axes). The Sub-body CoM position vector is denoted as $\prescript{}{j}{\mathbf{c}}_{i}$ (green dashed arrow) in the frame $\prescript{}{j}{f}_i$ and $\prescript{}{j}{\mathbf{r}}_{i}$ (orange dashed arrow) in the frame $\prescript{}{0}{f}_i$. The body has an overall CoM (red cross-circle) and its position vector is denoted as $\mathbf{r}_i$ (red dashed arrow) in the frame $\prescript{}{0}{f}_i$. (b) The deformed configuration of a body: sub-bodies move relative to each other with their CoM position vectors denoted as $\prescript{}{j}{\mathbf{r}}^{\prime}_{i}$ (orange dashed arrow). The body CoM is denoted as $\mathbf{r}_i^{\prime}$ (red dashed arrow).
• Figure 3: Multibody Representation. (a) The rigid multibody tree topology: each body (grey) is connected to its preceding body through a joint (green). The adjoint transform $\prescript{i}{}{\mathbf{X}}_{p(i)}$ (blue arrow) is defined by the joint $\mathbf{J}_i$. (b) The deformable multibody tree topology: each body is linked to its preceding body through a joint and a corresponding sub-body (blue). The adjoint transform $\prescript{i}{}{\mathbf{X}}_{p(i)}$ (blue arrow) is defined by the joint $\mathbf{J}_i$ and deformation (orange arrow).
• Figure 4: Control Framework Diagram. This paper addresses two control components (shown in red blocks); the centroidal predictive dynamics and MPC-based whole-body controller. The other control components (grey-shaded) and the corresponding signal pattern (dashed-arrow) are beyond the scope of this work.
• Figure 5: Trajectory of Centroid States. Tracking performance of 1-minute trot stepping is investigated between the rigid (blue) and compliant models, and between CCPDI-enabled and CCPDI-disabled MPC settings for the compliant model in four directions: height (top-left), roll angle (top-right), pitch angle (bottom-left), and yaw angle (bottom-right). Spinal compliance is configured as $k_s=36 N/m$ and $l_{rest} = 0.180 m$.