Design an optimal adaptive controller for robot manipulator
Mahdi souzanchi1, Aliasghar arab1, vahab khoshdel1
1- Department of Electrical & robotic Engineering, Shahrood University of Technology,
Shahrood, Iran
Email:[email protected], [email protected],[email protected]
Abstract
In this article, optimal adaptive method is used for
SCARA robot. One of the advantages of this method
is in order to control it, there is no need to have exact
parameters such as joints weight, and the controller
can reach itself to real parameters. According to the
extreme application of robots in industry such as
moving parts and joints, they cause weight changing
and change of moment of inertia that it would not be
an easy task for the controller to identify these
changes and adapt itself to them. Thus, we will use
the adaptive method until its controller adapts itself
with the system. Finally the parameters which have
been taken from the controller are converged with
constant values. First, we calculate the dynamic and
kinematics equations and then we simulate robot.
Next, we use the controller design which gives
necessary moments to the robot. We utilize the
adaptive method in this controller and we use the
genetic algorithm to optimization error and calculate
the γ and ̂ ( ) in adaptive control low.
Keywords: scara robot, adaptive control, genetic
algorithm, uncertainty
1 INTRODUCTION
Dynamics and kinematics of robots consists of highly
nonlinear couplings and uncertainties in model.
To conquest the nonlinearity and uncertainty of the
robot dynamics, the proportional–integral–derivative
(PID) is the most prevalent controller in industrial
settings. Although, some controllers such as PID
controllers are profitable for regulating goals, they
cannot work well for tracking purposes. Instead,
model-based controllers can work well. However, if
exact models are required they cause very large
models with low response and difficult to perform
[1]. In some applications, it is essential to determine
the motion in much more details than simply stating
the desired final position.
In trajectory tracking control, a model-based robot
controller that is calibrated to work perfectly using
exact models of the system may increase control
efficiency. While the parameters of the dynamic
model of the robot change unpredictably in time [2],
the assumption of having exact models of the robot
system cannot cause the adaption of robot to any
changes and uncertainties in its models and
environment [3]. For example, when a robot picks up
several tools of different sizes with unknown
orientations or gripping points, the overall dynamics
of robot changes and is therefore difficult to control
exactly. So, to have an acceptable level of control
system performance when large and unknown
changes in model parameters occur, an adaptive
control approach has to be considered.
In the past several years, various adaptive control
schemes have been proposed for nonlinear systems.
an adaptive control for nonlinear systems with
unmodelled dynamics [4], a system to improve
nonlinearly parameterized bounding functions [5], an
adaptive control scheme for a class of nonlinear
uncertainty [6] and several schemes that you can
surveyed in [7-13] are some application of adaptive
control for nonlinear systems.
Lyapunov method based adaptive control is one of
adaptive controls that cannot guarantee stability and
performance while using fixed controllers. This
method doesn’t support specific class of uncertainties
which could be unknown functional, unmodeled
dynamics, disturbances [14].
Recently, optimal control of robot manipulators has
attracted a great attention. Various optimal trajectory
control schemes have been developed for robot
manipulators with unknown parameters, such as
optimal sliding mode control, or optimal adaptive
robust control.
In this paper we control SCARA robot with adaptive
Tracking control algorithm that is designed by use of
Lyapunov theorem. This controller is robust for
parametric uncertainties. Genetic algorithm to design
an optimal adaptive controller for this robot is used.
This paper is organized as follows. Section 2 explains
modeling of the robot. Section 3 develops the
proposed control law. Section 4 describes the
adaptive method to estimate and compensate the
uncertainties. Genetic algorithm is explained in
Section 5. Section 6 illustrates the simulation results.
Finally, Section 7 concludes the paper.
2 Modeling
An electrical manipulator is motivated by electrical
motors and thus the manipulator is controlled
according to control laws. So, it is more acceptable to
consider the manipulator dynamics including the
motors. Dynamics of an electrically driven rigid
manipulator considering the dynamical model of dc
motors is used for tracking control [15]. The
manipulator dynamics is given by
( )
(
)
( )
( )
where
is a vector of generalized joint
positions, ( )
is the inertia matrix,
(
)
is a vector of generalized centripetal,
and Coriolis forces, ( )
is a vector of
generalized gravitational forces and
is a vector
of generalized joint forces. In order to deserve a high
controllability for tracking, a manipulator can be
driven by dc motors. Based on the motion equation of
permanent magnet dc motor, a matrix form of
dynamic equation for the motors is expressed as
̈
) ̇
(
( )
(
( ) ̈
(
)
( )
(
)
(
( )
( )
( ))
( )
̇)
( )
( )
( )
Unmodeled dynamics and external disturbances
should be included, so electrical manipulator
dynamics is formulated using the following:
( )
(
(
(
)
)
)
( )
( )
( )
d stands for unmodeled dynamics and external
disturbances.
In this paper modeled SCARA robot . SCARA stands
for Selective Compliant Articulated Robot for
Assembly. SCARA is a new compound and as the
name suggests, it is used for assembly operation.
The first and second joints are used to determine X
and Y coordinates and he third one for Z coordinate
like figure1 and the fourth one is a griper. SCARA
robot has a great maneuverability in X-Y axis. This
robot is exploited not only for assembly but also for
lifting and replacing from one place to another.
( )
Where
is a vector of motor angles and
vector of motor voltages,
are
×
constant diagonal
matrices of torque constant, back emf constant,
resistance, inertia, damping, and reduction gear ratio
of motors, respectively. In this model of motor,
numerically
.The joint angle vector is
related to the motor angle vector through the gear
reduction ratio as
is
a
Figure 1 work space of SCARA
( )
Substituting (3) and (2) into (1) yields dynamics of
robot including the actuators as follows:
(
(
( )) ̈
( )
(
̇)
A simple presentation of (4) is
) ̇
( )
3 Proposed control law
The robot manipulator is expressed by the multi
input/multi-output system. A torque control strategy
which uses the model of robot manipulator is
proposed as the control approach, and a nominal
model is chose to explain a control law, as follows:
̂( )
̂(
)
̂( )
(
)
The nominal model is known and proposed based on
the knowledge about the system such that the
nominal parameters ̂ ̂ , and ̂ are the estimations
of
, and , respectively. The dynamics of the
nominal model is simpler than the exact model
̇
(
)
(
)
(
̂(
))
)
(
e=
)
(13)
It is easy to show from (1), (12) that:
̂)
(
̂( )
̂)
(
̂)
(
(
(
)
̂ ( )
̂)
(
̂( )
(
(
̂ ( )
)
(
)
(
̂)
is
̂)
)
(
)
( ̂̇ )
∫(
(
̂ ( )
( ̂̇ ) (
̂)
)
+ ̂ ( )
̂)
(
(
)
)
where is the ̂ ( ) initial value.
)
(
( ̂̇ ) (
(
with is Hurwitz and given a positive definite
symmetric matrix , the positive definite symmetric
matrix is calculated from the Riccati equation as
follows:
( )
We can control ̇ by the first term of ̇ .therefor we
have:
)
If propose τ=
that is dynamics vector and
constant parameters vector then we have:
̂)
̇
The control low is:
̂ ( )(
̂( )
( ̂̇ ) (
)
5 GA (Genetic Algorithm)
4 Adaptive
uncertainty
estimation
of
the
parametric
We design an adaptive system to calculate ̂ for the
control law (15).
An adaptive controller which is designed by use of
Lyapunov theorem is employed. The difference
between the parameters given by
̂ in the closed
loop system affects the error of estimation (16).
Regulating ̂ reduces the
. Thus, using
(16), a positive definite function ( ) is suggested,
using the following:
(
̂) (
̂)
(
)
where is a positive scalar and the positive definite
symmetric matrix.
The direct method of Lyapunov is applied
to propose a robust control law. The state space form
of (16) is:
̇
( )
[ ]
̇
[
̂ ( )
]
(
̂)
(
[ ]
(
)
)
Genetic algorithm (GA) is a population based
optimization technique in which, according to
the concepts of natural selection, genetics and
evolution, the best solution of a given problem is
searched. An initial random population of
individuals is the starting point of the search. An
individual is a possible candidate solution for the
optimization problem. At each evolutionary step
(i.e., at the generation of a new population), a
fitness function evaluates the individuals. Three
kinds of operator make each evolution: breeding,
mutation and selection. Selection happens by
exclusion of a given proportion of the population
based on probabilistic ‘‘survival of the fittest’’.
Excluded ones are replaced by children, which
are born by breeding the remaining individuals
in the population. A child is bred just when two
parent individuals who get a more chance to
choose fitter individuals are selected. In order to
explore new areas of the response surface, a
random change in optimization variables
happens, through the mutation operator. GA
iteratively improves the set of solutions by using
the aforementioned stages to find a good
solution. In the traditional GA, only binary
variables (genes) that formed a string
(chromosome) were allowed. After that, the final
binary digits are decoded as original real
numbers. Beside, in a real-coded GA, all genes
are real numbers. Therefore, the real-coded GA
is more suitable than the binary-coded one for
solving engineering problems [16].
In this paper used GA to compute γ and ̂ ( ) in
adaptive control low and the cost function for
optimization was:
|∫
|
(
)
is the error of joint n
6 simulation
The proposed control law is applied to control an
articulated robot manipulator with a symbolic
representation in Fig. 2. The Denavit–Hartenberg
(DH) parameters of the articulated robot are given in
Table 1, where the parameters θi, di, ai , and αi are
Table 1 The Denavit–Hartenberg parameters
Table 2 The electric motor parameters
called the joint angle, link offset, link length, and link
twist, respectively.
Motor parameters are given in Table 2. The
maximum voltage of each motor is set to Umax =40V
The desired joint trajectory for the joint 1,2,4 are
shown in Fig. 3,4,5 . The desired trajectory should
be sufficiently smooth such that all its derivatives up
to the required order are bounded. The desired
trajectory starts from zero and goes to 1.6 rad in 5
sec. the desired joint set point for joint 3is shown in
Fig. 6. The error of joints is shown in Fig. 7
Table 3 optimal parameters
Figure 2:DH coordinate frame assignment for the SCARA manipulator.
teta desired joint1
teta joint 1
1.5
1
teta(rad)
teta(rad)
1.5
1
0.5
0.5
0
0
teta joint2
teta desired joint2
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time(sec)
Figure 3:desired joint trajectory for the joint 1
0
0
0.5
1
1.5
2
2.5 3 3.5 4 4.5 5
time(sec)
Figure 4:desired joint trajectory for the joint 2
2
1.4
teta desired joint4
teta joint4
1.2
1.5
d3 desired
d3
1
d(metr)
teta(rad)
1
0.5
0.8
0.6
0.4
0
0.2
-0.5
0
0.5
1
1.5
2
2.5 3
3.5
time(sec)
Figure 5: desired joint trajectory for joint4
4
4.5
0
0
5
0.5
1
1.5
2
2.5
3
time(sec)
3.5
4
4.5
5
Figure 6: desired joint set point for the joint 3
1.2
error joint 1
error joint 2
error joint 3
error joint 4
1
0.8
error
7 Conclusion
This paper has developed an optimal adaptive
control of electrically driven robot manipulators
using the torque control strategy. We have
used a GA to optimization of controller .Simulation
results have shown the effectiveness of the method.
this control approaches is robust with an exact
tracking performance and we can use other method
for optimization such PSO , memetic, etc.
0.6
0.4
0.2
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error2
0
-0.2
0
Refrence:
error3
error1
0.5
5.
error4
1
1.5
2
2.5
3
3.5
4
time(sec)
Figure 7: error of joints
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