Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 6, No. 2, 2023 

 

137 

Tracking Control of Wheeled Mobile Robot Based on 
MMPC 
Jiabao Xu1, 2, *, Guoliang Zhang1, 2 

1 School of Automation and Information Engineering, Sichuan University of Science & Engineering, Yibin 644000, China 
2 Artificial Intelligence Key Laboratory of Sichuan Province, Yibin 644000, China 
* Corresponding author: Jiabao Xu (Email: 1248217710@qq.com) 

 
Abstract: In order to solve the problems of low tracking accuracy and large steady-state error in track tracking of wheeled 
mobile robots, a MPC control scheme combining feedforward and feedback control is designed on the basis of Traditional Model 
Predictive Control (TMPC). First, when establishing the WMR kinematic error model, the linearization error is retained, and the 
control input and incremental constraints are considered. Secondly, a feedforward controller is designed for a given desired 
trajectory of wheeled mobile robot; At the same time, the multi constraint model predictive control (MMPC) strategy with speed 
optimization is added to design the feedback controller. Then, the stability of the designed MMPC controller is analyzed based 
on Lyapunov theory. Finally, MMPC control scheme is compared with TMPC control scheme and PID control scheme. The 
results show that compared with TMPC control scheme, MMPC control scheme reduces the lateral error by 17.87%, the 
longitudinal error by 17.12%, and the heading angle error by 9.81%; Compared with the PID control scheme, the lateral error is 
reduced by 20.82%, the longitudinal error is reduced by 23.26%, and the heading angle error is reduced by 5.87%. 

Keywords: Multi constraint model predictive control, Wheeled mobile robot, Track tracking, Stability. 
 

1. Introduction 
In recent years, the wide application of wheeled mobile 

robots in various fields has attracted great attention of 
scholars [1-2]. The WMR control tasks are mainly divided 
into path planning [3] and trajectory tracking. In order to 
realize WMR trajectory tracking, many scholars have 
proposed different control methods. Literature [4] designed a 
synovial film controller based on the WMR kinematics model, 
even if the WMR has a large initial error in the control, it can 
get better control. Literature [5] considered WMR parameters 
and non-parameters two kinds of uncertainty, combined with 
adaptive control and neural network methods, to solve the 
mobile robot center of gravity offset, model uncertainty and 
other factors. Literature [6] studied the trajectory tracking 
control of WMR on uneven ground, and designed a double 
closed-loop strategy; the external disturbance was estimated 
by the inner loop, and the expected speed was output by the 
outer loop, which realized the smooth turning of WMR. 
Literature [7] uses the enhanced variable structure based on 
the synovium to track the trajectory of WMR, which improves 
the robot's control ability, reduces the attitude error, and the 
linear velocity and angular velocity are controllable. 

Since WMR is a non-integrity constrained system, model 
predictive control has a feed-forward-feedback structure, 
which can handle nonlinear input-output systems with 
constraints [8-9]. Literature [10] proposes an MPC control 
scheme for nonholonomic constrained WMR, and solves 
linear MPC by using quadratic programming for continuous 
linearization of the WMR error model. Literature [11] 
developed a robust predictive control based on neural network 
optimization to stabilize WMR. Experiments under various 
dynamic conditions proved that the proposed scheme can 
improve the computational efficiency of MPC. Literature [12] 
realized the local path tracking of the robot based on the 
control scheme combining PID control and MPC. 

Based on the above analysis, when the traditional model 

predictive control scheme is applied to WMR trajectory 
tracking, there will still be problems of low tracking accuracy 
and large steady-state error. On the one hand, these TMPC-
based control schemes do not consider the error caused by the 
control increment limit and linearization, so the controller is 
designed through the control increment of WMR, so as to 
improve the WMR trajectory tracking accuracy. On the other 
hand, in WMR, the control quantity also has value constraints 
in the time domain, such as speed constraints, etc. These 
constraints are unavoidable, so designing a multi-constraint 
MPC controller is a problem worth studying. 

Considering the above problems, this paper proposes a 
multi-constraint model predictive control scheme with a 
combination of feedforward and feedback control on the basis 
of traditional model predictive control. First, the kinematic 
error model of WMR is linearized and high-order items are 
retained, while considering the WMR control input and its 
incremental constraints; second, the feedforward controller is 
designed by calculating the reference control input through 
the expected trajectory; then, in the feedback The speed 
optimization strategy is added to the control, the control 
incremental state space equation is used as the prediction 
model, the objective function is designed under multiple 
constraints, and the constraint optimization problem in the 
finite time domain is defined to obtain the MMPC trajectory 
tracking controller; then, the designed MMPC control Finally, 
the simulation and physical experiments verify that the 
MMPC control scheme has better performance than the 
TMPC control scheme and the PID control scheme in 
trajectory tracking. 

2. Problem Description 
WMR is a typical underactuated system with nonintegrity 

constraints. The kinematics model of the wheeled mobile 
robot is shown in Figure 1. 



 

138 

 
Figure 1. WMR Kinematic Model 

 
In Figure 1, there are two Cartesian coordinate systems,

{ , , }X O Y  is the global coordinate system fixed on the ground and

{ , , }x o y  is the local coordinate system on the wheeled mobile 
robot.  is the angle between the forward direction and X , and 

,v  are the translational and rotational velocities of the WMR 
respectively. It is assumed that the wheel is in a pure rolling motion 
without sliding in the plane. At the instant of motion, the velocity of 
the WMR is directed towards the main axis of the robot. The slave 
wheel acts as a support during the motion and its influence can be 
neglected . From the non-completeness constraint of the

( )sin ( ) ( )cos ( ) 0x t t y t t( )sin ( ) ( )cos ( ) 0x t t y t t( )sin ( ) ( )cos ()sin ( ) ( )cos (( ) ( ) (( ) ( )( ) ( )cos) os )(( )cos , the kinematic equation can be 
obtained as follows 

 
cos ( ) 0

( ) ( ( ), ( )) ( )sin ( ) 0
0 1

t
t f t u t u tt( ) () ()( f) () (  (1) 

 
In equation (1), ( ) [ ( ) ( ) ( )]Tt x t y t t   represents the 

position and orientation of the WMR in the global coordinate 
system { , , }X O Y   and ( ) ( ) ( ) Tt v t tu   represents the 
control input vector where the linear velocity is ( )v t  velocity 
is ( )t  

A state tracking error vector ( ) [ ( ) ( ) ( )]Te e e et x t y t t  
is defined in the local coordinate system { , , }x o y which is 
expanded in a Taylor series at the reference trajectory point 
( ( ), ( ))r rt tu to obtain a wheeled mobile robot error model, 
which is represented as follows

 
0 0 sin ( ) ( ) ( )

( ) 0 0 cos ( ) ( ) ( )
0 0 1 ( ) ( )

cos ( ) 0
( ) ( )

sin ( ) 0 ( )
( ) ( )

0 1

r r r

e r r r

r

r
r

r
r

v t x t x t
t v t y t y t

t t

t
v t v t

t t
t t

( )e ( )) 000000

d

      (2) 

 
In equation (2), ( )td  is the higher order term of the Taylor 

expansion of ( )e t( )e Considering the operational safety of the 
wheeled mobile robot, both need to keep v  and within a 
certain range, with the following constraints on the controller: 

 

 

min max

min max

min max

min max

v v v

v v v
 (3) 

The maximum linear and angular velocities are maxv  and

max  , respectively, and the minimum linear and angular 
velocities are minv  and min respectively. For each control 
cycle, there are constraints on the linear velocity increment 
v and the angular velocity increment  
The trajectory tracking problem studied in this paper is 

described as follows: design the controller to output suitable
v  and such that the tracking error of the wheeled mobile 
robot converges asymptotically, which can be expressed as 
the asymptotic convergence of the modes of [ , , ]Te e ex y to 

zero [14] i.e. lim [ , , ] 0T
t e e ex y

3. Control Scheme Design 
3.1. General design of the control scheme 

The overall design of the control scheme is shown in Figure 
2: 

 
Figure 2. System control block diagram 

 
The control input acting on the WMR system is ( )tu

and is expressed as 
 

 ( ) ( ) ( )rt t tu u u( )(  (4) 

 
( )r tu  and ( )tu( )u  are the feed-forward control output and 

the feedback control output at t  respectively. 
The feed-forward controller uses the desired trajectory at 

the current moment to calculate the reference control input. 
The feedback controller first optimizes the reference speed 
from the previous moment and then predicts the control input 
for the current moment via the MMPC controller. Finally the 
system applies the results of the two controllers together to 
the WMR, thus completing the trajectory tracking control. 

3.2. Feedforward controller design 
For a given reference trajectory ( ( ), ( ))r rx t y t  , the 

feedforward control input is ( ) [ ( ), ( )]Tr r rt v t tu When 
the wheeled mobile robot is tracking the reference trajectory, 
the linear velocity ( )rv t  is calculated as follows

 
2 2( ( ) ( 1)) ( ( ) ( 1))

( ) r r r r
r

x t x t y t y t
v t

t
 (5) 

 
The linear velocity is positive when the robot is moving in 

the positive direction of the trajectory and negative vice versa. 
The heading angle at each moment on the reference trajectory 
is  



 

139 

( ) ( 1)( ) arctan
( ) ( 1)
r r

r
r r

y t y tt
x t x t

 (6) 

 
In equation (6),   is the reference direction, 0  

represents the positive direction along the reference trajectory 
and 1   represents the reverse direction along the 
reference. The angular velocity is calculated as follows: 

 
( ) ( 1)( ) r r

r
t tt

t
 (7) 

 
The feedforward control law ( )r tu   can be obtained by 

deriving the control input ( ), ( )r rv t t

3.3. Speed optimization strategy design 
Variations in speed during the operation of a wheeled robot 

affect the lateral error. In order to make the error between the 
actual trajectory and the reference trajectory during the 
trajectory tracking of the WMR smaller, specifically in the 
case of possible large deviations at turns, the speed is 
optimally adjusted. 

The velocity optimization strategy consists of two main 
aspects, on the one hand, the system calculates the curvature 
of the WMR trajectory at the next moment based on the 
current reference trajectory, as shown in Figure 3. On the 
other hand, is the design of a function between the reference 
velocity and the curvature of the trajectory. 

 

M

N

1
2

3

4

5

I
A AA( )x , y B BB( )x , y

C

O

 
Figure 3. Curve Curvature Diagram 

In Figure 3, O is the center point of the curve. Using the 
point M as an example, the formula for calculating its angle

2  is as follows: 

1 M A B M M A B M
2 2 2 2 2

M A M A B M B M

( )( ) ( )( )cos
( ) ( ) ( ) ( )

x x x x y y y y

x x y y x x y y
 (8) 

 
If the calculated angle 2  is greater than the set standard 

angle 0 , then point A is determined to be the starting point 
of the curve, and a new path point M is selected as the starting 
point and again determined according to equation (8) until the 
calculated angle appears to be less than the standard angle. 
The radius of curvature of the curve is solved by calculating 
the coordinates of the start and end points of the curve cR

 

2 2
M O M O( ) ( )cR x x y y  (9) 

 
The chord length of the curve cI  is
 

2 2
M N M N( ) ( )cI x x y y  (10) 

 
Corresponding central angle of the curve c

 

12sin
2
c

c
c

R
I

 (11) 

 
The final calculated curvature of the curve K is: 
 

c

c
K

L
 (12) 

 
In equation (12), cL   is the length of the curved arc, and 

considering the WMR in a two-dimensional plane, the speed 
of the mobile robot needs to be limited. Depending on the 
curvature of the trajectory the WMR runs on, its speed ( )v t( )v(  
is set to 

( )
( ) rv t
v t

K
(

( ) rv (rv(
K

 (13) 

3.4. MMPC controller design 
For the feedback controller, the multi-constraint model 

predictive control strategy is designed by taking the sampling 
time T , approximating the discretization of equation (2) 
using the first-order difference quotient method, and 
obtaining the expressions for the rewritten discrete state space 
and output ( )y k  as follows: 

 
( 1) ( ) ( ) ( )

( ) ( )
e e

e

k A k B k k
k C k

u d
y

( ) (( ) () (( )( )( )( )( )
 (14) 

 
In equation (14) ( )e k  is a state variable, ( )ku( )u  is a 

control input variable, ( )ky  is a control output variable,
( )kd  is a measurable error variable, and 
 

1 0 sin 1 0 0cos 0
0 1 cos , , 0 1 0sin 0
0 0 1 0 0 10

r r r

r r r

v T T
v T T

T
A B C     (15) 

 
It is assumed that the system state variables are fully 

measurable and that the system output is fully controllable. 
To eliminate static errors, the following incremental variables 
are defined for equation (8): 

 
( ) ( ) ( 1)
( ) ( ) ( 1)
( ) ( ) ( 1)

e e ek k k
k k k
k k k

u u u
d d d

( ) ( ) ( 1)) ( ) () ( ) (( ) ( ) () ( ) (u( ) ( )( )) ( )) (( )( ) ( )) (( )( )  (16) 



 

140 

Where e  is the state increment and uu is the control 
increment. Bringing equation (16) into equation (14), the 
incremental model is obtained as 

 
( 1) ( ) ( ) ( 1) ( ) ( )

( ) ( )
e e ek k k k k

k k
A I A B u d

y C
( ) (( ) (( )( )( )( )( )

 (17) 

 
In equation (17), I   is the unit matrix. To design the 

MMPC controller, the future dynamics of the system is first 
predicted based on model (11) using the latest value of 

( )e k  as the initial condition. Set the prediction time domain 
to p  , the control time domain to m and m p  . To 
calculate the prediction equations for the MMPC, assume that 
the control quantities in the control time domain are constant. 
For the current moment k define the p   step predicted 
control output quantities as follows: 

 

1( 1| ) [ ( 1| ) ( | )]Tp pk k k k k p kY y y((((  (18) 

 
( )kU  is a sequence of control quantity increments as 

independent variables in a constrained optimization problem 
defined as 

 

1( ) [ ( ) ( 1)]Tmk k k mU u u( ) ( 1)]Tm( ) () (( ) () (( )( )( ) (((  (19) 

 
The number of vectors in the matrix is indicated under the 

matrix in Eqs. (18) and (19) and is not necessarily the 
dimension of the matrix. The output predicted for the next p  
steps of the system can be calculated from the following 
prediction equation: 

 

1 2( 1| ) ( ) ( 1) ( ) ( )p x x u dk k x k x k k kY S S S U S d  (20) 

 
in equation (20), 

 

2 2 2
1

0 1 1
1 2

1

0 1 1

( )

, ,

c c c

i i i
c c c
i i i

x x d

p p p
i i i

c c c
i i i

A

C A I C A C

C A C A C
S S S

C A C A C A

ddx dx2 ,2 d22 ,22 ,2
 

1 2

1 1

1 2

1 1 1

0 0
( ) 0

( ) ( )

( ) ( ) ( )

c

c c

m m
i i

c c cu
i i

p p p m
i i i

c c c
i i i

C B
C A I B C B

C A B C A B C BS

C A B C A B C A B
p

((c (((((

 

Further, the magnitude of the control movements of our 
wheeled mobile robot must not be too large, for defining the 
following objective function as 

2 2( ( 1| )) ( 1) ( )y p uk k k kJ Y R U  (21) 

 
In equation (21), y and u  are the weighting matrices at 

the current prediction moment and the reference trajectory 
sequence input is 

 

1( 1) ( 1) ( 2) ( ) Tpk k k k pR r r r((  (22) 

 
Due to the presence of constraints, the MMPC optimization 

problem is solved by numerical methods. Since the objective 
function is quadratic, the constrained optimization problem is 
converted to a quadratic programming problem, defined 

 

1 2( 1) ( 1) ( ) ( 1) ( )p x x dk k x k k kE R S S S d  (23) 

 
Bringing the prediction equation (20) into the objective 

function (21), the objective function becomes 
 

2 2( ( )) ( 1) ( )

( ) ( ) ( ) ( )

2 ( 1) ( ) ( 1) ( 1)

k y u u

T T T T T
u y y u u u

T T T T
y y y y

k k k

k k k k

k k k k

J S U E U

U S S U U U

E U E E
 (24) 

 
In equation (24), ( 1) ( 1)T T

y yk kE E  is not related to 
the control increment sequence ( )kU  , so the objective 
function is transformed into the following form 

 

min max

min max

min ( ) ( ) ( 1) ( )

. .
( ) ( ) ( ) ( )

T T
kU

I

k k k k

s t
k k k k

J U H U G U

U U U
U A U U U

 (25) 

 
Where T T T

u y y u u uH S S ( 1) 2 ( 1)T T
u y y pk kG S E

Since 0H , the quadratic programming problem is 
solvable for any weighting matrices y   u  . The 
control input increment for solving is denoted as 

 
* * *

1( ) [ ( ) ( 1)]Tmk k k mU u u ((*(*(*  (26) 
 
So * ( )kU  is a non-linear function of the state quantity
( )e k  in the control time domain and in the prediction time 

domain. The first element of the resulting control sequence is 
generally applied to the wheeled mobile robot system. That is 
 

*( ) ( 1) ( )k k ku u u( ) ( )( ) ( 1)) ( 1)) (( 1)u( ) ( 1)) ( 1)) (( 1)( 1)  (27) 
 
At the next moment, after entering the control cycle, the 

new state quantities are brought into the multi-constraint 
optimization problem, i.e. equation (19), and solved again. 
The closed-loop control law of the MMPC is therefore 
defined as

 
*( ) [ 0 0] ( )n nk I ku U]0]0]0]0]  (28) 



 

141 

3.5. MMPC controller stability analysis 
For the purpose of discussing the nominal stability of the 

MMPC closed-loop system, it is assumed that ( ) 0kR the 
system states are all measurable, i.e. only the case of state 
feedback is considered. 

In MMPC, the time k  converges to  after each moment 
of optimization, but when analysis the stability of the MMPC 
closed-loop system, the value solved for the optimization 
problem in each state does not guarantee the stability of the 
closed-loop system. 

From the Lyapunov definition of stability [15], the stability 
of the system is a concept that tends to infinity in time, so the 
stability of the MMPC is assumed to be in the infinite 
prediction time domain and infinite control time domain, i.e. 
the stability of the MMPC is discussed for the case of
p m  

Theorem 1 Consider the system model (17) if there exists 
a positive definite bounded objective function kJ  satisfying: 

(1) 0 0J  and for any 0k  exists 0kJ  
(2) the Near the equilibrium point (0) 0 kJ   is 

continuous, 
(3) )Along the trajectory of the system, there are 
 

1 || ( ) ||k k kJ J  (29) 

 
Where   is a   function the system is progressively 

stable. 
Proof: Suppose that the infinite time domain constrained 

optimization problem at k   has a solution whose value 
satisfies the control increment constraint and the control 
quantity constraint such that the optimized value 

 
12 2

0 0

( | ) ( )
p m

k p RQ
i i

k i k k iJ Y U  (30) 

 
is bounded, in equation (26), Q  is a weighting on the control 
output error and R  is a weighting on the control increment, 
compared to the objective function equation (21) with

,T T
y y u uQ R  

The sequence of control increments at 1k  is ( 1)kU , 
which is the vector corresponding to the infinite time domain 
interval [ 1, ]k  starting at 1k  for the optimal solution 
at k and the objective function value is calculated as : 

2 2
1

0 0
22

( 1 | 1) ( 1 )

|| ( ) || ( )

k p RQ
i i

k Q R

k i k k i

k k

J Y U

J C U

 (31) 

 
Similarly 1kJ is bounded. The control sequence at 1k is 

chosen to be a feasible solution to the infinite time domain 
constrained optimization problem. It is also deduced that

 
2

1 || ( ) ||k k QkJ J C  (32) 

When is (0) 0 ( ) 0ku then kU  is a feasible 

solution to the optimization problem, corresponding to 0J . 
From equation (28), we know that for any 0k we have 

0kJ Equation (29) shows that kJ  is monotonically 
decreasing and has a minimum at (0) 0 and is continuous, 
which proves Theorem 1. 

In summary, the objective function of the MMPC kJ is a 
Lyapunov function of the WMR system and the WMR system 
model (11) is nominally asymptotically stable. 

4. Experimental Analysis of The 
System 

The WMR was simulated by MATLAB and the MMPC 
controller module with feedforward control was built in 
Simulink for simulation experiments. The hardware processor 
for the simulation experiments is an AMD Ryzen7 4800U and 
the software platform is Matlab 2018a. 

Simulation experiments are carried out under circular 
trajectory tracking and compared using TMPC and positional 
PID algorithms. The error maxima and variance values of 
each index are used as the judging criteria for trajectory 
tracking, and the trajectory tracking effects of the three 
control schemes, MMPC, TMPC and PID, are compared. 
Taking the lateral displacement error as an example, the 
positional PID controller is shown in equation (33): 

1

( ) ( ( )) ( ) ( ( ) ( 1))
k

p i d
j

k K e k K e j K e k e k  (33) 

 
Where *( ) ( ) ( )e k x k x k  is the transverse error

p iK K   dK  are the proportional, integral and 
differential parameters respectively. In order to make the 
system control effect optimal, it is necessary to optimize the 
multi-constraint model prediction control parameters, after 
repeated debugging, a set of better parameters of the 
positional PID control scheme 0.9pK 0.01IK

0.45dK  , and also obtain the system simulation 
parameters of MMPC as shown in Table 1: 

 
Table 1. Parameter Comparison Data 

Parameters MMPC TMPC PID 
Control time domain m  60 60 - 
Predicted time domain 

p  30 30 - 

Status quantity  3 3 3 
Control volume u  2 2 2 
Relaxation factor 

weights  10 10 - 

Error weights y  10 (3 )peye N  10 (3 )peye N  - 
Control of incremental 

weights u  (2 )ceye N  (2 )ceye N  - 

4.1. Simulation of circular trajectory tracking 
A circular trajectory is given as follows: 
 

( ) 1.25sin(0.2 ), ( ) 1.75 1.25cos(0.2 )r rx t t y t t  (34) 

 
In equation (34), [0,50]t The constraints for the linear 



 

142 

velocity v   and angular velocity   are | | 1 /v m s   and
| | 0.2 /rad s MMPC respectively, and the incremental 
constraints for the MMPC control scheme are
| | 0.02 /v m s   and | | 0.02 /rad s . During the 
simulation, the initial starting point of the desired trajectory 

of the WMR is ( , , ) (0,0.5,0)r r rx y  and the initial error 
between the actual trajectory starting point and the desired 
trajectory starting point is set as the initial error, the initial 
starting point of the actual trajectory is set as 
( , , ) ( 0.5,0.5,0)x y  , and the running time is T 50s
the simulation results are shown in Figure 4. 

 

 
Figure 4. Simulation Diagram of Circular Trajectory Tracking 

 
It can be seen in Figure 4 that the MMPC control scheme, 

the TMPC control scheme and the PID control scheme are all 
able to achieve trajectory tracking. It can also be seen from 
Figure 4 that the MMPC has the best tracking effect with the 

presence of initial position error, the TMPC has the second 
best effect and the PID has a relatively large tracking error. 
The variation of tracking error for the three control schemes 
is shown in Figure 5(a-c). 

 

 
(a) Lateral displacement trajectory tracking error 

 
(b) Longitudinal displacement trajectory tracking error 

 
(c) Heading angle trajectory tracking error 

Figure 5. Simulation Diagram of Circular Trajectory Tracking Error 
 



 

143 

 
Figures 5(a-c) show the tracking error simulation, where 

WMR converges to zero faster than TMPC and PID under the 
MMPC control scheme. The regulation time to track to the 
desired trajectory is 20t s in the MMPC and 26t s  in 
the TMPC control scheme, while WMR has the longest 
regulation time using the PID control scheme at 31t s   . 
This is due to the MMPC control scheme with feed-forward 
control, which responds faster to the steering process when 
there is an initial error, improving the trajectory tracking 
accuracy of the WMR. After 31s , the WMR maintains good 
tracking performance for all three control schemes. 

 
Table 2. Comparison of circular track simulation  

Performance indicators MMPC TMPC PID 

Lateral error 

Maximum 
values 0.065 0.13 0.167 

Variance 
values 

0.15 0.16 0.163 

Longitudinal 
error 

Maximum 
values 

0.22 0.271 0.29 

Variance 
values 

0.07 0.089 0.102 

Heading angle 
error 

Maximum 
values 

0.29 0.33 0.349 

Variance 
values 

0.097 0.106 0.111 

 
Table 2 shows comparative performance data for the 

circular trajectory simulation tracking. It can be seen that 
MMPC has the smallest of all three performance metrics 
compared to the three control schemes. In terms of lateral 
error, the maximum error value for MMPC is 0.065m, while 
the maximum error for TMPC and PID is 0.13m and 0.167m 
respectively. 

The calculation shows that the MMPC control scheme 
reduces lateral error by 17.96%, longitudinal error by 6.53% 
and heading angle error by 4.13% compared to the TMPC 
scheme. Compared to the PID control scheme, the lateral 
error is reduced by 28.18%, the longitudinal error by 8.94% 
and the heading angle error by 6.09%. This indicates that the 
MMPC control scheme is superior to the TMPC and PID 
control schemes. 

 

 
Figure 6. Control Incremental Change of MMPC 

 
Figure 6 shows the MMPC control increment variation and 

it can be seen that both the linear velocity increment v and 
the angular velocity increment   satisfy the input 
constraints. With the velocity optimisation strategy, the 
velocity of the WMR stably stays within the constraint. The 
simulation results verify the effectiveness of the velocity 
optimisation strategy. 

4.2. Circular trajectory tracking experiment 
 

 
Figure 7. EAIBOT Mobile Platform 

 
Figure 7 shows the EAIBOT mobile platform. In order to 

verify the robustness and feasibility of the control scheme 
designed in this paper in a real environment, physical 
experiments are carried out using this mobile platform. To 
record the trajectory tracking data, a model was built using 
Simulink in conjunction with ROS as shown in Figure 8. 

 

 
Figure 8. Simulink and ROS Joint Simulation 

 
The MPC control scheme and the PID control scheme are 

still chosen for comparison. The algorithm parameters are the 
same as in Table 1. 

The constraints for the linear velocity v   and   the 
angular velocity are | | 1 /v m s   and | | 0.2 /rad s  
respectively. The control increment constraints are 
| | 0.1 /v m s   and | | 0.1 /rad s  .Without affecting 
the experimental results, the initial starting point of the 
desired trajectory of the WMR was set to
( , , ) (0.85,0.8,0)r r rx y  to track a circular trajectory and 
compare the performance of the three control schemes. 

 

 
Figure 9. Effect Picture of Circular Trajectory Tracking 

 
Figure 9 shows the effect of circular trajectory tracking. 

Under the physical experiment, the MMPC control scheme 
can complete the trajectory tracking experiment better and the 
tracking performance of the system is good, while the MPC 



 

144 

control scheme and the PID control scheme both show different degrees of deviation during the tracking trajectory. 
 

 
(a) Lateral displacement trajectory tracking error 

 
(b) Longitudinal displacement trajectory tracking error 

 
(c) Heading angle trajectory tracking error 

Figure 10. Rendering of Circular Track Tracking Error 
 

Figure 10(a-c) shows the circular trajectory tracking error 
diagram. During the trajectory tracking process, its tracking 
effect has a certain bias, taking into account two main factors: 

(1) Factors of the experimental site, such as the coefficient 
of friction and flatness of the ground causing the robot tyres 
to slip sideways or create a gap with the ground. 

(2) Insufficient accuracy of the EAIBOT moving platform 
sensors, resulting in deviations in the calculation of both 

linear and angular velocities at the previous moment. 
As a result of these two main factors, the tracking error of 

the robot does not tend to be zero, but the overall trend is as 
expected, i.e. the results are similar to those of the simulation. 
It can be seen that the MMPC control scheme has improved 
tracking accuracy and reduced steady-state error compared to 
the TMPC and PID control schemes

Table 2. Comparison of circular track performance 
Performance indicators MMPC TMPC PID 

Lateral error 
Maximum 

values 
0.025 0.047 0.051 

Variance values 0.013 0.027 0.033 

Longitudinal error 
Maximum 

values 
0.026 0.051 0.069 

Variance values 0.014 0.02 0.033 

Heading angle error 
Maximum 

values 
0.043 0.058 0.052 

Variance values 0.024 0.027 0.029 
 

Table 3 shows the comparative data of the circular 
trajectory tracking performance in the physical experimental 
environment. In terms of longitudinal error, the MMPC has a 
maximum error value of 0.026 m, while the TMPC and PID 

have a maximum error of 0.051 m and 0.069 m respectively. 
The calculation shows that the MMPC control scheme 
reduces the lateral error by 17.87%, the longitudinal error by 
17.12% and the heading angle error by 9.81% compared to 



 

145 

the TMPC scheme. Compared to the PID control scheme, the 
lateral error is reduced by 20.82%, the longitudinal 9 error is 
reduced by 23.26% and the heading angle error is reduced by 
5.87%. 

Figure 11 shows the variation of the MMPC control 
increments. Even in the real scenario, with the velocity 
optimization strategy, both the linear velocity increments v  
and the angular velocity increments   satisfy the input 
constraints. In summary, both the simulation and physical 
experiments validate the effectiveness and robustness of the 
MMPC control scheme. 

 

 
Figure 11. Control Incremental Change of MMPC 

 

5. Conclusion  
To address the problems of low WMR tracking accuracy 

and large steady-state error, this paper designs a MMPC 
control scheme combining feedforward control and feedback 
control. Firstly, the WMR kinematic model is established 
considering the control increment constraint and the error 
caused by the retained linearization. Then, the MMPC 
feedback controller with feedforward controller is designed 
with the addition of MMPC with speed optimization strategy, 
and the stability of the MMPC controller is demonstrated 
using Lyapunov theory. Finally, the feasibility of the control 
scheme designed in this paper is verified through simulation 
experiments and physical experiments. The results show that 
the MMPC control scheme based on the combination of 
feedforward control and feedback control is better than the 
TMPC control scheme and the PID control scheme in terms 
of tracking accuracy and tracking error. In future research, we 
will consider improving the real-time performance of the 
algorithm, using the WMR dynamics model, etc. On this basis, 
we will further improve the control scheme proposed in this 
paper in order to improve the trajectory tracking control 
performance. 

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