AN ADAPTIVE FUZZY SLIDING-MODE CONTROLLER FOR A RO ...
14th World Congress ofIFAC
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Copyright © 1999 IFAC 14th Triennial World Con.Q;ress, Beijing;, P.R. China
A.n i\..daptive Fuzzy Sliding-JYlode Controller for a Robot Arm R. G. Berstecher and R. Palm
H. Unbehauen
Siemens AG
Bochum Faculty of Electrical Engineering Control Engineering Laboratory 44780 Bochum, Germany email:
[email protected] Ruhr-University
Corporate Technologies, Dept. ZT IK 4 81730 Munich, Germany email: Rainer
[email protected]
Abstract: This paper deals with a new adaptive sliding mode controller and its practical applica.tion to a robot manipulator arm. The theory for this new fuzzy sliding mode controller approach a.nd for the heuristics-based linguistic adaptation is presented and a mathematical description is derived. Furthermore the application of this adaptive controller is shown for a simulated two-link robot arm and for the real time operation of a mobile three-link robot arm, respectively. The obtained results show the high efficiency of the new controller type. Copyright© 1999 IFAC Keywords: Adaptive control, fuzzy controi, sliding mode, robotics
1
Introduction
In this paper a linguistic, heuristics-based adaptation algorithm for a fuzzy sliding-mode controller (FSMC) is presented. The algorithm relies on linguistic knowledge in form of fuzzy if-then rules which reflect how an experienced operator would adapt the controller in order to obtain a desired closed-loop hehavior. The adaptation parameters are grouped into a parameter vector r. A linguistically defined adaptation law uses the switching variable SO" of the FSMC, its time derivative Sa, and the manipulated variable u in order to compute a resulting adaptation parameter change r. The analytic description of the adaptation law reveals similarities to conventional control theory and relates uncertainties in the system description with the expected adaptation parameter convergence. In Sect. 2, the basic principles of FSMCs are briefly reviewed. In Sect. 3, the heuristics-based motivation for the linguistic adaptation law is presented and an equivalent mathematical description is derived. In Sect. 4, the adaptation law is applied to the control of a two-link robot arm in the simulation and in Sect. 5 to the real time control of a three-link robot arm_ Concluding remarks are given in Sect. 6
2
The fuzzy sliding-mode controller
In the following, we consider a nonlinear n-th order system of the form x(n)
=
!(x,
x, ... , x(n-l)) + b(x, X, ... , x(n-l»)u.(1)
Let b( x, X, ... ~ x(n-l)) > 0 and the state x :::::: (x x ... x(n-l)T has to follow the desired trajectory (n_l»)T • Th en we d efi ne . Xd = ( Xd Xd ••• xd
_ n-1 ~ -6
(
n - 1 i
)
.A
i (n-i-i)
Re
(2)
,
i=O
(0 h were e -x - Xd -e e- ••• e (n_l»)T • Sa represents a switching surface in the state space. AR is a parameter specifying Sa and therewith the dynamics of the closed-loop behavior according to [Slotine 1991, Hung 1993, Drakunov 1992]. Fuzzy controllers which use the switching variable Sa for calculating the manipulated variable u are called FSMCs [Palm 1994, Sun 1996]. Their fuzzy rules have the form
R{ : If
Sa
is LSt Then
"U
is LUj
•
(3)
LSt, and LUj represent the linguistic values for the variables s" and "IL, respectively. defined by corresponding membership functions. The manipulated variable u is the output of the FS:>vIC while the switching variable Sa is its input. The shape of the nonlinear transfer characteristic of a FSMC depends not only on the values of tt, but also on the membership functions of the rule antecedents (if-parts) and consequents (then-parts) and the defuzzification method. In what followS,9fc represent.s the transfer characteristic of the FSMC and describes the dependence of the manipulated variable u on the switching variable Sa and the
500
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress of lFAC
AN ADAPTIVE FUZZY SLIDING-MODE CONTROLLER FOR A RO ...
adaptation parameters r = (rl, ... ,r;, ... ,rN). By choosing the center-of-sulllil defuzzification method, the output u of the FSMC is computed as N
2: J-l'(so )Mi
u = grc ( Bo,r )
=
;=1 '-N".,:;-----
By influencing the parameters ri only at the supporting points su,;, coupling effects between the different parameters are avoided. Figure 1 illustrates that a change in ri by Llr; does not alter the output of the FSMC at the supporting pointsa,j , j i= i.
(4)
I: /Ji (Sc )A;
;=1
where Jli is the degree to which the antecedent of the i-th fuzzy rule is satisfied by a particular crisp value of sc. )\1; and Ai are the moment and area of the membership function defining the linguistic value of u in the i-th rule. The point at which a triangular membership function has value 1 is called a supporting point. Here, the j\l points Sa,j, j = 1, ... , N, wi~h membership 1 to the membership functions defining the linguistic values of Su, are the supporting points in the (u , so) diagram (see Fig. 1). At the j-th supporting point the antecedents of the other IV - 1 rules have degrees of satisfaction equal to zero and therefore these rules do not fire. Thus, according to (4) and the j-th rule Ri in (6), the output of the FSMC at the j-th supporting point sr:r,j is
(5)
u
U i-I
Ui Ui+1
~c
Figure 1: Dependence of the FSMC characteristic on the parameter value ri at so,.
The points (scr,j, :!}fi) lie on the nonlinear transfer charJ acteristic of the FSMC in the (U, Ba) diagram.
I\,. U NB !
3
NB
The linguistic adaptation law
• the approach velocity s~ of e to the switching surface So = 0, and • the manipulated variable u in the antecedents of the fuzzy rules describing the linguistic adaptation strategy. Each fuzzy rule computes a parameter change ri for the i-th component r j of the parameter vector r. Thus, the L rules RI, defining the adaptation strategy for T', at su,;, thus have the form is LSt and
S(7
is LSt and u is LUj
Then
rj
is
LR{
Z
P
»B\" P \ ··'z>.. ·N··.·
N
Z P
• the distance Sa of the error vector e from the switching surface So = 0 in accordance with (2),
S(7
N
PB NB
-.,;,'
The adaptation law deals with the change of the values of the supporting points ro and is defined by heuristicsbased linguistic rules describing how an expert would adapt the FSMC to achieve a desired closed-loop behavior [Astrom 1995]. The adaptation law uses
If
s~
(6)
LSt ,LSt-;LU j and LW; represent the linguistic values for the variables involved in the rule antecedent and consequent, respectively.
PB
p ..:
-'
Z
Z
N .. ··NB
NB
.~
.'
N .. ···NB
NB
NB
N .. ··NB
NB
NB
NB
NB
NB
NB
NB
-..
NB
Figure 2: Linguistically defined adaptation algorithm for Ti where 'sa is P B' The switching variable So in Table 6 characterizes one supporting point of the transfer characteristic and therefore is identical for all adaptation rules. By representing Sa, U and ri as in Table 6 and omitting 8 u in the look-up table representation from Fig. 2, t he similarity of the adaptation algorithm with the structure of a FSMC becomes apparent [Berstecher (a) 1996, Berstecher (b) 1996]. This fuzzy rule base has the structure of a FSMC because, as shown in Fig. 2, in the cells parallel to the main diagonal (cells with values Z), the same output value ri
501
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress oflFAC
AN ADAPTIVE FUZZY SLIDING-MODE CONTROLLER FOR A RO ...
is always chosen. This FSMC in the adaptation loop is used here for the adaptation of the FSMC in the underlying control loop. The FSMC in the adaptation loop is used for the adaptation of the parameter So,i. In this case the switching line sa,; == 0 is called an ada.ptation line with respect to the parameter ri. In analogy to So == 0 the adaptation line Sa,i == 0 can analytically be described by
Here, index Q, i indicates the switching variable characterising the adaptation at the i-th supporting point whereas index 0" characterizes the switching variable of the FSMC in the control loop. In the above expression, ..\""; > 0 indicates the slope of the adaptation line sa,; == 0, and (Sa,i o' Ui o) defines a point on the adaptation line sa,; = O. The adaptation law can analytically be expressed by
(8) Apuzz == f(s", S,,' u) reflects the control characteristic obtained by the rules (6)
4
The two-link robot arm
The adaptation algorithm is firstly tested on a simulation example of a two-link robot arm. Let the basic equation describing the motion of a robot arm be
M(O)
·8+ N(O, 0) == u,
(9)
e
where 0 is a (k x 1) is the position vector, is a (k xl) velocity vector, 0 is a (k x 1) accelleleration vector, M(8) is a (k X k) matrix of inertia (invertable), N(O,O) is the (k xl) vector of damping, centrifugal, coriolis, gravitational forces, and u is the (k x 1) vector of generalized forces. The quality of the controller design, consisting of a FSMC and fuzzy sliding-mode adaptation block is examined for each link of a simulated twolink robot with the angles 0 1 and O2 • The results obtained by two adaptive fuzzy sliding-mode controllers are presented in the following. The sampling interval is chosen as 0.008s. Using planetary drives, the respective arms can be regarded as decoupled since they have a high inertia due to the drives. The two-link robot can then be regarded as being composed of two SISOsystems. Thus, every arm can be modelled effectively as a double integrator. Nonlinear effects in the form of position-dependent friction and remaining coupling effects are neglected and will be rejected during control. According to the kinematic properties of the robot, i.e. x = f(O),
the cartesian coordinates x have to be transformed into joint coordinates according to
(11) The function f describes the kinematic model of the robot and the geometric relations between cartesian coordinates and joint coordinates. An efficient approach to solve the inverse problem is the differential method where by differentiation of (10) with respect to time we obtain
x=
(12)
J(8)iJ.
~~ == J(B) is a m x n Jacobi-matrix, m is the number of cartesian coordinates, and n the number of joint coordinates. In order to avoid control osciallations, a continuously differentiable velocity profile is chosen
(13) In Fig. 3, the reference trajectory in the form of a square is given (--). The task of the controller is to make the system (-) follow the desired system trajectory from an initial state which does not lie on the reference trajectory. In Fig. 4, the output values of the two angles of the robot arm 0 1 ( - - ) and O2 (-) and their respective velocities are given. At the same time, the manipulated variables UI ( - - ) and U2 (-) are shown. In Fig. 5, the adaptation of the four parameters of the first fuzzy sliding-mode controller are shown. Although the initial adaptation rate is limited the parameters converge quickly. By using the fuzzy sliding-mode controller in the adaptation block, a robust adaptation behavior can he introduced while the adaptation rate is limited. At the same time, the parameter convergence is fast enough. In Fig. 6 the respective values for the second FSMC are given.
y[m]t
O.08 , - - - - - . . . , - - - - - , - - - - - - , - - - - - - - ,
0.06 0.04 0.02
o ·0.02 -0.04 -0.06
-0·'].O':-:5~--~O----:-O.~05----0:"".1-----:-'O.15 ~
x[m]
Figure 3: Reference (- -) and actual (-) trajectory
(10)
502
Copyright 1999 IFAC
ISBN: 0 08 043248 4
AN ADAPTIVE FUZZY SLIDING-MODE CONTROLLER FOR A RO ...
'. ',1=:·:= o
1
2
3
4
14th World Congress ofIFAC
1 -6 '.
5
'··;':~l "'·"'~~l o
J
2
3
4
.8
',
'
-10
j
-12
·14
-
-160'-----~--~--~--~4----"S
---
-O.5L~---~-.........---~--~---....1-
o
123
4
5
t !&j
[[sI
Figure 5: Parameter adaptation for the first controller Figure 4: State space and manipulated variahle for the arm
5
Real time experiments with a three-link robot arm
.6.r2c6.T2tAI13.Ar24j°.os 007 0.06
0.05
The described adaptive controller was also applied to a three-li.nk robot arm. This robot arm is installed on a mobile platform and used for environmental exploration as well as for shifting of obstacles. Thus the position control system for the end effector at the third link has to cope with widely varying operation conditions. Therefore a rohust adaptive controller was necessary. This is an ideal real time application for the here introduced control scheme of an adaptive fuzzy sliding mode controller. The robot arm is depicted in Fig. 7. The links are driven by special DC motors. The angles are me
0.2
0,25
0.3
0.35
x [m]
Figure 8: Reference trajectory (--) and measured trajectory (-)
-0.4 -0.5
o
2
4
6
10
12
14
~
t
Figure 11: Swiching variable layer
82
Is]
in the boundary
2.4
2.2
2,0
1.8
1.6 1.4 0':---=-----:-4-~6~-~S----:I":'Q-~12:----l14 ~
t[s]
Figure 9: Desired values (--) and actual values of the second link (-)
-0.3 -0.4
-0.5
o
2
4
6
10
12
i4 ~
t (s)
-3L....--~-~--~-~
o
2
6
__
~
___
10
Figure 12: Swiching variable layer
~_~
12
83
in the boundary
14 ~
t
Is]
Figure 10: Desired values (--) and actual values of the third link (-)
504
Copyright 1999 IF AC
ISBN: 008 0432484
AN ADAPTIVE FUZZY SLIDING-MODE CONTROLLER FOR A RO ...
6
14th World Congress ofIFAC
Concluding remarks
This paper has presented a new type of an adaptive fuzzy sliding mode controller. The mathematical description and the heuristics-based linguistic adaptation law as well as the practical application in simulation and real time experiments have been presented. Through the adaptation of the nonlinear controller characteristics the chattering of the controller has been suppressed and exellent performance of the tracking behavior has been achieved. Due to the very small sampling time the design is based on a quasi continuous approach. For larger sampling times, however, a discrete time approach should be developed in the future.
References [Astrom 1995] Astrom KJ., Wittenmark B. (1995). A survey of adaptive control applications. In Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, pages 649-654. [Berstecher (a) 1996J Berstecher R., Palm R., Unbehauen H. (1996). Direct fuzzy adaptation of a fuzzy controller. In Proceedings of the International Federation of Automatic Control, IFAC, 13 th World Congress, San Francisco, pages 39-44. [Berstecher (b) 1996] Berstecher R., Palm R., Unbehauen H. (1996). Linguistic adaptation of a fuzzy controller. In Proceedings of the Sixth IEEE International Conference on Fuzzy Systems, New Orleans, pages 1794-1799. [Hung 1993] Hung J.H., Gao W., Hung J.C. (1993). Variable structure control: A survey. IEEE Transactions on Industrial Electronics, 40(1):2-22. [Pahn 1994J Palm R. (1994). Robust Control by Fuzzy Sliding Mode. Automatica, 30(9):1429-1437. [Slotine 1991] Slotine J.-J., \V. Li (1991). Applied Nonlinear Control. Prentice Hall, Englewood Cliffs. [Sun 1996] Sun F.C., Sun Z.Q., Feng G. (1996). Design of adaptive fuzzy sliding mode controller for robot manipulators. In Proceedings 0/ the Sixth IEEE International Conference on Fuzzy Systems, New Orleans, pages 62-67. [Drakunov 1992] Young K.D., Drakunov S.V. (1992). Sliding mode control with chattering reduction. In Proceedings of the American Control Conference, Chicago, pages 1291-1292.
505
Copyright 1999 IF AC
ISBN: 008 0432484
