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Modeling, Simulation and Position Control of 3DOF Articulated Manipulator

Article  in  Indonesian Journal of Electrical Engineering and Informatics (IJEEI) · October 2014

DOI: 10.11591/ijeei.v2i3.119

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Indonesian Journal of Electrical Engineering and Informatics (IJEEI)
Vol. 2, No. 3, September 2014, pp. 132~140
ISSN: 2089-3272  132

Modeling, Simulation and Position Control of 3DOF

Articulated Manipulator

Hossein Sadegh Lafmejani1 , Hassan Zarabadipour2

Faculty of Technical and Engineering
Imam Khomeini International University, Qazvin, Iran, Islamic Republic Of
Tel: +98 (281) 8371753
1 2
e-mail: h_sadegh@ikiu.ac.ir , hassan.zarabadipour@gmail.com

Abstract
In this paper, the modeling, simulation and control of 3 degrees of freedom articulated robotic
manipulator have been studied. First, we extracted kinematics and dynamics equations of the mentioned
manipulator by using the Lagrange method. In order to validate the analytical model of the manipulator we
compared the model simulated in the simulation environment of Matlab with the model was simulated with
the SimMechanics toolbox. A sample path has been designed for analyzing the tracking subject. The
system has been linearized with feedback linearization and then a PID controller was applied to track a
reference trajectory. Finally, the control results have been compared with a nonlinear PID controller.

Key Words: Feedback linearization, kinematic, Manipulator, Robot, SimMechanics

1. Introduction
Industrial robots are widely used in various fields of application nowadays. So
production of them is increasing rapidly. Manipulators are a kind of industrial robots which has
been attracted so much attention of engineers, especially control and mechanics engineers.
Control engineering concentrate on designing the controllers in order to have the manipulators
operated with the best quality and less errors. Designing the controllers for manipulators has
several approaches like tracking and force control.
The control methods can be classified into three types: the first type is traditional
feedback control (PID and PD) [1-4]. The second type is adaptive control [5-11] and the third
type is the iterative learning control (ILC) [12-17]. Some other control methods, including the
robust control [18-20], inverse dynamics control [21-23], model based control, switching control,
and sliding mode-control can be in one or another way reviewed either as specialization and/or
combination of the three basic types, or are simply different names due to different emphasis
when the basic types are examined.
In this paper, the 3DOF articulated manipulator has been studied. The direct and
inverse kinematics has been obtained by geometrical calculates [21]. Then the dynamic model
of the system has been extracted by Lagrange method that is very powerful in modeling the
sophisticated mechanical systems. In this paper, a PID controller has been designed for 3DOF
robotic manipulator which has been linearized by feedback linearization and the results have
been compared with a nonlinear PID controller.

2. System Modeling
The first step of designing controllers for a system is modeling. In other word, we need
the physical characteristics or the mathematical equations of the system in order to design a
good controller. Modeling contains kinematics and dynamics. Kinematics is the motion science
that studies the position, velocity, acceleration and derivatives of them without regarding the
force and torque. Manipulator movement characteristics are studied in kinematics science for
robots and contain two main parts: forward kinematics and inverse kinematics. In other hand,
the relation between these movements and the force and torque is studied in dynamics science.

Received May 29, 2014; Revised August 3, 2014; Accepted August 26, 2014
IJEEI ISSN: 2089-3272  133

Figure 1. 3DOF articulated manipulator in Spherical coordinates for forward kinematic analysis

2.1. Direct Kinematics

If we state the end effector coordinates of manipulator based on the angles of the joints,
it means the forward kinematics. In other word, in forward kinematics, the measures of the joint
space are available and we want to determine the measures of coordinate space. In reality,
forward kinematics analyzing is a mapping from joint space to the coordinate space. According
to Figure 1 the forward kinematics of the 3DOF articulated manipulator has been determined as
below:

cos cos ∗ sin (1)

cos cos ∗ cos (2)

sin sin (3)

Where , and are the length of the links.

2.2. Inverse Kinematics

By inversing the forward kinematics definition we have inverse kinematics definition. By
these equations we can find the appropriate angles for the desired end effector coordinates.
According to the two definitions of kinematics, it is clear that the inverse kinematics is more
sophisticated than the inverse kinematics. According to the Figure 2 we have:

2  , (4)

(5)

(6)
 
(7)

(8)

2 , 2 cos , sin (9)

Modeling, Simulation and Position Control of 3DOF Articulated Manipulator (Hossein SL)
134  ISSN: 2089-3272

Figure 2. 3DOF articulated manipulator in Spherical coordinates for inverse kinematic analysis

2.3. Velocity Kinematics

In order to design a controller to track a path we need to have the relations between the
velocity of the joint and the velocity of the end effector that named velocity kinematics. In this
case, by differentiation of Eqs. (1) to (3) we have:

cos ∗ cos cos ∗ sin ∗ sin sin ∗

sin ∗ sin ∗ (10)

sin ∗ cos cos ∗ cos ∗ sin sin ∗

sin ∗ cos ∗ (11)

cos cos ∗ cos ∗ (12)

Regarding the vectors and θ , then we obtain:

(13)
0

Where S denotes to (Sin) and C denotes to (Cos) and J is the manipulator jacobian matrix and
determination of that is a base issue for all manipulators. The determinant of the 3DOF
manipulator is:

det sin sin sin sin 2 (14)

The roots of the above equation are the singular points of the manipulator. Singular points are
those in which the manipulator can’t move in a certain direction.

IJEEI Vol. 2, No. 3, September 2014 : 132 – 140

IJEEI ISSN: 2089-3272  135

Figure 3. Analytical modeling of 3DOF manipulator in matlab Simulink

2.4. Dynamic Modeling

The dynamical analysis of the robot investigates a relation between the joint
torques/forces applied by the actuators and the position, velocity and acceleration of the robot
arm with respect to the time. Dynamics of the robot manipulators is complex and nonlinear that
might make accurate control difficult. The dynamic equations of the robot manipulators are
usually represented by the following coupled non-linear differential equations which have been
derived from Lagrangians [21]:

Q , (15)

Where  is the inertia matrix,  , is the coriolis/centripetal matrix, is the gravity

vector, and Q is the control input torque. The joint variable  is an n-vector containing the joint
angles for revolute joints. The mentioned matrix of the 3DOF articulated manipulator can be
computed by:

1 1 1
1,1 cos cos cos
2 3 3
cos cos (16-1)

1,2 0 (16-2)

1,3 0 (16-3)

2,1 0 (16-4)

2,2 cos (16-5)

2,3 cos (16-6)

3,1 0 (16-7)

3,2 cos (16-8)

3,3 (16-9)

Modeling, Simulation and Position Control of 3DOF Articulated Manipulator (Hossein SL)
136  ISSN: 2089-3272

Figure 4. SimMechanics modeling of 3DOF manipulator in matlab simulink

1,1 sin 2 sin 2 sin 2

sin 2 cos sin (17-1)

2,1
sin sin   sin 2 sin 2
sin 2 sin 2   (17-2)

1
3,1 sin  
2
sin 2 cos sin   (17-3)

1,1 0 (18-1)

2,1 cos cos cos (18-2)

3,1 cos (18-3)

 
Figure 5. Comparing graph of analytical model with SimMechanics model

IJEEI Vol. 2, No. 3, September 2014 : 132 – 140

IJEEI ISSN: 2089-3272  137

Figure 6. modeling Error

3. SimMechanics Toolbox
SimMechanics is based on simulink, which is the research and analysis environment of
the controller and the object system in a cross-cutting / interdisciplinary [24]. Multi-body
daynamic mechanical systems can be analyze and modeled by SimMechanics and all works
such as control would be completed in the simulink envirement. This toolbox provides a plenty
number of corresponding real system components, such as: bodies, joints, constraints,
coordinate systems, actuators and sensors. Complex mechanical system can be created by
these modules in order to analyze the mechanical systems like manipulators. In this paper, the
toolbox has been used to analyze the 3DOF articulated manipulator.
For a validation of modeling of the system, the 3DOF manipulator has been designed in
SimMechanics and compared with the analytically modeled system. Figure 3 shows the simulink
design of the manipulator and Figure 4 shows the SimMechanics modeling of it. For this system
1  and 1  for all links.
As it was mentioned before, we used the SimMechanics toolbox for validation of
analytical modeling. The results of this study are brought in Figures 5 and 6. According to Figure
5, the outputs of two simulations are completely similar to each other and Figure 6 shows the
error between two simulations that is verified the accuracy of the analytical modeling.

4. Feedback Linearization and Control

In general condition, a manipulator with links is stated as a nonlinear system with
multi input and determining the feedback linearization conditions of them is more complex than
the single input systems, but has a similar idea. For a manipulator with degree of freedom we
regard the Eq. (15) and replace with a new variable . So we have:
 
, (19)

Modeling, Simulation and Position Control of 3DOF Articulated Manipulator (Hossein SL)
138  ISSN: 2089-3272

Figure 7. Inverse daynamic control block diagram

The is a positive definite matrix, so 0 and according to the Eqs. (15) and (19)
we have:

, , ⇒       (20)

Where is a auxiliary control input that would be designed. Of course, it is essential to mention
that, by this method the system could not be linear completely but also there is some
nonlinearity in it. The following control law has been used for designing controller:

(21)

Therefore:

0 (22)

By choosing the , , appropriately and using the Routh-Hurwitz stability criterion, we

have:

, , 0 ,  , , …, (23)

The block diagram of PID controller with feedback linearization named inverse dynamic
control has been brought in Fig. 7. For studying the operation of the inverse dynamics controller,
it has been compared with nonlinear PID controller [25]. For this goal, a circular reference path
has been regarded and the controllers have been test. The results of these two controllers have
been shown in Figs. 8 and 9. According to these figures, the inverse dynamics controller is
better than the other one. But it is important to mention that if we want to use inverse dynamics
controller, we'll need the all parameters of the manipulator accurately. It is clear that reaching
the parameter accurately is not possible practically and always we have some uncertainty in
system.

5. Conclusion
According to the paper the robot manipulator have complex nonlinear dynamic model
that makes its control so difficult. Although using the classic controllers are good but
uncertainties in manipulators is high. Thus, using the fuzzy controllers and intelligent method
like neural network is proposed for controlling these kinds of complex systems.

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IJEEI ISSN: 2089-3272  139

 
Figure 8. Inverse dynamic tracking control

Figure 9. PID tracking control

References
[1] Craig John J. Introduction to robotics: mechanics and control. Third Edition: Addison-Wesley. 1986.
[2] Qu ZH. Global stability of trajectory tracking of robot under PD control. Dynamics and Control. 1994;
4(1): 59-71.
[3] Kelly R. PD control with desired gravity compensation of robotic manipulators: a review. The
International Journal of Robotics Research. 1997; 16(1): 660-672.
[4] Chen Q, Chen H, Wang YJ, Woo PY. Global stability analysis for some trajectory tracking control
schemes of robotic manipulators. Journal of Robotic Systems. 2001; 18(2): 69-75.
[5] Craig John H. Adaptive control of mechanical manipulators: Addison-Wesley. 1988.
[6] Choi JY, Lee JS. Adaptive iterative learning control of uncertain robotic systems. IEE Proc. Control
Theory Appl. 2000; 147(2): 217-223.

Modeling, Simulation and Position Control of 3DOF Articulated Manipulator (Hossein SL)
 140  ISSN: 2089-3272

[7] Slotine JJ, Li W. On the adaptive control of robot manipulators. The International Journal of Robotics
Research. 1987; 6(3): 49-59.
[8] Li Q, Poo AN, Teo CL, Lim CM. Developing a neuro-compensator for the adaptive control of robots,
IEE Proc. Control Theory Appl. 1996;142(6): 562-568.
[9] Li Q, Poo AN, Teo CL. A multi-rate sampling structure for adaptive robot control using a neuro-
compensator. Artificial Intelligence in Engineering. 1996; 10(1): 85-94.
[10] Tomei P. Adaptive PD controller for robot manipulators. IEEE Trans Robot Automation. 1991; 7(4):
565-570.
[11] Li Q, Tso SK, Zhang WJ. Trajectory tracking control of robot manipulators using a neural-network-
based torque-compensator. Proceedings of the Institution of Mechanical Engineers, Part I. Journal of
Systems and Control Engineering. 1998; 212(5): 361-372.
[12] Arimoto S, Kawamura S, Miyasaki F. Bettering operation of robots by learning. Journal of Robotic
Systems. 1984; 1(2):123-140.
[13] Kawamura S, Miyazaki F, Arimoto S. Realization of robot motion based on a learning method. IEEE
Transactions on Systems, Man, and Cybernetics. 1988; 18(1):123-6.
[14] Tayebi A. Adaptive iterative learning control for robot manipulators. Proc. of the American control
conference, Denver, CO, USA: June 2003; 5: 4518-23.
[15] Kuc TY, Nam K, Lee JS. An iterative learning control of robot manipulators. IEEE Trans Robot
Automation. 1991; 7(6): 835-842.
[16] Chen YQ, Moore KL. PI-type iterative learning control revisited. Proc. of the American control
conference, Anchorage, AK, USA: May 2002; 2138-43.
[17] Yan XG, Chen IM, Lam J. D-type learning control for nonlinear time-varying systems with unknown
initial states and inputs. Trans Inst Measure Control. 2001; 23(2): 69-82.
[18] Cheung JWF, Hung YS, Robust learning control of a high precision planar parallel manipulator.
Mechatronics. 2009; 19(1): 42-55.
[19] Han J, Park J, Chung W. Robust coordinated motion control of an underwater vehicle-manipulator
system with minimizing restoring moments. Ocean Engineering. 2011; 38(10): 1197-1206.
[20] Li Z, Ge SS, Wang Z. Robust adaptive control of coordinated multiple mobile manipulators.
Mechatronics. 2008; 18(5-6): 239-250.
[21] Craig John J. Introduction to Robotics: Mechanics and Control, Third Edition: Prentice Hall, New York.
2005.
[22] Craig JJ, Hsu P, Sastry SS. Adaptive control of mechanical manipulators. The International Journal of
Robotics Research. 1987; 6(2): 10-20.
[23] Spong MW, Ortega R. On adaptive inverse dynamics control of rigid robots. IEEE Transactions on
Automatic Control. 1990; 35(1): 92-95.
[24] Zheng-wen LI, Guo-liang Zhang, Wei-ping Zhang, Bin JIN. A Simulation Platform Design of Humanoid
Robot Based on SimMechanics and VRML. Procedia Engineering. 2011; 15: 215-219.
[25] Cervantes I, Ramirez JA. On the PID tracking control of robot manipulators. Systems & Control
Letters. 2001; 42(1): 37-46.

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