Sample-efﬁcient Model Predictive Control Design of Soft Robotics by Bayesian Optimization Anuj Pal Tianyi He Wenpeng Wei

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Sample-efﬁcient Model Predictive Control Design of Soft Robotics by

Bayesian Optimization

Anuj Pal, Tianyi He∗, Wenpeng Wei

Abstract— This paper presents a sample-efﬁcient data-driven

method to design model predictive control (MPC) for cable-

actuated soft robotics using Bayesian optimization. Instead of

modeling the complex dynamics of the soft robots, the proposed

approach uses Bayesian optimization to search the best-guessed

low-dimensional prediction model and its associated controller

to minimize the objective function of closed-loop responses. The

prediction model is updated by Bayesian optimization from

the closed-loop input-output data in each iteration. A linear

MPC is then designed based on the updated prediction model,

and evaluated based on the closed-loop responses. Different

from directly searching controller parameters, the closed-loop

system stability, and inputs/outputs constraints can be easily

handled in the MPC design. After a few iterations, a convergent

solution of a (sub-)optimal controller can be obtained, which

minimizes the user-deﬁned closed-loop performance index. The

proposed method is simulated and validated by a high-ﬁdelity

simulation of a cable-actuated soft robot. The simulation results

demonstrate that the proposed approach can achieve desired

tracking controller for the soft robot without a prior-known

model.

I. INTRODUCTION

Soft robots are made of continuously deformable materials

or structures to mimic the biological continuum motions. The

distributed softness or continuum brings unique compliance

and ﬂexibility compared to the conventional rigid robots.

Therefore, the soft robots are showing advantages in the

applications of medical devices [1], human-robot interac-

tions [2]. However, the softness leads to complex dynamics,

which challenges the development of appropriate control

algorithms to exploit its advantages.

The soft robot is ideally an inﬁnite-dimensional system.

The continuum structure theory [3] can accurately describe

the dynamics using PDE. However, these models are inﬁnite-

dimensional, so they are hard to be used for controller design.

In the model-based control approach, the key challenge is

to develop a low-dimensional model but accurate enough to

achieve good control performance. Many modeling methods

of ﬁnite-dimensional approximations are reported, including

Piecewise Constant Strain (PCC) approach [4], Variable

Strain (VS) approach [5], Finite Element Model (FEM) [6]

and its following model-reduction methods [7] .

Anuj Pal and Wenpeng Wei are with the Department of Mechanical

Engineering, Michigan State University, East Lansing, Michigan, 48824,

USA. Emails: palanuj@msu.edu, weiwenpe@msu.edu.

Tianyi He is with the Department of Mechanical and Aerospace

Engineering, Utah State University, Logan, Utah, 84342, USA. Email:

tianyi.he@usu.edu.

* Corresponding author. This work has been submitted to ACC 2023.

With the appropriate models, different control techniques

can be applied. Model predictive control can effectively ad-

dress the constraints of the states and inputs. This technique

has been used in a pneumatic humanoid robot [8], [9]. To

account for the model uncertainty, robust H∞control is used

to control distributed actuators in a segmented soft robotic

arm [10]. A detailed review of the model-based control of

soft robotics can be found in [11]. It is widely recognized that

the performances of model-based methods heavily rely on the

accuracy of the model. Therefore, the model-based control

approach has another obvious challenge in the trade-off

between accuracy/performance and adaptability/complexity.

Recent advances in the data-driven (or learning-based)

control method provide an alternative approach to the model-

based control approach. Traditional machine learning (ML)

or deep learning (DL) methods can be applied to approximate

the complex model without a prior understanding of the

system dynamics. The input-output data will be used to

establish a black-box model. However, the black-box model

is hard to be implemented in the control design. The closed-

loop system stability is still an unsolved issue. Therefore, its

applications in real-time control are limited. A review paper

on the machine learning of soft robots provides more details

[12].

A promising middle-point method is to integrate data-

driven and model-based control to tackle the control of

complex soft robots. This emerging method is considered

a powerful candidate to address the complexity and non-

linearity. Some works of such methods are reported in the

literature. An iterative learning model predictive control can

improve the model accuracy by gradually updating the model

parameters using the data from repetitive processes [13].

Koopman operator theory offers a data-driven method to

generate a linear model in the lifted high-dimensional space

to approximate the nonlinear model. Linear control methods,

including linear MPC and LQG, can then be easily imple-

mented [14], [15]. However, both the ML/DL and Koopman

operator approaches need a large number of experiments,

collecting data to learn a good model. This drawback leads

to expensive costs in the learning process and hinders the

applications of soft robots.

Bayesian optimization is a promising approach that has

been proven to reduce the computational burden of identi-

fying the optimal parameters for any complex system. The

approach involves the use of a data-driven model and an

actual system to iteratively improve the system knowledge

as per the desired performance function. The approach has

been validated in easing the computational burden for various

arXiv:2210.08780v1 [cs.RO] 17 Oct 2022

problems ranging from parameter calibration and control

design. References [16], [17], [18], [19], [20], [21], [22]

implemented the Bayesian optimization framework in auto-

motive domain for performing the engine calibration. Apart

from automotive applications, the Bayesian optimization

approach has also been successfully implemented in other

applications such as analog/rf circuit design [23], ground-

water reactive transport model [24], actuator modeling [25],

and designing natural-gas liquefaction plant [26]. All these

works have shown the capability of data-driven approaches

for modeling a complex system with a relatively simple

model for computation.

In this paper, we present a sample-efﬁcient data-driven

method to design the MPC of a cable-actuated soft robot

by Bayesian optimization. Due to the nonlinearity and

complexity of the soft robot, the nonlinear mapping from

the model parameters and associated control parameters to

ultimate system performance is hard to model. We treat the

nonlinear mapping as a Gaussian process and use Bayesian

optimization to ﬁnd the best prediction model that matches

with online input-output data. The best-guessed prediction

model is then used to design a linear MPC. The closed-

loop data is collected to evaluate the control performance

and update the prediction model by Bayesian optimization.

An optimal solution of the best-guessed model and MPC will

be obtained after iterating a few experiments.

To the best knowledge of the authors, this is the ﬁrst time

that MPC for the soft robot has been designed using the

Bayesian optimization technique. The main contributions of

this work are three-fold: 1) Proposing a Bayesian optimiza-

tion framework for the soft robot in identifying the optimal

reduced-order system dynamics approximation 2) Integrating

linear MPC with Bayesian optimization to achieve accurate

tracking control of soft robots 3) Demonstrating the track-

ing control performance and computational burdens of the

proposed method.

The rest of this paper is organized as follows. Section II

formulates the problem and shows the overview of the

control scheme. Section III introduces Bayesian optimization

and the algorithm for control design. Section IV then presents

the simulation results in a high-ﬁdelity environment. At last,

conclusions are made, and future work is discussed.

II. PROBLEM FORMULATION

Consider a soft robot actuated by cables, which is a multi-

input-multi-output (MIMO) system. Its nonlinear dynamic is

described by the discrete-time nonlinear system (1)

xt+1=f(xt,ut)

yt=h(xt,ut)(1)

where xt,ut,ytdenote the state, control inputs and outputs at

time index t. The nonlinear function f(·)and h(·)are the

nonlinear dynamic model of the soft robot that is assumed

to be unknown.

The soft robot is actuated by three cables embedded in the

body, and the end point is installed by a laser. The end-point

(𝒙𝟏, 𝒙𝟐)

Fig. 1. Cable-actuated soft robot and positioning of its end-point on 2D

plane.

will point at the x−yplane, and the tracking controller is

expected to track the reference trajectory on the x−yplane.

The pre-deﬁned task is to track a given reference trajectory

rtin the time horizon (t=1,2,...,N), written as [1,N]. The

control law (2)

ut=g(yt,rt)(2)

is expected to enforce yttracks reference rt. The objective

is to ﬁnd a control law that minimizes the weighted tracking

errors and control inputs within the entire time horizon,

as indicated in the problem formulation in (3). Q,Rare

weighting matrices for tracking errors and inputs, and Q≥

0,R>0.

min

{ut}N

t=1

J=min

{ut}N

t=1

∑

t=1

(yt−rt)TQ(yt−rt) + uT

tRut(3a)

subject to: unknown : xt+1=f(xt,ut)(3b)

unknown : yt=h(xt,ut)(3c)

umin ≤ut≤umin (3d)

ymin ≤yt≤ymin (3e)

The optimal solution of the control law is denoted as

u∗

t=g∗(yt,rt). The unknown dynamics f(·)and h(·)make

the mapping from control law g∗impossible to be evaluated

by the unknown models. In the traditional model-based ap-

proach or Koopman operator approach, a tremendous amount

of data needs to be collected by conducting sufﬁciently many

experiments at operating points covering the whole range of

interest. Therefore, solving the optimal control law is very

expensive for the traditional methods that need to establish

the mapping from control inputs to the ultimate system

performance. However, a given control law can be evaluated

by conducting experiments/simulations, collecting the input-

output data, and analyzed in the cost function of (3). In other

words, the control law can be sampled and improved through

the online input-output data. The Bayesian optimization is a

suitable tool to achieve this control objective without precise

dynamic models.

Instead of using Bayesian optimization to directly auto-

tune the controller [27], Bayesian optimization is used in this

paper to improve the approximated prediction model based

on the best ’guess’ in each iteration. The well-studied MPC

will be used to design the controller, and the performance

index will be evaluated on the closed-loop system with MPC.

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Sample-efcientModelPredictiveControlDesignofSoftRoboticsbyBayesianOptimizationAnujPal,TianyiHe,WenpengWeiAbstractThispaperpresentsasample-efcientdata-drivenmethodtodesignmodelpredictivecontrol(MPC)forcable-actuatedsoftroboticsusingBayesianoptimization.Insteadofmodelingthecomplexdynamicsofthesoft...

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Sample-efﬁcient Model Predictive Control Design of Soft Robotics by Bayesian Optimization Anuj Pal Tianyi He Wenpeng Wei

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