Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 2, No. 2, 2022 

 

28 

Trajectory Tracking of Mobile Robot Based on 
Improved Hierarchical Sliding Mode Control 

Rong Wang*, Yongchun Liu 

School of Sichuan University of Science and Engineer a. College of Automation and Information Engineering, Sichuan, China. 

* Corresponding author: Rong Wang 

 

Abstract: A control algorithm based on PD control and non-singular terminal hierarchical sliding mode control is proposed 

for tracking control of mobile robot systems. First establish a dynamic model of mobile robot under cartesian coordinate system, 

the system can be divided into subsystems of posture and position, attitude subsystem by using PD control, location subsystem 

using non-singular terminal hierarchical sliding mode control, the introduction of the system state higher order term, improve 

the system in the convergence speed and stability of the sliding surface, the system reaches a steady state in a limited time; 

Finally, the stability of the system is proved theoretically, and the effectiveness of the control method is verified by simulation. 

Keywords: Mobile Robot; Hierarchical Sliding Mode Control; Terminal Sliding Mode Control; PD Control. 

 

1. Introduction 

In underactuated systems, the number of control variables 

is less than the number of degrees of freedom, so that the 

partial degrees of freedom of the system cannot be directly 

controlled; common underactuated systems include 

manipulators, mobile robots, quadrotor UAVs, etc.; among 

them, the mobile robot is a multi-input and multi-output 

system, and the research is relatively complex. The trajectory 

tracking technology is an important part of the mobile robot 

control field. The error between the trajectory of the mobile 

robot and the reference trajectory is an important indicator of 

the trajectory tracking performance [ 2,3]. 

Sliding mode control method is a simple and effective 

robust control method, and the response is fast. The control 

method designs the error-related sliding surface of the system 

error. When the system control quantity reaches the sliding 

surface, the system tends to be stable, so as to realize the 

trajectory tracking control of the mobile robot. Lee et al. 

proposed to apply sliding mode control to wheeled mobile 

robot, so that the mobile robot can run steadily on the target 

trajectory. However, the above method is only to construct a 

single-layer sliding mode. For multi-input multi-output 

system, it is difficult to control all variables of the whole 

system by constructing a sliding surface. Wang et al. [ 5, 6, 7] 

proposed a robust nonlinear controller based on hierarchical 

sliding mode control for a class of underactuated systems to 

achieve the balance and motion of the underactuated system; 

the stability of the closed-loop system is obtained by using 

Lyapunov stability criterion and Barbalat lemma. Pham et al. 

proposed a new trajectory tracking control algorithm for 

wheeled mobile robots, which combines hierarchical sliding 

mode and backstepping control methods. The tracking control 

guarantees the closed-loop stability and zero tracking error. 

However, the hierarchical sliding mode control in the above 

literature adopts ordinary linear sliding surface, which can 

only guarantee the asymptotic convergence of the system but 

cannot guarantee the convergence time. Man [ 9, 10] et al. 

when designing the sliding surface; the nonlinear sliding 

surface is used to replace the traditional linear sliding surface, 

and the terminal sliding mode control is proposed, which 

makes the convergence time of the system limited. However, 

singular problems will occur when the parameters are not set. 

Then a nonsingular terminal sliding mode control is proposed 

to solve the singular problem of general terminal sliding mode 

control. 

The control objective of this paper is to control the mobile 

robot system to complete the tracking control with the actual 

control input torque 𝜏1, 𝜏2 . The mobile robot can keep 

moving along the ideal trajectory. Linear sliding surface is 

used in the hierarchical sliding mode control method in the 

above literature. Although the convergence problem of the 

system is solved, the sliding surface does not converge in 

finite time, which makes the system chattering larger. In this 

paper, on the basis of Reference, the terminal sliding mode 

control instead of the ordinary sliding mode control can 

effectively improve the time of the system convergence to the 

stable state and the effect of the system to suppress the 

chattering problem that often occurs in the sliding film 

controller. Secondly, the non-singular terminal sliding mode 

control can avoid the singular point problem in the 

hierarchical sliding surface. At the same time, the 

convergence time of the system can be obtained. The fast 

power reaching law can improve the speed of the system 

convergence, effectively eliminate the chattering 

phenomenon and verify the asymptotic stability of the system.  

2. Dynamic modeling of mobile robots 

 
Figure 1. Mobile robots 

According to the structure of the nonholonomic mobile 

robot, point C is the geometric center between two wheels, 

point M is the focus of the robot, and 𝑞 =
[𝑥 𝑦 𝜃]𝑇represents the position coordinates of point M in 

Cartesian coordinate system. d is the distance between two 



 

29 

points, and θ is the direction angle of the robot. Assuming that 

the mobile robot can only move along the direction of the 

vertical drive wheel, it must meet the condition of pure rolling 

without sliding 

�̇� 𝑐𝑜𝑠 𝜃 − �̇� 𝑠𝑖𝑛 𝜃 − 𝑑�̇� = 0            (1) 

By nonholonomic constraints 

𝐴𝑇(𝑞)�̇� = 0                    (2) 

The constraint matrix is 

𝐴𝑇(𝑞) = [− 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 −𝑑]𝑇             (3) 

According to Euler-Lagrange principle, the dynamic model 

of the robot is[11,12] 

where 𝑀(𝑞) ∈ 𝑅3×3  is the inertia matrix, 𝐶(𝑞, �̇�) ∈
𝑅3×3 is a matrix of centripetal and oosnad forces, 𝐺(𝑞) ∈
𝑅3 is the gravity matrix, 𝐹(�̇�) ∈ 𝑅3 is friction, τ𝑑 ∈ 𝑅

3 For 

external disturbances,𝐴𝑇 ∈ 𝑅3×1is the constraint matrix; λ ∈
𝑅  is a Lagrange multiplier, 𝐵(𝑞) ∈ 𝑅3×2  is the input 

transformation matrix, τ ∈ 𝑅2 is the control input torque, 

where𝑀(𝑞), 𝐶(𝑞, �̇�), 𝐵(𝑞), 𝐺(𝑞), τ, λ  is defined as 

𝑀(𝑞) = [
𝑚 0 𝑚𝑑 𝑠𝑖𝑛 𝜃
0 𝑚 −𝑚𝑑 𝑐𝑜𝑠 𝜃

𝑚𝑑 𝑠𝑖𝑛 𝜃 −𝑚𝑑 𝑐𝑜𝑠 𝜃 𝐼
], 

𝐶(𝑞, �̇�) = [
0 0 −𝑚𝑑�̇� 𝑠𝑖𝑛 𝜃
0 0 𝑚𝑑�̇� 𝑐𝑜𝑠 𝜃
0 0 0

],𝐺(𝑞) = 0, 

𝐵(𝑞) =
1

𝑟
[
𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 𝜃
𝑅

𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 𝜃
−𝑅

],τ = [
τ𝑟
τ𝑙
], 

𝐹(�̇�) = 0,λ = −𝑚(�̇� 𝑐𝑜𝑠 𝜃 + �̇� 𝑠𝑖𝑛 𝜃)�̇�. 

where𝑚is the quality of the robot; 𝑟is the radius of the left 

and right wheels; 𝐼 is the moment of inertia; τ𝑟 , τ𝑙  are the 

control torque of the left and right wheels; Since the control 

input is the torque of the robot's left and right wheels, it is 

ordered 

{
𝑢1 = τ𝑟 + τ𝑙
𝑢2 = τ𝑟 − τ𝑙

               (4) 

In this paper, considering the case where the geometric 

center C and the center of gravity M coincide, that is d=0, the 

dynamic model of the robot can be obtained: 

{
 
 

 
 �̈� =

λ

𝑚
𝑠𝑖𝑛 𝜃 +

1

𝑚𝑟
𝑢1 𝑐𝑜𝑠 θ

�̈� = −
λ

𝑚
𝑐𝑜𝑠 θ+

1

𝑚𝑟
𝑢1 𝑠𝑖𝑛 θ

θ̈ =
𝑅

𝐼𝑟
𝑢2

               (5) 

3. Controller design 

Since the mobile robot is a typical underdrive system, the 

input control amount of the system is (𝑢1, 𝑢2) , and the 

controlled output amount is (𝑥, 𝑦, θ), In order to ensure that 

the system can be effectively controlled, the system is divided 

into a posture subsystem and a position subsystem, The 

attitude subsystem utilizes PD control, and the position 

subsystem utilizes improved hierarchical sliding mode 

control to ensure that the system reaches a stable state. 

3.1. Position subsystem controller 

In this paper, hierarchical sliding mode control method is 

chosen to make a single control input to control the two 

subsystems ideal stable state, so that the entire position 

control system to achieve the ideal state. 

3.1.1. Sub-section Headings 

Nonsingular terminal sliding mode control 

In the design of the sliding surface, the nonlinear function 

is used instead of the linear sliding surface, so that the 

tracking error of the system can converge to zero in a finite 

time. Thus, improving the dynamic performance of the 

system. The general form of nonsingular sliding surface is: 

𝑠 = 𝑥 +
1

𝑐
�̇�
𝑝

𝑞                     (6) 

where 𝑐 > 0, 𝑝, 𝑞 are positive odd, and 𝑝 > 𝑞. 

3.1.2. Hierarchical singular terminal sliding mode 

controller design  

By using hierarchical sliding mode control, the conversion 

from multi-objective control to single-objective control is 

realized, and the control design of the system is simplified. 

The above dynamic model can be represented by a general 

second-order underactuated system model: 

1 2

2 1 1

3 4

4 2 2

( ) ( ) ( )

( ) ( ) ( )

x x

x f x g x u t

x x

x f x g x u t

=


= +


=
 = +

             (7) 

where  

1 1( ) sin ( ) cosf x x
m mr

 
=  = g

 

𝑓2(𝑥) = −
λ

𝑚
𝑐𝑜𝑠 θ , g1(𝑥) =

λ

𝑚𝑟
𝑠𝑖𝑛 θ, 

𝑥 = [𝑥 𝑦 θ]𝑇 , 𝑢(𝑡) = 𝑢2(𝑡)  is the system input, the 

tracking error of two state variables of the position subsystem 

is defined as? 

1

2

3

4

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

d

d

d

d

e t x t x t

e t x t x t

e t y t y t

e t y t y t

= −


= −


= −
 = −

              (8) 

The parameter trajectory of. Design the first sliding surface 

for (8) : 

𝑠1 = 𝑒1 +
1

𝑐1
𝑠𝑖𝑔(𝑒2)𝛾1 , 𝑠2 = 𝑒3 +

1

𝑐2
𝑠𝑖𝑔(𝑒4)𝛾2    (9) 

where 𝑐1, 𝑐2 , 𝛾1, 𝛾2 are positive constant, and 0 < 𝛾1 <
1,0 < 𝛾2 < 1 . According to the equivalent control method, 

each equivalent control quantity of the position subsystem can 

be obtained as follows:  

𝑢𝑒𝑞1 = −
1

𝑔1(𝑥)
(𝑓1(𝑥) + 𝑐1

𝑞1

𝑝1
𝑒2
2−

𝑝1
𝑞1)            (10) 

𝑢𝑒𝑞2 = −
1

𝑔2(𝑥)
(𝑓2(𝑥) + 𝑐2

𝑞2

𝑝2
𝑒4
2−

𝑝2
𝑞2)        (11) 

Construct the second sliding surface: 

𝑆 = 𝛼𝑠1 + 𝛾𝑠2                    (12) 

w𝛼, 𝛾here is a normal number, and𝛼𝑔1(𝑥) + 𝛾𝑔2(𝑥) ≠ 0. 

In order to ensure that the two state variables of the position 

subsystem can slide stably along the respective sliding surface, 

the control input of the position subsystem needs to contain 

the equivalent control quantity on each sub-sliding surface. 

Therefore, the total control input of the position subsystem is: 

𝑢1 = 𝑢𝑒𝑞1 + 𝑢𝑒𝑞2 + 𝑢𝑠𝑤            (13) 

where 𝑢𝑠𝑤 is the switching control quantity of the position 

subsystem in the approach stage? In order to obtain the 

control input 𝑢1  of the position subsystem, the switching 

control part  𝑢𝑠𝑤 of the controller needs to be further 

determined.  

Where 𝑓1(𝑥), 𝑓2(𝑥), 𝑔1(𝑥), 𝑔2(𝑥)  will be respectively 

denoted by 𝑓1, 𝑓2, 𝑔1, 𝑔2 . The exponential approach law is 

adopted: 

�̇� = −𝜂𝑠𝑎𝑡(𝑆) − 𝑘𝑆              (14) 



 

30 

Where 𝜂, 𝑘  are the normal number, 𝑠𝑎𝑡(𝑆)  is the 

saturation function, Δis the boundary layer, 𝑘 =
1

Δ
 , and its 

specific definition is as follows: 

𝑠𝑎𝑡(𝑆) = {
1
𝑘𝑠
−1

𝑆 > Δ
|𝑆| ≤ Δ

𝑆 < −Δ

           (15) 

The switching control quantity of the position subsystem in 

the approach stage is: 

𝑢𝑠𝑤 = −
𝛾𝑔2(𝑥)𝑢𝑒𝑞1

𝛼𝑔1(𝑥)+𝛾𝑔2(𝑥)
−

𝛼𝑔1(𝑥)𝑢𝑒𝑞2

𝛼𝑔1(𝑥)+𝛾𝑔2(𝑥)
−

𝜂𝑠𝑎𝑡(𝑆)+𝑘𝑆

𝛼𝑔1(𝑥)+𝛾𝑔2(𝑥)
  (16) 

Then the control quantity of the position subsystem is: 

1 2

1 2

1 1 2

2 2
1 2

1 1 2 2 2 4

1 2

1 2 1 2 1 2

( ( ) ) ( ( ) )
( )

( ) ( ) ( ) ( ) ( ) ( )

eq eq sw

p p

q q

u u u u

q q
f x c e f x c e

p p sat S kS

g x g x g x g x g x g x

 


     

− −

= + +

+ +
+

= − − −
+ + +

   (17) 

The Lyapunov theorem is used to find the switching control 

quantity of the position subsystem, and the Lyapunov 

function is constructed as follows 

𝑉 =
1

2
𝑆2                (18) 

Take the derivative of the above equation 

1 2

1 2

1 2

1 2

1 2

1 2

1 2

2 2
1 2

2 2 2 4 4 4

1 1 2 2

2 2
1 2

2 2 1 1 1 4 4 2 2 1

1 1 2 2

2 2
1 2

2 2 1 1 1 2 4 4

1 1 2 2

( )

( ( ) ( ))

( ( ( )) ( ( )))

= ( ( ( ( ))) ( (

p p

q q

p p

q q

p p

q q

eq eq sw

V SS S s s

p p
S e e e e e e

c q c q

p p
S e e f g u e e f g u

c q c q

p p
S e e f g u u u e e f

c q c q

 

 

 

 

− −

− −

− −

= = +

= + + +

= + + + + +

+ + + + + + 2 2 1 2

2

( ))))

| | 0

eq eq swg u u u

S kS

+ + +

= − − 

 (19) 

3.1.3. Stability analysis 

Theorem 1 For the underactuated system shown in 

Equation (7), if the sub-sliding mode surfaces of each level 

are designed according to Equations (9) respectively, and the 

control law shown in Equation (17) is adopted, the sliding 

mode surface of the second layer of the position subsystem is 

asymptotically stable. 

Proof: According to Equation (19) : 

�̇� = −𝑆 ∗ (𝜂𝑠𝑎𝑡(𝑆) − 𝑘𝑆) = −𝜂|𝑆| − 𝑘𝑆2 ≤ 0  (20) 

Combining equation (18), it can be seen that 𝑉 > 0 . 

According to Lyapunov theorem, the position subsystem is 

stable. 

By integrating t on both sides of Equation (19), we can get: 

𝑉(0) = 𝑉(𝑡) + ∫ 𝜂
𝑡

0
|𝑆| + 𝑘𝑆2𝑑𝑡 ≥ ∫ 𝜂

𝑡

0
|𝑆| + 𝑘𝑆2𝑑𝑡 (21) 

Then according to Barbalat's theorem [13], when 𝑡 → ∞, 

𝑙𝑖𝑚
𝑡→∞

𝑆 = 0. The second sliding mode surface can be proved to 

be asymptotically stable. 

Theorem 2: For an underactuated system shown in Eq. (7), 

if the sub-surfaces of each level are designed according to 

Eq.(9), respectively, and the control law shown in Eq. (17) is 

adopted, the sum of the two first-level sliding surfaces of the 

position subsystem is asymptotically stable. 

From the barbalat theorem and Lyapunov theorem, it can 

be proved that Theorem 2 holds [7], and the position 

subsystem is asymptotically stable. 

Attitude subsystem controller 

For attitude subsystem 

�̈� =
𝑅

𝐼𝑟
𝑢2                 (22) 

Assume that the ideal Angle command is 𝜃𝑑 , and the 

tracking error is 𝑒𝜃 = 𝜃 − 𝜃𝑑 , For attitude subsystem, PD 

control method is adopted to design the control law as follows: 

𝑢2 = −𝐾𝑝𝑒𝜃 − 𝐾𝑑�̇�𝜃            (23) 

where 𝐾𝑝, 𝐾𝑑 is the normal number? 

Let's take the Lyapunov function as 

𝑉 =
1

2

𝐼𝑟

𝑅
�̇�𝑇�̇� +

1

2
𝐾𝑝𝑒

𝑇𝑒              (24) 

Since 𝐼, 𝑟, 𝐾𝑝 they are all normal numbers, it can be known 

that 𝑉 is positive definite. Then the derivative of the above 

formula can be obtained as follows: 

2

( )

0

p p d p

d

Ir
V ee K ee e K e K e K e

R

K e

= + = − − +

= − 

    (25) 

Since it is semi-negative definite and positive definite, the 

attitude subsystem is stable.  

4. Literature References 

In order to verify the effectiveness and robustness of the 

proposed control method, a trajectory tracking control 

experiment of wheeled mobile robot was conducted using 

Matlab/Simulink simulation environment. The parameters of 

the mobile robot are selected according to the reference [12]. 

The specific parameters of the mobile robot system are shown 

in Table 1: 

Table 1. Three Scheme comparing 

𝑚/𝑘𝑔 𝑅/𝑚 𝑟/𝑚 𝐼/(𝑘𝑔/𝑚2) 

4 0.2 0.04 2.5 

The parameters of non-singular terminal Layered Sliding 

mode controller (NTHSMC) are as follows: 𝛼 = 1, 𝛾 = 50, 

𝑐1 = 35 , 𝑐2 = 35 , 𝑞1 = 55 , 𝑝1 = 53 , 𝑞2 = 55 , 𝑝2 = 53 , 

𝜂 = 5 , 𝑘 = 5 , Δ = 0.005 . The parameters of the PD 

controller are as follows: 𝐾𝑝 = 5 , 𝐾𝑑 = 10 . Set the circle 

trajectory with the center of the ideal trajectory as the origin 

and the radius as 2: 

{

𝑥𝑑(𝑡) = 2 𝑐𝑜𝑠𝜃𝑑
𝑦𝑑(𝑡) = 2 𝑠𝑖𝑛𝜃𝑑

𝜃𝑑 = 0.1𝑡

 

Type of; The starting position of the mobile robot is set as 

𝑞 = [0 0
𝜋

2
]; The mobile robot is simulated and analyzed. 

The parameters of the layered sliding mode controller (HSMC) 

are 𝛼 = 1 , 𝛾 = 50 , 𝑐1 = 35 , 𝑐2 = 35 , 𝜂 = 5 , 𝑘 = 5 , Δ =
0.005. 

FIo. 2 shows the motion trajectory of the mobile robot. It 

can be seen from FIo. 2 that the actual trajectory can quickly 

track the given desired trajectory and keep the basic 

coincidence, which highlights the effectiveness of the 

controller. When the controller parameters are consistent, N 

TSMC has higher tracking performance than HSMC, but 

HSMC cannot achieve the tracking purpose.  FIo. 3 is the 

tracking error curve. It can be seen from the curve that around 

10s, the system gradually enters the steady state, and the 

mobile robot starts to run along the circular trajectory, and the 

tracking error all converges to zero, which has good trajectory 

tracking performance.  FIo. 4 shows the variation trend of 

each sliding mode surface under non-singular terminal 

Layered Sliding mode control (NTHSMC) and the total 

sliding mode surface after linear combination. It can be seen 

from Figure 4 that the sub-sliding surface and the combined 

sliding surface under NTHSMC can achieve a fast and stable 

trend and remain in a stable state, while effectively reducing 

chattering. Although the convergence speed is different, they 

all converge to zero in finite time. The convergence speed of 

each sliding mode surface can be achieved by adjusting the 

parameter size of the controller, or by improving the reaching 

law to accelerate the convergence speed. 

Figure 5 is the change curve track of the input control sum 

of the position subsystem and the attitude subsystem. It can 



 

31 

be seen that the convergence curve of the controller is 

gradually stable and finally converges to a stable state. 

  
Figure 2. Trajectory of the mobile robot  

 
Figure 3. The tracking error curve 

 
Figure 4. The sliding mode surface 

 
Figure 5. the input control curve 

5. Conclusion 

This paper presents the design and implementation of a 

controller based on mobile robot which combines non-

singular terminal sliding mode control and PD control. The 

first layer of the layered sliding mode controller is a non-

singular terminal sliding mode control structure for each 

variable of the position subsystem, and the second layer is a 

linear combination of the first layer. The advantage of the 

proposed control method is that it can reach the desired stable 

state in a given time. The simulation results show that the 

control method has good stability and robustness. 

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