Switched Adaptive Control for a Class of Robot Manipulators
Switched Adaptive Control for a Class of Robot Manipulators
NIU Ruishan,WANG Xia,ZHAO Jun?CHEN chao
State Key Laboratory of Synthetical Automation for Process Industries,Northeastern University, Shenyang,Liaoning, china110004
E-mail: niuruishan0530@https://www.sodocs.net/doc/365643702.html,
Abstract: A switched adaptive control scheme is proposed for robot manipulators to solve the problem caused by the load variation.The parameter jump caused by load changing is treated as the mode switching of the robot manipulator system.A switched controller, that switches synchronously with the system is designed.The update laws are designed in a way that the estimation value unchanged when its corresponding subsystem is inactive. Asymptotic state tracking under arbitrary switching is gained.An example of a two DOF robot arm is presented to validate the proposed method.
Key Words: Switched nonlinear system, Adaptive control, State tracking, Robot manipulators
1Introduction
Robotics is a comprehensive science with the rapid development. The control of robot is the most important part of this science. However model uncertainty in a robotic system has a strong impact on the control accuracy. Many approaches have been considered to solve this problem. The main purpose of tracking control in robot manipulators system is to control the driving torque of the joints to make the state, composed of position and speed, follow the given ideal trajectory.
Because the system of robot manipulators has many uncertainties, adaptive control method is widely used in this system [1-4].It is well known that there are two important properties of robot dynamics:its linear-in-parameter formulation and its passivity property. According to the first property, Craig et al presents an adaptive control method which ensures the global convergence of the tracking errors [l]. In [2], Liu Ming proposes a modified version of Craig's scheme. Others also put forward different improvements.
However, robots often do some monotone, frequent and repetitive job, and repeatedly pick up and lay down some specified loads. The abrupt change of load leads to the parameter jump in the robot dynamics. However, the conventional adaptive control demand that the uncertain parameters vary slowly, the control performances will deteriorate when the load change abruptly, because the adaption process needs time.
From the robot's working process, we know that the load is a major uncertainty in all uncertainties of robot system, and the uncertainty of load is switching, because the load can be changed. For example, the load can jump from 0 kilogram to 10 kilogram instantaneously. So using only one adaptive controller cannot meet the performance. Therefore, we consider the switched adaptive controller.
In this paper, we describe the load change as the switch of subsystems in a switched model and present a switched adaptive control scheme to solve the problem of jumping loads.Assume that the robot only takes some specific loads, which is the most general case.Every kind of the load has its own sub-controller, and when the weight of the load changes,
*This work was supported by the National Natural Science Foundation of China under Grants 61174073. the controller switches to the corresponding sub-controller at the same time. The scheme results in the global convergence of tracking errors. Using the conventional adaptive design approach and linear-in-parameter property of robot motion equations mentioned above, a switched parameter adaptation algorithm for adaptive controller is obtained.
One problem of this method is the coupling between the individual update laws. In [5], different update laws are designed for not only active period but also inactive period of each subsystem. The coupling can be avoided by keeping the estimation value unchanged when the corresponding subsystem is inactive. This method is used in the robot systems to overcome the coupling problem resulted by that all the subsystems share a common state. For the error dynamics, the common Lyapunov design scheme is used to obtain the parameter update law.The algorithm ensures the global convergence of the tracking error
The results in this paper have two features. First of all, a switched model is used to formulate the load change of robot manipulator.Secondly, the parameters of different sub-controllers are estimated individually.
The remainder of the paper is organized as follows. The overall system structure and its properties are given in section 2.The switched adaptive controller, update law and stability analysis are investigated in section 3.An example is presented in section 4 to validate the method. Finally, in section 5, the conclusions are given.
2System Structure
In this paper we consider the rigid connection of robot manipulators. In the absence of friction and other disturbances, the dynamics of a robot manipulator with n DOFs can be described by the Lagrange equation [12]:
,
)(
),(
)(W
q
G
q
q
q
C
q
q
D
(1) where
n
R
q is the robot arm joint angle vector;
n
R
W is the control torque;
n
n
R
q
D u
)
(is the inertia matrix of the mechanical arm;
n
R
q
q
C
)
,
( is the centrifugal force and Coriolis force;
n
R
q
G
)
( is the gravity items.
The mechanical arm has three dynamic features:
3URFHHGLQJV RI WKH VW &KLQHVH &RQWURO &RQIHUHQFH -XO\ +HIHL &KLQD
Feature 1),(2)(q q C q D
is the diagonal symmetric matrix.
Feature 2 The inertia matrix )(q D is the symmetric positive definite matrix. It is satisfied with the inequality
,
)(2
22
1x m x q D x x m T
d d wher
e 1m and 2m are positive numbers.
Feature 3 There is a parameter vector which depends on the parameter of the mechanical arm to make the following equation true
,),,,()(),()(M U -U -q q W q G q q C q D (2)
where ),,(q q
q W is the function matrix as the r n u regression matrix, and M
is the 1u r physical parameter
vector of the robot.
Depending on the different weight of the load, we build different models. The switched dynamics of the robot manipulator is:
,)(),()(V V V V W q G q q q C q
q D (3) where N V is the index of subsystems, as well as the
index of load types.
For the system ?3?,we design switched adaptive controller. Every sub-system has its unique sub-controller.When the weight of loads changes,the controller will switches to the corresponding sub-controller.The design of the sub-controller is based on the adaptive theory,and the adaptive law is the piecewise function.When the Nth subsystem active,parameters of the Nth model will change regularly, while if the Nth subsystem do not active,the Nth model’s parameters will not change.
3Switched Adaptive Controller Design
The switched adaptive controller with adaptive laws is
designed in this section. The stability and error convergence are analyzed using the common Lyapunov method.
3.1Switched Adaptive Controller
For dynamic equation (3) we propose the following control torque:
),(?),(?))((?q G q q q C e K e K q q D p v d V
V V V W (4) where V
V V G C D ?,?,? is the estimate of V V V G C D ,,. The adaptive law is:
PE B q D W T i T i i T )(?)(~1 * M ,V i 0~ i
M
.V z i (5) Theorem 1 For dynamic system (3), controller (4) with the update law (5), guarantees the global convergence of the tracking error under arbitrary switching.
Block diagram of robot system with the proposed controller is
3.2The Proof of Theorem 1
According to Feature 3 of the mechanical arm, we have
.?),,()(?),(?)(?M V
V V V q q q W q G q q q C q q D (6)It follows from (4) and (6) that
.?),,())((?M W V V V
q q q W e K e K q q q D p v d (7) Note that M M M M W V V ?~,),,(, q q q W q q e d
,we have
.~),,()(?1M V
V q q q W q D e K e K e p v (8) Denote
??o
??a ??
o?
?a ??o
??a n n v p n n I B K K I A e e 0,0, H , (9) (8) turns into
.~),,()(?1M V
V q q q W q D B A H H (10)We choose v K p K , such that ,
T A p pA Q where p and Q are positive matrices.
Select a common Lyapunov function
],[1)*) T T tr p V H H (11)
where
,~~1??
??o????a )N M M ????
o????a** * 111
1N with 0!*i ?N i 2,1 .
In view of (9) and the adaptive law (5) the time derivative of the V becomes
.
])(?)(~[~2)(11H H H H H Q p B q D W pA p A V T
T i T T i T i i i T T * M M
Because 0d H H Q V
T , we can conclude that 0~ ??
o
??aM H
is stable. However, we cannot prove Fig 1 Block diagram of robotic system
with the proposed controller
q
q q
asymptotically stable. In order to prove 0o E , we have to make further discussion. By the definition and monotonicity of V, we have inequality
f d d f d )()()(00t V t V V ,,
00t t t t ,)(~)()(022f d * d t V P m m M
O O H where m O is the smallest eigenvalue of the matrix. Because
f
f L
L M ,H and
2
)(H H H Q Q V m
T
O d , we get the equation .)()
(1
)]()([)
(1)(100
2
00
f
d f
d 33
f f
t V Q V t V Q dt V Q dt m m t m t O O O H Notice that 2
L H , we have f
L q q q d d d
,,,~,M H and f
L q q q W ,,, . Then,
.f ??o??a ??o??a L q q q q e
e
d d H In summary, 2
L H and f L H lead to 0o H , i.e.
d d q q
q q o o ,as f o t . 4
Numerical Simulation
Consider a two DOF robot arm given in Fig 2[12].
,cos )(sin cos sin 2cos sin 2cos 221431212121222222??
o
??a ??o??a ??
o??a??o??a W W K H E D K H E K H E K H E K H D Y Y q e e Y Y q q
q
q q q q q where
.
/,
),sin(cos ),cos(sin ),
sin(cos cos 2),cos(sin sin 21221111111212221
42122213212222221221222
22211l g e l m I l l m e q q e q q
Y q q e q q
Y q q e q q q q q Y q q e q q q q q Y c K H E D ,,,are the unknown parameters of robot arm
.
sin ,cos ,,
1212212
2111e ce e e ce e ce e e e ce e e c l l m l l m l m I l m l m I l m I G K G H E D
Table 1: The Physical Parameters of the Robot
1m 1
1l 11
c l 1/21
I 1/12e m 2.8ce l 0.9e
I 0.37e
G 0
Assuming that there is only one kind of load and all the
loads weigh 0.2kg. The parameters are 3,e m 1
ce l ,
2/5e I with this load. The load of arm changes as shown
in Fig. 2 in which “1” stands for having the load and “0” represents no load. More specifically, Fig. 2. represents the switching signal of System (3).
Fig. 2 Switching Signal.
W and M can be expressed as
?
1.2
2.3
t
1
(t V
????
?????
??
o
?????????
??a )sin(cos sin )
sin(cos cos 2sin sin 2)
cos(sin cos )cos(sin sin 2cos cos 2cos 0
cos ),,,(21221121212
22221222212122112121222221
222212122121q q e q q q q q
q q e q q q q q q
q q q q q q e q q q q q q q e q q q q q q q q q q q q q e q q e q q q q
q W d d d d d d d d d d d d d d d d d d T ,
].
~~~~[~K H E D M T We specify the reference trajectories and control parameters as 12()sin ,
()sin .d d q t t q t t S S 121v v k k ,
121p p k k .
With the control law (4) and the adaptive law (5),all the conditions of Theorem 1 are satisfied. The tracking errors starting from the initial values 12(0)[(0),(0)][1,1]T e e e are shown in Fig. 3.
As claimed in Theorem 1the tracking errors converge to zero under arbitrary switching.
time(s)
p o s i t i o n t r a c k i n g o f l i n k 1
time(s)p o s i t i o n t r a c k i n g o f l i n k 25Conclusion This paper presents a switched adaptive control scheme
for the robot manipulators system in which the load changes
instantaneously. The update law works only if the
corresponding subsystem is activated, which avoids the coupling between different controllers. The method ensures the global convergence of the tracking errors. References
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NIU Ruishan is a Master candidate at Northeastern University, and her major is control theory and control engineering. Her research interests include adaptive control, switched systems and robotics. E-mail: niuruishan0530@https://www.sodocs.net/doc/365643702.html,.
WANG Xia is a Ph.D. candidate at Northeastern University, and her major is control theory and control engineering. Her research interests include adaptive control and switched systems. E-mail: wangxiahbu@https://www.sodocs.net/doc/365643702.html,.
ZHAO Jun is a professor at the College of Information Science and Engineering, Northeastern University. From February 1998 to February 1999, he was a Senior Visiting Scholar at the Coordinated Science Laboratory, University of Illinois, Urbana-Champaign. From November 2003 to May 2005, he was a Research Fellow in the Department of Electronic Engineering, City University of Hong Kong. His main research interests include switched systems, hybrid control, nonlinear systems and robust control. E-mail: zhaojun@https://www.sodocs.net/doc/365643702.html,.