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A Fuzzy Variable-Structure-Adaptive Control Design for Robot Manipulators
A Fuzzy Variable-Structure-Adaptive Control Design for Robot Manipulators Dongbiao Zhao, Jianying Zhu (1) Dept. of Mechanical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, P. R. China Received on January 4,1995
Abstract The increasing demand on robotic system performance leads to the use of advanced control strategies. This paper proposes a method of nonlinear feedback control introducing fuzzy inference into variable structure adaptive control for the nonlinear robot manipulator systems. The fuzzy inference is introduced to treat the nonlinearities of the control systems. The fuzzy logic control approach enhances the proposed variable structure adaptive control since it gives robust property against system uncertainties and external disturbances ,and also expert's experience can be added into the controller in parallel. The controller is an intelligent one. An automated design technique to numerically optimize membership functions for the fuzzy controller is discussed. In addition, the method is capable of handling the chattering problem inherent to variable structure control simply and effectively. A simulation study of a robot manipulator with three degrees of freedom is presented to demonstrate these features of the method. Keywords :robotic manipulators, fuzzy logic, optimization
1. Introduction It has been reported"j that real-time control of a manipulator based on a detailed mathematical model is difficult to achieve. Furthermore, as the model is both complex and nonlinear ,a relatively long computation is needed to determine the required control signal. To overcome the difficulties mentioned above, viz. , accurate dynamic modeling and computational time constraints,a new control design is proposed for robotic manipulators based on the combining of fuzzy inference and variable structure control theory (VSC). The controller structure is changed continuously by fuzzy logic such that if the error is large or its rate is large, then the controller makes the system respond quickly and vice versa in order to obtain a robust controller that is insensitive to both the system noise and observation noise. Fuzzy control has been applied to several practical ~ y s f e m ~ ~ ~ ~because " ' ~ ' " ~it enables us to deal with nonlinearities of control systems and to introduce the knowledge of experts to control rules. However, there are also some problems, for example, lack of systematic design procedures of fuzzy controller and the stability analysis. The control algorithm using the theory of Among them, VSC have been there still exist some nontrivial difficulties in the Unfavorable feature of the existing
.
Annals of the ClRP Vol. 44/1/1995
VSC designs is a large amount of chattering which is associated with the excessive control torques (forces). Essentially, the VSC uses discontinuous control action to drive state trajectory toward a specific switching hyperlane until the origin of the state space is reached. This principle provides guidance to design a fuzzy controller for system stability, while the system is still an open problem in the study of fuzzy controller design'"'. On the other hand, the fuzzy controller inherently possesses the robustness of system performance[":. Therefore the combination of the two control principles, which is called fuzzy variable structure control (FVSC) in this paper, provides a new tool to design a robust controller for nonlinear systems. The fuzzy logic control approach enhances the proposed variable structure adaptive control since i t gives robust property and also human being's experience can be added into the controller in parallel. The controller is an intelligent one. An automated design technique to numerically optimize membership functions for the fuzzy controller is discussed. In addition. the method is capable of handling the chattering problem inherent to variable structure control simply and effectively. A simulation study of a robot manipulator with three degrees of freedom is presented to demonstrate these features of the method.
363
2. Robotic Manipulator Model The dynamic equations of motion for a general rigid link manipulator having n degrees of freedom can be described as follows (1) D(q)Q F(q,q)q G ( q ) = u(t) where q E R" is joint displacements, u E R' is applied joint torgues (or forces), D ( q ) = DT(q) > O , D ( q ) E R"""is the inertia matrix, F ( q , c i ) i E R" is the centripetal and Coriolis torque vector, and G ( 9 ) E R" is the gravitational torgue (force) vector. Defining x E R2"to be the state vector
+
+
x=
r;]
Equation (1 1 can be written in a state variable form x = Ax Bu (2) In model-referenced adaptive control design , the desired behavior of the robotic system is expressed through the use of a reference model driven by a reference input. For robotic application, we select a reference model of the form : (3) x, = A,x, R,u, where:
+
+
With A, = - diag ( a , 1, A , = - diag ( a l , ).uO, > O.a,, > O;u, E R" is an external reference input. Note that ,with the above choice of the reference model, the structure conditions for the "perfect model following" are always satisfied whichever are the values of the manipulator and reference model parametersCB2.
3. Variable Structure Adaptive Control Law Based on Fuzzy Inference 3. 1. Variable Structure Control
Variable structure model-reference adaptive control (VSMRAC) systemsC13are characterised by discontinuous control action which changes structure upon reaching a set n switching hypersurfaces s, , in the tracking error state spaceLg3 S = (sI,s2,-*,s.)= Ce (4) where C E RnX2" is the matrix to be determined. The control law has the form u.? , if s,(e) > 0 (5) u l = {u;, if s , ( p ) 0 where u, is the ith component of u . The condition for the sliding motion to occur on the ith switching is lim i, < O and lim s, > 0 (6)
<
.,-0+
s, - 0 -
or equivalently s,iz < 0 . The above pair of inequalities are a sufficient condition for sliding mode to existCg1.In the sliding mode, the system satisfies the equations
364
s i ( e ) = 0 and s i ( e ) = 0 (7) Suppose that the sliding mode exists on all the switching surfaces ; then the equation governing the error system can be obtained by substituting for an equivalent control u, for the original control .r9:
According to the equations ( 2 ) and (31,the tracking error vector is e ( t ) = x - x, (8) It can be easily shown that 4 = Arne ( A - A,)x - B,u, Bu (9) From (9) and (71,we have S = C[A,e ( A - A,)x BU - B,u,] = 0 (10) Assuming CB is non-singular ,then, urP= - (CB)-'C[A,e ( A - A , ) x - B,u,] (11) u, is equivalently the average value of u which maintains the state on the switching surfaces S ( e ) = 0 . The actual control u consists of a lowfrequency (average) component u , ~and a highfrequency (chattering) component A u as follows u = u,, au
+ +
+
+
+
+
where A u , is discontinuous on the switching surfaces s i ( e ) = 0 . Nonlinear compensation for u, reduces the amplitude of vibrational input in sliding motion compared with ( 5 1. Namely, this compensation mitigates the extent of chattering due to sliding mode control. since it removes the main interactions. In summary, if the control input is so designed that the inequality s,;, < 0 is satisfied, together with the properly chosen switching hyperplane, then the state will be driven toward the origin of the state space along the hyperplane from any given initial state. This is the way of the VSC to guarantee asymptotic stability of the system.
3. 2. Fuzzy Variable Structure Control As described in the section 3. 1,the objective of the VSC is to design a control law so that the condition on the sliding mode, ( 7 ) , is satisfied. In this section we will develop a fuzzy logic control law in an attempt to accomplish this objective. The inputs of the proposed fuzzy controller are S and CS which are the fuzzified variables of s and , respectively. The output of the fuzzy controller is AU which is the fuzzified variable of Au (change of u ). All the universes of discourse of S,CS and AU are ranged from - 6 to 6: thus, the range of nonfuzzy variables s,s and A u must be scaled to fit the universe of discourse of fuzzified variables S,CS and AU with scaling factors K,,K,and K, ,respectively. A selection of the scaling factors is not wholly subjective ,since their magnitudes are a compromise between the speed of response and steady-state accuracy.
In implementation, CS is approximated with ( S, - S h - , )/ A T where A T is the sampling period. Each fuzzy variable is quantized into seven qualitative fuzzy variables (see table 1 ). For simplicity of optimization, a triangular type membership function is chosen for the above stated variables,as shown in Figure 1. Based on the above qualitative analysis, we are able to design the control input u in an attempt to satisfy the inequality s; < 0 . The resulting control rules are shown in Table 1, by which the center of gravity method is used to form a look-up table ( a decision table) which directly relates the controller inputs s and i with the controller output change Au. The control strategy could be summarized by a conceptually conditional statement IF more far from the switching hyperplairr THEN tnorc Zarge feedOack guin (13) This linguistic rule should play an important role in the design procedure of the FVSC. For handling more informations of system dynamic. the FVSC not only uses S as the inputs * but also includes the change of CS as the other inputs. In other words, the FVSC is designed based on the state feedback technique which is the same as in VSC, and provides fuzzy rules to obtain the feedback gains. By the statement (13) and the experience of VSC design, the fuzzy control rules of the FVSC can be estabilished. Table 1 shows the rule base. Note that the control rules are in the form of ( 1 4 ) and we can use (36) and ( 1 7 ) to construct an overall relation , R. A simple fuzzy control consists a set of linguistic rules ,such as 5 IS S' and IS CS' L":IF THEN Au IS A w l (14) where L" denotes the i j t h process rule ( i = 1,2 * - , t n and j = 1,2,-,1 1. S',CSJand AW'are the fuzzy sets, defined by membership functions. The membership function can have many forms ,but in this study,we use a triangular function of the form (Fig. 1) ( x - a ) / ( ( , - a ) if a < x < b p ( z ) = (x - c ) / ( b - c ) if b < x < c others i 0 (15) where x is an input, and a,D and c satisfy the condition Q < b < c By appropriately varying the values of a , 6 and c , fuzziness of the physical variable, x ,can be described in different ways. Applying the inputs ( s and 5 1 as conditions to the decision rules yield a set of actions (Au ) with various degree of membership to the fuzzy sets, In general, the construction of the fuzzy control is based on the physical properties of the system, the input-output data , and the corresponding empirical knowledge and so on. A fuzzy rule can be implemented by a fuzzy
.
relation
R" = (S' x CS') x AU' (16) and the fuzzy relations of the overall rules R ,can be written *.I
K =UK"
(17)
'.I
Now ,if the fuzzy sets of inputs s and s are respect to S' and CS' ,then the correspondance CW' is given by AU' = (S' X CS')oK (18) The defuzzification process is defined as
'=I
where u is the defuzzified control input, s is the number of fuzzy sets corresponding to AU' The defuzzification process is selected such that the defuzzified control input u can be obtained with less computation effortL5:.
.
3. 3. Optimal Design of the Membership Functions Design of membership functions and specification of decision rules must be done on basis of thorough knowledge of the system to be controlled. Although expoitation of expert knowledge in control decision making is the most advantageous aspect of the fuzzy control approach. the lack of effective design techniques is a limiting factor. Most fuzzy controllers are similar in their basic structure ,but a large number of parameters must be specified in the design of a given controller. Evaluation and tuning of the controller parameters are typically done in trialand-error manner through simulation or actual implementation. This task requires significant effort. If a simulation model is available, numerical optimization i s an efficient tool for designing membership functions, and for tuning and automated testing of the controller. Suppose a membership function p ( z ) has the form of eq. (15). The design objective is to find values for the membership function parameters. a,D and c so that appropriate fuzziness of the particular physical variable can be described. Values can be optimally chosen if a cost function. which represents the performance of the controller, is optimized over a wide range of membership function parameters, through a course of repeated simulations. Nonlinear and time-varying robotic dynamics are easily handled ,given appropriate simulation models. The approach is to apply a high-dimensional simulated annealing optimization algorithm"22.The resulting algorithm is suitable for a high dimensional numerical optimization problem and is known to avoid local minima. As an indication of the controller performance, we consider the cost function consisting of the squared norm of the state error
.
365
(20)
where T is a simulation length. Having set up the simulation model ,membership the controller functions boundaries , and performance index, the optimization problem can now be started. Defining parameters as a vector, ZI ,and a constraint set, V ,then the object is min J ( T ) (21) V Ev
4. Simulation and Results A numerical simulation study has been performed to investigate the effectiveness of robotic manipulator robust control design via the proposed FVSC for nonlinear time varying systems. A manipulator with three degrees of freedom is considered with the following set of parameters: 7nI =30kg, 7n2 = 18. 6kgl I, = f 2 = 0. 5m, where tn, ,fi ,i = 1,2 denote the mass and the length of the ith link (see Fig. 2 ). The actuators were assumed with no dynamics and no power limitations. The reference model has been chosen for each linkage of the robotic system, with aOi= 100 and a,<= 14.14. The purpose of this tracking following experiments is to demonstrate the performance of the control system in following a specified trajectory for both position and velocity of the tip of the manipulator. In order to exercise control of the manipulator in terms of joint variables we solved for the 9,'s at points along the path specified originally in Cartesian coordinates. The circular path is a 141. 4mm diameter horizontal circle centered at 5OOmm. 500mm. and 200mm. The simulation was first performed with a sampling interval of 0.02 sec. The results given in Figs. 3 and 4 indicate that the tip of the arm follows the desired circular path from the start point where the tracking error in X-direction is 10mm. The required applied torque are shown in Fig. 5. Finally, the responses of the manipulator control system have been evaluated when Okg, 5kg, and lOkg payload is respectively considered. Reference model responses and manipulator responses are indistinguishable, which implies that the system is insensitive to payload variations ,i. e. ,to parameter variations. In order to demonstrate the external disturbance rejection capability of the FVSC, a random external disturbance is applied on the first joint of the robotic manipulator. Its amplitude and frequency are 2.9deg , and 50Hz, respectively. Fig. 8 and 9 show the simulation results. The corresponding state and position responses are almost unaffected by the external disturbance. In variable structure control state trajectories chattering along a switching hyperplane, as shown in Figure 6, is inevitable in
366
practice because the control input is discontinuous in the vicinity of the sliding hyperplane. Chattering is a serious drawback of VSC and has been an important problem in the field of VSCcll*clO:. Here, we want to demonstrate that the FVSC is able to handle this chattering problem effectively by adjusting the control input near the sliding hyperplane. The simulation results in Figure 7 show the state trajectory of system response for the robotic manipulator. No significant chattering along the sliding hyperplane is observed in the trajectories.
5. Conclusion This paper proposed a method of nonlinear feedback control introducing fuzzy inference into variable structure control for the nonlinear robot manipulator systems, in which the membership functions were created using an optimization algorithm. The attractive features of the control method are summarized in the following ( 1 1 By constructing a fuzzy-switching hyperplane, the dynamic behavior of FVSC is predictable. ( 2 ) The FVSC is robust against the robotic parameter uncertainties and external disturbances. ( 3 ) The objective of the control rules is simple and clear. That is, the nonlinear compensation control input AU should only be adjusted to satisfy s,;, < 0 . This is a way of the FVSC to guarantee asymptotic stability of the system. (4) The optimal membership function in the controller can be obtained using a highdimensional simulated annealing optimization algorithm. ( 5 ) The chattering problem inherent with variable structure control can be handled effectively without involving sophisticated mathematics. The fuzzy logic control approach enhances the proposed variable structure adaptive control since it gives robust property and also human being's experience can be added into the controller in parallel. The controller is an intelligent one. Furthermore, extension of an optimization approach to the design and evaluation of a rule base is currently under study. Acknowledgment The authors wish to acknowledge supports of this study by the National Nature Science Foundation of China and of the Postdoctorial Science Foundation of China. References
[ 11
Shudong Sun and Jianying Zhu, Variable
structure Model Reference Adaptive Control Design for Robot Manipulators. Annals of the CIRP, Vol. 38, no. 1, 1989, pp475-479. Nedungadi , Ashok, Application of Fuzzy Logic to Solve the Robot Inverse Kinematic Problem, Fourth World Conference on Robotics Research, Sept. 17-19, 1991, Pittsburgh, Pennsylvania, USA. Dubowsky, S. and DesForges, D. T. ,The Application of Model-Referenced Adaptive Control to Robotic Manipulators ,Journal of Dynamic Systems, Measurement, and Control ,Vol. 101,Sept. 1979. Lim ,C. M. ,and Hiyama, T. ,Application of Fuzzy Logic Control to a Manipulator ,IEEE Transactions on Robotics and Automation, Vol. 7.110. 5,1991. Chen. C. -L. ,Chen, P. -C. ,and Chen, C. K. ,Analysis and Design of Fuzzy Control System, Fuzzy sets and Systems ,Vol. 53, 1993,pp125-140. [6] Chan,Y.T. ,Perfect Model Following with a Real Model, Proc. Joint Automat. Contr. Conf. ,1973 ,pp287-293. [ 7 1 Wakileh, B. A. M. and Gill, K. F. ,Use of Fuzzy Logic in Robotics, Computers in Industy,Vol. 10.1988.pp35-46. [83 Young, K. -K. D. , Design of Variable Structure Model-Following Control Systems, IEEE Transactions on Automatic Control,Vol. AC-23,170. 6,Dec. 1978. [9] Utkin ,V. I. ,Variable Structure Systems with Sliding Modes, IEEE Transactions on Automatic Control ,Vol. AC-22 ,no. 2.1977. [ 103 Hung, J. Y. , et al, Variable Structure Control : A Survay, IEEE Transactions on Industrial Electronics, Vol. 40,no. 1 1993, pp2-22. [ 11] Sugeno, M. , An Introductory Survey of Fuzzy Control, Information science , Vol. 36 19.85 9~~59-83. Karimi ,A. ,Sebald,A. ,V. ,and Isaka,S. , [12] Use of simulated Annealing in design Design of Very High Dimensioned Minimax Adaptive Controllers ,Proceedings of 1989 Asilomar Conference on Circuits and Systems.
Table.
Rule base of the FVSC
S I
I
PL PM PS PO NO NS NM NL
Au
PL PL PM PM PS PS 0
PL PM PS 0 NS NM NL
cs
PL PM PM PS PS
PM PM PS PS 0 0 NS NS NS
PM PS PS 0 NS NS NM
PM PS PS 0 NS NS NM
PS PS 0 NS NS NM NM
PS 0 NS NS NM NM NL
0 NS NS NM NM NL NL
t Fig. 1 A triangular-shaped membership function
z1 /
-
Y
Fig. 2
Configuration of a manipulator with three degrees of freedom
x-y phase
0.4' 0.4
I
0.5
0.6
X Fig. 3
Tracking a circular path
367
t-xyz error
x-y phase OS6*
0.4 0.4 I
E-
f
0.5
i
I
0.6
X
Fig. 8 Tracking the circular path with a random
0
disturbance acting on the first joint
I
g200' 0
I
I
1
2
I
I
1-xyz error
I
3
4
0.021
i
1
I
0
1
2
3
t Fig, 5
Input torques for tracking the circular path
states
5! I
I
I
!
i
I
!
11
4
t Fig. 9
1
2
3
4
Fig. 6 Trajectories of the state variables in VSC
states
0 Fig. 7
368
1
2 t
3
4
Trajectories of the state variables in the FVSC
Tracking error of the end-effector of the manipulator with a random disturbance acting on the first joint
Abstract The increasing demand on robotic system performance leads to the use of advanced control strategies. This paper proposes a method of nonlinear feedback control introducing fuzzy inference into variable structure adaptive control for the nonlinear robot manipulator systems. The fuzzy inference is introduced to treat the nonlinearities of the control systems. The fuzzy logic control approach enhances the proposed variable structure adaptive control since it gives robust property against system uncertainties and external disturbances ,and also expert's experience can be added into the controller in parallel. The controller is an intelligent one. An automated design technique to numerically optimize membership functions for the fuzzy controller is discussed. In addition, the method is capable of handling the chattering problem inherent to variable structure control simply and effectively. A simulation study of a robot manipulator with three degrees of freedom is presented to demonstrate these features of the method. Keywords :robotic manipulators, fuzzy logic, optimization
1. Introduction It has been reported"j that real-time control of a manipulator based on a detailed mathematical model is difficult to achieve. Furthermore, as the model is both complex and nonlinear ,a relatively long computation is needed to determine the required control signal. To overcome the difficulties mentioned above, viz. , accurate dynamic modeling and computational time constraints,a new control design is proposed for robotic manipulators based on the combining of fuzzy inference and variable structure control theory (VSC). The controller structure is changed continuously by fuzzy logic such that if the error is large or its rate is large, then the controller makes the system respond quickly and vice versa in order to obtain a robust controller that is insensitive to both the system noise and observation noise. Fuzzy control has been applied to several practical ~ y s f e m ~ ~ ~ ~because " ' ~ ' " ~it enables us to deal with nonlinearities of control systems and to introduce the knowledge of experts to control rules. However, there are also some problems, for example, lack of systematic design procedures of fuzzy controller and the stability analysis. The control algorithm using the theory of Among them, VSC have been there still exist some nontrivial difficulties in the Unfavorable feature of the existing
.
Annals of the ClRP Vol. 44/1/1995
VSC designs is a large amount of chattering which is associated with the excessive control torques (forces). Essentially, the VSC uses discontinuous control action to drive state trajectory toward a specific switching hyperlane until the origin of the state space is reached. This principle provides guidance to design a fuzzy controller for system stability, while the system is still an open problem in the study of fuzzy controller design'"'. On the other hand, the fuzzy controller inherently possesses the robustness of system performance[":. Therefore the combination of the two control principles, which is called fuzzy variable structure control (FVSC) in this paper, provides a new tool to design a robust controller for nonlinear systems. The fuzzy logic control approach enhances the proposed variable structure adaptive control since i t gives robust property and also human being's experience can be added into the controller in parallel. The controller is an intelligent one. An automated design technique to numerically optimize membership functions for the fuzzy controller is discussed. In addition. the method is capable of handling the chattering problem inherent to variable structure control simply and effectively. A simulation study of a robot manipulator with three degrees of freedom is presented to demonstrate these features of the method.
363
2. Robotic Manipulator Model The dynamic equations of motion for a general rigid link manipulator having n degrees of freedom can be described as follows (1) D(q)Q F(q,q)q G ( q ) = u(t) where q E R" is joint displacements, u E R' is applied joint torgues (or forces), D ( q ) = DT(q) > O , D ( q ) E R"""is the inertia matrix, F ( q , c i ) i E R" is the centripetal and Coriolis torque vector, and G ( 9 ) E R" is the gravitational torgue (force) vector. Defining x E R2"to be the state vector
+
+
x=
r;]
Equation (1 1 can be written in a state variable form x = Ax Bu (2) In model-referenced adaptive control design , the desired behavior of the robotic system is expressed through the use of a reference model driven by a reference input. For robotic application, we select a reference model of the form : (3) x, = A,x, R,u, where:
+
+
With A, = - diag ( a , 1, A , = - diag ( a l , ).uO, > O.a,, > O;u, E R" is an external reference input. Note that ,with the above choice of the reference model, the structure conditions for the "perfect model following" are always satisfied whichever are the values of the manipulator and reference model parametersCB2.
3. Variable Structure Adaptive Control Law Based on Fuzzy Inference 3. 1. Variable Structure Control
Variable structure model-reference adaptive control (VSMRAC) systemsC13are characterised by discontinuous control action which changes structure upon reaching a set n switching hypersurfaces s, , in the tracking error state spaceLg3 S = (sI,s2,-*,s.)= Ce (4) where C E RnX2" is the matrix to be determined. The control law has the form u.? , if s,(e) > 0 (5) u l = {u;, if s , ( p ) 0 where u, is the ith component of u . The condition for the sliding motion to occur on the ith switching is lim i, < O and lim s, > 0 (6)
<
.,-0+
s, - 0 -
or equivalently s,iz < 0 . The above pair of inequalities are a sufficient condition for sliding mode to existCg1.In the sliding mode, the system satisfies the equations
364
s i ( e ) = 0 and s i ( e ) = 0 (7) Suppose that the sliding mode exists on all the switching surfaces ; then the equation governing the error system can be obtained by substituting for an equivalent control u, for the original control .r9:
According to the equations ( 2 ) and (31,the tracking error vector is e ( t ) = x - x, (8) It can be easily shown that 4 = Arne ( A - A,)x - B,u, Bu (9) From (9) and (71,we have S = C[A,e ( A - A,)x BU - B,u,] = 0 (10) Assuming CB is non-singular ,then, urP= - (CB)-'C[A,e ( A - A , ) x - B,u,] (11) u, is equivalently the average value of u which maintains the state on the switching surfaces S ( e ) = 0 . The actual control u consists of a lowfrequency (average) component u , ~and a highfrequency (chattering) component A u as follows u = u,, au
+ +
+
+
+
+
where A u , is discontinuous on the switching surfaces s i ( e ) = 0 . Nonlinear compensation for u, reduces the amplitude of vibrational input in sliding motion compared with ( 5 1. Namely, this compensation mitigates the extent of chattering due to sliding mode control. since it removes the main interactions. In summary, if the control input is so designed that the inequality s,;, < 0 is satisfied, together with the properly chosen switching hyperplane, then the state will be driven toward the origin of the state space along the hyperplane from any given initial state. This is the way of the VSC to guarantee asymptotic stability of the system.
3. 2. Fuzzy Variable Structure Control As described in the section 3. 1,the objective of the VSC is to design a control law so that the condition on the sliding mode, ( 7 ) , is satisfied. In this section we will develop a fuzzy logic control law in an attempt to accomplish this objective. The inputs of the proposed fuzzy controller are S and CS which are the fuzzified variables of s and , respectively. The output of the fuzzy controller is AU which is the fuzzified variable of Au (change of u ). All the universes of discourse of S,CS and AU are ranged from - 6 to 6: thus, the range of nonfuzzy variables s,s and A u must be scaled to fit the universe of discourse of fuzzified variables S,CS and AU with scaling factors K,,K,and K, ,respectively. A selection of the scaling factors is not wholly subjective ,since their magnitudes are a compromise between the speed of response and steady-state accuracy.
In implementation, CS is approximated with ( S, - S h - , )/ A T where A T is the sampling period. Each fuzzy variable is quantized into seven qualitative fuzzy variables (see table 1 ). For simplicity of optimization, a triangular type membership function is chosen for the above stated variables,as shown in Figure 1. Based on the above qualitative analysis, we are able to design the control input u in an attempt to satisfy the inequality s; < 0 . The resulting control rules are shown in Table 1, by which the center of gravity method is used to form a look-up table ( a decision table) which directly relates the controller inputs s and i with the controller output change Au. The control strategy could be summarized by a conceptually conditional statement IF more far from the switching hyperplairr THEN tnorc Zarge feedOack guin (13) This linguistic rule should play an important role in the design procedure of the FVSC. For handling more informations of system dynamic. the FVSC not only uses S as the inputs * but also includes the change of CS as the other inputs. In other words, the FVSC is designed based on the state feedback technique which is the same as in VSC, and provides fuzzy rules to obtain the feedback gains. By the statement (13) and the experience of VSC design, the fuzzy control rules of the FVSC can be estabilished. Table 1 shows the rule base. Note that the control rules are in the form of ( 1 4 ) and we can use (36) and ( 1 7 ) to construct an overall relation , R. A simple fuzzy control consists a set of linguistic rules ,such as 5 IS S' and IS CS' L":IF THEN Au IS A w l (14) where L" denotes the i j t h process rule ( i = 1,2 * - , t n and j = 1,2,-,1 1. S',CSJand AW'are the fuzzy sets, defined by membership functions. The membership function can have many forms ,but in this study,we use a triangular function of the form (Fig. 1) ( x - a ) / ( ( , - a ) if a < x < b p ( z ) = (x - c ) / ( b - c ) if b < x < c others i 0 (15) where x is an input, and a,D and c satisfy the condition Q < b < c By appropriately varying the values of a , 6 and c , fuzziness of the physical variable, x ,can be described in different ways. Applying the inputs ( s and 5 1 as conditions to the decision rules yield a set of actions (Au ) with various degree of membership to the fuzzy sets, In general, the construction of the fuzzy control is based on the physical properties of the system, the input-output data , and the corresponding empirical knowledge and so on. A fuzzy rule can be implemented by a fuzzy
.
relation
R" = (S' x CS') x AU' (16) and the fuzzy relations of the overall rules R ,can be written *.I
K =UK"
(17)
'.I
Now ,if the fuzzy sets of inputs s and s are respect to S' and CS' ,then the correspondance CW' is given by AU' = (S' X CS')oK (18) The defuzzification process is defined as
'=I
where u is the defuzzified control input, s is the number of fuzzy sets corresponding to AU' The defuzzification process is selected such that the defuzzified control input u can be obtained with less computation effortL5:.
.
3. 3. Optimal Design of the Membership Functions Design of membership functions and specification of decision rules must be done on basis of thorough knowledge of the system to be controlled. Although expoitation of expert knowledge in control decision making is the most advantageous aspect of the fuzzy control approach. the lack of effective design techniques is a limiting factor. Most fuzzy controllers are similar in their basic structure ,but a large number of parameters must be specified in the design of a given controller. Evaluation and tuning of the controller parameters are typically done in trialand-error manner through simulation or actual implementation. This task requires significant effort. If a simulation model is available, numerical optimization i s an efficient tool for designing membership functions, and for tuning and automated testing of the controller. Suppose a membership function p ( z ) has the form of eq. (15). The design objective is to find values for the membership function parameters. a,D and c so that appropriate fuzziness of the particular physical variable can be described. Values can be optimally chosen if a cost function. which represents the performance of the controller, is optimized over a wide range of membership function parameters, through a course of repeated simulations. Nonlinear and time-varying robotic dynamics are easily handled ,given appropriate simulation models. The approach is to apply a high-dimensional simulated annealing optimization algorithm"22.The resulting algorithm is suitable for a high dimensional numerical optimization problem and is known to avoid local minima. As an indication of the controller performance, we consider the cost function consisting of the squared norm of the state error
.
365
(20)
where T is a simulation length. Having set up the simulation model ,membership the controller functions boundaries , and performance index, the optimization problem can now be started. Defining parameters as a vector, ZI ,and a constraint set, V ,then the object is min J ( T ) (21) V Ev
4. Simulation and Results A numerical simulation study has been performed to investigate the effectiveness of robotic manipulator robust control design via the proposed FVSC for nonlinear time varying systems. A manipulator with three degrees of freedom is considered with the following set of parameters: 7nI =30kg, 7n2 = 18. 6kgl I, = f 2 = 0. 5m, where tn, ,fi ,i = 1,2 denote the mass and the length of the ith link (see Fig. 2 ). The actuators were assumed with no dynamics and no power limitations. The reference model has been chosen for each linkage of the robotic system, with aOi= 100 and a,<= 14.14. The purpose of this tracking following experiments is to demonstrate the performance of the control system in following a specified trajectory for both position and velocity of the tip of the manipulator. In order to exercise control of the manipulator in terms of joint variables we solved for the 9,'s at points along the path specified originally in Cartesian coordinates. The circular path is a 141. 4mm diameter horizontal circle centered at 5OOmm. 500mm. and 200mm. The simulation was first performed with a sampling interval of 0.02 sec. The results given in Figs. 3 and 4 indicate that the tip of the arm follows the desired circular path from the start point where the tracking error in X-direction is 10mm. The required applied torque are shown in Fig. 5. Finally, the responses of the manipulator control system have been evaluated when Okg, 5kg, and lOkg payload is respectively considered. Reference model responses and manipulator responses are indistinguishable, which implies that the system is insensitive to payload variations ,i. e. ,to parameter variations. In order to demonstrate the external disturbance rejection capability of the FVSC, a random external disturbance is applied on the first joint of the robotic manipulator. Its amplitude and frequency are 2.9deg , and 50Hz, respectively. Fig. 8 and 9 show the simulation results. The corresponding state and position responses are almost unaffected by the external disturbance. In variable structure control state trajectories chattering along a switching hyperplane, as shown in Figure 6, is inevitable in
366
practice because the control input is discontinuous in the vicinity of the sliding hyperplane. Chattering is a serious drawback of VSC and has been an important problem in the field of VSCcll*clO:. Here, we want to demonstrate that the FVSC is able to handle this chattering problem effectively by adjusting the control input near the sliding hyperplane. The simulation results in Figure 7 show the state trajectory of system response for the robotic manipulator. No significant chattering along the sliding hyperplane is observed in the trajectories.
5. Conclusion This paper proposed a method of nonlinear feedback control introducing fuzzy inference into variable structure control for the nonlinear robot manipulator systems, in which the membership functions were created using an optimization algorithm. The attractive features of the control method are summarized in the following ( 1 1 By constructing a fuzzy-switching hyperplane, the dynamic behavior of FVSC is predictable. ( 2 ) The FVSC is robust against the robotic parameter uncertainties and external disturbances. ( 3 ) The objective of the control rules is simple and clear. That is, the nonlinear compensation control input AU should only be adjusted to satisfy s,;, < 0 . This is a way of the FVSC to guarantee asymptotic stability of the system. (4) The optimal membership function in the controller can be obtained using a highdimensional simulated annealing optimization algorithm. ( 5 ) The chattering problem inherent with variable structure control can be handled effectively without involving sophisticated mathematics. The fuzzy logic control approach enhances the proposed variable structure adaptive control since it gives robust property and also human being's experience can be added into the controller in parallel. The controller is an intelligent one. Furthermore, extension of an optimization approach to the design and evaluation of a rule base is currently under study. Acknowledgment The authors wish to acknowledge supports of this study by the National Nature Science Foundation of China and of the Postdoctorial Science Foundation of China. References
[ 11
Shudong Sun and Jianying Zhu, Variable
structure Model Reference Adaptive Control Design for Robot Manipulators. Annals of the CIRP, Vol. 38, no. 1, 1989, pp475-479. Nedungadi , Ashok, Application of Fuzzy Logic to Solve the Robot Inverse Kinematic Problem, Fourth World Conference on Robotics Research, Sept. 17-19, 1991, Pittsburgh, Pennsylvania, USA. Dubowsky, S. and DesForges, D. T. ,The Application of Model-Referenced Adaptive Control to Robotic Manipulators ,Journal of Dynamic Systems, Measurement, and Control ,Vol. 101,Sept. 1979. Lim ,C. M. ,and Hiyama, T. ,Application of Fuzzy Logic Control to a Manipulator ,IEEE Transactions on Robotics and Automation, Vol. 7.110. 5,1991. Chen. C. -L. ,Chen, P. -C. ,and Chen, C. K. ,Analysis and Design of Fuzzy Control System, Fuzzy sets and Systems ,Vol. 53, 1993,pp125-140. [6] Chan,Y.T. ,Perfect Model Following with a Real Model, Proc. Joint Automat. Contr. Conf. ,1973 ,pp287-293. [ 7 1 Wakileh, B. A. M. and Gill, K. F. ,Use of Fuzzy Logic in Robotics, Computers in Industy,Vol. 10.1988.pp35-46. [83 Young, K. -K. D. , Design of Variable Structure Model-Following Control Systems, IEEE Transactions on Automatic Control,Vol. AC-23,170. 6,Dec. 1978. [9] Utkin ,V. I. ,Variable Structure Systems with Sliding Modes, IEEE Transactions on Automatic Control ,Vol. AC-22 ,no. 2.1977. [ 103 Hung, J. Y. , et al, Variable Structure Control : A Survay, IEEE Transactions on Industrial Electronics, Vol. 40,no. 1 1993, pp2-22. [ 11] Sugeno, M. , An Introductory Survey of Fuzzy Control, Information science , Vol. 36 19.85 9~~59-83. Karimi ,A. ,Sebald,A. ,V. ,and Isaka,S. , [12] Use of simulated Annealing in design Design of Very High Dimensioned Minimax Adaptive Controllers ,Proceedings of 1989 Asilomar Conference on Circuits and Systems.
Table.
Rule base of the FVSC
S I
I
PL PM PS PO NO NS NM NL
Au
PL PL PM PM PS PS 0
PL PM PS 0 NS NM NL
cs
PL PM PM PS PS
PM PM PS PS 0 0 NS NS NS
PM PS PS 0 NS NS NM
PM PS PS 0 NS NS NM
PS PS 0 NS NS NM NM
PS 0 NS NS NM NM NL
0 NS NS NM NM NL NL
t Fig. 1 A triangular-shaped membership function
z1 /
-
Y
Fig. 2
Configuration of a manipulator with three degrees of freedom
x-y phase
0.4' 0.4
I
0.5
0.6
X Fig. 3
Tracking a circular path
367
t-xyz error
x-y phase OS6*
0.4 0.4 I
E-
f
0.5
i
I
0.6
X
Fig. 8 Tracking the circular path with a random
0
disturbance acting on the first joint
I
g200' 0
I
I
1
2
I
I
1-xyz error
I
3
4
0.021
i
1
I
0
1
2
3
t Fig, 5
Input torques for tracking the circular path
states
5! I
I
I
!
i
I
!
11
4
t Fig. 9
1
2
3
4
Fig. 6 Trajectories of the state variables in VSC
states
0 Fig. 7
368
1
2 t
3
4
Trajectories of the state variables in the FVSC
Tracking error of the end-effector of the manipulator with a random disturbance acting on the first joint