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Design of Fuzzy Learning Compensators and Controllers for Autonomous. Redundant Robot Manipulators
Design
of
Fuzzy
Learning
Compensators
and
Controllers
for
Autonomous.
Redundant Robot Manipulators Dr. Ranjan Vepa and Hrs. Ann Now' Queen Hary and Westfield College. London University. London. U.K. El 4NS. Abstract.
In this paper we present two techniques for learning rules for
implementing
fuzzy
controllers.
the
first
of
which
is
based
on
the
application of a suitable control strategy. The second seeks to generate the appropriate control rules to drive the plant towards the desired response. Both techniques are discussed in the context of their application to robot manipulators. Key \lords.
Fuzzy Control.
Learning Systems.
Sliding Hode Control.
Robot
Hanipulators 1.
INTRODUCTION In the third approach the control actions are known to be the result of applying a suitable control strategy ( Karr [1991a. 1991b], Bersini [1992]). A typical strategy is the minimisation of a certain cost functional. A major difficulty is the proper choice of the cost functional . The choice is important as it dictates the performance achieved by the controller . the puprpose of a learning controller is to identify the best cost metric that results in the desired performance. Thus if one has a good performance metric i t is possible in principle to update the cost index to achieve good performance . Karr's method is essentially a based on minimization of a quadratic cost index using a genetic approach . His method is a typical fuzzy optimal control technique rather than a learning technique .
In recent years the study of control has received considerable attention by researchers from diverse and different fields like physiologists. ecologists. economists. engineers and bussiness executives . Although each of their application areas is different the realization that control is basic to an extraordinary array of applications has led to a number of new paradigms for its implementation . The study of control as an universal process was first initiated by Norbert Weiner [1954] in an early seminal publication. Fuzzy Logic control (Mamdani [1974], Assilian and Mamdani [1975]) is one of the new paradigms being used in many control applications . It combines to large extent the advantages of both Fuzzy sets (Zadeh 11965]) and Rule based control (Zadeh [1973]. Michie [1989]) . However a major difficulty with fuzzy logic control is that their must be available a set of Complete rules . Completeness implies that the rules cover all possible situations that the plant would encounter during its operation life. It also implies that the rules would provide the appropriate actions that must be implemented by the controller in order to achieve the desired performance objective. If rules of this nature are not available it is imperative that such rules are established in the first place . One may identify four different approaches to this problem. in the context of real-time control .
In the fourth approach the desired performance features. rather than a control strategy. may be known explicitly . Barto. Sutton and Anderson 11983] and Barto [1990] were the first to recognise the need for a separate assessment of performance of the controller independent of the control strategy . This they do by using a reinforcement type learning technique. It is vital to understand the differences between adaptive. self-organising and learning controllers. Adaptive control seeks to modify the controller parameters in order to emulate the response of a model. Self-organisers on the other hand also modify the controller but in order to meet certain performance requirement . The assessment of performance may involve the application of some additional and relevant assessment techniques. Learning controllers may then be considered to be those which modify the control strategy in order to achieve a desired performance.
In the first approach it is assumed that a set of valid and acceptable continuous time contol actions are known . These must then be described linguistically as a set of situation-action rules. The rule learning problem here is essentially an approximation problem . A number of neural network and genetic algorithm based methods belong to this category. In the second approach it is assumed that the desired response of the plant is known rather than the control action required to generate it. Self organising controllers ( Procyk [1977]. Procyk and Mamdani [1979]. Yamazaki and Mamdani 11982]. Handic et. al. [1985], Unkens and Hasnain [1991]) essentially seek to adjust the rules so as to achieve the desired response.
In this paper we present two learning techniques. the first of which is based on the application of a suitable control strategy. The second seeks to generate the appropriate control rules to drive the plant towards the deSired response . Both techniques are discussed in the context of their application to robot manipulators.
312
2.
aJ4P~SATION
AND CXlNTROL OF ROBOT MANIPULATORS
Robot aanipulators are generally classified as serial or parallel chain kinematic linkages or as closed and open kinemaUc chains (ViJay Kumar and Gardner (1989), Salisbury and Craig (1982) . However most practical robot systems including walking machines, multifingered grippers and cooperating arms consist of two or more articulated serial chains which act in parallel on an object . In most cases they include one or more closed kinematic chains and usually have one or more redundant actuators. On the other hand serial kinematic chains in the more conventional robot manipulators are characterised by kinematic redundancies where the the number of actuators exceed the dimensions of the task space . Thus the degree of redundancy of a robot manipulator is determined relative to the task it is designed to perform. In such redundant articulated robot manipulators it is not enough if one resolves the redundancy; rather it should be exploited fully to perform the task at hand dextorously and in a versatile manner. It is therefore important to evaluate relevant performance metrics to achieve this objective . A number of techniques have been proposed in the literature for the resolution of the redundancy based on a variety of concepts . (Seraji [1989]]. One of the more recent techniques proposed is based on configuration control where the redundancy of the manipulator was utilized to achieve an optimum redundant configuration directly in the task space , thereby also avoiding the inverse kinematic transformation from the task to the joint space . While the concept has been an extremely reasonable and realistic one , it is almost always essential that the optimal configuration must be specified in some form or the other by the user. Thus there is a need for an auxil iary or addi tional set point or task trajectory. The generation of this trajectory is not autonomous . In most space-based robot s ystems, such as those envisaged for use in a space station , the concept is not entirely practical and it is essential that this additional task be set autonomously . Further the control techniques suggested in the literature for use with redundant manipulators have almost universally been model based. The first of the learning techniques we propose is a model free and autonomous approach to configuration control based on a Fuzzy Learning algorithm proposed earlier by the second author . A model free approach is also adopted for the primary task trajectory con t rol . The autonomous configuration control scheme is based on the idea of Fuzzy compensation and is illustrated in Fig . 1.
First we consider the application of the concept in joint space . The fuzzy compensation is based on the concept of 'safety' where the compensator seeks to function in such a way that system's operating point in the state space has the widest possible spectrUlll of control actions available for implementation. Since some points in the multi-dimensional state space are constrained by the need for the task trajectory to be followed the other points are so chosen such that a performance metric, which is a measure of the control actions available is maximized . The fuzzy compensator thus ensures that the robot Is operatlng at a point of maxlmUlll safety thus ensurlng that it is in a stable configuratlon. When the concept is · applied in the task space rather than in joint space It is Important to consider the ' manlpulabillty' of the kinematic chain (Yoshihara (1985) ) . The concept is applied to a dual link model of the UNIHATION PUMA 560 robot manipulator . This manipulator can be represented by an inverted double link configuration and it is shown that it can be 'safely' maintained in a vertical configuration . Alternate configurations with the task trajectory defined in the cartesian space have also been investigated. The relationship between the classical concepts of controllability and observability in control theory and manipulability in kinematics to the concept of 'safety' is clearly shown and a suitable performance metriC, for autonomous configuration control in the task space, is proposed . 3. THE NOTION OF SAFETY The state of a system is said to be 'safe' when the system is not very sensitive to disturbances from the environment and consequently not easily disturbed from this state. The learning algorithm which is the basis of the fuzzy learning compensation methodology locates the 'safe' regions of the phase plane and tunes the controller so that it remains in the safest region of the phase plane . This is the basic compensation strategy involved. A quantitative measure of safety is developed and the controller drives the system so as to maximize this performance metric . Thus the stability of equilibrium is ensured provided the quantitative measure of safety is a maximum only at the corresponding point in the phase plane . The quantitative measure is also chosen so as to reflect the need for a wide spectrum of control actions being available to the controller at the safest point in the state space . The quanti ta ve measure embodies both the concepts of stability and controllability and can be learned recursively . Further details of the concept can be found in Nowe (1992). 4. THE MEASURE OF SAFETY
Feedback Although "safety" is essentially a property of the state of the system it is assessed entirely within the controller . Thus the measre of safety is defined in terms of the inputs to and outputs of the fuzzy controller . The fuzzy controller is modelled as a function, F, which maps a fuzzy region modelled by a fuzzy set over a region of the state space 1 to a spectrum of
Plant
J
control actions modelled by another fuzzy set over a region of possible control actions O . The image J
Fe 1 / of each fuzzy region 1 J expresses a fuzzy restriction OJ on the possible control actions for that region.
Fuzzy Compensator
The safety of a fuzzy region is reflected in the spectrum of the corresponding image . The wider the spectrum the safer the region . It is therefore proposed to measure the safety of a fuzzy region 1 by taking the integral of the membership
Redundancy Resolution by Fuzzy Compensation
Fig . 1 The Fuzzy Compensator Concept
J
313
~OJ
functlon
utrice. uy then be deterained. The solution uy be coapactly represented as,
of the correspond1ng laage 0J'
J~
S( I ) •
0
J
P' du
z
(1 )
DJ
where the integration is with respect to the control action, u, and is over the entire UnIverse of dIscourse, O.
• J:TTe-A(t-T)T yTB BTT TTe-AT(t-T)T dT o
implies that det (P;) .. O. A necessay condit10n 1s therefore obta1ned from the fact, that,
I~I ~II (state) x S(II)
(state)
Recogniz1ng that 1n a non11near s1tuat10n T 1s the Jacobian, J, of the non11near transformat1on 1t follows that for controllabi11ty 1n the transformed doma1n a necessary cond1t1on 1s given by,
I
where N 1s the number of fuzzy regions cover1ng the state space . Before presenting the learning algorithm we briefly present an equivalent def1nition of 'safety' based on linear theory. In the next sect10n we apply the algor1thm for comput1ng safety to a linear second order system and compare the features of the two measures.
det(JJT) .. 0 and th1s 1s the cond1tion for manupulability. Thus ensur1ng that the system is controllable in a particular reference frame guarantees manupulab1l1 ty.
5. LINEAR SECOND ORDER SYSTEMS
We assume without loss of genera11ty that PI and P are d1agonal. If this is not so they may
We consider a linear second order system of the form,
x
A
+
X
+
z
be d1agona11zed by suitably transforming the state vectors.
(3)
BU
C X
which may be rewritten as a pair of coupled first order equations as.
We now consider the quantity, V "' -
d
(4)
dt
Hence it follows that, PI2 P
+
P~2
- P
=0 T
C
",
0 or P
2 A P + P AT '" B BT. 2
I
2
2
12
= pT12 T
"' P C I
c
B
u.
P
IJ
[~
-
I}
P
X
J
(t) + P
( X
X (t) 1
2J J
J
(t)
+
c
JJ
X (t) J
)1
lj p/tl 1
(5)
are the diagonal elements of PI' c jj the d1agonal elements of C which 1s also assumed diagonal and PJ(t) is defined by,
'" 0 ) C P and
I
J
I
(6)
Equation (6) for the Grammian matrix P is also the 2 equation for the Grammian matrix corresponding to the eguation, 2 + A 2
c
where P
( or P
I}
IJ PIJ
The controllability Grammian matrix satisfies the steady state equation ,
]
(10)
det(TTT) .. O.
(2)
I~I ~I
(9)
det(P' ) • [ det(TTT)]3 det(P ) z z Thus we requ1re that,
H
H
(8)
Controllabil1ty 1n the transformed doma1n requ1res that all eigenvalues of P; be positive and this
Dur1ng the learn1ng we also need to measure the safety of a cr1sp state. S1nce a cr1sp state generally belongs to several fuzzy regions, the safety measure is def1ned as the we1ghted average of the safety measures of all the fuzzy regions to which the cr1sp state belongs.
S(state)
• J:e-A' (t-T) B' B,T e-A'T(t_~) dT o
(7)
The above equations also indicate that the Grammian matrix corresponding to the states X is linearly related to the Grammian matrix corresponding to the states X. This is an extremely 1mportant feature of linear second order systems. The controllability Grammian matrix expresses the relative controllability of the states, relative to each other . Thus if the jlh eigenvalue of Grammian
~~~~iX t~:t la~~: r~t~ti~~a~~ t~: o~~~~~i~!l:mp!~~: controllable. For purposes of robot trajectory planning it is often required to transform from a set of Cartesian space coordinates to a set of joint space coordinates. The effect of these transformations on the matrices A. Band C isTthat they transform to A~, B' and C' where, A' • TAT, C' • T CT and B' • T B. The corresponding equations for the Grammian
Pj(t).
Xj(tl + CjjXj(t)
The quant1ty V is a linear function of the states which reflects the relative contribUtions to the various states of the control input. Further we assume that the states have been quantized and deflne safety as, S • G
J
dV
(11)
DIJ where the 1ntegration 1s along a trajectory 1n the phase-space over a particular domain of quant1zation 1n the phase-space and G 1s an arb1trary constant. If the system trajectory is ent1rely w1th1n the quant1zed equi11brium state dur1ng the time window under consideration S would be max1mum for this state and zero for all other points in the phase-space. We note that V is a constant along the straight lines Pj(t) an arbitrary constant and this entirely reflects the fact that all state pairs along these lines are equally realizable by applying a sultable control 1nput to the system in an initial quiescent state . On the other hand 1f S 1& a max1mum at a set of
314
states located along a line It Indicates that the point where S Is a aaxlaUII represents an equilibrium point. For the cla.s of second order systems considered above it can be shown that this Is in fact the case. Thus the quantity S embodies both the concepts of controllability and stability and aax1mizes S implies that the system is both controllable and stable In the viCinity of S • S..x The aeasure of safety defined above is precisely the same as the one defined for in the fuzzy si tuaUon if one assumes all the relevant membership functions to be equal to 1.
The learn1na al,orltha which reinforces the control actions that transfer the system to a safer state and thus learns the .mbership functions J.i OI
corresponding to the suaaarlzed .s follows:
+
a x
+ c x
•
based on our notion of safety was constructed for three pairs of values of a and c. The safety is a maximum along the line s • 0, when compared to values in both the directions of the axes, where, s ~ x(t) + ~ x(t)·
ay
be
• Choose an ini Ual state, i~ at random within the doain under consideration .nd repeat the following :
(12)
b u
DJ'
• Repeat the following steps until convergence 18 reached:
The fuzzy safety eap for a linear one degree of freedom second order system given by, x
consequents
Select random,
a
control
action
at
0,
•
• apply the control action during a time period At to the system, • record the new state i ~+1
0 • update the membership grade of
Along the line s & 0 it is a maximum at the origin of the phase plane and the i t tends to fall off gradually in a direction orthogonal to s • O. The rate at which it falls off is inversely proportional to the parameter 'a' which is also the damping in the system. The slope ~ is a function of 'c' and 'a' and the time step. Thus the analysis clearly illustrates the features of the safety metric defined by equation (1) and these are similar to those that would be expected of the metric defined by equation (11) .
0
•
of
all consequents 0J by AOJ(O.) which is defined as: AOJ(os) •
~I (i~)
x
~
J
0) x [R(i t +,1 .
-
0 (0 )] J.
and,
0 if out of domain under consideration R( i
\,+1
,0) s
S(1
{
min[ 1,
~+1
)
--------~~--~------~I.-I, •• xl •• l expect.tion (or J
otherwise
• Until trajectory is outside the domain under consideration .
6 . THE LEARNING ALGORITHM Prior to learning the phase space is quantized into a number of discrete domains and overlapping fuzzy membership functions are defined over each domain, I J' and its neighbours . All the membership functions are assumed to be triangular shaped and overlapping below a grade equal to 0.5 . These membership functions defining the antecedants in the fuzzy rules are fixed . The control action is also quantized and covered by overlapping membership functions . Initially they are assumed to be same and equal to 0 . 5 over the entire domain of the control action, 0 which is made up of N discrete domains, OJ' j & 1, 2, 3, ... , N.
In the above algorithm ~ E [ 0, 11 is the learning rate which may be set to zero after the completion of the learning cycle. The maximal expectation of I J is a maximum of a table of safety measures of the n most recent states i i~
for which
~I (i~)
<+1
reached from a state
> 0 . 5 and n is taken to be
J
twice the number of elements in the discretised output set O. The details may be found in Nowe (1992).
The algorithm was applied to a typical UNIMATION PUMA 560 robot manipulator. In a typical case the specification for the inverted double pendulum configuration was that the upper pendulum be maintained at an angle 9 • 0 . 4. A fuzzy rule 1
The learning algori thm seeks to adjust the membership functions of the conclusions 0 of 1
the M fuzzy rules such that : • A control action 0 which transfers the current state to a new state outside the domain under consideration is excluded. Thus a membership grade equal to 0 . 0 is assigned to the corresponding domains . The best control action 0 for a fuzzy region I J is the action which brings the system into the safest region that can be reached from there. It is rewarded with a membership grade equal to 1 in the corresponding image Of The membership grade of other control actions is between 0 and 1, and is related to the state the system is tranfered to from a partlcular ini tial state. The larger the safety of the new state, the larger the membership grade .
315
based controller was used to maintain the upper pendulum while the lower pendulum was allowed to settle down to a safe position. The resulting stable angular position for the lower pendulUm 9
2
• -0 . 14 .
7. FUZZY LEARNING SLIDING MODE CONTROL Many Fuzzy systems have been designed on the basis of rules defined in a 'phase plane' with the error and change in error between the set point and the output as the reference axes . A related general approach based on variable structure controllers is to divide the phase plane into two half- planes by means of a switching line and the control action signals in the two semi planes can be easily defined provided the controller is defined along the switching line. The basic idea is to drive the system towards the switching line and then along it to the origin. Such controllers have been referred to as sliding mode controllers ( Slotine (1985) l. where the sliding mode refers to
of
the dynaaic response svitcb1na 11ne.
the
the
Artificial Neural Nets have the unique ability to construct system inverses. Thus liven that a system has as its output, a certain deaanded value plus a known error signal, the input required to ,enerate the output can be constructed using a neural net. Several controllers generate the control signal entirely from a knowledge of the error and in these cases it is possible to construct the error-control signal relationship when the dynamics of the error Signal is completely defined irrespective of the plant dynamics . Fortunately it 1& possible to define the dynaaics of the error completely along a switching line by choosing the dynamical equation defining the sliding mode . Thus it is a straight forward matter to construct the control rules along the switching line and this can be done by simulating the error dynamics independent of the plant. The problem now reduces to an appropriate choice of the model for the sliding mode and the parameters defining it, whlch are of course no longer fuzzy. Once the control rules are established along the switching line i t is a fairly stralght forward matter to define the rules in the two seml planes on either slde of the swltching line. Thls is the baslc learning strategy adopted to construct the fuzzy Loglc control rules . The concept of a fuzzy Slldlng Hode Controller 15 not new (Palm (1989). However to the authors knowledge it has not been used as a learning strategy for syntheslzlng control rules. Thus the technique adopted here to learn the control rules for implementing a fuzzy controller involves a high degree of synergism . The method is applied to an inverted pendulum and is shown to work satisfactorily without any prevlous knowledge about the control rules . The inverse modelling was implemented based on two approaches . The first was based on an analytical nonlinear differential equation model of the pendulum . The second was based on Neural Net modelling. Resul ts based on the first method are reported in the paper . The results based on the neural net inverse model will be presented elsewhere .
8. SLIDING MODE THEORY AND THE LEARNING STRATEGY The theory of sliding mode controllers can be most easily explained using the second order model considered earlier , i . e ., x
+
a x
+
c x
b u
( 12 )
The objective of the controller is to drive the system towards a sliding line ( or a sliding surface in the case of a mul ti-degree of freedom system ) and then drlve the system along the line towards the equilibrium state . In the above situation the equation of the sliding line is assumed to be: s
z
~(t) • ~ x(t) • 0
(13)
's' represents the normal distance in the state space from the current state to the sUding line. Hence if one ~ishes to drive the plant towards the sliding line s must either be negative when 5 15 positive or be positive when 5 is negative; i. e . ,
s s < O.
In order to satisfy choose 'u' to be,
thls
u "' (a ~(t) • c x(t)
reqUirement
we
first
•
a x
c x
a x
1
c x
-~ x •
s •
+
and,
ss·
SUI'
Thus u must be chosen such that, 1
s u
1
< O.
u 1 is interpretated as the control action required to drive the plant to the sliding line while (u u 1 ) Is the control action required to drive the plant along the line to the origin of the phase plane. We may choose u to be, 1 u • - sign(s) U 1 o where U is a positive constant. further details of o sliding mode theory as well as its robustness properties may be found in Slotine (1985). for our purpose this baslc Introduction wlll sufflce. Sliding mode theory can be adapted to form the basls of a learning algorithm for controlling a plant when no control rules whatever are available a pr jor 1. The method relies on the fact that the sliding mode can be defined independent of the plant to be controlled . Thus it is possible to simulate the error along the sliding line . The simulated error is subtracted from the desired set point to compute the output of the plant along the sliding line . If an Inverse model of the plant , based on a neural network or otherwlse 15 available, then 1t may be used to compute the control actlon necssary to produce the output of the plant along the sl1dlng llne . The control action, the error and the change in error are then fuzzifled to establish a set of control rules along the sllding line. The entire process is an offline process and resul ts In a basic set of control rules. Once all the rules along the slidlng llne are found the plant 15 set in operatlon, and durlng operation additlonal rules are establlshed at each polnt In the phase plane not covered by the existing rules . The procedure for dolng thls first Involves Identifying the current state of operation in the phase plane . A perpendlcular Is drawn from this point towards the slidlng line . The control action, u. 1 ' at the foot of the perpendlcular on the slidlng line Is noted and the control actlon at the current state is found by using the formula : u "' u. 1 - Uo slgn(s) where ' s' 15 the perpendicular distance between the current state and the sliding line . As an example the control of an Inverted pendulum with no initial control rules and only the knowledge of an Inverse model a set of control rules were contructed. Every rule generated by the learning algorithm has the structure :
where Term E, Term C and Term U are appropriate linguistic -terms . The linguistic terms used are Positive Big (PB) , Positlve Medium (PM), Positive Small (PS), Positlve Zero (PZ), Negative Zero (NZ), Zero (ZRJ. Negative Small (NS), Negative Medium (NH) and Negative Big (NB), The linguistic terms Term_E, Term_C and Term_U corresponding to each rule are 11sted below in Table 1. 9 . CONCLUSIONS Two fuzzy learning technlques for compensation and control have been presented in this paper .
~ ~(tJ • u )/b.
The fuzzy learning compensatlon technlque is currently being applied to multl-degree of freedom robotic systems and the resul ts are currently being extensively evaluated .
The equation of motion, (12), then takes the form, x
Hence,
u. 1
316
Table 1 Fuzzy rule base aenerated T__U
TenoJ:
t ...._c
1
PB
2
lIB
III OR 101 OR lIS PBOR I'll OR PS
3
I'll
lIS
PB
4
101 PS
PS NZ PZ NZ PZ NZ PZ NZ PZ
lIB
Jlul. 110.
, 5
lIS
7
pz
,•
NZ
10
11 12 13 14
I'll
101 NZ PZ PZORNZ PZORNZ
PB ZR PSORI'IIORPB IISORIIIORDORPB I'll
PS IOIORIISORPB I'll OR PS OIIDORZR PB III
lIS
PS
The fuzzy learning sliding ~ode technique is currently being ~od1fied for use with coupled higher order systems . For such systems rather than define swi tching surfaces in a higher dimensional phase plane, an initial decoupling feedback is being synthesized using a neural net approach . The system is then decoupled into a set of second order nonlinear systems and above technique is then implemented for each mode separately. The entire methodology is expected to be very useful for several aerospace applications.
Michie, Donald, (1989) 'Application of Machine learning to recognition and control ', University of Wales Review of Science and Technology, No. 5, pp 23-28. Nowe, A. , (1992) 'A Self-Tuning Robust Controller', Technical report 92-1, University of Brussels, February 10 .
and Mamdani, E. H. (1979), 'A Procyk, T. J. Linguistic Self-Organizing Process controller,' Automatica, ~ , 15-30.
Barto , A. G., Sutton, R. S. and Anderson C. IJ., (1983) 'Neuron-like Adaptive elements that can solve difficul t learning control problems,' IEEE Trans . Syst . Man Cybern, Vol. SMC - 13, pp 834 846 .
Procyk T. K. , Cl 977) 'Self Organising Controller for dynamic processes', PhD thesis, Queen Mary College , U. K. ,
Barto, A. G., (1990), 'Connectionist Learning for Control : An Overview,' In : Miller, 11. T. , Sutton R. and IJerbos, P. (Eds . ), (1990) 'Neural Networks for Control', MIT Press, Cambridge, U.S . A. , pp. 5-58.
Salisbury, J . K. , and J . K. Craig, (1982) , 'Analysis of mul tifingered hands: Force control and Kinematic Issues,' Int . J. Robotics Res. ,vol . 1., no . I, pp 4-17.
Bersini, H., (1992), 'Generation Automatique de Systemes de Commande Floue par les Methodes de Gradient et les Algorithmes Genetiques', Presente aux Deuxlemes Journees Nationales sur les Applications des Ensembles Flous. Nimes, 2-3 November, 1992 . for
Fuzzy
Karr , C. L. (1991), 'Design of Adaptive Fuzzy Logic Controller using A Genetic Algorithm', In: Belew, R. and Booker, L. (Eds . ) (1991) Proceedings of the Fourth International Conference on Genetic Algorithms , Morgan Kaufmann . Llnkens, D. A. and Hasnain, S. B. (1991), 'Self Organizing fuzzy logic control and application to muscle relaxant anaesthesia, 1££ Proc.-D , ~, 3, 274-284 . (1974), ' Application of Fuzzy Mamdani, E.H . , Algorithms for Control of a Simple Dynamic plant ' , Proc. 1£££, lll, pp 1585-1588. Mamdani, E. H. and Assllian, S. (1975), ' A Fuzzy Logic Controller for a Dynamic Plant ' , IntJ. J . Han-Hachine Stud . , 1, pp 1-13. Handic , N. J ., Scharf, E. M. and Mamdani, E. H. (1985), 'Practical Application of a heuristic fuzzy rule based controller to the dynamic control of a robot arm,' 1£££ Proceedings-D, Special Issue on Robotics, ~,4, 190-203.
Fuzzy INFO,
Palm, Rainer (1991) , ·Sliding Mode Fuzzy Control ", Paper presented at the IEEE' 92 Colloquium, San Diego, U. S. , 8 - 13 March .
10. References
Karr, C., (1991) 'Genetic Algorithms Controllers ', AI Expert, February.
PB lIB
SeraJi, Homayoun , (1989), 'Configuration Control of Redundant Manipulators: Theory and implementation,' IEEE Trans. Robiotics Automat., vol . 5 , no . 4, pp 472-490. Slotine, J . J . F. (1985), "The Robust Control of Robot Manipulators," The International Journal of Robotic Research, Vol . 4, No . 2, pp 49-64 . ViJay Kumar and John F. Gardner, (1990), 'Kinematics of Redundantly actuated closed chains,' IEEE Trans . Robiotics Automat.. vol. 6 . ,no . 3, pp 269-274. (1954) , ' The Human Use lie iner, N., Beings,' Doubleday, New York, 1954.
of
Human
Yamazaki, T. and Mamdani, E.H . (1982), 'On the performance of a rule-based self organising controller,' 1£££ Conf. on ApplJcation of Adaptive and Hultivariable Control, Hull, 50-55 . Yoshihara, T., (1985), 'Manipulability of Robot Mechanisms, ' Int . J. Robotics Res ., vol. 4. , no . 2, 1985. Zadeh, L. A. (1965), 'Fuzzy Sets', Control, ~, 28-44.
Information and
Zadeh, L.A. (1973), 'Outline of a new approach to the analysis of complex systems and deCision processes', 1£££ Trans. , SMC-3, 338-353 .
317
of
Fuzzy
Learning
Compensators
and
Controllers
for
Autonomous.
Redundant Robot Manipulators Dr. Ranjan Vepa and Hrs. Ann Now' Queen Hary and Westfield College. London University. London. U.K. El 4NS. Abstract.
In this paper we present two techniques for learning rules for
implementing
fuzzy
controllers.
the
first
of
which
is
based
on
the
application of a suitable control strategy. The second seeks to generate the appropriate control rules to drive the plant towards the desired response. Both techniques are discussed in the context of their application to robot manipulators. Key \lords.
Fuzzy Control.
Learning Systems.
Sliding Hode Control.
Robot
Hanipulators 1.
INTRODUCTION In the third approach the control actions are known to be the result of applying a suitable control strategy ( Karr [1991a. 1991b], Bersini [1992]). A typical strategy is the minimisation of a certain cost functional. A major difficulty is the proper choice of the cost functional . The choice is important as it dictates the performance achieved by the controller . the puprpose of a learning controller is to identify the best cost metric that results in the desired performance. Thus if one has a good performance metric i t is possible in principle to update the cost index to achieve good performance . Karr's method is essentially a based on minimization of a quadratic cost index using a genetic approach . His method is a typical fuzzy optimal control technique rather than a learning technique .
In recent years the study of control has received considerable attention by researchers from diverse and different fields like physiologists. ecologists. economists. engineers and bussiness executives . Although each of their application areas is different the realization that control is basic to an extraordinary array of applications has led to a number of new paradigms for its implementation . The study of control as an universal process was first initiated by Norbert Weiner [1954] in an early seminal publication. Fuzzy Logic control (Mamdani [1974], Assilian and Mamdani [1975]) is one of the new paradigms being used in many control applications . It combines to large extent the advantages of both Fuzzy sets (Zadeh 11965]) and Rule based control (Zadeh [1973]. Michie [1989]) . However a major difficulty with fuzzy logic control is that their must be available a set of Complete rules . Completeness implies that the rules cover all possible situations that the plant would encounter during its operation life. It also implies that the rules would provide the appropriate actions that must be implemented by the controller in order to achieve the desired performance objective. If rules of this nature are not available it is imperative that such rules are established in the first place . One may identify four different approaches to this problem. in the context of real-time control .
In the fourth approach the desired performance features. rather than a control strategy. may be known explicitly . Barto. Sutton and Anderson 11983] and Barto [1990] were the first to recognise the need for a separate assessment of performance of the controller independent of the control strategy . This they do by using a reinforcement type learning technique. It is vital to understand the differences between adaptive. self-organising and learning controllers. Adaptive control seeks to modify the controller parameters in order to emulate the response of a model. Self-organisers on the other hand also modify the controller but in order to meet certain performance requirement . The assessment of performance may involve the application of some additional and relevant assessment techniques. Learning controllers may then be considered to be those which modify the control strategy in order to achieve a desired performance.
In the first approach it is assumed that a set of valid and acceptable continuous time contol actions are known . These must then be described linguistically as a set of situation-action rules. The rule learning problem here is essentially an approximation problem . A number of neural network and genetic algorithm based methods belong to this category. In the second approach it is assumed that the desired response of the plant is known rather than the control action required to generate it. Self organising controllers ( Procyk [1977]. Procyk and Mamdani [1979]. Yamazaki and Mamdani 11982]. Handic et. al. [1985], Unkens and Hasnain [1991]) essentially seek to adjust the rules so as to achieve the desired response.
In this paper we present two learning techniques. the first of which is based on the application of a suitable control strategy. The second seeks to generate the appropriate control rules to drive the plant towards the deSired response . Both techniques are discussed in the context of their application to robot manipulators.
312
2.
aJ4P~SATION
AND CXlNTROL OF ROBOT MANIPULATORS
Robot aanipulators are generally classified as serial or parallel chain kinematic linkages or as closed and open kinemaUc chains (ViJay Kumar and Gardner (1989), Salisbury and Craig (1982) . However most practical robot systems including walking machines, multifingered grippers and cooperating arms consist of two or more articulated serial chains which act in parallel on an object . In most cases they include one or more closed kinematic chains and usually have one or more redundant actuators. On the other hand serial kinematic chains in the more conventional robot manipulators are characterised by kinematic redundancies where the the number of actuators exceed the dimensions of the task space . Thus the degree of redundancy of a robot manipulator is determined relative to the task it is designed to perform. In such redundant articulated robot manipulators it is not enough if one resolves the redundancy; rather it should be exploited fully to perform the task at hand dextorously and in a versatile manner. It is therefore important to evaluate relevant performance metrics to achieve this objective . A number of techniques have been proposed in the literature for the resolution of the redundancy based on a variety of concepts . (Seraji [1989]]. One of the more recent techniques proposed is based on configuration control where the redundancy of the manipulator was utilized to achieve an optimum redundant configuration directly in the task space , thereby also avoiding the inverse kinematic transformation from the task to the joint space . While the concept has been an extremely reasonable and realistic one , it is almost always essential that the optimal configuration must be specified in some form or the other by the user. Thus there is a need for an auxil iary or addi tional set point or task trajectory. The generation of this trajectory is not autonomous . In most space-based robot s ystems, such as those envisaged for use in a space station , the concept is not entirely practical and it is essential that this additional task be set autonomously . Further the control techniques suggested in the literature for use with redundant manipulators have almost universally been model based. The first of the learning techniques we propose is a model free and autonomous approach to configuration control based on a Fuzzy Learning algorithm proposed earlier by the second author . A model free approach is also adopted for the primary task trajectory con t rol . The autonomous configuration control scheme is based on the idea of Fuzzy compensation and is illustrated in Fig . 1.
First we consider the application of the concept in joint space . The fuzzy compensation is based on the concept of 'safety' where the compensator seeks to function in such a way that system's operating point in the state space has the widest possible spectrUlll of control actions available for implementation. Since some points in the multi-dimensional state space are constrained by the need for the task trajectory to be followed the other points are so chosen such that a performance metric, which is a measure of the control actions available is maximized . The fuzzy compensator thus ensures that the robot Is operatlng at a point of maxlmUlll safety thus ensurlng that it is in a stable configuratlon. When the concept is · applied in the task space rather than in joint space It is Important to consider the ' manlpulabillty' of the kinematic chain (Yoshihara (1985) ) . The concept is applied to a dual link model of the UNIHATION PUMA 560 robot manipulator . This manipulator can be represented by an inverted double link configuration and it is shown that it can be 'safely' maintained in a vertical configuration . Alternate configurations with the task trajectory defined in the cartesian space have also been investigated. The relationship between the classical concepts of controllability and observability in control theory and manipulability in kinematics to the concept of 'safety' is clearly shown and a suitable performance metriC, for autonomous configuration control in the task space, is proposed . 3. THE NOTION OF SAFETY The state of a system is said to be 'safe' when the system is not very sensitive to disturbances from the environment and consequently not easily disturbed from this state. The learning algorithm which is the basis of the fuzzy learning compensation methodology locates the 'safe' regions of the phase plane and tunes the controller so that it remains in the safest region of the phase plane . This is the basic compensation strategy involved. A quantitative measure of safety is developed and the controller drives the system so as to maximize this performance metric . Thus the stability of equilibrium is ensured provided the quantitative measure of safety is a maximum only at the corresponding point in the phase plane . The quantitative measure is also chosen so as to reflect the need for a wide spectrum of control actions being available to the controller at the safest point in the state space . The quanti ta ve measure embodies both the concepts of stability and controllability and can be learned recursively . Further details of the concept can be found in Nowe (1992). 4. THE MEASURE OF SAFETY
Feedback Although "safety" is essentially a property of the state of the system it is assessed entirely within the controller . Thus the measre of safety is defined in terms of the inputs to and outputs of the fuzzy controller . The fuzzy controller is modelled as a function, F, which maps a fuzzy region modelled by a fuzzy set over a region of the state space 1 to a spectrum of
Plant
J
control actions modelled by another fuzzy set over a region of possible control actions O . The image J
Fe 1 / of each fuzzy region 1 J expresses a fuzzy restriction OJ on the possible control actions for that region.
Fuzzy Compensator
The safety of a fuzzy region is reflected in the spectrum of the corresponding image . The wider the spectrum the safer the region . It is therefore proposed to measure the safety of a fuzzy region 1 by taking the integral of the membership
Redundancy Resolution by Fuzzy Compensation
Fig . 1 The Fuzzy Compensator Concept
J
313
~OJ
functlon
utrice. uy then be deterained. The solution uy be coapactly represented as,
of the correspond1ng laage 0J'
J~
S( I ) •
0
J
P' du
z
(1 )
DJ
where the integration is with respect to the control action, u, and is over the entire UnIverse of dIscourse, O.
• J:TTe-A(t-T)T yTB BTT TTe-AT(t-T)T dT o
implies that det (P;) .. O. A necessay condit10n 1s therefore obta1ned from the fact, that,
I~I ~II (state) x S(II)
(state)
Recogniz1ng that 1n a non11near s1tuat10n T 1s the Jacobian, J, of the non11near transformat1on 1t follows that for controllabi11ty 1n the transformed doma1n a necessary cond1t1on 1s given by,
I
where N 1s the number of fuzzy regions cover1ng the state space . Before presenting the learning algorithm we briefly present an equivalent def1nition of 'safety' based on linear theory. In the next sect10n we apply the algor1thm for comput1ng safety to a linear second order system and compare the features of the two measures.
det(JJT) .. 0 and th1s 1s the cond1tion for manupulability. Thus ensur1ng that the system is controllable in a particular reference frame guarantees manupulab1l1 ty.
5. LINEAR SECOND ORDER SYSTEMS
We assume without loss of genera11ty that PI and P are d1agonal. If this is not so they may
We consider a linear second order system of the form,
x
A
+
X
+
z
be d1agona11zed by suitably transforming the state vectors.
(3)
BU
C X
which may be rewritten as a pair of coupled first order equations as.
We now consider the quantity, V "' -
d
(4)
dt
Hence it follows that, PI2 P
+
P~2
- P
=0 T
C
",
0 or P
2 A P + P AT '" B BT. 2
I
2
2
12
= pT12 T
"' P C I
c
B
u.
P
IJ
[~
-
I}
P
X
J
(t) + P
( X
X (t) 1
2J J
J
(t)
+
c
JJ
X (t) J
)1
lj p/tl 1
(5)
are the diagonal elements of PI' c jj the d1agonal elements of C which 1s also assumed diagonal and PJ(t) is defined by,
'" 0 ) C P and
I
J
I
(6)
Equation (6) for the Grammian matrix P is also the 2 equation for the Grammian matrix corresponding to the eguation, 2 + A 2
c
where P
( or P
I}
IJ PIJ
The controllability Grammian matrix satisfies the steady state equation ,
]
(10)
det(TTT) .. O.
(2)
I~I ~I
(9)
det(P' ) • [ det(TTT)]3 det(P ) z z Thus we requ1re that,
H
H
(8)
Controllabil1ty 1n the transformed doma1n requ1res that all eigenvalues of P; be positive and this
Dur1ng the learn1ng we also need to measure the safety of a cr1sp state. S1nce a cr1sp state generally belongs to several fuzzy regions, the safety measure is def1ned as the we1ghted average of the safety measures of all the fuzzy regions to which the cr1sp state belongs.
S(state)
• J:e-A' (t-T) B' B,T e-A'T(t_~) dT o
(7)
The above equations also indicate that the Grammian matrix corresponding to the states X is linearly related to the Grammian matrix corresponding to the states X. This is an extremely 1mportant feature of linear second order systems. The controllability Grammian matrix expresses the relative controllability of the states, relative to each other . Thus if the jlh eigenvalue of Grammian
~~~~iX t~:t la~~: r~t~ti~~a~~ t~: o~~~~~i~!l:mp!~~: controllable. For purposes of robot trajectory planning it is often required to transform from a set of Cartesian space coordinates to a set of joint space coordinates. The effect of these transformations on the matrices A. Band C isTthat they transform to A~, B' and C' where, A' • TAT, C' • T CT and B' • T B. The corresponding equations for the Grammian
Pj(t).
Xj(tl + CjjXj(t)
The quant1ty V is a linear function of the states which reflects the relative contribUtions to the various states of the control input. Further we assume that the states have been quantized and deflne safety as, S • G
J
dV
(11)
DIJ where the 1ntegration 1s along a trajectory 1n the phase-space over a particular domain of quant1zation 1n the phase-space and G 1s an arb1trary constant. If the system trajectory is ent1rely w1th1n the quant1zed equi11brium state dur1ng the time window under consideration S would be max1mum for this state and zero for all other points in the phase-space. We note that V is a constant along the straight lines Pj(t) an arbitrary constant and this entirely reflects the fact that all state pairs along these lines are equally realizable by applying a sultable control 1nput to the system in an initial quiescent state . On the other hand 1f S 1& a max1mum at a set of
314
states located along a line It Indicates that the point where S Is a aaxlaUII represents an equilibrium point. For the cla.s of second order systems considered above it can be shown that this Is in fact the case. Thus the quantity S embodies both the concepts of controllability and stability and aax1mizes S implies that the system is both controllable and stable In the viCinity of S • S..x The aeasure of safety defined above is precisely the same as the one defined for in the fuzzy si tuaUon if one assumes all the relevant membership functions to be equal to 1.
The learn1na al,orltha which reinforces the control actions that transfer the system to a safer state and thus learns the .mbership functions J.i OI
corresponding to the suaaarlzed .s follows:
+
a x
+ c x
•
based on our notion of safety was constructed for three pairs of values of a and c. The safety is a maximum along the line s • 0, when compared to values in both the directions of the axes, where, s ~ x(t) + ~ x(t)·
ay
be
• Choose an ini Ual state, i~ at random within the doain under consideration .nd repeat the following :
(12)
b u
DJ'
• Repeat the following steps until convergence 18 reached:
The fuzzy safety eap for a linear one degree of freedom second order system given by, x
consequents
Select random,
a
control
action
at
0,
•
• apply the control action during a time period At to the system, • record the new state i ~+1
0 • update the membership grade of
Along the line s & 0 it is a maximum at the origin of the phase plane and the i t tends to fall off gradually in a direction orthogonal to s • O. The rate at which it falls off is inversely proportional to the parameter 'a' which is also the damping in the system. The slope ~ is a function of 'c' and 'a' and the time step. Thus the analysis clearly illustrates the features of the safety metric defined by equation (1) and these are similar to those that would be expected of the metric defined by equation (11) .
0
•
of
all consequents 0J by AOJ(O.) which is defined as: AOJ(os) •
~I (i~)
x
~
J
0) x [R(i t +,1 .
-
0 (0 )] J.
and,
0 if out of domain under consideration R( i
\,+1
,0) s
S(1
{
min[ 1,
~+1
)
--------~~--~------~I.-I, •• xl •• l expect.tion (or J
otherwise
• Until trajectory is outside the domain under consideration .
6 . THE LEARNING ALGORITHM Prior to learning the phase space is quantized into a number of discrete domains and overlapping fuzzy membership functions are defined over each domain, I J' and its neighbours . All the membership functions are assumed to be triangular shaped and overlapping below a grade equal to 0.5 . These membership functions defining the antecedants in the fuzzy rules are fixed . The control action is also quantized and covered by overlapping membership functions . Initially they are assumed to be same and equal to 0 . 5 over the entire domain of the control action, 0 which is made up of N discrete domains, OJ' j & 1, 2, 3, ... , N.
In the above algorithm ~ E [ 0, 11 is the learning rate which may be set to zero after the completion of the learning cycle. The maximal expectation of I J is a maximum of a table of safety measures of the n most recent states i i~
for which
~I (i~)
<+1
reached from a state
> 0 . 5 and n is taken to be
J
twice the number of elements in the discretised output set O. The details may be found in Nowe (1992).
The algorithm was applied to a typical UNIMATION PUMA 560 robot manipulator. In a typical case the specification for the inverted double pendulum configuration was that the upper pendulum be maintained at an angle 9 • 0 . 4. A fuzzy rule 1
The learning algori thm seeks to adjust the membership functions of the conclusions 0 of 1
the M fuzzy rules such that : • A control action 0 which transfers the current state to a new state outside the domain under consideration is excluded. Thus a membership grade equal to 0 . 0 is assigned to the corresponding domains . The best control action 0 for a fuzzy region I J is the action which brings the system into the safest region that can be reached from there. It is rewarded with a membership grade equal to 1 in the corresponding image Of The membership grade of other control actions is between 0 and 1, and is related to the state the system is tranfered to from a partlcular ini tial state. The larger the safety of the new state, the larger the membership grade .
315
based controller was used to maintain the upper pendulum while the lower pendulum was allowed to settle down to a safe position. The resulting stable angular position for the lower pendulUm 9
2
• -0 . 14 .
7. FUZZY LEARNING SLIDING MODE CONTROL Many Fuzzy systems have been designed on the basis of rules defined in a 'phase plane' with the error and change in error between the set point and the output as the reference axes . A related general approach based on variable structure controllers is to divide the phase plane into two half- planes by means of a switching line and the control action signals in the two semi planes can be easily defined provided the controller is defined along the switching line. The basic idea is to drive the system towards the switching line and then along it to the origin. Such controllers have been referred to as sliding mode controllers ( Slotine (1985) l. where the sliding mode refers to
of
the dynaaic response svitcb1na 11ne.
the
the
Artificial Neural Nets have the unique ability to construct system inverses. Thus liven that a system has as its output, a certain deaanded value plus a known error signal, the input required to ,enerate the output can be constructed using a neural net. Several controllers generate the control signal entirely from a knowledge of the error and in these cases it is possible to construct the error-control signal relationship when the dynamics of the error Signal is completely defined irrespective of the plant dynamics . Fortunately it 1& possible to define the dynaaics of the error completely along a switching line by choosing the dynamical equation defining the sliding mode . Thus it is a straight forward matter to construct the control rules along the switching line and this can be done by simulating the error dynamics independent of the plant. The problem now reduces to an appropriate choice of the model for the sliding mode and the parameters defining it, whlch are of course no longer fuzzy. Once the control rules are established along the switching line i t is a fairly stralght forward matter to define the rules in the two seml planes on either slde of the swltching line. Thls is the baslc learning strategy adopted to construct the fuzzy Loglc control rules . The concept of a fuzzy Slldlng Hode Controller 15 not new (Palm (1989). However to the authors knowledge it has not been used as a learning strategy for syntheslzlng control rules. Thus the technique adopted here to learn the control rules for implementing a fuzzy controller involves a high degree of synergism . The method is applied to an inverted pendulum and is shown to work satisfactorily without any prevlous knowledge about the control rules . The inverse modelling was implemented based on two approaches . The first was based on an analytical nonlinear differential equation model of the pendulum . The second was based on Neural Net modelling. Resul ts based on the first method are reported in the paper . The results based on the neural net inverse model will be presented elsewhere .
8. SLIDING MODE THEORY AND THE LEARNING STRATEGY The theory of sliding mode controllers can be most easily explained using the second order model considered earlier , i . e ., x
+
a x
+
c x
b u
( 12 )
The objective of the controller is to drive the system towards a sliding line ( or a sliding surface in the case of a mul ti-degree of freedom system ) and then drlve the system along the line towards the equilibrium state . In the above situation the equation of the sliding line is assumed to be: s
z
~(t) • ~ x(t) • 0
(13)
's' represents the normal distance in the state space from the current state to the sUding line. Hence if one ~ishes to drive the plant towards the sliding line s must either be negative when 5 15 positive or be positive when 5 is negative; i. e . ,
s s < O.
In order to satisfy choose 'u' to be,
thls
u "' (a ~(t) • c x(t)
reqUirement
we
first
•
a x
c x
a x
1
c x
-~ x •
s •
+
and,
ss·
SUI'
Thus u must be chosen such that, 1
s u
1
< O.
u 1 is interpretated as the control action required to drive the plant to the sliding line while (u u 1 ) Is the control action required to drive the plant along the line to the origin of the phase plane. We may choose u to be, 1 u • - sign(s) U 1 o where U is a positive constant. further details of o sliding mode theory as well as its robustness properties may be found in Slotine (1985). for our purpose this baslc Introduction wlll sufflce. Sliding mode theory can be adapted to form the basls of a learning algorithm for controlling a plant when no control rules whatever are available a pr jor 1. The method relies on the fact that the sliding mode can be defined independent of the plant to be controlled . Thus it is possible to simulate the error along the sliding line . The simulated error is subtracted from the desired set point to compute the output of the plant along the sliding line . If an Inverse model of the plant , based on a neural network or otherwlse 15 available, then 1t may be used to compute the control actlon necssary to produce the output of the plant along the sl1dlng llne . The control action, the error and the change in error are then fuzzifled to establish a set of control rules along the sllding line. The entire process is an offline process and resul ts In a basic set of control rules. Once all the rules along the slidlng llne are found the plant 15 set in operatlon, and durlng operation additlonal rules are establlshed at each polnt In the phase plane not covered by the existing rules . The procedure for dolng thls first Involves Identifying the current state of operation in the phase plane . A perpendlcular Is drawn from this point towards the slidlng line . The control action, u. 1 ' at the foot of the perpendlcular on the slidlng line Is noted and the control actlon at the current state is found by using the formula : u "' u. 1 - Uo slgn(s) where ' s' 15 the perpendicular distance between the current state and the sliding line . As an example the control of an Inverted pendulum with no initial control rules and only the knowledge of an Inverse model a set of control rules were contructed. Every rule generated by the learning algorithm has the structure :
where Term E, Term C and Term U are appropriate linguistic -terms . The linguistic terms used are Positive Big (PB) , Positlve Medium (PM), Positive Small (PS), Positlve Zero (PZ), Negative Zero (NZ), Zero (ZRJ. Negative Small (NS), Negative Medium (NH) and Negative Big (NB), The linguistic terms Term_E, Term_C and Term_U corresponding to each rule are 11sted below in Table 1. 9 . CONCLUSIONS Two fuzzy learning technlques for compensation and control have been presented in this paper .
~ ~(tJ • u )/b.
The fuzzy learning compensatlon technlque is currently being applied to multl-degree of freedom robotic systems and the resul ts are currently being extensively evaluated .
The equation of motion, (12), then takes the form, x
Hence,
u. 1
316
Table 1 Fuzzy rule base aenerated T__U
TenoJ:
t ...._c
1
PB
2
lIB
III OR 101 OR lIS PBOR I'll OR PS
3
I'll
lIS
PB
4
101 PS
PS NZ PZ NZ PZ NZ PZ NZ PZ
lIB
Jlul. 110.
, 5
lIS
7
pz
,•
NZ
10
11 12 13 14
I'll
101 NZ PZ PZORNZ PZORNZ
PB ZR PSORI'IIORPB IISORIIIORDORPB I'll
PS IOIORIISORPB I'll OR PS OIIDORZR PB III
lIS
PS
The fuzzy learning sliding ~ode technique is currently being ~od1fied for use with coupled higher order systems . For such systems rather than define swi tching surfaces in a higher dimensional phase plane, an initial decoupling feedback is being synthesized using a neural net approach . The system is then decoupled into a set of second order nonlinear systems and above technique is then implemented for each mode separately. The entire methodology is expected to be very useful for several aerospace applications.
Michie, Donald, (1989) 'Application of Machine learning to recognition and control ', University of Wales Review of Science and Technology, No. 5, pp 23-28. Nowe, A. , (1992) 'A Self-Tuning Robust Controller', Technical report 92-1, University of Brussels, February 10 .
and Mamdani, E. H. (1979), 'A Procyk, T. J. Linguistic Self-Organizing Process controller,' Automatica, ~ , 15-30.
Barto , A. G., Sutton, R. S. and Anderson C. IJ., (1983) 'Neuron-like Adaptive elements that can solve difficul t learning control problems,' IEEE Trans . Syst . Man Cybern, Vol. SMC - 13, pp 834 846 .
Procyk T. K. , Cl 977) 'Self Organising Controller for dynamic processes', PhD thesis, Queen Mary College , U. K. ,
Barto, A. G., (1990), 'Connectionist Learning for Control : An Overview,' In : Miller, 11. T. , Sutton R. and IJerbos, P. (Eds . ), (1990) 'Neural Networks for Control', MIT Press, Cambridge, U.S . A. , pp. 5-58.
Salisbury, J . K. , and J . K. Craig, (1982) , 'Analysis of mul tifingered hands: Force control and Kinematic Issues,' Int . J. Robotics Res. ,vol . 1., no . I, pp 4-17.
Bersini, H., (1992), 'Generation Automatique de Systemes de Commande Floue par les Methodes de Gradient et les Algorithmes Genetiques', Presente aux Deuxlemes Journees Nationales sur les Applications des Ensembles Flous. Nimes, 2-3 November, 1992 . for
Fuzzy
Karr , C. L. (1991), 'Design of Adaptive Fuzzy Logic Controller using A Genetic Algorithm', In: Belew, R. and Booker, L. (Eds . ) (1991) Proceedings of the Fourth International Conference on Genetic Algorithms , Morgan Kaufmann . Llnkens, D. A. and Hasnain, S. B. (1991), 'Self Organizing fuzzy logic control and application to muscle relaxant anaesthesia, 1££ Proc.-D , ~, 3, 274-284 . (1974), ' Application of Fuzzy Mamdani, E.H . , Algorithms for Control of a Simple Dynamic plant ' , Proc. 1£££, lll, pp 1585-1588. Mamdani, E. H. and Assllian, S. (1975), ' A Fuzzy Logic Controller for a Dynamic Plant ' , IntJ. J . Han-Hachine Stud . , 1, pp 1-13. Handic , N. J ., Scharf, E. M. and Mamdani, E. H. (1985), 'Practical Application of a heuristic fuzzy rule based controller to the dynamic control of a robot arm,' 1£££ Proceedings-D, Special Issue on Robotics, ~,4, 190-203.
Fuzzy INFO,
Palm, Rainer (1991) , ·Sliding Mode Fuzzy Control ", Paper presented at the IEEE' 92 Colloquium, San Diego, U. S. , 8 - 13 March .
10. References
Karr, C., (1991) 'Genetic Algorithms Controllers ', AI Expert, February.
PB lIB
SeraJi, Homayoun , (1989), 'Configuration Control of Redundant Manipulators: Theory and implementation,' IEEE Trans. Robiotics Automat., vol . 5 , no . 4, pp 472-490. Slotine, J . J . F. (1985), "The Robust Control of Robot Manipulators," The International Journal of Robotic Research, Vol . 4, No . 2, pp 49-64 . ViJay Kumar and John F. Gardner, (1990), 'Kinematics of Redundantly actuated closed chains,' IEEE Trans . Robiotics Automat.. vol. 6 . ,no . 3, pp 269-274. (1954) , ' The Human Use lie iner, N., Beings,' Doubleday, New York, 1954.
of
Human
Yamazaki, T. and Mamdani, E.H . (1982), 'On the performance of a rule-based self organising controller,' 1£££ Conf. on ApplJcation of Adaptive and Hultivariable Control, Hull, 50-55 . Yoshihara, T., (1985), 'Manipulability of Robot Mechanisms, ' Int . J. Robotics Res ., vol. 4. , no . 2, 1985. Zadeh, L. A. (1965), 'Fuzzy Sets', Control, ~, 28-44.
Information and
Zadeh, L.A. (1973), 'Outline of a new approach to the analysis of complex systems and deCision processes', 1£££ Trans. , SMC-3, 338-353 .
317