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Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control
Daniel Bruder, Brent Gillespie, C. David Remy, Ram Vasudevan
TL;DR
Soft robots are difficult to control precisely because constructing models suitable for model-based control is challenging. This paper develops a data-driven Koopman identification method and applies the resulting explicit linear model to MPC; across trajectory-following tasks, the Koopman-based controller outperformed a linear-state-space MPC benchmark.
Problem
Soft robots require precise control, but their modeling challenges make models suitable for model-based control difficult to construct.
Method
The paper extends Koopman system identification for sparse, outlier- and noise-resistant models and applies the identified model to MPC on a physical soft robot.
Results
Across Tasks 1, 2, and 3, K-MPC achieved 1.26 cm average tracking error versus 2.45 cm for L-MPC.
Takeaways & Limitations
The Koopman-based method produced an explicit control-oriented model that commanded a soft robot to follow reference trajectories better than another linear data-driven MPC model.
Takeaways & Limitations
Further work is needed to make the method feasible for higher-dimensional robotic systems and to account for external loading and contact forces.
Abstract
from arXiv · showhide
Controlling soft robots with precision is a challenge due in large part to the difficulty of constructing models that are amenable to model-based control design techniques. Koopman Operator Theory offers a way to construct explicit linear dynamical models of soft robots and to control them using established model-based linear control methods. This method is data-driven, yet unlike other data-driven models such as neural networks, it yields an explicit control-oriented linear model rather than just a "black-box" input-output mapping. This work describes this Koopman-based system identification method and its application to model predictive controller design. A model and MPC controller of a pneumatic soft robot arm was constructed via the method, and its performance was evaluated over several trajectory following tasks in the real-world. On all of the tasks, the Koopman-based MPC controller outperformed a benchmark MPC controller based on a linear state-space model of the same system.
I. INTRODUCTION
Soft robots offer safer interaction but are difficult to model for precise, model-based control because they deform continuously and lack canonical state variables. The paper uses Koopman theory to construct explicit linear models from data and applies them to convex MPC.
- Soft robots safely interact with delicate objects and adapt passively to unstructured environments, supporting wearable, assistive, and medical applications.
- Continuous deformation and infinite degrees of freedom leave soft robots without a canonical state representation.Existing representations therefore rely on restrictive simplifying assumptions.
- Black-box machine-learning models can predict soft-robot behavior but do not provide explicit models suitable for established model-based control techniques.
- Koopman theory lifts nonlinear dynamics into a linear representation that can be controlled using established linear methods.
- Data-driven identification captures input-output behavior while avoiding ambiguity in selecting states for systems with infinite degrees of freedom.
- The paper extends Koopman identification to improve sparsity and robustness to outliers and noise, then applies the model to MPC for a physical soft robot.
II. LINEAR SYSTEM IDENTIFICATION
The identification procedure approximates the infinite-dimensional Koopman operator in a finite basis using data-driven linear regression. Delays, basis functions, and snapshot pairs define the finite-dimensional lifted model.
- Koopman theory represents nonlinear system flow linearly in an infinite-dimensional space of observables.The observables are functions whose evolution follows the system trajectories.
- Because the exact Koopman operator is infinite-dimensional, the method projects it onto a finite-dimensional subspace and estimates it by modified EDMD regression.
- A lifting function maps the state into a vector of basis-function values, defining the finite-dimensional lifted state and its image manifold.
- The finite-dimensional Koopman matrix is chosen to best approximate observable evolution on the selected subspace in the L2-norm sense.
- Identification uses measured snapshot pairs, which are lifted and assembled into data matrices for least-squares estimation.
- Delays can improve model accuracy by enlarging the snapshot domain, after which the identification procedure remains unchanged.
C. Building Linear System from Koopman Operator
The input-extended Koopman model produces a discrete linear state-space representation with lifted-state dynamics and a projection back to the physical state. Keeping inputs unlifted makes the model suitable for real-time convex feedback optimization.
- The controlled Koopman model uses lifted dynamics z[j + 1] = Az[j] + Bu[j] and reconstructs the physical state as x[j] = Cz[j].
- The initial lifted state is obtained from the physical initial condition, while C projects lifted states back into the original state space.
- Inputs are deliberately left unlifted so they enter the identified model linearly.This preserves the control-oriented structure needed for convex optimization.
- The model’s A and B matrices are embedded in the transpose of the identified Koopman matrix and isolated by partitioning it.
- Iterating the lifted linear dynamics can leave the manifold of legitimate lifted states, so projection onto that manifold reduces deviation and improves predictive performance.
D. Practical Considerations: Overfitting and Sparsity
The method uses LASSO regularization to reduce sensitivity to noise and overfitting, while promoting sparsity in the Koopman model. A projection operator then keeps simulated lifted states near the valid lifted-state space, balancing sparsity against prediction accuracy.
- Overfitting and sparsity: LASSO regularization reduces susceptibility to outliers and noise during Koopman operator identification.The L1 penalty also addresses overfitting in least-squares regression.
- Overfitting and sparsity: L1 regularization drives matrix elements to zero, producing a sparser Koopman operator and associated model matrices.Sparsity reduces storage requirements and supports higher-dimensional lifting functions.
- Identification procedure: The identification pipeline lifts data, combines lifted states with inputs, estimates the Koopman operator, extracts A and B, and applies a projection operator.The output is ˆA := PA and ˆB := PB.
- Projection and accuracy: The desired model is the sparsest one that minimizes the distance of each predicted lifted state from the valid lifted-state space.Without this constraint, Aψ(a[j]) + Bu[j] may leave that space and cause inaccurate simulations.
III. MODEL PREDICTIVE CONTROL
Koopman-based MPC repeatedly optimizes future inputs using a finite-dimensional linear model in lifted coordinates. Its convex formulation incorporates prediction dynamics, costs, and state and input constraints, enabling efficient receding-horizon control.
- MPC formulation: MPC optimizes control inputs over a finite horizon, applies the first input, and repeats the optimization at each timestep.The controller uses model predictions to select inputs for the task.
- MPC formulation: The optimization includes quadratic state and input costs together with polyhedral state and input constraints.The cost and constraint matrices depend on the prediction horizon and lifted-state formulation.
- MPC formulation: Koopman-based MPC initializes predictions from the current lifted state ψ(x[k]) and enforces z[i + 1] = ˆAz[i] + ˆBu[i].The lifted linear dynamics connect predicted states and inputs across the horizon.
- Computational properties: Because the optimization problem is convex, it has a unique globally optimal solution and can be efficiently solved without initialization.The formulation contrasts with nonlinear MPC, which involves nonlinear constraints or costs.
IV. EXPERIMENTS
The experiments evaluate the paper’s modeling and control methods on a physical soft robot through multiple real-world tasks. The setup includes pressure regulation of pneumatic actuators and camera-based tracking of a laser-dot output.
- Experimental program: The experiments demonstrate the modeling and control methods described in the paper on a physical soft robotic system.Several tasks from the final experiment were also recorded in supplementary video.
- Experimental setup: Three pressure regulators control the pneumatic actuators, while a camera tracks the laser dot attached to the end effector.The arrangement provides pressure inputs and an observable position output for the experiments.
A. Robot Description: Soft Arm with Laser Pointer
The experimental platform is a suspended pneumatic soft arm with two bending sections and a laser pointer at its end effector. Its output is measured as the laser-dot position on a board, and repeated responses exhibit variability of up to 2 cm.
- Robot platform: The robot is a suspended soft arm with a laser pointer attached to its end effector and a webcam measuring the projected dot.The dot lands on a 50 cm × 50 cm board positioned 34 cm below the relaxed pointer tip.
- Robot platform: The arm has two sections of three pneumatic artificial muscles connected so that the lower section bends oppositely to the upper section.Only three pressure lines are required because the muscles in each corresponding arrangement are internally connected.
- System variability: Identical sinusoidal inputs can produce output trajectories varying by up to 2 cm across trials.The figure summarizes the mean response and trajectory distribution within two standard deviations.
B. Characterization of Stochastic Behavior
Electronic pressure regulators contribute stochastic behavior that limits predictive precision in pneumatically driven soft robots. In repeated sinusoidal trials, nearly all observed laser-dot positions stayed within 1 cm of the mean trajectory, establishing an approximate best-case control precision.
- Electronic pressure regulators can cause different outputs from identical inputs and states, limiting predictive capability in pneumatic soft robots.
- The robot’s stochastic behavior was quantified from period-to-period output variation under sinusoidal inputs applied to its three actuators.
- Nearly all observed laser-dot positions across 210 periods fell within 1 cm of the mean trajectory.
- The observed stochasticity implies an expected best-case control precision of approximately 1 cm relative to a desired trajectory.
C. Data Collection and Model Identification
The study collected randomized input-output data across the robot’s operating range, then identified Koopman and linear state-space models from the resulting measurements. The selected Koopman representation minimized prediction error while making 70% of its lifted system matrix entries zero.
- Sixteen trials of approximately 20 minutes each used smoothly varying randomized inputs to sample the robot across its operating range.The lookup table contained uniformly distributed values between zero and ten, with transition periods varying from 5 to 10 seconds.
- The Koopman model and a four-dimensional linear state-space baseline were identified from the same collected data.The Koopman identification used 191,000 snapshot pairs with a single delay.
- The Koopman basis contained 330 monomials with maximum degree 4, and models were evaluated across λ values from 0 through 50.
- 70% of the selected lifted system matrix entries were zero because the chosen model minimized prediction error while promoting sparsity.Increasing λ reduced matrix density but increased prediction error; the selected model balanced these effects by minimizing prediction error.
D. Experiment 1: Model Prediction Comparison
The experiments compared Koopman-based predictions and model predictive control against linear-model baselines using sinusoidal prediction and three trajectory-following tasks. The Koopman-based controller achieved lower tracking error than the linear MPC controller in the reported tasks.
- Model prediction comparison: Over a 2.5-second horizon under identical sinusoidal inputs and initial conditions, Koopman model predictions were more accurate than the linear model’s predictions.
- Controller setup: Both controllers operated at 10 Hz with a 2.5-second prediction horizon and penalized reference-trajectory deviations using running and terminal costs.
- Controller comparison: The visual comparison placed K-MPC results in row 1 and L-MPC results in row 2, with reference trajectories and a two-standard-deviation noise buffer overlaid.
- Trajectory-following evaluation: The evaluation used three trajectory-following tasks: Pacman for 90 seconds, Star for 180 seconds, and block letter M for 300 seconds.Tracking error was measured as Euclidean distance from the reference trajectory at each time step.
- Controller comparison: K-MPC achieved 1.26 cm average tracking error compared with 2.45 cm for L-MPC in the reported tasks.
V. CONCLUSION
The work applied Koopman-based modeling and MPC to a soft robot, achieving better trajectory following than another linear data-driven MPC controller. The authors identify higher-dimensional systems and external loading or contact forces as directions requiring further work.
- The Koopman-based MPC controller commanded a soft robot to follow reference trajectories better than an MPC controller based on another linear data-driven model.
- The method makes explicit control-oriented models of soft robots easier to construct, supporting rapid development of new control strategies and applications.
- The reported results are preliminary, and further work is needed to make the methods feasible for higher-dimensional robotic systems.
- Future work includes promoting sparsity, selecting effective observable bases, and modeling external loading and contact forces.