1 Safety-based Speed Control of a Wheelchair using Robust Adaptive Model Predictive Control

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Safety-based Speed Control of a Wheelchair using

Robust Adaptive Model Predictive Control

Meng Yuan, Ye Wang, Lei Li, Tianyou Chai, Wei Tech Ang

Abstract—Electric-powered wheelchair plays an important role

in providing accessibility for people with mobility impairment.

Ensuring the safety of wheelchair operation in different ap-

plication scenarios and for diverse users is crucial when the

designing controller for tracking tasks. In this work, we propose

a safety-based speed tracking control algorithm for wheelchair

systems with external disturbances and uncertain parameters at

the dynamic level. The set-membership approach is applied to

estimate the sets of uncertain parameters online and a designed

model predictive control scheme with online model and control

parameter adaptation is presented to guarantee safety-related

constraints during the tracking process. The proposed controller

can drive the wheelchair speed to a desired reference within

safety constraints. For the inadmissible reference that violates

the constraints, the proposed controller can steer the system

to the neighbourhood of the closest admissible reference. The

effectiveness of the proposed control scheme is validated based

on the high-ﬁdelity speed tracking results of two tasks that involve

feasible and infeasible references.

Index Terms—Model predictive control, speed tracking, robotic

wheelchair, safety constraints.

I. INTRODUCTION

WHEELCHAIRS are essential devices in providing mo-

bility for elderly and physically impaired people in-

cluding patients with a spinal cord injury and stroke patients

[1]–[3]. Among different types of wheelchairs, the electric-

powered wheelchair has seen increasing popularity due to its

convenience compared to manual wheelchairs. According to a

survey study in [4], around 80% of electric wheelchair users

rely on the joystick to maneuver the wheelchair. However, the

unmodiﬁed reference generated by the joystick may result in

unexpected high speed and acceleration with safety issues, and

advanced speed control is generally required.

As a system with nonholonomic constraints and a clear

model of the mechatronics system, the tracking control of

wheelchairs has been widely studied by many researchers [5]–

[7]. Depending on the type of model used and the level of

control implemented, the speed control of wheelchairs can

be divided into two categories. The ﬁrst category is related

to the kinematic model of the system where the motion

of interest fails to satisfy Brockett’s necessary conditions

[8]. Given the desired Cartesian position and orientation of

the wheelchair, the control objective is to design the linear

and angular velocities such that the wheelchair tends to the

given position with the required orientation. In [8], a ﬁnite-

time tracking of the wheelchair based on cascaded control

architecture and sliding model control was presented. In [9],

an adaptive tracking controller was designed for a wheelchair

with the input-to-state stability guarantee.

Although some existing algorithms, e.g. the timed elas-

tic band-based method, can minimize the execution time of

trajectory while considering the kinodynamic constraints, the

tracking control at the kinematic level is under the assumption

of perfect velocity tracking at the dynamics level, which may

not be realistic in practical applications [10]. The ignored

disturbances and system uncertainties can cause severe con-

straint violation even if the velocity and acceleration tolerances

are considered at the kinematic level when conducting both

planning and control.

The second class of tracking control for nonholonomic

systems involves the dynamics of actuators. The objective of

this type of control is to design the current or voltage at the

dynamics level to ensure the convergence of the system state

to desired Cartesian position or wheel velocities. In [10], the

trajectory tracking of nonholonomic system is achieved by

force control using cascaded and back-stepping techniques.

Later, the authors extend their controller for motion tracking

of a mobile robot at the voltage level with a simpliﬁcation

based on a linear relation between velocity and voltage [11].

Based on Lyapunov and back-stepping methods, [12] proposed

an adaptive control law for stabilizing and tracking of nonholo-

nomic robots with unknown system parameters.

However, none of these methods discussed above guarantee

the safety-related constraints of wheelchairs in the presence of

disturbances. This is an important and common problem that

can be found in scenarios when the wheelchair is required to

operate within some given speed and acceleration constraints

while the daily-life tasks introduce external disturbances such

as driving the wheelchair on inclined ramps. Moreover, the

advent of rentable or sharing wheelchairs at big-city hospitals

makes it desirable to design a controller that can ensure safety-

related constraints at the dynamics level while considering the

variation of system parameters due to the change of users [13].

Model predictive control (MPC), as an optimal control strat-

egy capable of explicitly handling the operation constraints,

has been widely used in applications with strict demands

on state and input constraints [14], [15]. To ensure robust

constraint satisfaction for systems with parameter variations,

in [16], a robust tube MPC is designed for a linear parameter

varying (LPV) system and the online computation load is

reduced by constructing a terminal set involving the norm

bounds of tube parameters. In [17], a recursive least square-

based adaptive MPC is designed for constrained systems with

unknown model parameters. The tube-based adaptive MPC

provides less conservative performance compared with the

robust tube MPC. For systems with both parameter variations

and external disturbances, robust adaptive MPC with the set-

arXiv:2210.02692v1 [eess.SY] 6 Oct 2022

membership approach is adopted to provide robust constraint

satisfaction with online parameter update [18]–[21]. However,

most of the existing works of robust adaptive MPC were

designed for the regulation problem and few works can be

found for the tracking problem of systems with constraints in

the presence of external disturbances and uncertain parameters.

Inspired by the timely needs of operating wheelchairs

safely in different application scenarios and for diverse users,

the main contribution of this work is to present a safety-

based constrained tracking control algorithm for wheelchair

systems with external disturbances and uncertain parameters

at the dynamics level. Under the assumption of unknown-but-

bounded disturbances, the set-membership approach is adopted

to provide an updated set of uncertain parameters which is

used in the subsequent MPC design. For any infeasible refer-

ence that violates the safety constraints, an optimization-based

method is utilized to compute the closest admissible reference

for tracking. The state and input constraints are robustly

satisﬁed in the proposed MPC framework with reference-

dependent and tube-based constraints in vertex representation

of states. The recursive feasibility and input-to-state stability

of the wheelchair system with the proposed MPC controller

are guaranteed. The effectiveness of the proposed safety-based

control algorithm is validated by two speed tracking tasks on

the high-ﬁdelity model of a practical wheelchair.

The remainder of this paper is organized as follows: In

Section II, the dynamics of the wheelchair system at actuator

level is described and the system in the LPV form with

unknown parameters is discussed. The proposed robust adap-

tive tracking controller with details in parameter estimation,

feasible reference generation, robust constraint satisfaction and

tracking MPC formulation are presented in Section III. The

tracking results of the proposed safety-based control algorithm

are demonstrated in Section IV. Section V concludes this

work.

Notations: When deﬁning the variable, we follow the rule

that capitalized letters are for matrices and small letters are

for vectors or scalars. Rand Zare the sets of real and integer

numbers. Given two integers a,b∈Z,Za+,{i∈Z|i≥a}

and Z[a,b],{i∈Z|a≤i≤b}. The i-th row of matrix

Xand the i-th element of vector xare represented by X[i]

and x[i], respectively. The i-th vertex of x∈ X is denoted

by x(i). A non-negative matrix is denoted by X≥0. The

positive deﬁnite and semi-deﬁnite matrices are represented as

X0and X0, respectively. For a matrix X∈Rn×n,

the smallest eigenvalue is denoted by λ(X). The identity

matrix of dimension nis denoted by Inand an m-dimension

vector with all elements as 1 is denoted by 1m. The m×n

matrix with all elements as zero is denoted by 0m,n. A

diagonal matrix with main diagonal elements a1, . . . , anis

denoted by diag(a1, . . . , an). The following sets are deﬁned:

Sn={X∈Rn×n:X=X>},Sn

0={X∈Sn:X0}

and Sn

0={X∈Sn:X0}. A convex polyhedral set of x

is deﬁned as Px(Fx, bx) = {x|Fxx≤bx}. With P∈Sn

0, an

ellipsoidal set of xis deﬁned as E(P, 1) = {x|x>P x ≤1}.

For a vector x∈Rnwith a matrix Q,kxkdenotes the 2-norm

and kxkQstands for px>Qx. The vector xi|krepresents the

predicted value of xat a sampling time instant k+ibased on

measurement at k. The vector x(k)stands for the measured

value of xat a sampling time instant k. A continuous function

f: [0, a)→[0,∞)belongs to class K∞if a=∞and

f(r)→ ∞ as r→ ∞.

II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION

A. Wheelchair Dynamics

With implicit variable change for rotary to translational

movement, the electrical and mechanical subsystems of elec-

tric powered wheelchair are given as follows [12]:

L˙

ic+Ric+Kev=u, (1a)

M˙v+Dv +wf=Ktic,(1b)

where v= [v1, v2]>is the vector of the linear velocities, u=

[u1, u2]>is the vector of the motor voltages, ic= [ic1, ic2]>

is the vector of the currents and wf= [w1, w2]>is the vector

of the lumped disturbance torques on the right and left wheels,

respectively. Mand Dare the equivalent mass and damping

coefﬁcient matrices, which can be expressed as

M=m11 m12

m21 m22 , D = diag(d1, d2),

where m11,m12,m21 and m22 are scalars. The non-zero

matrix Mcouples the dynamics of left and right wheels,

and d1and d2are the damping coefﬁcients for the right

and left wheels, respectively. Furthermore, L= diag(l1, l2)

and R= diag(r1, r2)are the inductance and resistance

matrices of system, respectively. Ke= diag(ke1, ke2)and

Kt= diag(kt1, kt2)are the back electromotive force constant

and torque constant matrices, respectively.

For system without current feedback ic, the dynamics in

(1a) and (1b) can be reformulated as

¨v+ (M−1D+L−1R) ˙v+ Γv=M−1L−1Ktu+w, (2)

where w=−M−1˙wf−M−1L−1Rwfand Γ =

M−1L−1RD +M−1L−1KtKeare the lumped terms.

Let x,[v>,˙v>]>be the state vector. By using the Euler

forward approximation, the discrete-time state-space model

with a sampling time interval Tscan be formulated as

x(k+ 1) = Ax(k) + Bu(k) + Ew(k)

=I2TsI2

−TsΓI2−Ts(M−1D+L−1R)x(k)

+02,2

TsM−1KtL−1u(k) + 02,2

TsI2w(k).

(3)

All the states x(k)and inputs u(k)are required to satisfy the

following constraints to guarantee the safety of the wheelchair:

Gx(k) + Hu(k)≤b, (4)

with given matrices G∈Rnc×4,H∈Rnc×2and b∈Rnc.

Remark 1. The safety constraints in (4) are deﬁned in a

general form, where bis not necessarily a vector with all

elements of 1. This form offers more degrees of freedom

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1Safety-basedSpeedControlofaWheelchairusingRobustAdaptiveModelPredictiveControlMengYuan,YeWang,LeiLi,TianyouChai,WeiTechAngAbstractElectric-poweredwheelchairplaysanimportantroleinprovidingaccessibilityforpeoplewithmobilityimpairment.Ensuringthesafetyofwheelchairoperationindifferentap-plicationscena...

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1 Safety-based Speed Control of a Wheelchair using Robust Adaptive Model Predictive Control

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