Indonesian Journal of Electrical Engineering and Informatics (IJEEI)
Vol. 1, No. 3, September 2013, pp. 99~108
ISSN: 2089-3272, DOI: 10.11591/ijeei.v1i3.80
99
Modeling, Simulation and Position Control of 3 Degree
of Freedom Articulated Manipulator
Hossein Sadegh Lafmejani*, Hassan Zarabadipour
Imam Khomeini International University
e-mail: h_sadegh@ikiu.ac.ir
Abstract
In this paper, the modeling, simulation and control of 3 degree 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.
Keywords: feedback linearization, forward kinematic, inverse kinematic, manipulator, robot,
SimMechanics-toolbox
1. Introduction
Industrial robots are widely used in various fields of application now days. 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 tree
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 emphases when the basic types are examined. In this paper,
the 3 DOF 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 March 29, 2013; Revised July 26, 2013; Accepted August 25, 2013
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ISSN: 2089-3272
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 3 DOF articulated manipulator has been determined as
below:
Figure 1. 3DOF articulated manipulator in Spherical coordinates for forward kinematic analysis
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:
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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 equations (1-3) we have:
(10)
(11)
(12)
Modeling, Simulation and Position Control of 3 Degree of … (Hossein Sadegh Lafmejani)
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where S denote to (Sin) and C denote 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:
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.
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]:
Where
is the inertia matrix,
is the coriolis/centripetal matrix,
is the
gravity vector, and
is the control input torque. The joint variable is an n-vector containing
the joint angles for revolute joints. The mentioned matrix of the 3 DOF articulated manipulator
can be computed by:
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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.
Figure.3. Analytical modeling of 3DOF manipulator in matlab simulink
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 show the simulink
design of the manipulator and Figure 4 shows the SimMechanics modeling of it. For this system
and
for all links.
Figure 4. SimMechanics modeling of 3DOF manipulator in matlab Simulink
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As it was mentioned before, we used the SimMechanics toolbox for validation of
analytical modeling. The results of this study are brought in Figure 5 and Figure 6. According to
the 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.
Figure 5. Comparing graph of analytical model with SimMechanics model
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
regard the equation (15) and replace
with a new variable
degree of freedom we
. So we have:
Figure 6. Modeling error
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Figure 7. Inverse daynamic block diagram
The block diagram of PID controller with feedback linearization named inverse dynamic
control has been brought in Figure 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 showed in Figure 8 and Figure 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 the parameter of the manipulator accurately.
It is clear that reaching the parameter accurately is not possible practically and always we have
some uncertainty in system.
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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 uncertainty
in manipulators is high. Thus using the fuzzy controllers and intelligent method like neural
network is proposed for controlling these kinds of complex systems.
Figure 8. Inverse dynamic tracking control
Figure 9. PID tracking control
f
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