International Journal of Scientific Engineering and Research (IJSER) www.ijser.in ISSN (Online): 2347-3878
Volume 2 Issue 3, March 2014

Trajectory Tracking of a 2-Link Robotic Manipulator
Using Adaptive Terminal Sliding Mode Controller
Saurav Chanda1, Pranjal Gogoi2
1

Dept. of Electronics & Communication Engineering, Dibrugarh University Institute of Engineering & Technology
Dibrugarh University, Assam, India
2
Dept. of Electronics & Communication Engineering, Dibrugarh University Institute of Engineering & Technology
Dibrugarh University, Assam, India

Abstract: This paper proposes an adaptive terminal sliding mode controller which combines adaptive control and sliding mode control to control a 2-link nonlinear robotic manipulator with uncertain parameters. We use an adaptive algorithm based on the concept of sliding mode control to alleviate the chattering phenomenon of control input. Adaptive laws are developed to obtain the gain of switching input and the boundary layer parameters. The stability and convergence of the robotic manipulator control system are guaranteed by applying the Lyapunov stability theorem. Simulation results demonstrate that the chattering of control input can be alleviated effectively. The proposed controller scheme can assure robustness against a large class of uncertainties and achieve good trajectory tracking performance.
Keywords: Robotic manipulators, sliding mode control, adaptive law, Lyapunov stability theorem

1. Introduction
A chattering free adaptive terminal sliding mode controller for uncertain systems is proposed in this paper. The actual control law is obtained by integrating the discontinuous derivative control signal and hence it is continuous. The sliding mode control is based on the design of a high-speed switching control law that drives the system’s trajectory onto a user-chosen hyperplane in the state space, also known as sliding surface.
The main feature of sliding mode control are the following: (1) fast response and good transient performance; (2) robustness against a large class of perturbations or model uncertainties; and (3) the possibility of stabilizing some complex nonlinear systems which are difficult to stabilize by state feedback control laws. Chattering phenomenon can cause some problems such as saturation and heat for mechanical parts of robot manipulators or drivers. To reduce or eliminate the chattering, various papers have been reported by many researchers and classified in two most important methods, namely, boundary layer saturation method and estimated uncertainties method.
Non linear control methodologies are more general because they can be used in linear and non linear systems. These controllers can solve different problems such as, invariance to system uncertainties and resistance to the external disturbance.
The most common non linear methodologies that have been proposed to solve the control problem consist of the following methodologies: •
•
•
•
•
•

Feedback Linearization Control Methodology
Passivity-Based Control Methodology
Sliding Mode Control Methodology
Robust Lyapunov-Based Control Methodology
Adaptive Control Methodology
Artificial Intelligence-Based Methodology

The requirement of prior knowledge about the uncertainty bands for designing terminal sliding mode controllers is not a

Paper ID: J2013152

necessary requirement in the proposed controller. Trajectory tracking of a 2-link robotic manipulator which is a non-linear system with mismatched uncertainties also considered in our simulation study. Simulation results demonstrate that the proposed control strategy is successful in eliminating the undesired chattering in the control input while ensuring satisfactory stabilization as well as tracking performances.
Hence the proposed controller is suitable for practical applications. .

2. Description
2.1 Tracking Control of a Robotic Manipulator
Dynamics of an n-link robotic manipulator can be expressed as,
(1.0)
M(q) + C(q, ) + G(q) = τ + represent the position, velocity and where q, , ∈ acceleration of the joints respectively,
M(q) =
(q) + ΔM(q) ∈ stands for the inertia matrix,
C(q, ) = (q, ) + ΔC(q, ) ∈
(q) + ΔG(q) ∈ is the centripetal Coriolis matrix, G(q) = is the joint torque vector and is the gravitational vector, τ ∈
∈
is the disturbance torque vector. Here
(q),
(q, ), (q) are the nominal terms and ΔM(q), ΔC(q, ),
ΔG(q) represent the perturbations in the system matrices.
Then the dynamic model of the robotic manipulator can be written as,
(q) + (q, ) +
(q) = +
+ F(q, , ) (1.1) where F(q, , ) = −ΔM(q) − ΔC(q, ) − ΔG(q) ∈ is the lumped system uncertainty which is bounded by the following function
||F(q, , )|| ≤ + ||q|| + || ||2 (1.2) and are positive constants. where ,
Suppose the control objective is to make the robotic manipulator track a reference trajectory. Let and q be the
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International Journal of Scientific Engineering and Research (IJSER) www.ijser.in ISSN (Online): 2347-3878
Volume 2 Issue 3, March 2014 desired and actual position vectors. The tracking error and its derivatives are defined as e=q− , = − and = −
Using (1.1),
=
[ +
+ F(q, , ) – (q, ) –
(q)] –
(1.3)

2.2 Adaptive Terminal Sliding Mode Controller where, =

+

[

The time derivative of (1.3) yields
[ −

=
(q, ) −

(q, )

(q)] −

) +
(

(

) +

(q)]

[τ −

(q))] +

+

+
(q,

+

+
F(q,

,

)=

[τ −

(q))]+

(q, ,
[
(q,

)

−

(q,

,

) =

such that,
(q, , ) ≤
Here

,

(q,

,

) +

F(q, , )

+

||q|| +

and

Defining

(1.5)

( −

adaptation

error

the

(2.2)

=

(2.3)
)

(2.4)

Where positive tuning parameters. The dead zone technique is used to modify the adaptive tuning law as =

(

0,
={
0,

)+c

The nonsingular terminal sliding manifold (NTSM) for an nlink robotic manipulator is chosen as σ=s+β (1.7)

where Here β = diag(

,

, ....,

) is a design matrix.

,
0,
(2.5) positive constant. where Now, the adaptive terminal sliding mode control law for the robotic manipulator is obtained as,

Taking the derivative of (1.7) yields
=

+

+

(2.6)

+ β(p/q)

= β(p/q)

as

are to be estimated by using the

)

are positive constants.

where c = diag( , ...,
) is a design matrix. The first and second derivative of (1.6) can be obtained as,
= +c

) is a positive matrix.

and

parameters adaptation law.

Let us consider the linear sliding surface as, s = + ce (1.6)

+c =

the

=

Remark 5:5: The assumptions in the above inequalities are valid as the input disturbance is assumed to be bounded, i.e.
|| || < χ where χ is a positive constant. Furthermore, the modeling uncertainity F(q, , ) is also bounded by the assumption
+ ||q|| +
.
||F(q, , )|| ≤

=

(2.1)
……….

Here
+

+

+

−

+ F(q, , ) (1.4)

where

(2.0)

( + (q/p)

(1.8)

For an n-link robotic manipulator (1.0), if the NTSM manifold is chosen as (1.7), then the tracking error e will converge to zero if the time derivative of the control input is selected as,
=
+
(1.9)

Paper ID: J2013152

A Lyapunov function is defined as V = control law it is easy to find that,
=
=
=

and using the

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International Journal of Scientific Engineering and Research (IJSER) www.ijser.in ISSN (Online): 2347-3878
Volume 2 Issue 3, March 2014

Clearly,

the

above equation implies that if, the control law forces the sliding manifold σ to zero in finite time.

physical parameters of the two-link robotic manipulator considered in the simulation study

Using Lyapunov stability criterion,
The NTSM manifold σ can be shown to possess finite time reachability to zero which ensures that the tracking error of the converges to zero in finite robotic manipulator e = q − time. 3. Simulation Results
The proposed chattering free adaptive terminal sliding mode
(TSM) controller is applied for trajectory tracking of a two-link rigid robotic manipulator shown in Fig.5.4. For the above twolink manipulator, the dynamic equation (5.27) has the following parameters,
M(q)=

Table 1.0: Physical parameters of two-link robotic manipulator
The

reference

as disturbances considered are and proposed controller signals

are and . The inertial states are selected
. The external

are

. The parameters of the selected as and .
C(q, ) =
The simulations are carried out in MATLAB – Simulink platform by using ODE 4 solver with a fixed step size of
0.005sec.

G(q) =

The tracking response and control inputs obtained by using the
NTSM controller.

Figure 1.1: Configuration of a two-link robotic manipulator is the angular position vector where
Here q(t) =
(t) are the angular positions of joints 1 & 2. M(q) is the inertia matrix, C(q) is the centripetal Coriolis matrix. G(q) is the applied torque. The is the gravity vector and two-link robotic manipulator has four inner states
,
,
,
two output states and and two inputs
. Friction terms are ignored. Table 1.0 lists the and Paper ID: J2013152

Figure 1.2: Output tracking response of Joint 1 & Joint 2 with the controller

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International Journal of Scientific Engineering and Research (IJSER) www.ijser.in ISSN (Online): 2347-3878
Volume 2 Issue 3, March 2014

Figure 1.5: Control input of Joint 1 & Joint 2 with proposed controller Figure 1.3: Control input of Joint 1 & Joint 2 with proposed controller Simulation results obtained by applying the proposed adaptive
TSM control laws (2.0) and (2.6) are shown in Fig. 1.4 - Fig.
1.7. It is observed from Fig. 1.4 that both the joints 1 and 2 track the reference trajectory faithfully. The control inputs applied to both the joints show no chattering as is evident in
Fig. 1.5. The convergence plots of the estimated parameters
,
are shown in Fig. 1.6. The sliding surfaces and the sliding manifolds are plotted in Fig. 1.7 which confirms that these converge to zero quickly.

&
Figure 1.6: Estimated parameters proposed adaptive tuning method

using the

Figure 1.4: Output tracking response of Joint 1 & Joint 2 with the controller

Paper ID: J2013152

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International Journal of Scientific Engineering and Research (IJSER) www.ijser.in ISSN (Online): 2347-3878
Volume 2 Issue 3, March 2014 manipulator which is a nonlinear system with mismatched uncertainty is considered which demonstrates the efficiency of the proposed control strategy. Simulations performed on a twolink robotic manipulator demonstrate the effectiveness of the proposed controller.

5. Future Scope
Discrete Sliding Mode Controllers will be easier to implement as compared to microcontrollers and digital signal processors
(DSPs) which can also be used to control a huge number of continuous systems. The proposed design method may be extended by using intelligent controllers based on Neural
Networks and Fuzzy Logic to incorporate flexibility and intelligence. References

Figure 1.7: Sliding surfaces and sliding manifolds using the proposed controller
The output and input performances of the proposed adaptive
TSM controller as well as the controllers designed by Feng et al. for the two-link robotic manipulator are tabulated in Table
1.2. It is noted that the proposed adaptive TSM controller offers comparable tracking performance by applying a smoother control input having minimal total variation as compared to the controllers designed by Feng et al. Moreover, the overall control energy spent in the case of the proposed adaptive TSM controller is not more than those in the other methods. Table 1.1: Comparison of controller performance
Controller Performance
Types Of Controller

Feng et al.

Proposed adaptive
TSM controller

Joint
1
Joint
2
Joint
1
Joint
2

IAE

Total
Variation

2-norm of input 14.8
5

205.01

72.22

7.22

127.92

207.79

7.01

114.29

114.61

4.05

74.02

140.03

4. Conclusion
An adaptive terminal sliding mode (TSM) controller is proposed where the nonsingular terminal sliding manifold guarantees fast and finite time convergence. The proposed adaptive TSM controller is successfully applied for stabilization. Trajectory tracking of a two-link robotic

Paper ID: J2013152

[1] V. I. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electromechanical Systems. Taylor & Francis, London,
UK, 1999.
[2] Y. Feng, X. Yu, and Z. Man, “Non-singular terminal sliding mode control of rigid manipulators,” Automatica, vol. 38, pp. 2159 –2167, 2002.
[3] V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Transactions on Automatic Control, vol. 22
(2), pp. 212–222, 1977.
[4] K. K. Young, Variable Structure Control for Robotics and
Aerospace Systems. Elsevier, Amsterdam, The
Netherlands, 1993
[5] G. Bartolini, A. Ferrara, and E. Punta, “Multi-input second-order sliding-mode hybrid control of constrained manipulators,” Dynamics and Control, vol. 10, (3), pp.
277–296, 2000.
[6] V. I. Utkin, Sliding Modes in Control and Optimization.
Springer, Berlin, 1992.
[7] W. Gao and J. C. Hung, “Variable structure control of nonlinear systems: A new approach,” IEEE Transactions on Industrial Electronics, vol. 40, (1), pp. 45–50, 1993.
[8] I. Boiko, L. Fridman, and M. I. Castellanos, “Analysis of second-order sliding-mode algorithms in the frequency domain,” IEEE Transactions on Automatic Control, vol.
49, (6), pp. 946–950, 2004.
[9] K. D. Young, V. I. Utkin, and ¨Umit ¨Ozg¨uner, “A control engineer’s guide to sliding mode control,” IEEE
Transactions on Control System Technology, vol. 7, (3), pp. 328–342, 1999.
[10] Z. Qu and D.M Dawson, Robust Tracking Control Of
Robot Manipulators. Piscataway, NJ: IEEE Press, 1996.
[11] M.W. Spong, “On the robust control of robot manipulators”, IEEE Transactions on Automatic Control,
Vol. 37, no. 11, pp. 1/82-1/86, 1992
[12] Harashima, F., H. Hashimoto and K. Maruyama, 1986.
Practical robust control of robot arm using variable structure system. IEEE Conference, pp: 532-539.
[13] Slotine, J.J.E. and S.S. Sastry, 1983. Tracking control of nonlinear systems using sliding surfaces, with application to robot manipulators. Int. J. Control, 38 (2): 465-492.
[14] Curk, B. and K. Jezernik, 2001. Sliding mode control with

22 of 23

International Journal of Scientific Engineering and Research (IJSER) www.ijser.in ISSN (Online): 2347-3878
Volume 2 Issue 3, March 2014 perturbation estimation: Application on DD Robot
Mechanism, Robotica, pp: 641-648.
[15] Hsu, F.Y. and L.C. Fu., 1995. Nonlinear control of robot manipulators using adaptive fuzzy sliding mode control.
IEEE Conference on Intelligent Robots and Systems, pp:
156-161.
[16] Yu, W., A.S. Poznyak and E.N. Sanchez, 1999. Neural adaptive control of two link manipulator with sliding mode compensation. IEEE Conference on Robotics and
Automation, pp: 3122-3127.
[17] Lu, X., 2007. An investigation of adaptive fuzzy sliding mode control for robotic manipulators, M.S. Thesis,
Carleton Univ., Ottawa, Ontario.
[18] Sharma, R. and M. Gopal, 2008. A Markov game-adaptive fuzzy controller for robot manipulators. IEEE Trans. On
Fuzzy Systems, 16 (1): 171-186.
[19] J. J. Slotine and S. S. Sastry, “Tracking control of nonlinear systems using sliding surfaces with application to robot manipulators,” International Journal of Control, vol. 38, pp. 465 – 492, 1983.
[20] R.-J. Wai and L.-J. Chang, “Adaptive stabilizing and tracking control for a nonlinear inverted-pendulum system via sliding-mode technique,” IEEE Transactions on
Industrial Electronics, vol. 53(2), pp. 674–692, 2006.
[21] Y. J. Huang, T.-C. Kuo, and S.-H. Chang, “Adaptive sliding-mode control for nonlinear systems with uncertain parameters,” IEEE Transactions on Systems, Man, And
Cybernetics-Part B:Cybernetics, vol. 38, (2), pp. 534–539,
2008
[22] P. Li and Z.-Q. Zheng, “Robust adaptive second-order sliding-mode control with fast transient performance,” IET
Control Theory and Applications, vol. 6, (2), pp. 305–312,
2012.
[23] Y. Feng, X. Han, Y. Wang, and X. Yu, “Second-order terminal sliding mode control of uncertain multivariable systems,” International Journal of Control, vol. 80, (6), pp.
856 – 862, 2007.
[24] H. Sira-Ramirez, “Non-linear discrete variable structure system in quasisliding mode,” International Journal of
Control, vol. 54, (5), pp. 1171–1187, 1991.
[25] F. Plestan, Y. Shtessel, V. Bregeault, and A. Poznyak,
“New methodologies for adaptive sliding mode control,”
International Journal of Control, vol. 83, (9), pp. 1907–
1919, 2010.

Author Profile
Saurav Chanda is pursuing 3rd Year B.Tech in Electronics &
Communication Engineering from Dibrugarh University
Institute of Engineering & Technology, Dibrugarh University,
Assam. He is quite interested in the field of Robotics & Non
Linear Control from the very outset.
Pranjal Gogoi is pursuing 3rd Year B.Tech in Electronics &
Communication Engineering from Dibrugarh University
Institute of Engineering & Technology, Dibrugarh University,
Assam. He is quite interested in the field of Automatic Control
System & Stochastic Control from the very beginning.

Paper ID: J2013152

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