Multi-segment Soft Robot Control via Deep Koopman-based Model Predictive Control

TL;DR

Soft robots present high-dimensional, nonlinear, time-varying dynamics that challenge precise control. This paper introduces Deep Koopman-based Model Predictive Control (DK-MPC), which learns a deep Koopman operator to lift the nonlinear dynamics into a linear latent space and then uses Model Predictive Control for trajectory tracking. The approach combines a deep autoencoder for lifting with linear operators $A$ and $B$, yielding a global linear model $z_{k+1} = A z_k + B u_k$ in the latent space. Real-world experiments on the 3-segment soft robot 'Chordata' demonstrate high-precision tracking and moving-target tracking, markedly outperforming a baseline K-MPC. The work suggests that DK-MPC offers a practical, data-driven path toward dexterous soft-robot control.

Abstract

Soft robots, compared to regular rigid robots, as their multiple segments with soft materials bring flexibility and compliance, have the advantages of safe interaction and dexterous operation in the environment. However, due to its characteristics of high dimensional, nonlinearity, time-varying nature, and infinite degree of freedom, it has been challenges in achieving precise and dynamic control such as trajectory tracking and position reaching. To address these challenges, we propose a framework of Deep Koopman-based Model Predictive Control (DK-MPC) for handling multi-segment soft robots. We first employ a deep learning approach with sampling data to approximate the Koopman operator, which therefore linearizes the high-dimensional nonlinear dynamics of the soft robots into a finite-dimensional linear representation. Secondly, this linearized model is utilized within a model predictive control framework to compute optimal control inputs that minimize the tracking error between the desired and actual state trajectories. The real-world experiments on the soft robot "Chordata" demonstrate that DK-MPC could achieve high-precision control, showing the potential of DK-MPC for future applications to soft robots.

Figures (6)

• Figure 1: Illustration of the proposed Deep Koopman-based Model Predictive Control (DK-MPC) framework for a multi-segment soft robot. The deep Koopman operator maps both the reference state $x^{{ref}}$ and the state $x$ into a high-dimensional linear latent space. Based on the latent states and the linear dynamics, an MPC controller is employed to generate the optimal control signals $u^*$, ensuring the end-effector of the soft robot follows the reference trajectory.
• Figure 2: Design of deep Koopman operator architecture in Sec. 2.C. (a) Deep auto-encoder for learning lifting function $\varphi$ and its inverse function $\varphi^{-1}$. (b) Illustration of the learning process for linear operator $A$ and control-affine matrix $B$.
• Figure 3: Design of the multi-segment soft robot “Chordata”. (a) Overview of the robot, which has $3$ independent segments and a total length of $450 mm$. The addition of bones along the joints enhances stability and stiffness. (b) Detailed segment design: a flexible Stewart platform with three bellows in a circular array, constrained by a central rigid pivot. (c) Experimental platform setup, including a stereo camera (MicronTracker H3-60) for tip position tracking, a pneumatic drive controlling air pressure, and PC controller for data processing, system identification and control.
• Figure 4: Visualization of trajectory tracking. (a) Task execution of the soft robot, where the green dashed arrow is the direction and the elliptical arc is the trajectory. (b) The results and errors of the proposed DK-MPC and K-MPC controllers during circular trajectory tracking.
• Figure 5: The results of DK-MPC and K-MPC in 'T', 'H', and 'U' trajectory tracking tasks. The reference trajectories are displayed in purple, and the actual trajectories are shown in orange.