Modeling, Embedded Control and Design of Soft Robots using a Learned Condensed FEM Model

TL;DR

The paper presents a learning-based condensation of FEM for soft robots that reduces full FEM states to a low-dimensional constraint-space via matrices $W$ and $oldsymbol{ delta}^{free}$. A supervised MLP learns these mechanical quantities from actuator and contact states, enabling fast direct/inverse modeling, real-time embedded control, and differentiable design calibration and optimization. The approach is demonstrated on a Soft Finger and Soft Gripper with contact interactions and on the Stiff-Flop pneumatically actuated robot, achieving substantial speedups (up to ~1 kHz prediction) and enabling both open- and closed-loop control with comparable accuracy to full FEM. The framework supports single learned models across design variations, multi-robot control, and future extensions to dynamics, multi-point contacts, and online learning, offering a practical path toward Embodied Intelligence in soft robotics.

Abstract

The Finite Element Method (FEM) is a powerful modeling tool for predicting soft robots' behavior, but its computation time can limit practical applications. In this paper, a learning-based approach based on condensation of the FEM model is detailed. The proposed method handles several kinds of actuators and contacts with the environment. We demonstrate that this compact model can be learned as a unified model across several designs and remains very efficient in terms of modeling since we can deduce the direct and inverse kinematics of the robot. Building upon the intuition introduced in [11], the learned model is presented as a general framework for modeling, controlling, and designing soft manipulators. First, the method's adaptability and versatility are illustrated through optimization based control problems involving positioning and manipulation tasks with mechanical contact-based coupling. Secondly, the low memory consumption and the high prediction speed of the learned condensed model are leveraged for real-time embedding control without relying on costly online FEM simulation. Finally, the ability of the learned condensed FEM model to capture soft robot design variations and its differentiability are leveraged in calibration and design optimization applications.

Figures (15)

• Figure 1: Illustration of the proposed framework and its applications. The FEM model of a robot is projected in the constraint space, and the corresponding matrices are learned using a neural network. The learned matrices can be used in different applications like A) real-time embedded control, B) inverse control involving predefined contact points, C) control of multiple identical robots from a single learned model, and D) both design optimization and calibration applications.
• Figure 2: Illustration of the condensed FEM model for a Soft Finger with contacts. The Soft Finger is actuated by 2 cables and has 3 fixed contact points located at the tip. The FEM model of the robot (left) is used to learn a condensed model (middle). The condensed model is based on the projection of the compliance matrix into the constraint space, resulting in drastically reducing the number of characteristic dimensions describing a robot state. The learned model is a data-driven supervised MLP model that predicts the projected compliance $W$ and the characteristic constraint distances $\delta^{free}$, based on the active constraint state (actuation, contact) and the initial values $W_0$ and $\delta_0^{free}$. On the right, two control scenarios are considered: A) a Soft Finger robot pushing a button; and B) a Soft Gripper robot manipulating a cube. The cube effector is represented by a green frame.
• Figure 3: Control results for the Soft Finger robot pressing a button. The reached horizontal position (mm) of the cube is compared on a trajectory of 9 successive goals (orange dots) when using learned mechanical matrices (blue cross). The contact force exerted at the interface between the Soft Finger and the button as well as the sum of actuation forces exerted on the two cables during the trajectory are also displayed. 3 different positions of the Soft Finger robot met during this trajectory are shown as examples. In this representation, the effector is a green frame, and the goal is a red frame.
• Figure 4: Control results for the Soft Gripper robot. The positions of the manipulated cube (mm) are compared on a trajectory for 20 different goals (orange dots) when using learned mechanical matrices (blue cross) or mechanical matrices computed from the full simulation (red cross).
• Figure 5: Illustration of the physical prototype and the condensed FEM model for the Stiff-Flop robot. The robot is actuated by 6 pneumatic cavities represented by blue cylinders. A location sensor is located at its tip. The reduced FEM-based modeling from ChaillouRobosoft2023 is also displayed.