Journal of Robotics and Control (JRC) Volume 7.
Issue 2.
March 2026 ISSN: 2715-5072.
DOI: 10.
18196/jrc.
Dead-zone Compensation for Robotic Manipulators using Neural Networks Vu Thi Yen 1*.
Pham Van Cuong 1.
Bui Van Huy 1.
Ngo Manh Tung 1.
Nguyen Dang Toan 1 Email:
Hanoi University of Industry.
Hanoi.
Vietnam yenvt@haui.
vn, cuongpv0610@haui.
vn, 3 huybv@haui.
vn, 4 tung_nm@haui.
vn, 5 toannd@haui.
*Corresponding Author AbstractAiThis paper introduces an adaptive neural network (ANN.
based Radial Basis Function (RBF) for robotic manipulators in order to solve unknown dynamics, external disturbances and dead-zone compensation, thereby enhancing the accuracy of tracking control.
To address these problems, two Radial Basis Function neural networks (RBFNN) are designed: the RBFNN is employed to estimate the nonlinearities in the actuator, and the other RBFNN is utilized to compensate for dead-zone in the systemAos feedforward channel.
The RBFNN is a method that delivers good control performance for systems with uncertain models because of its fast-learning algorithm and strong approximation capability.
The design of online adaptive control training laws and dead-zone estimation is caried out using Lyapunov stability theory together with approximation Besides, a robust control plays an auxiliary controller to guarantee the robustness and stability under various Simulation results demonstrate the proposed controller achieves significant improvements in tracking accuracy, reducing the steady-state tracking error by up to 90% comparing with baseline methods, while ensuring smooth convergence and robust performance under parameter variations and disturbances.
The dead-zone compensation effectively eliminates oscillations and overshoots that appear in the uncompensated case.
These results validate the effectiveness and reliability of the proposed approach for high-precision robotic tracking control.
KeywordsAi Sliding model control.
Robotic manipulator.
Dead-zone compensation.
Adaptive control.
RBF neural
INTRODUCTION
Robots are now commonly employed in industrial production lines.
Therefore, control methods for robot systems continue to attract attention and remain interesting research topics.
Robottis are one type of multi-input multioutput object with the uncertain nonlinear dynamics, and uncertain parameters.
Designing a suitable control that provides efficient control of a robotic is a challenge to be To handle this problem, a variety of control methods have been proposed such as backstepping control in .
, adaptive control .
, sliding mode control (SMC) .
, intelligent control based on fuzzy logic .
, etc.
Backstepping control is a powerful nonlinear control technique that has been widely applied to robot manipulators due to its ability to explicitly handle system nonlinearities and strong dynamic coupling.
Starting from the EulerAeLagrange dynamic model of robotic manipulators, the backstepping approach designs the control law recursively by introducing virtual control inputs and Lyapunov functions at each step to guarantee system stability.
By systematically Austepping backAy from the kinematic level to the dynamic level, the method ensures accurate trajectory tracking and asymptotic stability .
Adaptive control for robot manipulators addresses uncertainties in system dynamics by adjusting control parameters online based on real-time feedback .
These approaches enabled accurate trajectory tracking and ensures system stability despite unknown parameters, external disturbances, and varying payloads.
Consequently, adaptive control has become a fundamental technique for enhancing the robustness and performance of robotic manipulators in complex and dynamic environments.
However, adaptive controllers have slow convergence speed, are sensitive to noise, computationally complex and requires existing knowledge of the robotAos dynamics and predetermined control gain parameters.
Moreover, pure adaptive control often does not handle nonlinear phenomena well such as dead-zone, saturation, backlash, static friction.
SMC is a simple yet robust nonlinear control approach with high effectiveness.
It exhibits low sensitivity to parameter variations, strong resistance to disturbances, and fast dynamic The sliding mode method with multiple parameters structure have proposed to improve the tracking performance of robot manipulators .
However, the design of sliding mode control requires prior knowledge of the systemAos mathematical model as well as the upper bounds of model Moreover, it inherently suffers from the chattering phenomenon, manifested as high-frequency oscillations around the sliding surface.
Recently.
NNs have been widely utilized in the control of robotic manipulators .
because the unknown dynamic of robot and nonlinearities were approximated by NNs.
In .
, the authors were used the adaptive neural network control in order to approximate the uncertain nonlinear dynamics of In .
, to improve trajectory tracking performance under various environments, an adaptive control based on RBFNNs was proposed to control CDRM.
The proposed controller applied the RBF neural network to approximate nonlinear robot dynamics.
In .
, to deal with the unknown dynamics of an industrial robot manipulator, a robust adaptive control combined with the RBF neural network was Two kinds of adaptive controllers based on RBFNNs for an uncertain robot system were proposed in .
to enhance the control performance and improve approximation accuracy.
The NNs technique has the ability to robustly approximate nonlinear systems with high It is particularly suitable for robot systems with complex, nonlinear dynamics that are difficult to model.
Journal Web site: http://journal.
id/index.
php/jrc Journal Email: jrc@umy.
Journal of Robotics and Control (JRC) ISSN: 2715-5072 Weights of the network can be adjusted to adapt to changes in the system and environment.
The above studies focused on unknown dynamic of robot without considering dead-zone.
In fact, in practical control system commonly present deadzone nonlinearity and it often deteriorates overall system In robotic control, dead-zone typically leads to issues such as excessive steady-state error, degraded transient response, and significant overshoot.
Hence, developing effective dead-zone compensation techniques is crucial to improve the performance of robotic systems.
In the past, many various controllers proposed to deal the dead-zone .
proposed to improve the control system performance.
Furthermore, these approaches presuppose that some parameters of the dead-zone function, notably its width and slope, are available.
In reality, obtaining accurate values for these parameters is challenging, particularly due to their timevarying nature.
Due to the significance of dead-zone.
In this paper, an adaptive robust control scheme by neural network technique is proposed to address dead-zone compensation, external disturbances, and unknown dynamics in robotic system, thereby improving tracking performance, learning speed and approximation capability.
This proposed method combines the advantage of adaptive control.
SMC, and NNs for robot manipulators.
In this framework, the unknown robot dynamics are approximated by neural networks, while the dead-zone nonlinearity is effectively Furthermore, all controller parameters are adaptively tuned according to Lyapunov stability theory.
a result, the effectiveness, and robustness of the control system are ensured.
The organization of this paper is as follows: Section 2 discusses the preliminaries.
Section 3 details the control design and stability analysis.
Section 4 reports the simulation results on the two-link robot, and Section 5 presents the TABLE I.
MAIN SYMBOLS USED IN THE MANUSCRIPT
Variables A Ea R nC1 AAEa R nC1 aEa R nC1 M NN (A ) Ea R nCn C (A .
AA) Ea R nCn
GNN (A ) Ea R nC1
( )
A smc output approximation signal of RBF neural network II.
PRELIMINARIES
Mathematical model of robotic manipulators The dynamics of an n- link robot are as following:
( )
M NN (A ) a CNN A .
A A GNN (A ) FNN A = A Oe A 0 The vector A = EEA1 A 2 A n EE Ea R nC1 represents the joint a = EEA1 A 2 .
A n EE Ea R nC1 , a = EEA1 A 2 .
A n EE Ea R nC1 is M NN (A ) Ea R
( )
nC n C NN A .
AA Ea R nC n expresses the vector of the Coriolis and centripetal forces.
GNN (A ) Ea R nC1 expresses the vector of the ( ) FNN AA Ea R nC1 is the vector of the frictions.
A0 Ea R nC1 denotes the vector of the unknown disturbances, and A Ea R nC1 expresses the vector of joints torque signal In designing the controller, several properties of the robot dynamics .
are assumed as follows.
Property 1: The inertial matrix M NN (A ) is a symmetric and bounded as follows as:
C xT M NN (A ) x C a2 x .
Ax Ea R n .
( ) is skew Property 2: The matrix MA NN (A ) Oe 2C NN A .
AA Joint velocity vector.
symmetry, in which:
Joint acceleration vector.
( )
xT E MA NN (A ) Oe 2C NN A .
AA E x = 0 inertial matrix FNN AA Ea R nC1 vector of the frictions Control input before the deadzone hidden layer output modeling error and approximation sliding mode control AE .
AEA
in which a1 and a 2 are specified as known position constants Physical meaning Joint position vector vector of the Coriolis centripetal forces vector of the gravity a j , bj center vector and width of the basis function.
W ji
weight connecting values of the jth hidden node to the ith output
( )
( ) are satisfied:
A .
GNN (A ) .
FNN A
Property 3: CNN A , a ( ) ( ) C Fk .
CNN A , a A C Ck A .
GNN (A ) C Gk .
FNN A
with Ck .
Gk .
Fk are constants and Ck .
Gk .
Fk A 0 .
Property 4: the unknown disturbance yua0 OO ycIycu is bounded as given below:
A 0 C A k ,A k A 0 Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks .
Journal of Robotics and Control (JRC) ISSN: 2715-5072 Dead zone Controller Aycc Dead zone Robotic yc Fig.
Structure of the robotic control system dead-zone compensation Based on the assumptions in .
The dead-zone function is illustrated in Fig.
1 and described as follows as:
hr ( u ) for u A d r A = D .
) = E
for dl C u C d r hl ( u ) for u A dl here, y denotes the control input, u represents the robotAos control input after the dead-zone compensation.
A denotes the control signal after the dead-zone.
Neural network control Structure The neural network control structure is shown in Fig.
where n is the number link of industrial robot manipulator.
m is the number nodes of hidden layer.
The hidden layer output is expressed as follows:
NN I
Eayc .
ycuycAycA
NN II
yccyco yc yccyc yc yc Eayco .
parameters of the dead zone (DZ), with hr .
), hl .
) being the unknown smooth functions.
The control input before the DZ is represented by yc, while yua represents the control input after the DZ.
Assuming that hr .
), hl .
) are monotonically reversible increasing functions on the defined interval.
Therefore, the inverse of the DZ is expressed as follows:
Eh Oe1 A 0 u A 0
Oe1
D (A ) = E 0
Eh Oe1 A 0 u A 0 ( Oe1 (A )) = A E Oe ( s Oe a )2 E A j ( s ) = exp E E 2b j with a j , b j are the center vector and width of the basis According to .
, the output signal of the neural network is determined as follows:
f j ( s ) = Eu WjiA j ( s ) , j = 1,.
, m j =1 W ji is the weight connecting values of the jth hidden node to the ith output node, and Equation .
can be written as:
We have:
D D
yua Fig.
The neural network compensator structure Fig.
Dead zone model Here, d r A 0, dl A 0 Dead zone To address the unknown nonlinear DZ, the neural network compensator is positioned at the front of the control input.
The overall structure of the robotic manipulator control system is presented in Fig.
f ( s ) = WT A ( s ) AE W is the weight of the NN, and A ( s ) = AuA1 .
A2 ,.
Am Ay .
AE is the modeling error of f ( s ) .
Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks Journal of Robotics and Control (JRC) ISSN: 2715-5072 The NN output signal is applied to approximate the unknown dynamic of robot, which is calculated as follows:
f ( s ) = W AT A ( s ) AEA W is the ideal NN weight.
AE is the approximation error.
The modeling error AE is bounded as:
AEA C AE 0
with AE 0 is a position constant fI ( s ) = WC A ( s ) .
y = D ( D Oe1 ( y ) ) = D ( y y NN ) Substituting Equation .
, and Equation .
into Equation .
, we have:
W T AAE NN ( y ) AE ( y y NN ) Define A = y WI NN ANN ( y ) WANN ANN ( y ) AE NN ( y ) , and A 0 = y WI NN ANN ( y ) .
A Oe A 0 = WANN ANN ( y ) AE NN ( y ) Make f (A ) = A (A ) , according to the first- order Taylor where R (An ) is remainder of the first- order expansion An is a value between (A 0 .
A ) .
R (An ) = .
= A (An ) (WA A ( y ) AE ( y ) ) .
y NN = WNN ANN ( y ) AE NN ( y ) .
where AE ( u ) .
AE NN ( y ) are neural network errors.
W .
WNN
are the ideal weights of neural network.
A ( u ) .
A N ( y ) are the outputs of the radial basis function.
DC ( u ) and yI NN are the estimation of the ideal NN and they NN NN
(A (A 0 ) A ' (A 0 )(A Oe A 0 ) R (An )) AE y y NN
( )
( )(A Oe A 0 ) W R (An ) AE ( y y NN ) T ' = W A A0 W A A0 = W A y WI NN A NN ( y ) ) (WA A ( y ) AE ( y )) T ' W A y WI NN A NN ( y ) DI ( u ) = WI A ( u ) .
W R (An ) AE ( y y NN ) yI NN = WI NN ANN ( y ) .
according to Equation .
Equation .
and Equation .
, the dead zone output can be determined as:
are calculated as:
WA = W Oe WI .
WANN = WNN Oe WI NN A (An 0 )(A Oe A 0 ) R (An ) = R WANN , y D ( u ) = W T .
A ( u ) AE ( u ) y = W T A y WI NN ANN ( y ) WANN ANN ( y ) according to Equation .
we have:
expansion, f (A ) = f (A 0 ) f (A 0 )(A Oe A 0 ) R (An ) and y NN is calculated as:
EhrOe1 ( y ) Oe y y A 0 y NN = E E h Oe1 ( y ) Oe y y A 0 E f From Equation .
, and Equation .
, we have:
The inverse of the dead zone can be written as:
D Oe1 ( y ) = y y NN = W T A ( y yI NN ) AE ( y yI NN ) = W T A y WNN ANN ( y ) AE NN ( y ) AE ( y y NN ) Dead Zone compensation neural network structure Neural network compensator is designed to includes two RBFNN, the RBFNN is used to estimate the nonlinear link in the actuator, and the other RBFNN is applied to compensate for Audead zoneAy of the system feedforward channel.
The neural network structure compensator is shown in Fig 4.
u = y yI NN ( y ) A = D .
) = W TA .
) AE .
) y = W T A ( y y NN ) AE ( y y NN ) The estimation signal of f ( s ) is designed as:
NN NN
y = WT A y WI NN ANN ( y ) WT A ' y WI NN ANN ( y ) WANN ANN ( y ) c ( t ) Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks .
Journal of Robotics and Control (JRC) ISSN: 2715-5072 c ( t ) = W T A ' y WI NN ANN ( y ) AE NN ( y ) WT R WANN , y AE ( y y NN ) .
C W A ' ( A ) AE NN ( y ) AE ( u ) From Equation .
Equation .
, and Equation .
, we NN NN
NN NN
A 2 A (A) 2
A (A) W
W A " (A) W
NN A NN ( A ) AE NN ( y ) NN NN
NN NN
I A ( y WI A ( y ) )WA A ( y ) a ( y WI A ( y ) )W A ( y ) W
y AE ( u ) = WT A y WI NN
ANN ( y ) c ( t ) AE ( u ) I T A ' ( u )WAT A ( y ) =WT A ( u ) W
NN NN
T '
W A ( u )W A ( y ) c ( t ) AE ( u ) I A ( u )WA A ( y ) y AE .
) = W A .
) W
A A ( u )W A ( y ) c ( t ) A 2 A (A) 2
A (A) W
WM A " ( A ) W
ANN ( A ) AE FNN
A T A ' ( u )WI T A ( y ) AE ( u ) Oe W
NN NN
A T A ' ( u ) WI T A ( y ) d ( t ) NN NN A ( A ) AE NN ( w ) AE ( u ) AA 2 b W b C b2 W Theorem 1: The norm of the model mismatch term d ( t ) is bounded, and its upper bound is given by.
a b W 2 b W b d ( t ) C b1 W where b1 , b2 , b3 and b4 are constant, which can be computed.
From Equation .
, d ( t ) can be rewritten as:
) AE .
) A A (A) W
C W
NN A NN ( A )
) AE N A b c .
) AE
C W
where b1 = A ( A ) WNN
ANN ( A )
from Equation .
, we have:
A ( A ) ANN ( A ) ,
b3 = WM A " ( A ) ANN ( A ) AE FNN , b4 = WM A ' ( A ) AE FNN WM AE .
) A ( A ) AE NN ( y ) Theorem 1 actually gives an upper bound on the norm of the modeling mismatch term d ( t ) , and this conclusion is used to prove the stability of the entire closed-loop system.
The adaptive rule for the neural network compensator is formulated as follows:
A A ' (A) W
) C W
NN F A NN ( A )
b2 = WM A T A ' ( u )W T A ( y ) Oe c ( t ) AE ( u ) d .
) = OeW
NN NN
From Equation .
, and Equation .
we have:
A = y Oe WC T A ' ( u ) WANN
ANN ( y ) A ( A ) AE NN ( y ) c ( t ) C WM A ' ( A ) AE FNN
NN NN
I T A ( y ) AE ( y )E c ( t ) = WT EA ' y W
NN NN
A , y AE.
y ) WT R W E W=I AoA ' ( u ) W I TA ( .
r Oe k Ao r W
E A
IT '
NN = AoA NN ( y ) rW A ( u ) Oe k1Ao r WNN Oek2 Ao r W here k1 , k 2 A 0 .
Ao = AoT .
Ao = Ao T i.
CONTROL DESIGN AND STABILITY ANALYSIS
The tracking error vector and the sliding mode function are defined as follows:
e ( t ) = Ad Oe A .
r = eA A e Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks Journal of Robotics and Control (JRC) ISSN: 2715-5072 where A = diag ( A1 .
A2 ,.
An ) is the constant gain matrix.
By taking derivative .
and substituting .
into we have:
a= Oe r A d A e Use Equation .
, we have:
M NN rAA= M NN a d A e Oe C NN r C NN A d A e GNN FNN A 0 Oe A .
AA CNN A d A e GNN FNN f ( .
= M NN a d Ae represents model information.
In fact, f ( .
of the model is unknown.
Therefore, it is necessary to determine the approximate value.
In this paper, an RBF neural network is employed to estimate f ( .
The a E x= e Ad Ad Ad The ideal control signal is determined as follows:
y = fI kr r A smc fI is the output approximation signal of RBF neural network.
A smc is the sliding mode control, and which is defined as follows:
E kW 2 E r A smc = E y E k p sgn ( r ) E 4 E r L .
) = 1 T Oe1
1 T Oe1
r M NN r Wa
y k y Wy W Ao W
1 T Oe1
WAA
NN Ao WNN La ( t ) = r T M NN rA r T M NN r WyT k yOe1WAy AA Oe Wa Ao W W T Ao Oe1W La ( t ) = r T ( M NN rA CNN r ) WyT k yOe1WAy .
T Oe1 AA Wa Ao W WNN Ao Oe1WNN Substituting Equation .
into Equation .
We have:
LAA
( t ) = r T Oekr r WCT A ' ( u )WNNT ANN ( y ) A T A ' ( u ) WI T A ( y ) r T OeW
NN NN
Oe r T d ( t ) Oe f fI A smc Oe A 0
T Oe1 a T Oe1
Oe1
y k y W y W Ao W WNN Ao WNN
LAA
( t ) = Oe r T kr r Oe r T WT A ' ( u )WI NNT ANN ( y ) .
The adaptive rule of the neural network is formulated as .
Substituting Equation .
, and Equation .
into Equation .
, we have:
T Oe1 A r T WCT A ' ( u )Wa NN A NN ( y ) Wy k y W y T Oe1 AA Ao W WNN Ao Oe1W Oe r T d ( t ) r T fA Oe r TA r TA Substituting Equation .
, .
and Equation .
into Equation .
, we have:
LAA
) = Oe r T kr r r T ( AE 0 A 0 ) W yT k r WI y
I WT k r W
WAA2 k1 r W
NN 1
M NN rA = OeC NN r Oe kr r WCT A ' ( u )WANN ANN ( y ) A T A ' ( u )WI T A ( y ) Oe d ( t ) f
OeW
NN NN
Using property 2 into Equation .
, we have:
with k p C AE 0 A 0 WI y = k yA ( r ) r T Oe kk y r WI y The derivative of ya.
along to time is:
= OeC NN r Oe A f A 0
WANN
k2 r W Oe fI Oe A smc A 0 Theorem 2: Consider the dynamics of an n- link robot .
, and assuming that the neural networks of the ideal weight Wy C WyM .
W C WM .
WNN C WNNM , controller is selected in Equation .
, the online learning laws are given in Equations .
, .
, and a sliding mode control is signed in Equation .
, then the tracking error of the control system is eventually uniformly bounded and stable, and the tracking error decreases with increasing gain of k r A 0 .
E kW y2 r Oer T E k p sgn ( r ) E E 4 r ) = Oer T kr r k r tr EW y Wy Oe Wy E k1 r tr EWAA W Oe W E Oe r d ( t ) E Oer kW y2 r tr WAA NN k 2 WCNN By using The Lyapunov function is designed as follows:
Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks r tr WAA NN k1 WNN Oe WNN ( WNN Oe WNN ) ) .
Journal of Robotics and Control (JRC) ( ( a y Wy Oe Wy
ISSN: 2715-5072
) ) C Wy Wy Oe W y Substituting Equation .
into Equation .
, we have:
AA W Oe W
C W
( (
a WOeW
)) C W W Oe W
a NN k1
( WNN Oe WNN ) ) W E
LAA
( t ) C Oe r kr r Oe k r EE Wy Oe y EE
NNM
k1 r W
tr Wa
NN k 2 WCNN
AA W
Oek2 WNN Oe W
Oe r
WNNM Oe k2 WNN
Oe r kr r
k1 r W
( g ( WA NN ) Oe b4 ) ( WaNN ) = k2 WNN ( k1 Oe 2k2 WNNM Oe b2 ) WNN
Oe k1WNNM k 2 WNNM b3
( WyM Oe Wy ) AA W Oe W Oe r T k1 r W
) WA
( WNNM Oe WNN ) Oe r d .
) ) = g ( WNN ) C We have:
kW y A ( WAANN ) .
WNN A 0A ,
) WA NN ,
C = inf g
r E k2 W
WNNM E
NNM
NNM k 2 WiM b3 We have:
Oe r k2 W
Oe r E Oe k1 E WM E E
E 4 EE
k1 EE E
a W 2 2k W
C k2 W
NNM
( t ) C k r Wy
(( k W
b EE
A Oe 1 E W 1 EE
Oe r k1 E W
2 EE M k1 EE E
C k2 WNNM W
WNN WNNM
NNM k 2 WNN
k2 WNNM W
LA( t ) C Oe r k r r
AA C W
Ie Oe WNN Oe W
NNM Oe WNN
Oe r ( k1 Oe 2k2 WNNM Oe b2 ) W
WNN Oe W
NN C WNN Oe WNN C WNN Oe WNNM
WNN
AA Oe k W 3 EE
r EE ( k1WM b1 ) W
AA W C W u
With WNN Oe W
tr WAA
WNN Oe WNN
NN k 2 WCNN
NNM Oe
LAA
( t ) C Oe r EE kr r k1 W EE
E W
E A NN WNN Oe W
C k W OeW
a b W 2 b W
Oe r EE b1 W
NN F
NN b4 E
( WNN Oe WNN ) ) = k2 WNN Oe WNN
tr (WAA NN ( WNN Oe WNN ) ) a tr W T ( W Oe W ) C k W Oe W
k2 WNN Oe W
( NN NN NN ) 2 NN NN
a 3 k r W W Oe W
Oe r k2 W
C k1 EE WNN WNN Oe WNN EE
Ck W
Oe W
AA W 2 2k W 2 W
r EE k2 W
NNM
NNM E
AA W Oe W
C W
L ( t ) C Oe r kr r k1 E W Oe E WM E E E
k1 E E E
E 1 E
Oe r E Oe k1 E WM E h W
E 4 E
r ( C b4 ) So LA( t ) C 0 with:
Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks E ) EE .
Journal of Robotics and Control (JRC) ISSN: 2715-5072
The parameters of the two-link robots are given r C
bE 1 E
k1 E WM 1 E C b4
A Oe 1 EE W b1 EE E C 1 k E W b1 E
k1 E W
k1 E E
C b4
M12 E
C12 EE
E 11
E .
NN EC
E 21 M 22 E
E 21 C22 E
E G1 E
E F1 E
EA 01 E
GNN = E E .
FNN = E E .
A = E
EG E
EF E 0
E 02 E
E M11
M NN = E
M = ( m1 m1 ) l 2 m l 2 2m l l cos (A2 ) = q1 q2 q3 cos (A2 ) ( ) A bE 1 E C k1 E WM 1 E C b4 Then the convergence condition of the system determined as:
bE 1 E k1 E WM 1 E C b4 k1 E kr min ( ) ( ) ( )( ( )( C = m l l sin (A ) a= q sin (A ) A .
V22 = 0
C = Oem l l sin A a = Oeq3 sin A A .
a C = Oem l l sin A A A = Oeq3 sin A A A 2 1 2
2 1 2
( )
( )(
= q cos (A ) q cos (A )(A A ) q cos (A ) 2 1 2
G = m gl cos (A )(A A ) = q cos (A )(A A ) 2 1 2
2 1 2
G = m m gl cos A m gl cos A A A 2 1 2
E WM 1 E
A C 1 k E W b1 E C b4
( )
M = M = m l 2 m l l cos A = q2 q3 cos A .
M = m l = q2
FNN
( )
( )
( )
E AA
E3A1 0.
5sign A1 E EA E E 0.
5sin 20t E
E A = E 01 E = E
E 0
EA E E 0.
5sin 20t E
E3a
5sign A1 E
AA = E
where m1 is the masses of link 1, and m2 is the mases of link
EE E 1 E
E EE
Oe1 WNN C max Eh E k1 E WM E C b4 E E .
EE EE 4 E
IV.
SIMULATION RESULTS
The ANNs is simulated on Matab - Simulink for two-link robots to demonstrate the effectiveness of the proposed l1 is length of link1, and l2 is length of link2.
g = 9,8.
/ s 2 ) is acceleration of gravity.
The robotic q = Au q1 q2 q3 q4 q5 Ay = Au 2,92 0, 78 0,88 3, 05 0,87 Ay A = AuA1 A 2 Ay are the positions of link1, link2.
The simulation steps are as follows:
Set A .
Ao.
Ao , k1 , k2 , kr , k p , k = 0.
01, k y = 20 Step1: Initial the parameter of the NNs with random values.
yco2 , yco2
Step 2: Update the NNs inputs x = e
A dT a A dT EE
I W
WI y .
Step 3: Calculate the output signal of RBFNN via .
and the Control input before the dead zone via .
Step 4: Update the weight of the controller via .
, .
yco1 , yco1 Step 5: Calculate the control input Step 6: Return to step 3 Fig.
The two-link robotics Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks Journal of Robotics and Control (JRC) ISSN: 2715-5072 The desired position trajectories of the robotic will be chosen by: A d = [A d 1 A d 2 ]T = .
, 2A t ) cos.
, 2A t )]T and Initial position joints and initial velocities of joints respectively A 0 = [A 01 A 02 ]T = .
, 0 0,.
T .
ad ) and a = [A A ]T = .
, 0 0,.
T .
Parameters of dead zone will be chosen dl = Oe10.
d r = 10.
) = u Oe d r .
) = u d l The structure of NNs is characterized by n=7 nodes.
The center vector is defined as: a= [-0.
1 0 0.
1 0 0.
1 0 0.
1 0 0.
1 0 0.
and width of the Gaussian basis function is b = 10 .
The initial weights of the neural networks are W = 15* eye.
Fig.
Tracking positions, control inputs, and position tracking errors of ANNs without compensation Dead-zone.
AF with compensation Dead- zone, and ANNs with compensation Dead-zone.
Dead-zone and Dead-zone estimation in the first simulation Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks Journal of Robotics and Control (JRC) ISSN: 2715-5072 The structure of NNs compensate for dead zone.
N=7 nodes.
E Oe40 Oe25 Oe10 0 10 25 40 E the center vector: c = E E Oe40 Oe25 Oe10 0 10 25 40 E and width of the basis function d = 100 , initial weights of neural network W ( 0 ) = Au 0 0 0 0 0 0Ay The proposed controller parameters are chosen as:
E200 0 E
k1 = 0.
k2 = 0, 0001.
Ao = EE 0 200EE .
Ao= E500 0 E EE 0 500EE .
kr = 20.
A = 5.
k p = 0,1.
Fig.
Tracking positions, control inputs, and position tracking errors of ANNs without compensation Dead-zone.
AF with compensation Dead- zone, and ANNs with compensation Dead-zone.
Dead-zone and Dead-zone estimation in the second simulation case Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks Journal of Robotics and Control (JRC) ISSN: 2715-5072 The mean square error (MSE) of the position tracking response is defined to quantify the respective control performance as:
MSE =
Eu EA .
) Oe A d .
) EE T t =1 E Fig.
Tracking positions, control inputs, and position tracking errors of ANNs without compensation Dead-zone.
AF with compensation Dead- zone, and ANNs with compensation Dead-zone.
Dead-zone and Dead-zone estimation in the third simulation case In which.
T denotes the total number of sampling The normalized mean square error (NMSE) of the Vu Thi Yen.
Dead-zone Compensation for Robotic using Neural Networks Journal of Robotics and Control (JRC) ISSN: 2715-5072 position tracking response, calculated using a per-unit value of 1 rad, is employed to evaluate the control performance.
To verify the effectiveness of the proposed controller, numerical simulations were conducted under three distinct Case 1 (Nominal Operatio.
: The system and robot parameters are selected as above Case 2 (Parameter Uncertaint.
: To test robustness, the controller parameters remained fixed while the robotAos q = Au q1 q2 q5 Ay = Au3,84 1 1,5 3,92 1,96Ay Case 3 (Disturbance Rejectio.
: The system setup was identical to case 1.
However, the system was subjected to a square Ae wave disturbance with an amplitude of A5Nm, a period of 0.
1s, and a pulse with 50% of period, starting from t=2s.
To evaluate the effectiveness and robustness of the proposed controller, we have conducted comparisons the proposed controller with two other methods such as ANNs without CDZ (Ablation study to verify the necessity of dead zone compensatio.
and AF with CDZ (Adaptive fuzzy controller as a benchmar.
To ensure statistical rigor, the Normalized Mean Square Error (NMSE) was calculated for the steady- state phase ( t Ea Au 2 C 20Ay .
to exclude transient initialization effects.
The quantitative results for all three scenarios Nominal Operation (Case .
Parameter Variations (Case .
, and External Disturbances (Case .
Aiare summarized in Table 1.
TABLE II.
SIMULATION RESULT COMPARISON OF ANNS WITHOUT CDZ,
ANNS WITH CDZ.
AND AF WITH CDZ.
NMSE
ANNs
CDZ
ANNs
CDZ
CDZ
Case 1 Case 2 Case 3 Link Link Link Link Link Link By comparing the results of the proposed method (ANNs with CDZ) with the ANNs without CDZ in Fig 7 and Table 1, the necessity of the specific dead zone compensation scheme is evident.
Quantitative analysis in Table 1 (Case .
shows that the lack of compensation results in a significant Oe4 steady Ae state error (NMSE) of link 1 of 9, 594 C 10 .
contrast, the proposed controller reduces this error by two Oe4 orders of magnitude to 0, 070 C 10 .
Furthermore.
Fig 7 reveals that the ANNs without dead zone compensation suffers from limit cycle oscillations around the equilibrium This phenomenon is caused by the discontinuous nature of the dead-zone when the joint velocity crosses zero.
The proposed controller effectively eliminates these oscillations by learning the inverse dead-zone model, thereby linearizing the actuator behavior for the tracking controller.
In addition, the superiority of the proposed controller is most evident in Case 2.
While the AF with CDZ degrades significantly, causing the link 2 NMSE to rises to Oe4 2, 330 C 10 , the ANNs with CDZ maintains superior Oe4 precision at 0,167 C 10 .
The control input in Fig.
8 reveals that AF with CDZ controller suffers from severe high frequency chattering to cope with the changed model.
contrast, the proposed controller achieves better accuracy with a smoother control signal.
This confirms that the ANNs with CDZ helps the system achieve higher accuracy.
In Case 3, when the system operates for 2s, disturbances are From Table 1 and Fig 9, we see that the NMSE of ANNs with CDZ method remains very low and more stable than the ANNs without CDZ and AF with CDZ.
Furthermore, even with disturbances, the dead zone estimation process still yields good results.
CONCLUSIONS
An adaptive controller using a neural network for a robotic system is presented to address unknown dynamics, external disturbances, and neural network dead-zone compensation, thereby improving tracking accuracy.
To cope with the uncertainties in the robot dynamics, a neural network compensator is designed by using two RBF neural networks.
One RBF neural network estimates the nonlinearities in the actuator, while the other compensates for the dead-zone in the system feedforward channel.
The online adaptive training laws and dead-zone estimation are derived from Lyapunov stability theory and approximation theory.
Simulation results on a 2-DOF robot demonstrate that the proposed method significantly reduces tracking errors and eliminates control chattering compared to AF with CDZ and ANNs without CDZ, particularly under parameter uncertainty.
However, several limitations of this study must be acknowledged such as: the validation is currently restricted to a simulation and the effectiveness of the RBFNN relies on the proper selection of center vectors and widths.
inappropriate choice may lead to local minima or slow Therefore, future work will focus on two areas:
first, implementing the proposed method on a physical 2DOF industrial robot to check the robustness and efficiency of controller.
Second, researching learning techniques to automatically tune the RBF parameters, thereby reducing the dependency on manual initialization.
FUNDING STATEMENT
This research was supported by Hanoi University of Industry.
Hanoi.
Vietnam, under Project Grant No.
14-2025-RD/HaaHCN.
REFERENCES