In this paper an efficient controller is presented for controlling SCARA robot in present of undesirable noises in variable states. To do so, feedback
linearization is considered as the main controller and kalman filter estimator is utilized to estimate the state variables from the noisy output signals. In
order to evaluate the performance of the purposed method, a simulation test is performed to apply this controller on the SCARA robot. The simulation
is done for both noisy and not noisy signals. The results show that the kalman filter observer has accomplished a good state estimation and feedback
lineation controller has tracked the desired signals perfectly as well.
The importance of using robot in daily life is shown, when
human life is at risk. For instance, in high-risk industrial places
such as working alongside the melting furnaces or neutralize
mines on the battlefield and such high-risk areas. In additional,
human accuracy is reduced when doing repetitive work. Mentioned
reasons, the use of robots has become prevalent in many diverse
industries and locations [1-8]. SCARA robot is a kind of usual robots
which is used for welding, painting, assembly and material handling
. This robot consists of three degree of freedom. Economic
justification for the cost of installing a robot must be considered
some important things. The robot control system must have some
necessities like short time of cycling, continuous and accurately
tracking. To achieve the short time cycling; the robot needs to have
enough fast response, to get to high tracking accuracy, the tracking
error should be very low . The importance problem in using of
the robot is accurate control method to guiding the end effector to
the desired position quickly.
There are various applicable controller methods that can be
divided in to different branches such as model-base or none modelbase
[11-21]; (Figure 1 & 2), adaptive [22-28], and intelligence [29-
36] methods. Tuning the parameters of a controller method which
contains a high number of tunable parameters, is very crucial and
can be very cumbersome. To accomplish this process, one way which
is very common, is employing an efficient optimization method
[37-95]. By determining a proper cost function, the optimization
algorithm can tune the controller parameters optimally. It worth
nothing that, selecting a power full optimization method is very
vital and must be done regarding the number of tunable parameters
and the complexity of the ultimate cost function. Recently, some
novel and effective population optimization algorithms have been
presented that can be used for this goal.
The aim of this paper is to introduce an applicable and effective
controller for controlling a SCARA robot in present of disturbance.
To do so, feedback linearization (FL) method which is a nonlinear
and effective controller is selected. Sensitivity to noise is a
disadvantage of FL. To overcome this problem using a kalman filter
to estimate the state variables is suggested.
This paper complained of several sections. Section 2 described
problems, parameter and explained the assumptions. In section 3
is reported the dynamic modeling direct kinematic of SCARA robot.
4th section depicted the controller. Stochastic feedback linearization
LQG with Kalman filter is used to control the SCARA manipulator
robot. This controller can estimate parameter and identified
original signal in comparison of disturbance signals. In section 5
demonstrated the results simulation by Figure 1 and curves. Figure
1 of tracking is pictured in this section also, which validated the
successful performance of controller. Finally, in ending section
explained the conclusions of paper.
In this paper the SCARA robot with 3 degree of freedom is
controlled. This robot is RRP type and is assumed all of arms is
rigid body. The friction in each joint is negligible. The original mass
moment of inertia is in Z-direction. There is no mass moment of
inertia in other plane. The final joint has a force in Z-direction which
is shown in Figure 1. G1 and G2 are manipulator center of mass.
In this section the dynamic equation of SCARA robot is
obtained. For this purpose, the Euler-Lagrangian method is used to
derive dynamic equations. So, the kinematic and potential energy
of the system must be computed to use in Lagrangian equation. The
kinematic and potential energy of the system is obtained as :
Where xci and yci are the center of mass position of ith
manipulator, m1,m2 and m3 are mass of manipulators,
Ic1 and Ic2are the moment of inertia around center of mass
manipulator 1 and 2 respectively and i è is the angel of the th i manipulator which is
shown in Figure 2, g is the gravity acceleration and z is the height of
center of mass manipulator 3.
yci are :
So, the velocity of each manipulator is
Where Lci and Lci are the distance between center of mass ith
manipulator and ith joint.
So, the Lagrangian of the system is obtained as [4,11,34]:
L = K −U
Where L is the Lagrangian function. By using the Euler-Lagrange
formulation the dynamic equation of the system is obtained as
In these equations q is states of the system and defined as:
This deferential equation can be expressed in matrix form as:
Where the matrices M, C, and G are represented the 3×3 inertia
matrix, 3×1 Centripetal-Coriolis
Matrix terms, and 3×1gravity matrix, respectively and T are
3×1 matrix which is defined as below:
In this section the stochastic LQE feedback linearization
controller with Kalman filter, is designed. Feedback linearization:
In feedback linearization method by equaling the nonlinear
system in to a stable linear system, the control law can be derived
as nonlinear system as below [1,16,26]. The stable linear system
has been considered as below:
Where matrix A and B be are defied as:
Where E is the vector of state variable of the linear system and
V is system input vector. The nonlinear system can be obtained also
Where X is the state variable of the nonlinear system, F and G
are derived from equations 14 to 16.
With considering E as error we have:
So, the controller law for nonlinear system derived as:
In control theory, the Linear-Quadratic-Estimator (LQE)
control problem is one of the most fundamental optimal control
problems . It concerns uncertain linear systems disturbed by
additive white Gaussian noise, having incomplete state information
and undergoing control subject to quadratic costs. Moreover,
the solution is unique and constitutes a linear dynamic feedback
control law that is easily computed and implemented (Figure
3). Finally, the LQE controller is also fundamental to the optimal
control of perturbed non-linear systems.
The linear system is considered as below:
In this problem C is 6x6 identity matrix.
Where the vector E is state variable, the vector V is control
input, the vector Y is measurement output and the vector n are
additive white Gaussian system noise and w is additive white
Gaussian measurement noise. The LQE controller is specified by
the following equations:
Where the matrix K is Kalman gain which is obtained from
Kalman filter equation.
Where in the above differential equations W the covariance of
w and N is the covariance of n.
To regulate the angle system by controller 1, 2 θ θ and z are
main parameters which are considered. The simulation results are
Investigation in two Conditions, one with present disturbance and
the other without consider disturbance. Also, the Measured and
Estimated states are illustrated in Figure 4a-4f for regulation and in
Figure 5a-5f for tracking purpose.
In this investigation, the SCARA robot control is presented.
SCARA robot is controlled by a stochastic feedback linearization
LQE with Kalman filter. This controller can control the robot
successfully and identified original signals in presence of the
disturbance’s signals. In feedback linearization method by equaling
the nonlinear system in to a stable linear system, the control law can
be derived as nonlinear system. LQE controller concerns uncertain
linear systems disturbed by additive white Gaussian noise, having
incomplete state information and undergoing control subject to
quadratic costs. This combination of useful control methods is
already set to lead the system to the desired position in presence
of disturbance as is evidenced in the simulation results and Figure
Figure 1:Schematic of front view of SCARA robot .
Figure 2:Schematic of left view of SCARA robot .
Figure 3:Schematic Stochastic controller.
Figure 4:Estimation of the state variables, a) θ1without noise, b) θ1with noise, c) θ2 without noise, d) θ2 with noise, e) z without
noise, f) z with noise.
Figure 5:Regulation for variable state, a) θ1 without noise, b) θ1 with noise, c) θ2 without noise ,d) θ2 with noise, e) z without
noise ,f) z with noise.