Piston-Driven Pneumatically-Actuated Soft Robots: modeling and backstepping control

Abstract

Actuators' dynamics have been so far mostly neglected when devising feedback controllers for continuum soft robots since the problem under the direct actuation hypothesis is already quite hard to solve. Directly considering actuation would have made the challenge too complex. However, these effects are, in practice, far from being negligible. The present work focuses on model-based control of piston-driven pneumatically-actuated soft robots. We propose a model of the relationship between the robot's state, the acting fluidic pressure, and the piston dynamics, which is agnostic to the chosen model for the soft system dynamics. We show that backstepping is applicable even if the feedback coupling of the outer on the inner subsystem is not linear. Thus, we introduce a general model-based control strategy based on backstepping for soft robots actuated by fluidic drive. As an example, we derive a specialized version for a robot with piecewise constant curvature.

Paper

This repository contains the simulations as presented in the paper Piston-Driven Pneumatically-Actuated Soft Robots: modeling and backstepping control by Maximilian Stölzle and Cosimo Della Santina published in the IEEE Control Systems Letters: https://doi.org/10.1109/LCSYS.2021.3134165.

Please cite our paper if you use our method in your work:

@article{stoelzle2022piston,
  title={Piston-Driven Pneumatically-Actuated Soft Robots: modeling and backstepping control},
  author={Stölzle, Maximilian and Della Santina, Cosimo},
  journal={IEEE Robotics and Automation Letters},
  publisher={IEEE},
  year={2022},
  volume={6},
  pages={1837-1842},
  doi={10.1109/LCSYS.2021.3134165}
}

Control Scheme

Simulation

We evaluate our proposed backstepping controller on the example of a planar three-segment soft robots modelled through the Piecewise Constant Curvature (PCC) approximation. The simulation of the system model and the controller is implemented in Simulink with access to symbolically-derived equations of motion. Additionally, the behaviour of baseline controllers such as end-to-end and coupling-aware PID can be simulated.

Structure

The following folder structure is used:

• main.slx: contains the main simulink model which implements both the controller and the system model. The Initialisation block contains physical variables of the simulated robot, the acting gravity vector and the initial robot state. The Control block includes both the backstepping and the baseline PID controllers. Please make sure to re-connect the f_p signal to the output of a different controller such as the Pneumatic Actuator model-aware PID or the Full-system model-unaware PID if needed. The Plant block implements the PCC forward dynamics and equations of motion and the actuator dynamics. It takes the control action (e.g. the force on the piston f_p) as an input.
• startup.m: runs config.m and adds folders to the MATLAB path.
• config.m: contains the configuration parameters for the derivation of dynamics and some of the simulation run-time parameters. This contains namely the number of segments, the length and mass density of each segment and the equilibrium state of the system.
• derive_dynamics.m: this script can be used to derive the element of the Equations of Motion and the conservative forces using the MATLAB Symbolic Toolbox.
• data/in_qref_ts.mat: contains the sequence of commanded configuration used for set-point regulation.

Usage

Below you can find a guide on how to run the code and re-produce the results in the paper.

1. Run the startup.m file to set up the environment
2. Derive the dynamics for the planar soft robotic arm through the PCC assumption with the script derive_dynamics.m. The resulting symbolic functions for evaluating the dynamics at a specific robot state are automatically saved in the folder funs.
3. Generate a control reference sequence with the script generate_control_ref_sequence.m. The sequence is automatically saved in data/in_qref_ts.mat.
4. Run the simulink model main.slx to simulate the system.
5. Animate the PCC system using the script plotting/animate_pcc_system.m. It automatically uses the time-series data from the simulation which is saved in the out variable of the workspace.
6. Save the out variable in a mat-file and save it in the data folder.
7. Generate time-series and Cartesian evolution plots using the scripts plot_cartesian_evolution.m and plot_time_series_v2.m in the plotting folder.

GIFs

Videos of simulations of a planar soft robot consisting of three segments. Gravity is pointing downwards in negative y-direction.

We simulate the response of the closed-loop generated by all controllers to a sequence of step references. The segments are initialised at the equilibrium configuration. At 10s, the reference is moved to the straight configuration q ̄ = 0. After another 30s, we change it again to q_bar = [0.6981 rad −0.5236 rad 0.1745 rad].

Coupling-aware PID

Backstepping