Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 12, No. 1, 2025 

 

172 

Constrained	Manipulator	Force	Position	Hybrid	Control	
Based	on	Fuzzy	Neural	Network	
Guogang Wang, Qilong Sun 

School of Jilin Institute of Chemical Technology, Jinlin Jilin, 132022, China 

 

Abstract: Aiming at the problem that the end motion of a robot arm is constrained and cannot be accurately modeled when it 
is in contact with the working environment, a reduced order fuzzy neural network sliding mode control method is designed to 
realize the high-precision tracking of the Angle, angular velocity and contact force of each joint. Firstly, the model of constrained 
double-joint manipulator is reduced to simplify the dynamic model and reduce the computational difficulty. Then, fuzzy neural 
network is used to approximate the friction and nonlinear control quantity generated during the movement of constrained 
manipulator. Finally, sliding mode control method is used to improve the robustness of the controller and the stability of the 
system is proved by Lyapunov function. The simulation results show that the tracking error of each joint of the constrained 
manipulator is controlled within 0.003. 

Keywords: Constrained Robotic Arm; Fuzzy Neural Network; Sliding Mode Control Algorithm; Model Reduction. 
 

1. Introduction 
In today's industrial automation production practice, the 

robotic arm is undoubtedly one of the core roles, widely 
involved in all kinds of production operations. When the robot 
arm only undertakes simple handling, welding and spraying 
tasks, it only needs to accurately control its position and carry 
out position tracking control. However, once complex product 
assembly, fine grinding, or tube grasping operations in the 
chemical field are involved, it is necessary not only to 
precisely plan the joint position of the robotic arm and the end 
effector, but also to fine-regulate the force exerted on the 
constraint surface during the operation of the robotic arm, in 
order to complete these relatively difficult production tasks 
[1,2]. Conventional control methods mainly include neural 
network control[3], robust control[4], fuzzy control[5], 
sliding mode control[6] and PID control[7]. The hybrid 
force/bit control technology of constrained manipulator has 
become a key research topic in the field of industrial 
automation. 

Naveen Kumar et al.[8]designed an intelligent optimal 
control method for constrained reconfigurable manipulator 
position/force control under uncertainty, which overcomes 
the uncertainty of complex systems and can achieve the 
required performance. Aiming at the manipulator based on 3-
DOF series elastic actuator (SEA) under partial state 
constraints, Yangchun Wei et al.[9]designed a model-free 
hybrid force/position controller (PPMHC) with specified 
performance, which improved the control accuracy and 
realized the fitting of Angle and angular velocity. 
Mohammad-Hossein Ghajar et al.[10]designed an intelligent 
hybrid position/force controller for a constrained robotic 
manipulator, which greatly reduced the tracking error of each 
joint of the manipulator despite the presence of large 
parameters and dynamic uncertainties. Tuo Yaoyao et 
al.[11]designed an event-triggered adaptive specified 
performance controller, which reduces the update frequency 
of the control signal and improves the resource utilization rate. 
Ruchika et al.[12], aiming at the problem that deformable 
links cause frequent changes in shape, which makes it 
difficult to model and control the longitudinal, proposed a 

segmented control method in which a neural network system 
is used to estimate nonlinear components, and the friction 
term and binding force of each joint are compensated by 
adaptive control, thus improving the control performance. 

This chapter focuses on the problem of position and force 
control of the manipulator, takes the dynamic equation of the 
constrained manipulator as the research object, integrates the 
position control and force control in the same controller, and 
combines the advantages of fuzzy neural network algorithm, 
sliding mode control algorithm and order reduction method. 
A design scheme of constrained mechanical arm force/bit 
hybrid controller based on fuzzy neural network reduced 
order sliding mode algorithm is proposed 

2. Dynamic Model of Constrained 
Manipulator 

The dynamic equation of the double-joint robotic arm is as 
follows: 

( ) ( , ) ( ) ( , ) fM q q C q q q G q f q q d τ τ          (1) 

Where, 
*( )( ) n nM q R  is the inertia matrix; 

*( )( , ) n nC q q R  is the centrifugal force and Coriolis force 

matrix, ( ) nG q R   is the gravity term vector, ( , )f q q  is 

friction link,d is unknown external disturbance torque, fτ  

is the binding vector; τ   is input torque vector error for 
control! No reference source found. 

Let x be the position of the end effector of the robotic arm, 
then the constraint equation is 

Φ( ) ( ( )) 0q h q              (2) 

The Jacobian matrix of the constraint equation is 

 
Φ( ) ( ) ( ) ( )

( )
q x x x h q

J q
q x q x q

    
  

    
 (3) 

Since the dynamic model of the robot arm is constrained, 
the double-joint robot arm is constrained by a force, so two 
degrees of freedom of the double-joint robot arm become one, 

and 2q is expressed by 1q  as: 



 

173 

 2 1( )q ψ q                   (4) 

then 

 

1
1

1 11
12

1

( )Ψ( )

q
q

q L q qq
qq

q

 
            




 


       (5) 

 1 1 1 1( ) ( )q L q q L q q                 (6) 

In the formula, 1 1

1

( ) Ψ( )

1

L q q

q

 
   

  

 

Therefore, the dynamic equation of a constrained two-
degree-of-freedom manipulator is: 

T
1 1 1 1 1 1 1 1 1 1 1( ) ( , ) ( ) ( ) ( )M q q C q q q G q f q d J q λ τ          (7) 

In the formula, 

 

l l l

1 1 1 1 1

1 1

T T
1

( ) ( ) ( )

( , ) ( ) ( ) ( , ) ( )

( ) ( )

( ) ( )

M q M q L q

C q q M q L q C q q L q

G q G q

J q J q 



 




 
 

Multiply equation (1.7) by 1( )TL q   left to obtain a 

reduced order constrained manipulator model: 
T T T

1 1 1 1 1 1 1 1( ) ( , ) ( ) ( )+ ( )L L LM q q C q q q G τdq L f q L L q        (8) 

Among them: 

 
T

1 1 1

T T
1 1 1 1 1 1 1 1 1

T T T
1 1 1 1 1 1

( ) ( ) ( ) ( )

( , ) ( ) ( , ) ( )( ( ) ( ) ( , ) ( ))

( ) ( ) ( ) ( ) ( )

L

L

L

M q L q M q L q

C q q L q C q q L q D q L q C q q L q

G q L q G q L q J q λ



  

 

  
 

The dynamic model in the formula has the following 
properties: 

Property 1:
T T

1 1 1 1( ) ( ) ( ) ( ) 0J q L q L q J q    

Property 2: 1 1 1( ) 2 ( , )L LM q C q q   is a skew symmetric 

matrix. 

3. Fuzzy Neural Network Sliding Mode 
Control Algorithm 

3.1. Controller Design 

Let ( )dq t 1 be the expected Angle of motion of the robot 

arm, and d
fτ  be the expected binding force on the end of the 

robot arm. In a double-joint robot arm, since 2 ( )q t   is a 

function of 1( )q t , the actual output of the system ( )q t  is 

1( )q t  , and the system satisfies ( ) 0dq   ,
T ( )d

f d dτ J q λ . Definition: 

 1 d1 1 dλe q q e λ λ   ，             (9) 

The design sliding surface is: 

 

1

1 1 1

1

[ ]L
λ λ

s e k e

e k e
s

e k e

 









              (10) 

Construct the auxiliary function as: 

 

1 1 1 1

1 1 1 1

r d

r d

r d λ λ

q q k e

q q k e

λ λ A e


 



 

 
                (11) 

Substituting formula (11) into formula (7) yields: 

 
1

T
1 1 1

1 1 1

( )

r r

M s τ F C s J q λ

M Cq fqF G d



    

   
 

       (12) 

Where F is a nonlinear function containing the friction term.  
In this paper, a four-layer Mamdani-type fuzzy neural 

network is adopted for design to approximate the nonlinear 
links of the robotic arm system. 

The first layer is the input layer, ( = 1,... ,n1) is the input 
term representing the fuzzy neural network, and 1 is the 
number of inputs. 

The second layer is the fuzzy membership degree 
calculation layer, which adopts the Gaussian function and is 
expressed by the formula as follows: 

 

2( ( ) )

( )

j
i i
j
i

x c

bj
i ix e




               (13) 

In the formula, j
ic , j

ib (i=1,... n1; j=1,... n2) represent the 

center and width of the fuzzy basis functions (FNNBF), 
respectively. The center value and width of the basis functions 
are as follows: 

 
2 2 2

1 2 1 2 1 2
1 1 2 2 1 1

1 1 1
1 1 2 2 1 1

[ ,..., , ,... ,..., ,... ]

,..., , ,..., ,..., ,...,

,n n n T
n n

Tn n n
n n

c c c c c c c

         
 

The third layer is the fuzzy reasoning normalization layer, 
where corresponding fuzzy reasoning tuning regulations are 
set to perform fuzzy reasoning operations on the signal: 

 
1
1

2 1
1 1[ ]

n j
i i

j n n j
j i i

h 

 



 




              (14) 

The fourth layer is the output layer: The output of each 
node in the fourth layer is the weighted sum of all input 
signals of that node as follows:  

 
2

1

n
j

i i j
j

y h


                   (15) 

Among them, )1, 2, , 1(iy i n    is the output of the 

fuzzy neural network. 
Fuzzy neural network algorithm is used to approximate it: 

 = +F WH                   (16) 

Where, W，   is bounded, that is, 

|| ||< || ||max maxW W H < H，  , max maxW H，   are constants, 

W  and H  are respectively the ideal weight matrix of the 
fuzzy neural network and the output basis vector of the fuzzy 

basis function,   is the error vector, and Ls  is the network 

input value when the fuzzy neural network approximates. 
Since the actual value of 8 is unknown, the estimated values 
of the vector parameters are set to be W c k、 、  

respectively, the estimated errors are ˆW W W   ,
ˆc c c   , ˆκ κ κ    and the approximation error of the 

nonlinear function F is: 



 

174 

 

ˆ ˆ

ˆ ˆ

ˆ ˆ ˆ Δc x

F F WH

WH ε WH

WH Wh c Wh κ F

 

  

   



  
      (17) 

therein, ,c k

H H
h h

c k

 
 
 

,and 

 ˆ ˆΔ c k nF W c W κ Wo ε             (18) 

The design control rate is 

 Tˆ ˆ sgn( )d L rr Lτ WH k s k J λs         (19) 

Where, ˆ ˆWH   is the output of the fuzzy neural network 

and d rk k，  is the positive definite matrix. 

The adaptive law of fuzzy neural network design is as 
follows: 

 

w

c

k

ˆ

ˆˆ

ˆ ˆ

f

T
c

T
k

W Γ LsH

c Γ hW Ls

k Γ h W Ls

  

  


 






           (20) 

Among them, w c kΓ Γ Γ、 、   are positive definite 

diagonal matrices. 
Substituting formula (15) (16) into formula (12) yields: 

1

1

1

T
1 1

T T
1 1

1

T
1

ˆ ˆ s )

( )

= ( ) (

ˆ ˆ ( )( )

ˆ

(

gn( ) - - -

ˆ sg

ˆ

n )-

ˆ sgn( )-

rd r

d r

d r

c x r

c x

M

s

W

WH k s

s τ F C s J q λ

J q λ J q λ

W Wh c h κ ΔF d J q λH λ

W Wh c Wh κ

k s WH ε C s

k s k s C s

H k s kΔF s Cd



 



   

      



   

 





     



  
  

                 (21) 

 

By multiplying both sides of the formula (18) by 1( )TL q  left at the same time, we get: 
     

ˆˆ sgn( )- -ˆT T T T T T T
L c d r LkM ss L W L Wh c L Wh κ L ΔF L L LH k s k s ε C                        (22) 

 

Theorem: Considering the constrained manipulator 
dynamics model (4.6) and fuzzy neural network adaptive 
sliding mode controller (4.11) and adaptive rate (4.12) in a 

closed loop system, if the ideal trajectory dq q q λ ， ， ，  

converges and is bounded, the sliding mode function s  
converges, and the fuzzy neural network estimation error is 

continuous, if *
r Fk    is satisfied, the system can be 

proved stable. 

3.2. Stability Analysis of Control System 
Define the Lyapunov function as: 

1 1 1
1

1 1 1

1 1 1 1
( )

2 2 2 2
1 1 1 1

( )
2 2 2 2

T T T T
L L w c k

T T T T
L w c k

V s M s tr W W c c k k

s M s tr W W c c k k

  

  

   

   

    

    

  

  

 (23) 

Derivation of formula (20) is obtained 
This shows that the closed-loop system is stable. 

4. Simulation Analysis 

4.1. Simulation Parameter Setting 
In order to verify the effectiveness of the design method, a 

double-joint robotic arm is selected for simulation, and the 
dynamic equation is as follows: 

 
 

1 1 1

1 1 1
1 1

1 1

1
Γ Γ Γ

2
1

Γ Γ Γ
2

ˆ ˆ

Γ Γ

sgn

ˆ( sgn( )- - )

( )(

T T T T T
L L w c

T T T T T T T
w c

T T
c x

T T
c

T T
d r

d r LH k s k s C

V s M s s M s tr W W c c

s L M s s L M s tr W W c c

s L W Wh c Wh F d

c c

s L K

s

s K s

  

  

 

    

    

     

 

   

 

       

      

 

 

 

 
 







 

 



 



 

 

1

1

*

1

( sgn( ) )

( sgn( ) )

sgn( )+ )

sgn( )+

ˆ ˆ ˆ

ˆ

Δ ) Γ

ˆ ˆ

Γ

| |

TT T T T T

c k

T

d r

T

d r

T T

d r

T T T

d r

T
c w

T T T T
w

T

n
T T T

i
i

T T T

c h s k h s

s k s k s F d

s k s k s F d

s k s s k s F

s k s s k s s F

W Wh c Wh k d F tr W W

W L W L

L tr W W s L W

L

L

H

L

H

L L s

L L







 

    

    

   

   

    



  





  

 



 







*

*

)

0

sgn( )

( )sgn( )T T

d

T T
rs k s s F

s

L L k s   





              (24) 



 

175 

 
 ( ) ( , ) ( ) ( , ) fM q q C q q q G q f q q d τ τ          (25) 

Therein, 

 
2 2 2

2 2 1 2 2 2 1 1 2 2 2 1 2 2 2

0 2 2

2 2 1 2 2 2 2 2

2 2 2 2 1 2 2 1 2

0

1 2 2 1 2

2 2 12 1 2 1 1

0

2 2 12

1 1 2 1

2 ( )
( )

1

2( , )
1

0
2

( )
( )

sin( ) sin(
( )T

l m l l m c l m m l m l l m c
M q

l m l l m c l m

m l s q l l m s q

C q q

l l m s q

m l gc m m l gc
G q

m l gc

l q l q q
J q

   




 




 


 
 
 
 
 
 
 
  
 
 

  





 





2

2 1 2

)

sin( )

3sin(4 )
[ ]
2sin(5 )

l q q

t
d

t

 
   





 

Among them, the simulation expected value is set to 

1 sin( ), 10sin( )dqd t λ t  , the initial state is [0,0] , the 

length of the mechanical arm connecting rod is 

1 21 1l m l m ， , the mass is 1 21 1m kg m kg ， , the 

acceleration of gravity is 9.8g   , and the controller 

parameter is 
6 4 2 0 2 4 6

[ ]
6 4 2 0 2 4 6

j
ic

  

  

 ,

2 2 2 2 2 2 2
[ ]
2 2 2 2 2 2 2

k   ,The adaptive law 

parameter is =5000 =50 50w c kΓ I Γ I Γ I， ，  ,

(20,10) (200,80)r dk diag k diag ，  , 1 10k   ,

10λA  . 

Take 1 2 1l l    to get it 

     1 1 1 2 1 2 1 1 2cos cos cos cos 0q l q l q q q q q         

As can be seen from the figure: 

 1 2 1 2,0 ,0
2

π
q q π q q π            (26) 

and, 

 

 1 1 2

1 1 2

2 1

cos( - =

-

) cos

-

2

π q q q

π q q q

q π q



            (27) 

 

1q

2q

2m

1m

1l

1l

2x

1x  
Fig 1. Structure diagram of constrained manipulator 

 
Since 2 1Ψ( )q q , then 1 1Ψ( ) 2q π q  , yields: 

 1 1

1

)

1
1

)
2

((L

q

q Ψ q

 
          

            (28) 

Thus, the properties of constrained robotic arm 1 can be 
proved. 

 

 1 1 1 1 2 1 2 2 1 2

1 1 2 1 2 2 1 2

1 1 2 1 2

1 1 2 1

1
( ) ( ) sin sin( ) sin( )

2

sin sin( ) 2 sin( )

sin sin( )

sin sin( )

0

J q L q l q l q q l q q

l q l q q l q q

l q l q q

l q l π q



 
        
     
   
   


 (29) 

4.2. Analysis of Simulation Results 
 

 
Fig 2 (a) 



 

176 

 
Fig 2 (b) 

Fig 2. joints 1 Angle angular velocity tracking curve 
 

Fig. 2 shows the simulation curves of the Angle variable 
and the angular velocity variable of the constrained robotic 
arm joint 1. The simulation results show that the fuzzy neural 

network sliding mode control can make the actual trajectory 
track the expected trajectory well. 

 

 

 
Fig 3. Contact force tracking and tracking error curve 

 
Fig. 3 shows the contact force tracking errorless simulation 

curve of the constrained manipulator joint. The simulation 
results show that the fuzzy neural network sliding mode 
controller can make the tracking error converge to 0.001 
within 0.2s. It can be seen that the design method improves 
the tracking accuracy of the mechanical arm in force control. 

5. Summary 
A hybrid force position control system based on fuzzy 

neural network is designed in this paper. Firstly, the 
constrained double-joint robot arm is reduced in price, and 
then the fuzzy neural network is designed to approximate the 
uncertainties in the robot arm system. The sliding mode 
controller is used to increase the stability and robustness of 
the control system, and finally the high-precision control of 
the constrained robot arm is realized. The simulation results 
show that the controller involved in this paper can ensure the 
accurate control of angular velocity and contact force in the 
presence of friction and other uncertainties, so that the system 
can maintain good performance. 

Acknowledgments 
PhD Project (No.: 222012312010). 

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[3] Wang J, Cui Y .Adaptive neural network tracking control for 
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