Fuzzy Control of a 2-DOF Direct Drive Robot Arm by Using ... - CiteSeerX

Fuzzy Control of a 2-DOF Direct Drive Robot Arm by Using a Parameterized T-Norm M. Önder Efe, O. Hasan Dagci, Okyay Kaynak Bogazici University, Mechatronics Research and Application Center, Bebek, 80815, Istanbul, Turkey Abstract – This paper presents the fuzzy control of a two degrees of freedom direct drive robot by using a parameterized T-norm. The main objective of the paper has been to prove that the T-norm studied can effectively be used as an aggregation operator for adaptive fuzzy controller synthesis. In the conventional approach, algebraic product is used as the aggregation operator, whereas the operator considered in this study uses a multiplication of ordinary minimum operator with a continuous function, which is a function of membership grades and which is differentiable with respect to the adjustable parameters. The results presented emphasize that a satisfactory tracking precision and a prominent cost minimizing activity could also be achieved by the use of aggregation operator discussed. 1. INTRODUCTION Fuzzy systems have become a widely used design framework in many engineering sciences. The outcome of ever-increasing interest to the design of fuzzy systems has shown itself in the form of products being used even in the daily life. Therefore it is no more surprising to see that many products or systems employing Fuzzy Inference Systems (FIS) in performing a task which is either from daily life or from industry. Washing machines, sewing machines, cameras, intelligent controllers, identifiers and pattern/image recognition systems are typical examples of such systems, which could somehow incorporate a degree of intelligence provided by fuzzy inference mechanisms. The technological endeavors in the 20th century have been in pursuit of systems capable of providing accuracy, optimality as well as safety. In this respect, fuzzy logic, with the property of representing expert knowledge in the form of IF-THEN statements is most appropriate for translating the experience of the human operator. There are several reasons that make the use of fuzzy systems so attractive. Firstly, the procedure enables the designer to impose his/her feelings, intuitions or beliefs into the task space. Secondly, the fuzzy system models possess a great flexibility in choosing the methods of fuzzification, defuzzification and construction of rule base. Thirdly, the adaptability of the architecture makes it useful in fine-tuning of the parameters. Much of the results, which have been reported so far, have focused on the improvement of fuzzy system performance. In [1], the architecture of a standard adaptive fuzzy system is described. The structure introduced by there by Wang [1], uses the product inference rule and Gaussian membership functions. The tuning is performed on the parameters of the Gaussians and the weights used in the defuzzification procedure. In [2], an extended version of fuzzy system construction called Adaptive Neuro Fuzzy Inference Systems (ANFIS) is developed. This architecture has greatly improved the realization performance of fuzzy system and has extensively been used for identification and control purposes [3], [4]. The output of ANFIS is a linear function of its augmented input vector. Therefore, by appropriately setting the defuzzifier parameters, one can easily obtain the standard fuzzy system described in [1]. Hence, the dimension of parameter space in ANFIS approach is greater than the standard approach. Another mechanism is proposed by Takagi and Sugeno [5], and is known as Takagi-Sugeno (TS) fuzzy model. The model analyzed in [5] performs an interpolation between linear system models on the basis of fuzzy activation. Therefore, TS fuzzy models can make use of the prominent features of conventional design methodologies in analyzing the critical issues such as stability and robustness.

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The approaches mentioned so far have been the most notable architectural developments in fuzzy system theory. The recent studies are addressed to the improvement of aggregation methods. In [6], Batyrshin and Kaynak propose parameterized T-Norms, which offer an additional adaptive learning possibility in aggregating the premises of rules. Introducing design flexibility through the adjustment of aggregation method is of a crucial importance in cases where nondifferentiable membership functions are used or when membership functions do not have adjustable parameters. What makes the adjustment of only the T-Norm parameters instead of tuning the entire set of parameters so attractive is the observation of accurate state tracking precision. Therefore the use of the proposed algorithm may be much preferable in industrial applications such as control of mechatronic systems or biochemical processes where high tracking precision is sought. In this paper, the fuzzy system model introduced in [1] has been chosen as thearchitecture. In the literature, typically the parameters of defuzzifier and the parameters of the membership functions are tuned. However, the approach presented in this paper uses parameterized T-Norms and the adaptation mechanism tunes only the T-Norm parameters instead of what commonly adopted in the literature. The paper is organized as follows: The next section describes the plant to be controlled. In the third section, fuzzy system model is briefly introduced and the learning algorithm is derived. The following section is devoted to the design of a fuzzy controller using the standard architecture in which bellshaped membership functions are used with parameterized T-Norms and standard defuzzification procedure. The fourth section describes the control strategy utilizing the proposed approach. Simulation results are presented in the fifth section. The concluding remarks on the system performance are given in the last section. 2. PLANT MODEL In this study, a two degrees of freedom direct drive robotic manipulator, which is illustrated in Figure 1, has been used as the test bed. Since the dynamics of such mechatronic systems is modeled by coupled and complicated differential equations, being in pursuit of output tracking precision becomes a tedious work due to the strong interdependency between the variables involved. Besides, the ambiguities concerning the friction related dynamics in the plant model make the design much more complicated if one utilizes the formalism described in conventional design techniques, which defy accurate analytical modeling. Therefore the methodology adopted must be intelligent in some sense.

Figure 1. Physical View of the Direct Drive Robotic Manipulator The general form of robot dynamics is described by (1) where M(θ), V(θ,θ’), τ(t) and f stand for the state varying inertia matrix, vector of coriolis terms, applied torque inputs and friction terms respectively. The plant parameters are given in Table 1 in standard units.

M (θ )θ ′′ + V (θ , θ ′) = τ (t ) − f

211

(1)

Table 1. Manipulator Parameters Motor 1 Rotor Inertia Arm 1 Inertia Motor 2 Rotor Inertia Motor 2 Stator Inertia Arm 2 inertia Payload Inertia Motor 1 Mass Arm 1 Mass Motor 2 Mass Arm 2 Mass

0.2670 0.3340 0.0075 0.0400 0.0630 0.0000 73.000 9.7800 14.000 4.4500

I1 I2 I3 I3C I4 IP M1 M2 M3 M4

Payload Mass Arm 1 length Arm 2 length Arm 1 CG Arm 2 CG Axis 1 Friction Axis 2 Friction Torque Limit 1 Torque Limit 2

0.000 0.359 0.240 0.136 0.102 5.300 1.100 245.0 39.20

Mp L1 L2 L3 L4 F1 F2

If the angular positions and angular velocities are described as the state variables of the system, four differential equations being coupled and first order can define the model. In (2) and (3), the terms seen in (1) are given explicitly.

 p + 2 p3cos(θ 2 ) p2 + p3cos(θ 2 ) M (θ ) =  1  p2  p2 + p3cos(θ 2 ) 

(2)

 − θ ′ ( 2θ ′ + θ 2′ ) p 3 sin(θ 2 ) V (θ , θ ′) =  2 12  θ 1′ p 3 sin(θ 2 )  

(3)

where p1 = 2.0857, p2 = 0.1168, p3 = 0.1630. 3. FUZZY SYSTEM MODEL The fuzzy system model analyzed in this part has been used as the controller. As introduced in the second section, the manipulator has two control inputs and each torque input is evaluated by the use of architecture discussed next. For each link, a separate Fuzzy Logic Controller (FLC) is used with the relevant error and the rate of error as the input variables. In this respect, it should be emphasized that the coupling effects are not taken into consideration by the controller. Clearly, this is a difficulty, which is to be alleviated by the controller, with the adjustment carried out on the aggregation procedure adopted. A. Architecture of the Fuzzy System Model The fuzzy system employed in this study has the following type of rules in the rule base. In this representation, lowercase variables denote the inputs, uppercase variables stand for the fuzzy sets corresponding to the domain of each linguistic label. IF THEN

u1 is U1 AND u2 is U2 AND … AND um is Um f = yi

The structure of fuzzy system is illustrated in Figure 2. In this study, R=9, m=2 which describe the number of rules in the rule base and number of inputs of each fuzzy controller respectively. As the membership functions, bell-shaped functions are used as described by (4).

212

Rule #1

u1

µ11

u2

µ12 T-Norm

y1

µ1m

um

yR

µR1

Σ

µR2

a T-Norm

µRm

Σ

b

F=a/b

F

Rule #R

Figure 2. Structure of the Fuzzy System

µij (u j ) =

1 1+

u j − cij

2bij

(4)

σ ij

The overall realization performed by FLC can now be formulated as follows: R

F=

∑ yi wi

i =1 R

(5)

∑ wi

i =1

where, the firing strengths denoted by w are evaluated through the use of following aggregation operator proposed by Batyrshin and Kaynak [6].

wi = T ( µi1 , µi 2 , pi ) = min(µi1 , µi 2 )(µi1 + µi 2 − µi1µi 2 ) pi

(6)

The described T-Norm is clearly commutative but not associative. B. Derivation of the Learning Algorithm In deriving the parameter update mechanism, gradient descent rule has been utilized. Therefore the derivation starts with the evaluation of the sensitivity derivatives involved. The objective of the algorithm is to minimize the cost function described by (7). The parameter update rule for pi is given by (8) in which pi is the parameter of ith rule’s aggregator.

J=

1 1 (d − F ) 2 = e 2 2 2

213

(7)

∆pi = − ηi

∂J ∂F ∂ F ∂ wi = ηi e = ηi e ∂ pi ∂ pi ∂ wi ∂ pi

(8)

In (7) d is the target output, and in (8) i runs from 1 to R with η being the learning rate. In all simulations η has been set to the normalized firing strength of related rule. The motivation behind this selection is twofold: first the normalized firing strength is between zero and unity, which is compatible with the common understanding on learning rate selection, second, the mostly activated rule needs greater steps in order to converge to the cost minimizing state of update procedure. Therefore the ith entry of η vector is evaluated as given by (9).

ηi =

wi

R

(9)

∑wj j =1

With these definitions, the partial derivatives are evaluated as given in (12) through (17).

y −F ∂F = Ri ∂ wi ∑wj

(10)

∂ wi = wi ln (µi1 + µi 2 − µi1µi 2 ) ∂ pi

(11)

j =1

For the controllers used for both axes, the set of adjustable parameters contains only the T-Norm parameters. 4. FUZZY CONTROL OF DIRECT DRIVE SCARA ROBOT The control mechanism utilized in this study uses the described fuzzy system directly as the controller. Two FLCs are employed in the design where the controllers use the related error and the rate of error as the input variables for each link. The scheme is an ordinary feedback control method illustrated in Figure 3. The internal structure of the FLC is depicted in Figure 4.

r

+

Σ _

FLC

τ

ROBOT

x

Figure 3. Pure Fuzzy Control Scheme

eb eb′

Base FLC

τ1

ee ee′

Elbow FLC

τ2

Figure 4. Internal Structure of Fuzzy Controller

In Figure 4, eb and eb′ denote the observed position and velocity error observed for base axis respectively, ee and ee′ stand for the same quantities for elbow axis. One can directly infer from Figure 4 that the coupling effects are not taken into consideration by the controller, i. e., the base axis

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controller does not use the data observed from elbow link and vice versa. Therefore, the state tracking precision becomes a challenge that is to be achieved by the controller. 5. SIMULATION RESULTS The fuzzy model used in the simulations has the membership functions described by (4) and illustrated in Figure 5. The parameters of the membership functions have not been adapted in order to emphasize the learning property introduced by T-Norm adaptation. Another set of model parameters are the weight vectors (y1, y2) used in the defuzzifier. Those parameters are also kept constant throughout the simulations and the values are given by (13) and (14). In this paper, a trapezoidal velocity profile is used as the reference signal for both axes. In Figure 6 the reference trajectories used in the simulations are illustrated. Figure 7 shows the discrepancy between reference signals and the actual state vector of the manipulator. In Figure 8, time behavior of the parametric cost, which is defined by (12), is illustrated. During the early phases, as indicated by a dotted rectangle, the excessive cost minimizing activity introduced by the methodology followed reduces the parametric changes. The results observed in the sense of tracking precision and the cost minimization clearly imply that the learning can be achieved through the adjustment of parameters of aggregation method presented in this paper.

  Rules   Rules J (t ) =  ∑ ∆p j (t ) 2  +  ∑ ∆p j (t ) 2   BASE  ELBOW  j =1  j =1

(

)

(

)

AXIS

(12)

AXIS

1.2

POS

1

ZERO

NEG

0.8

µ

0.6

0.4

0.2

0

-0.1

-0.05

0

0.05

0.1

Error or Rate of Error

Figure 5. Definitions of Membership Functions

y1 = [− 16.50 − 6.75 3.00 − 9.75 0.00 9.75 − 3.00 6.75 16.50]T

(13)

y 2 = [− 6.75 − 3.00 0.75 − 3.75 0.00 3.75 − 0.75 3.00 6.75]T

(14)

215

Base Link Position Reference

Elbow Link Position Reference

2

2

1.5

1.5

1

1

0.5

0.5

0

1

0

5

10

0

15

Time (sec) Base Link Velocity Reference

1

0.5

0.5

0

0

-0.5

-0.5

-1

0

5

10

0

-1

15

5

10

15

5

10

15

Time (sec) Elbow Link Velocity Reference

0

Time (sec)

Time (sec)

Figure 6. Reference Trajectories used in the Simulations

Base Link Positional Error -3

3

Elbow Link Positional Error -4

x 10

5

x 10

0

2

-5 1 -10 0

-1

-15

0

5

10

15

-20

Time (sec) Base Link Velocity Error

0.02

0.05

0

0

-0.01

-0.05

0

5

10

15

-0.1

Time (sec)

5

10

0

5

10

Time (sec)

Figure 7. State Tracking Errors Observed During the Simulations

216

15

Time (sec) Elbow Link Velocity Error

0.1

0.01

-0.02

0

15

10

10

10

10

10

10

Time Behavior of Parametric Cost

5

0

-5

-10

-15

-20

10

0

10

1

10

2

10

3

Iteration Number (Equivalent to 16 secs) Figure 8. Time Behavior of the Cost Function Defined by (12) In the literature, various fuzzy models are proposed and studied in motion control problems. Most of them consider the adaptation of fuzzifier and defuzzifier parameters. The contribution of this paper is to demonstrate whether a satisfactory performance could be observed or not if one imposes learning by tuning the aggregation operator. Obviously, the results confirm that the tracking precision, which can also be achieved by the conventional tuning strategies, can be observed by the approach discussed. 6. CONCLUSIONS In this study, the fuzzy control of a two degrees of freedom direct drive robotic manipulator is presented. The fuzzy system chosen in the work has the standard architecture, which utilizes a parameterized T-Norm in aggregating the rule premises. The learning algorithm has been derived for the use of discussed T-Norm. An interesting feature of the design presented in this paper is that the FLC blocks use solely the error and the rate of error for each link regardless of the coupling effects. Briefly, the approach presented in this paper achieves the tracking precision, and compensates the deficiencies caused by poor modeling. On the other hand, it has some undesirable characteristics in the sense of losing the physical meanings of fuzzy inference mechanism due to the adjustment performed on aggregation of rule premises. The debate on the use of such aggregation methods has not ended yet but the research on the correspondence between this action and the physical domain is still continuing. The work is in progress through the stability and robustness analysis of fuzzy control scheme with new aggregation operator proposed in this paper.

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7. REFERENCES [1] [2] [3] [4] [5] [6]

Wang, L., Adaptive Fuzzy Systems and Control, Design and Stability Analysis, PTR Prentice Hall, 1994, pp. 29-31. Jang, J.-S. R., C.-T. Sun and E. Mizutani, Neuro-Fuzzy and Soft Computing, PTR Prentice Hall, 1997. Efe, M. O. and O. Kaynak, “A Comparative Study of Soft Computing Methodologies in Identification of Robotic Manipulators,” Proc. 3rd Int. Conf. on Advanced Mechatronics, ICAM’98, 3-6 August, vol. 1, pp. 21-26, Okayama, Japan, 1998. Efe, M. O. and O. Kaynak, “A Comparative Study of Neural Network Structures in Identification of Nonlinear Systems,” Int. Journal of Mechatronics, (accepted for publication) 1998. Takagi T., M. Sugeno, “Fuzzy Identification of Systems and Its Applications to Modeling and Control,” IEEE Trans. On Systems, Man and Cybernetics, pp. 116-132, January 1985. Batyrshin I., O. Kaynak, I. J. Rudas,“Generalized Conjunction and Disjunction Operators for Fuzzy Control,” Proc. EUFIT, Sep. 7-10, pp. 52-57, Aachen, Germany, 1998. ACKNOWLEDGMENTS

This work is supported in part by a grant of Foundation for Promotion of Advanced Automation Technology and Bogazici University Research Fund, Project No: 97A0202 and 99A202.

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