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Application of fuzzy control to an inserting operation
I
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sets and systems Fuzzy Sets and Systems 66 (1994) 267-281
ELSEVIER
Application of fuzzy control to an inserting operation Pi-Cheng Tung*, Yen-Pu Hsu Department of Mechanical Engineering, National Central University, Chung-Li 32054, Taiwan
Received February 1994; revised April 1994
Abstract Because of the difficulty of dynamic equations for the inserting operation and the advantage of fuzzy control for easy execution without knowledge of the mathematical models. We used the force feedback signal measured by a force sensor and a fuzzy controller to regulate the position of the peg. The control strategies include two control loops; in one a PID position controller move the peg vague position near to the hole, and in the other the fuzzy controller regulates the final position according to contact force. Between the two control loops we use an index to activate a switch. To verify the proposed control strategy, we performed experiments with a Cartesian manipulator system driven by AC servo-motors. The experimental results reveal that we can insert a peg into a hole with a fitting clearance less than 0.01 mm. Keywords: Fuzzy control; Inserting operation; Contact force; Clearance
1. Introduction A property of modern control theory is that it is usually based on a mathematical model of system, whereas most systems being complicated, we cannot specify all related variables, nor point out exactly their importance. Under such conditions, we are interested to design a controller with an unknown mathematical model. Industrial robots are used extensively in manufacturing. The assembly operation is considered highly technological and difficult work, and raises new problems. The traditional Robot end-effector cannot fulfill applied needs in the task level position or trajectory, so engineers add a vision device or tactile sensor to it to make the robot more sensitive to the
* Corresponding author.
environment. For the typical process of insertion, for example, the tracking error must be small. Most present control strategy may achieve precise tracking, but can attain only a small or moderate range of speed. F r o m the automatic viewpoint, one might require a reasonable speed that is greater than a safe speed to improve the performance. If the speed of insertion is too small, it might rather be done by traditional manpower than by automation that becomes inefficient. Even though there is a high rate of success in the inserting operation, efficiency is also a worthy objective. In the literature related to assembly by manipulators, Whitney [21] reported a force analysis for various inserting processes and discussed a RCC (remote compliance center) device. G o t o et al. [4] discussed congestion during inserting and developed a sequence controller for this purpose.
0165-0114/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0165-01 14(94)00151-V
268
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
There are two basic approaches to the problems of controlling an inserting process - passive control and active control. The former uses a small stiffness of a mechanism with which robots can endure the disturbances. RCC is such a mechanism. The main advantage of RCC is to protect the pegs from jamming during insertion; it is a spring device that is helpful for assembling with a tiny clearance. In active control there is a feedback loop to produce compliance. The signals obtained from the force/torque sensor are sent to the controller. Craig and Railbert [2] and Craig [1,1 in several papers about robot control introduced the method of hybrid position and force control. Goldenberg and Bazerghi [31 mentioned the synthesis of robot control for an assembly process and regulation was illustrated using simulation results. In the application of fuzzy control, Mamdani and Assilian [14] applied the linguistic synthesis of fuzzy control to steam engines. Kouatli and Jones [12,1 introduced a convenient and efficient method for designing fuzzy controller in welding. Huang and Tomizuka I-7,1 designed a fuzzy controller to track a plane trajectory for the Cartesian manipulator. The results of simulation are better than that of traditional PD control. Li and Lau [13] combined the two decision tables (fine and coarse) in a fuzzy controller and applied the algorithms in a microprocessor-based servo-motor controller. Hara and Yokogawa [6,1 applied a fuzzy controller to an inserting operation by using a SCARA (selective compliance assembly robot arm), a force/torque sensor, and an RCC device. Many good papers, for example [5, 8, 11, 15, 20, 22"1, with an interesting application can be found in Kandel and Langholz [10,1. Like the human inserting operation, vision is first used to move the peg close to the hole; at the moment that the peg contacts the chamfer, the operator, having regard to the rule that "If the contact force is strong when the peg is inserted to the right side, then the peg should be moved a little bit to the left side", tries to insert progressively the peg into the hole. In this work, we used the linear scale to replace the vision sensor, and designed a PID controller to control the vague position of the peg. When the clearance between the peg and the hole is larger, it is possible to use position
control only. Having a good position controller and sensor is not enough to perform an inserting operation precisely; the precision of the machine is also important. Otherwise, even though there is no error on any axis, the steady-state error may exist at the tip of the gripper. It is difficult to resolve a problem of this kind using position control in a precise inserting operation. In order to resolve this difficulty, we add a force feedback with a force sensor. Then, with the aid of the strength and direction of the contact force between the peg and the chamfer, we can perform the inserting operation precisely. We sought to take advantage of force feedback directly to design a fuzzy controller to accomplish insertion. The reason that we chose a fuzzy controller is that we can easily describe the experiences and knowledge of operators in linguistic form for an inserting operation. Once we are able to describe the control system linguistically, then the rules of fuzzy control are obtained. Conversely, if we use traditional control skill, it is not proper to apply the human linguistic description. Not only its structure is on the basis of a crisp set, but also it needs a precise mathematical model. Because of the difficulty of the dynamic equation during the process of insertion and the main advantage of fuzzy control for easy execution without knowledge of the mathematical model, we designed a fuzzy controller in the force feedback loop.
2. Fuzzy control algorithm There are major procedures in fuzzy control algorithms: I. Choose membership functions; II. design control rules; III. decide methods of fuzzy reasoning, composition and defuzzification. A block diagram of fuzzy controller is shown in Fig. 1. First, the shape of the membership functions is arbitrary and depends on the user's preference. For simplicity, triangular, bell and trapezoidal shapes are commonly used. The design of control rules is based on the operator's understanding of the behavior of the process instead of its detailed mathematical model. In general, the control rules should be expressed in an " I F . . . T H E N . - . " statement. For
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
INPUT
f
FUZZIFICATION
MATCHING
I
I O~FUmF~'noN
COMPC~TION MAX,-MIN
269
-> OUTPUT
CONTROL RULE
Fig. 1. Block diagram of fuzzy controller.
example,
uniquely determined. The other procedure is the "center-of-gravity" method that we express as:
Anti: IF x is A T H E N y is B
N
Ant2: IF x is A'
(1)
V = ~, (l~n(x) x U n ) / ~ , # n ( x )
(4)
1
Con: y is B', in which x and y are names of objects, and A, A', B and B' are fuzzy concepts represented by fuzzy sets in universes of discourse U and V, respectively. The statement (1) may represent a certain relationship between A and B. From this point of view, several methods were proposed for this relation. We quote the fuzzy relation Re from Mizumoto and Zimmermann [16] as follows. Rc = A x B
~(u) ^ ~(v)/(u, v). 3~
(2)
xV
For any input of the fuzzy controller, fuzzy control is obtained via a compositional rule of inference that we use max-min composition. B' = A'o Ro.
(3)
The output of the fuzzy controller is a fuzzy set of control. Therefore, we need a defuzzification process that converts the fuzzy variable of the inference engine into a corresponding value of the universe discourse (crisp set). There are several ways to tackle these problems. The first and simplest method relies on choosing the control value for which the membership function attains a maximum. A difficulty arises when more than one element of/~(x) processes this maximal value and thus the defuzzification result is not
in which V: defuzzified result, /~n(X) membership function value, n: number of contributions, U~: universe of discourse. We choose the latter because the "center-of-gravity" method gives a more reliable decision than the "maximum" operation. The rule is simple but computation is complicated. If a multi-variable input or membership function distributes closely, control rules become numerous; then duration of execution becomes excessive. A servo control system requires for application a sampling duration more and more brief. If the algorithm is too complicated, it is difficult to implement in actual control [7]. Thus, in much literature [9, 17, 18, 19] are reports of input and output built into a decision table with a fuzzy control algorithm in advance; then, all output restores by looking up the table. The advantage of this method is its speed, but it is limited by the memory of any computer that cannot be expanded indefinitely. The hardware is becoming more efficient so that duration of calculations is decreasing gradually. In the experiment, the signal is measured by a force/torque sensor and transmitted to computer by a serial port at a transmission rate of RS-232C (9600 Baud rate) too small for the sampling frequency of systems to be increased. Hence a fuzzy controller output is computed directly requiting less than the transmission period of the RS-232C interface.
270
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
~LINEAR SCALER I
>[PIDPOSITION
r e ~ e c position
CONTROLLER
PLANT 1
INDEX
re~erence force
~ FUZZYFORCE
+
CONTROU/~
[ FORCE SENSOR t Fig. 2. Control block diagram.
ipeg
y
~holo
< X
T
chamfer
Fig. 3. Coordinate of peg and hole.
3. Insertion control algorithm In all inserting processes, the control strategy includes two control loops. A block diagram is shown in Fig. 2. One has a PID position controller to move the peg near the chamfered hole; the other employs the fuzzy controller to regulate the final position by contact force. Between the two control loops, an index is used to actuate a switch. The index is decided by the chamfer and the clearances between the peg and hole (Fig. 3). IXpeg -- Xhole14 I n d e x ,
(5)
[Ypeg -- Yhole[ ~< I n d e x ,
(6)
Index = (h - d)/2 + c = (clearance)/2 + c,
(7)
where Xpe,, ype,: peg position in the X, Y plane, Xho~, Yho~,: hole position in the X, Y plane, h: hole diameter, d: peg diameter, c: projective length of the chamfer in the X, Y plane.
The procedure is intended to ensure that the first point of contact is in the range of the chamfer. If the point is outside the range, the measured contact force has no direction and leads to insertion failure. With regard to force feedback, we compare the two fuzzy controllers as follows: (I) only force feedback, (II) force and variation of force feedback. 3.1. Only force f e e d b a c k in f u z z y controller
When the peg collides with the chamfer, the force/torque sensor measures the contact force in three directions. The fuzzy controller regulates the peg position depending on the strength and direction of the contact force. For example, if the contact force measured by the force/torque sensor is large in the positive X axis direction, the position of peg is deviated from the center of the hole. If we intend to insert successfully, the peg has to move much in the opposite direction. Physical sense of this kind is accepted easily by human beings, but is difficult to
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281 Table 2 Control rules in Y-axis direction
U NB
NM
NS
2~0
271
P8
PM
/
i PB t
h > -FMAX
Ah PB
PM
PS
ZO
NS
NM
NB
PB
PB
PB
PB
PB
PB
PB
PM PS ZO NS NM
PB PM PS ZO NS
PM PS PS NS NM
PM PS ZO NS NM
PM ZO ZO ZO NM
PM PS ZO NS NM
PM PS NS NS NM
PM PS ZO NS NM NB
NB
NM
NB
NB
NB
NB
NB
NB
NM
NB
F
FMAX
u
NM
Ns
Ps
pM
< -FMAX
f"
FMAX
>
F
U
NM
Ns
Ps
Table 3 Control rules in Z-axis direction
PM
< -VM~O(
VMAX
>
V
Fig. 4. (a) Membership function of linguistic values of force; (b) Membership function of linguistic value of force; (c) Membership function of linguistic values of voltage.
Table 1 Control rules in X-axis direction
fx
PB
PM
PS
ZO
NS
PB
NB
NB
NB
NB
NB
NB
PM PS ZO NS NM
NB NM NS ZO PS
NM NS NS PS PM
NM NS ZO PS PM
NM ZO ZO ZO PM
NM NS ZO PS PM
NM NS PS PS PM
NM NS ZO PS PM PB
NB
PM
PB
PB
PB
PB
PB
PB
Afx PB
PM
PS
ZO
PB
PB
PM PS ZO NS NM NB
PB PM PS ZO NS NM
PB
PB
PB
PM PS PS NS NM NB
PM PS ZO NS NM NB
PM ZO ZO ZO NM NB
NS
NM
NB
PB
PB
PM PS ZO NS NM NB
PM PS NS NS NM NB
PM PS ZO NS NM NB NB
m e m b e r s h i p functions s h o w n in Fig. 4(b). Because the s e r v o - m o t o r driver is of the velocity c o m m a n d type, we define the o u t p u t m e m b e r s h i p function s h o w n in Fig. 4(c). According to experience a n d skill d u r i n g the inserting operation, we design c o n t r o l rules in linguistic form as follows: X axis direction:
accomplish by a t r a d i t i o n a l controller. Because the c o n t a c t force is "large" a n d moves "much", it belongs to the field of description, a n d is n o t easy to describe exactly. W e a t t e m p t to a p p l y fuzzy sets to p r o m o t e this c o n d i t i o n . T h e first step is to build the i n p u t a n d o u t p u t m e m b e r s h i p function. We b u i l d seven i n p u t m e m b e r s h i p functions in the universe s h o w n in Fig. 4(a), or b u i l d bell-shape
I f f x is PB then Vx is PB
(8a)
I f f x is P M then Vx is P M
(8b)
I f f x is PS then Vx is PS
(8c)
I f f x is Z O then Vx is Z O
(8d)
I f f x is N S then Vx is N S
(8e)
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
272
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Iffx is NM then Vx is NM
(80
If efz is ZO then Vz is ZO
(lOd)
Iffx is NB then Vx is NB
(8g)
If efz is NS then Vz is NS
(lOe)
If efz is NM then Vz is NM
(lOf) (lOg)
Y axis direction:
Iffy is PB then Vy is PB
(9a)
If efz is NB then Vz is NB
Iffy is PM then Vy is PM
(9b)
Iffy is PS then Vy is PS
(9c)
Iffy is ZO then Vy is ZO
(9d)
Iffy is NS then Vy is NS
(9e)
in which PB (positive big), PM (positive medium), PS (positive small), ZO (zero), NS (negative small), NM (negative medium), NB (negative big); efz = f z d - f z ; f z d : desired force in the Z-axis direction; fx, fy, fz: contact force in the three axis directions measured with the force/torque sensor. There are twenty-one rules in total.
Iffy is NM then Vy is NM
(9f)
Iffy is NB then Vy is NB
(9g)
Z axis direction: If efz is PB then Vz is PB
(10a)
If efz is PM then Vz is PM
(lOb)
If efz is PS then Vz is PS
(lOc)
3.2. Force and variation offorce feedback in a fuzzy controller Because signals from the force/torque sensor vary largely in the Z-axis direction, we add a fuzzy variable for variation of contact force feedback to improve the system response. The control rules in each axis are shown in Tables 1-3, in which Afx, Afy, Aefz: variation offx, fy, efz, respectively.
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
273
Fig. 6. The main experimental equipments.
Table 4 Hole character hole diameter
chamfer type
A
10.30
cl
B
10.025 + 0.0025 -
C
0
10.0075 + 0.0025 -- 0
cl cl
4. Experiment and results The block diagram of the system is illustrated in Fig. 5 and the main experimental equipment is shown in Fig. 6. It can be divided into several parts as follows:
(1) Robot: cartesian manipulator, repeat precision is 0.05 mm. (2) Linear scale: 0.01 mm/count (3) Force/torque sensor: Lord 30/100 type. (4) work parts (unit:mm), peg: diameter 10.002, hole: There are three sizes as shown in Table 4. (5) Controller: PC-AT. The software program is prepared in the C language. The purpose of this work was to attain the greatest precision inserting with desired contact force under various conditions of clearances. In order to compare controller performances, we performed the following tests and experiments. With only the input and output of fuzzy controller, the characteristic relation between single input and output is shown in Fig. 7. The effects are insensitive to triangular and bell-shaped membership function. If there is another input to the fuzzy controller, i.e., force and variation of force, the
274
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
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Table 5 The figures of force feedback only in Fuzzy controller Hole
Position
Force
A B C
Fig. 8(a) Fig. 9(a) Fig. 10(a)
Fig. 8(b) Fig. 9(b) Fig. 10(b)
Table 6 Modified control rules in the Z-axis direction efz
PB PM PS ZO NS NM NB
Aefz PB
PM
PS
ZO
NS
NM
NB
NB NB NM NM ZO ZO NS
NB NB NM ZO ZO ZO ZO
NB NM NM ZO PS PM PB
NB NM NS ZO PM PB PB
NM NM NS ZO PM PM PB
NS ZO ZO PS PM PB PB
ZO ZO PS PM PM PB PB
differences are also very small. Therefore, we use the function of bell-shape because they are easier to implement. Although the sensor element of the force/torque sensor is of a strain gauge type and it is an analog device. We can read only digital data that have been converted by a force/torque sensor controller. Its unit force of resolution is ¼ oz near 7.8 g. Because in c o m p u t e r p r o g r a m integer calculation is faster than that in floating point, we calculate with unit force in the experiment. F o r the force feedback in a fuzzy controller as mentioned in Section 3.1, the desired force is - 2 0 0 unit force, F M A X = 100 unit force, V M A X = 0.15 voltage. The peg was inserted into A, B and C, three holes with diameters shown in Table 4. The results of position and force responses are shown in the figures and presented in Table 5. F o r the position responses shown in Figs. 8(a), 9(a) and 10(a), the unit of the vertical axis of the coordinate is 0.01 m m and the position is relative to the peg's final position. The unit of vertical axis is near 7.8 g in response in Figs. 8(b), 9(b) and 10(b).
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
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Fig. 8. (a) Position response of the peg inserted to hole A; (b) force response of the peg inserted to hole A.
For the force and variation of force feedback in the fuzzy controller mentioned in Section 3.2, the desired force is also - 2 0 0 unit force and e F M A X = 100. By performing the experiment, we
found that the response in the Z axis is unsati@factory. Through trial and error, we found the final control rules in Table 6. We insert peg into the A, B and C holes again; the results of position and
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
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force response are shown in the figures and presented in Table 7. Comparing Tables 3-6, we see that they differ appreciably. Hence the control rules, which are
designed according to operational experience, must be revised again (o become suitable to implement. In the similar conditions, such as FMAX, VMAX, the response improved on adding the
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Fig. 10. (a) Position response of the peg inserted to hole C; (b) force response of the peg inserted to hole C.
variation of force feedback in the fuzzy controller. The experimental results reveal that the minimal fitting clearance is 6.8 IJm. This value is obtained from the average diameter of several measured pegs
subtracting the hole diameter that is the average of two critical pin gauges. F r o m Figs. 8(a) and 8(b) we observed that the P I D controller switched to the fuzzy controller at
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
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time near 0.2 s. At that time the peg did not contact the chamfer, but each axis produced the bias force. Because the clearance between holes and peg was 0 . 3 m m in this example, without chamfering the
hole we inserted the peg successfully. O n l y a slight c o r r e c t i o n was r e q u i r e d when t = 6.5, t = 7, a n d t = 12.3. W h e n t = 16s, the peg t o u c h e d the bottom; each axis m a k e s a little oscillation. The steady
279
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
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state of the X and Y-axis force decays to 0, and Z-axis inclines to - 2 0 0 unit force. Except for hole A with the clearance 0.3 mm, the other holes need to be chamfered more or less.
According to Figs. 9(a), 9(b), the peg position is regulated by contact force. In Figs. 10(a), 10(b), the clearance for hole C is 6.8lam. After t---10s, Z-axis force increased
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994.) 267-281
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I 2
I 4
I' t;
I 8
I 10
t
I' 12
I 14
I 18
I 1:0
I 20
I 22
I~'
24
(see)
Fig. 13. (a) Position response of the peg inserted to hole C; (b) force response of the peg inserted to hole C.
gradually. The peg had a tendency to jam. When the peg reached the bottom, the force produced a large oscillation, and the response was poor.
Figs. 13(a) and 13(b) show the responses of inserting peg into holes C using force and variation of force feedback in the fuzzy controller. The response was better than that of simple force feedback.
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281 Table 7 The figures of force and variation of force in Fuzzy controller Hole
Position
Force
A B C
Fig. ll(a) Fig. 12(a) Fig. 13(a)
Fig. ll(b) Fig. 12(b) Fig. 13(b)
During all insertion processes, the maximum position error in the X and Y-directions that relates to the final position is 0.1 mm. However, as the measured clearance is just 6.8 pm, it means that the real position of peg differs from the value measured by the position sensor. In these circumstances, if we perform only by the position controller, even if the manipulators have good compliance or if we use RCC device, some system becomes damaged.
5. Conclusion
In the control strategy of precision insertion, force feedback is necessary especially in tiny clearances. Using the control method of this work, we need no precise position control and we can achieve successfully precision insertion. The experimental results reveal that by using the proposed control strategy we can insert a peg into a hole with fitting clearance less than 0.01 mm.
References [1] J.J. Craig, Introduction to Robotics: Mechanics and Control (Addison-Wesley, Reading, MA, 1989). F2] J.J. Craig and M. Railbert, Hybrid position/force control of manipulators, Trans. ASME Dynam. Systems Measurement Control 102 (1981) 126-133. [3] A.A. Goldenberg and A. Bazerghi, Synthesis of robot control for assembly process, Mechanism and Machine Theory 21(1) (1986) 43~2. [4] T. Goto, K. Takeyasu and T. lnoyama, Control algorithm for precision insert operation robots, IEEE Trans. Systems, Man, Cybernetics 10(1) (1980) 19-25. [5] E.A. Gutierrez and L.C. Rosa, A fuzzy logic based approach to machine tool control optimization, in: A. Kandel and G. Langholz, Eds., Fuzzy Control Systems (CRC Press, Boca Raton, 1994) 523-539.
281
[6] K. Hara and R. Yokogawa, Application of fuzzy reasoning to precision inserting operation, Fuzzy Sets and Systems 39 (1991) 205-214. [7] L. Huang and M. Tomizuka, A self-paced fuzzy tracking controller for two-dimensional motion control, IEEE Trans. Systems, Man, Cybernetics 20(5) (1990) 1115-1123. [8] T. Iokibe and T. Kimura, Human friendly fuzzy transportation system, in: A Kandel and G. Langholz, Eds., Fuzzy Control Systems (CRC Press, Boca Raton, 1994) 459-473. [9] A. Kandel, Fuzzy Mathematical Techniques with Applications (Addison-Wesley, Reading, MA, 1986). [10] A. Kandel and G. Langholz, Fuzzy Control Systems (CRC Press, Boca Raton, 1994). [11] M. Komori and T. Kitamura, Applications of a fuzzy control technique to superconducting actuators using high-Tc superconductors, in: A. Kandel and G. Langholz, Eds., Fuzzy Control Systems (CRC Press, Boca Raton, 1994) 499-521. [12] I. Kouatli and B. Jones, An improved design procedure for fuzzy control systems, Internat. J. Mach. Tools Manufact. 31(1) (1991) 107-122. [13] Y.F. Li and C.C. Lau, Development of fuzzy algorithms for servo systems, IEEE Control systems Magazine 9 (1989) 65-72. [14] E.H. Mamdani and S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man-Machine Studies 7 (1975) 1-13. [15] M.A. Manzoul, Fuzzy controllers on semi-custom VLSI chips, in: A. Kandel and G. Langholz, Eds., Fuzzy Control Systems (CRC Press, Boca Raton, 1994) 551-560. [16] M. Mizumoto and E.-J. Zimmermann, Comparison of fuzzy reasoning method, Fuzzy Sets and Systems 8 (1982) 253-283. [17] W. Pedrycz, Fuzzy Control and Fuzzy Systems (Research Studies Press, Wiley, New York, 1989). [18] P.A.S. Ralston and T.L. Ward, Fuzzy Control and Industrial Process, in G.W. Evans et al., Eds., Applications of Fuzzy Set Methodologies in Industrial Engineering (Elsevier, Amsterdam, 1989) 29-45. [19] E.H. Scharf and N.J. Mandic, The application of a fuzzy controller to the control of a multi-degree-offreedom robot arm, in: M. Sugeno, Ed., Industrial Applications of Fuzzy Control (Elsevier, Amsterdam, 1985) 41-61. [20] H.N. Teodorescu and A. Brezulianu, General analysis of fuzzy-controlled phase-locked loop, in: A. Kandel and G. Langhoiz, Eds., Fuzzy Control Systems (CRC Press, Boca Raton, 1994) 523-539. 1-21] D.E. Whitney, Quasi-static assembly of compliantly supported rigid parts, Trans. ASME Dynam. Systems Measurement Control 104 (1982) 65-77. ['22] S. Yoshida, A fuzzy logic controller for rigid risk drive, in: A. Kandel and G. Langholz, Eds., Fuzzy Control Systems (CRC Press, Boca Raton, 1994) 579-599.
ZZY
sets and systems Fuzzy Sets and Systems 66 (1994) 267-281
ELSEVIER
Application of fuzzy control to an inserting operation Pi-Cheng Tung*, Yen-Pu Hsu Department of Mechanical Engineering, National Central University, Chung-Li 32054, Taiwan
Received February 1994; revised April 1994
Abstract Because of the difficulty of dynamic equations for the inserting operation and the advantage of fuzzy control for easy execution without knowledge of the mathematical models. We used the force feedback signal measured by a force sensor and a fuzzy controller to regulate the position of the peg. The control strategies include two control loops; in one a PID position controller move the peg vague position near to the hole, and in the other the fuzzy controller regulates the final position according to contact force. Between the two control loops we use an index to activate a switch. To verify the proposed control strategy, we performed experiments with a Cartesian manipulator system driven by AC servo-motors. The experimental results reveal that we can insert a peg into a hole with a fitting clearance less than 0.01 mm. Keywords: Fuzzy control; Inserting operation; Contact force; Clearance
1. Introduction A property of modern control theory is that it is usually based on a mathematical model of system, whereas most systems being complicated, we cannot specify all related variables, nor point out exactly their importance. Under such conditions, we are interested to design a controller with an unknown mathematical model. Industrial robots are used extensively in manufacturing. The assembly operation is considered highly technological and difficult work, and raises new problems. The traditional Robot end-effector cannot fulfill applied needs in the task level position or trajectory, so engineers add a vision device or tactile sensor to it to make the robot more sensitive to the
* Corresponding author.
environment. For the typical process of insertion, for example, the tracking error must be small. Most present control strategy may achieve precise tracking, but can attain only a small or moderate range of speed. F r o m the automatic viewpoint, one might require a reasonable speed that is greater than a safe speed to improve the performance. If the speed of insertion is too small, it might rather be done by traditional manpower than by automation that becomes inefficient. Even though there is a high rate of success in the inserting operation, efficiency is also a worthy objective. In the literature related to assembly by manipulators, Whitney [21] reported a force analysis for various inserting processes and discussed a RCC (remote compliance center) device. G o t o et al. [4] discussed congestion during inserting and developed a sequence controller for this purpose.
0165-0114/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0165-01 14(94)00151-V
268
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
There are two basic approaches to the problems of controlling an inserting process - passive control and active control. The former uses a small stiffness of a mechanism with which robots can endure the disturbances. RCC is such a mechanism. The main advantage of RCC is to protect the pegs from jamming during insertion; it is a spring device that is helpful for assembling with a tiny clearance. In active control there is a feedback loop to produce compliance. The signals obtained from the force/torque sensor are sent to the controller. Craig and Railbert [2] and Craig [1,1 in several papers about robot control introduced the method of hybrid position and force control. Goldenberg and Bazerghi [31 mentioned the synthesis of robot control for an assembly process and regulation was illustrated using simulation results. In the application of fuzzy control, Mamdani and Assilian [14] applied the linguistic synthesis of fuzzy control to steam engines. Kouatli and Jones [12,1 introduced a convenient and efficient method for designing fuzzy controller in welding. Huang and Tomizuka I-7,1 designed a fuzzy controller to track a plane trajectory for the Cartesian manipulator. The results of simulation are better than that of traditional PD control. Li and Lau [13] combined the two decision tables (fine and coarse) in a fuzzy controller and applied the algorithms in a microprocessor-based servo-motor controller. Hara and Yokogawa [6,1 applied a fuzzy controller to an inserting operation by using a SCARA (selective compliance assembly robot arm), a force/torque sensor, and an RCC device. Many good papers, for example [5, 8, 11, 15, 20, 22"1, with an interesting application can be found in Kandel and Langholz [10,1. Like the human inserting operation, vision is first used to move the peg close to the hole; at the moment that the peg contacts the chamfer, the operator, having regard to the rule that "If the contact force is strong when the peg is inserted to the right side, then the peg should be moved a little bit to the left side", tries to insert progressively the peg into the hole. In this work, we used the linear scale to replace the vision sensor, and designed a PID controller to control the vague position of the peg. When the clearance between the peg and the hole is larger, it is possible to use position
control only. Having a good position controller and sensor is not enough to perform an inserting operation precisely; the precision of the machine is also important. Otherwise, even though there is no error on any axis, the steady-state error may exist at the tip of the gripper. It is difficult to resolve a problem of this kind using position control in a precise inserting operation. In order to resolve this difficulty, we add a force feedback with a force sensor. Then, with the aid of the strength and direction of the contact force between the peg and the chamfer, we can perform the inserting operation precisely. We sought to take advantage of force feedback directly to design a fuzzy controller to accomplish insertion. The reason that we chose a fuzzy controller is that we can easily describe the experiences and knowledge of operators in linguistic form for an inserting operation. Once we are able to describe the control system linguistically, then the rules of fuzzy control are obtained. Conversely, if we use traditional control skill, it is not proper to apply the human linguistic description. Not only its structure is on the basis of a crisp set, but also it needs a precise mathematical model. Because of the difficulty of the dynamic equation during the process of insertion and the main advantage of fuzzy control for easy execution without knowledge of the mathematical model, we designed a fuzzy controller in the force feedback loop.
2. Fuzzy control algorithm There are major procedures in fuzzy control algorithms: I. Choose membership functions; II. design control rules; III. decide methods of fuzzy reasoning, composition and defuzzification. A block diagram of fuzzy controller is shown in Fig. 1. First, the shape of the membership functions is arbitrary and depends on the user's preference. For simplicity, triangular, bell and trapezoidal shapes are commonly used. The design of control rules is based on the operator's understanding of the behavior of the process instead of its detailed mathematical model. In general, the control rules should be expressed in an " I F . . . T H E N . - . " statement. For
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
INPUT
f
FUZZIFICATION
MATCHING
I
I O~FUmF~'noN
COMPC~TION MAX,-MIN
269
-> OUTPUT
CONTROL RULE
Fig. 1. Block diagram of fuzzy controller.
example,
uniquely determined. The other procedure is the "center-of-gravity" method that we express as:
Anti: IF x is A T H E N y is B
N
Ant2: IF x is A'
(1)
V = ~, (l~n(x) x U n ) / ~ , # n ( x )
(4)
1
Con: y is B', in which x and y are names of objects, and A, A', B and B' are fuzzy concepts represented by fuzzy sets in universes of discourse U and V, respectively. The statement (1) may represent a certain relationship between A and B. From this point of view, several methods were proposed for this relation. We quote the fuzzy relation Re from Mizumoto and Zimmermann [16] as follows. Rc = A x B
~(u) ^ ~(v)/(u, v). 3~
(2)
xV
For any input of the fuzzy controller, fuzzy control is obtained via a compositional rule of inference that we use max-min composition. B' = A'o Ro.
(3)
The output of the fuzzy controller is a fuzzy set of control. Therefore, we need a defuzzification process that converts the fuzzy variable of the inference engine into a corresponding value of the universe discourse (crisp set). There are several ways to tackle these problems. The first and simplest method relies on choosing the control value for which the membership function attains a maximum. A difficulty arises when more than one element of/~(x) processes this maximal value and thus the defuzzification result is not
in which V: defuzzified result, /~n(X) membership function value, n: number of contributions, U~: universe of discourse. We choose the latter because the "center-of-gravity" method gives a more reliable decision than the "maximum" operation. The rule is simple but computation is complicated. If a multi-variable input or membership function distributes closely, control rules become numerous; then duration of execution becomes excessive. A servo control system requires for application a sampling duration more and more brief. If the algorithm is too complicated, it is difficult to implement in actual control [7]. Thus, in much literature [9, 17, 18, 19] are reports of input and output built into a decision table with a fuzzy control algorithm in advance; then, all output restores by looking up the table. The advantage of this method is its speed, but it is limited by the memory of any computer that cannot be expanded indefinitely. The hardware is becoming more efficient so that duration of calculations is decreasing gradually. In the experiment, the signal is measured by a force/torque sensor and transmitted to computer by a serial port at a transmission rate of RS-232C (9600 Baud rate) too small for the sampling frequency of systems to be increased. Hence a fuzzy controller output is computed directly requiting less than the transmission period of the RS-232C interface.
270
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
~LINEAR SCALER I
>[PIDPOSITION
r e ~ e c position
CONTROLLER
PLANT 1
INDEX
re~erence force
~ FUZZYFORCE
+
CONTROU/~
[ FORCE SENSOR t Fig. 2. Control block diagram.
ipeg
y
~holo
< X
T
chamfer
Fig. 3. Coordinate of peg and hole.
3. Insertion control algorithm In all inserting processes, the control strategy includes two control loops. A block diagram is shown in Fig. 2. One has a PID position controller to move the peg near the chamfered hole; the other employs the fuzzy controller to regulate the final position by contact force. Between the two control loops, an index is used to actuate a switch. The index is decided by the chamfer and the clearances between the peg and hole (Fig. 3). IXpeg -- Xhole14 I n d e x ,
(5)
[Ypeg -- Yhole[ ~< I n d e x ,
(6)
Index = (h - d)/2 + c = (clearance)/2 + c,
(7)
where Xpe,, ype,: peg position in the X, Y plane, Xho~, Yho~,: hole position in the X, Y plane, h: hole diameter, d: peg diameter, c: projective length of the chamfer in the X, Y plane.
The procedure is intended to ensure that the first point of contact is in the range of the chamfer. If the point is outside the range, the measured contact force has no direction and leads to insertion failure. With regard to force feedback, we compare the two fuzzy controllers as follows: (I) only force feedback, (II) force and variation of force feedback. 3.1. Only force f e e d b a c k in f u z z y controller
When the peg collides with the chamfer, the force/torque sensor measures the contact force in three directions. The fuzzy controller regulates the peg position depending on the strength and direction of the contact force. For example, if the contact force measured by the force/torque sensor is large in the positive X axis direction, the position of peg is deviated from the center of the hole. If we intend to insert successfully, the peg has to move much in the opposite direction. Physical sense of this kind is accepted easily by human beings, but is difficult to
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281 Table 2 Control rules in Y-axis direction
U NB
NM
NS
2~0
271
P8
PM
/
i PB t
h > -FMAX
Ah PB
PM
PS
ZO
NS
NM
NB
PB
PB
PB
PB
PB
PB
PB
PM PS ZO NS NM
PB PM PS ZO NS
PM PS PS NS NM
PM PS ZO NS NM
PM ZO ZO ZO NM
PM PS ZO NS NM
PM PS NS NS NM
PM PS ZO NS NM NB
NB
NM
NB
NB
NB
NB
NB
NB
NM
NB
F
FMAX
u
NM
Ns
Ps
pM
< -FMAX
f"
FMAX
>
F
U
NM
Ns
Ps
Table 3 Control rules in Z-axis direction
PM
< -VM~O(
VMAX
>
V
Fig. 4. (a) Membership function of linguistic values of force; (b) Membership function of linguistic value of force; (c) Membership function of linguistic values of voltage.
Table 1 Control rules in X-axis direction
fx
PB
PM
PS
ZO
NS
PB
NB
NB
NB
NB
NB
NB
PM PS ZO NS NM
NB NM NS ZO PS
NM NS NS PS PM
NM NS ZO PS PM
NM ZO ZO ZO PM
NM NS ZO PS PM
NM NS PS PS PM
NM NS ZO PS PM PB
NB
PM
PB
PB
PB
PB
PB
PB
Afx PB
PM
PS
ZO
PB
PB
PM PS ZO NS NM NB
PB PM PS ZO NS NM
PB
PB
PB
PM PS PS NS NM NB
PM PS ZO NS NM NB
PM ZO ZO ZO NM NB
NS
NM
NB
PB
PB
PM PS ZO NS NM NB
PM PS NS NS NM NB
PM PS ZO NS NM NB NB
m e m b e r s h i p functions s h o w n in Fig. 4(b). Because the s e r v o - m o t o r driver is of the velocity c o m m a n d type, we define the o u t p u t m e m b e r s h i p function s h o w n in Fig. 4(c). According to experience a n d skill d u r i n g the inserting operation, we design c o n t r o l rules in linguistic form as follows: X axis direction:
accomplish by a t r a d i t i o n a l controller. Because the c o n t a c t force is "large" a n d moves "much", it belongs to the field of description, a n d is n o t easy to describe exactly. W e a t t e m p t to a p p l y fuzzy sets to p r o m o t e this c o n d i t i o n . T h e first step is to build the i n p u t a n d o u t p u t m e m b e r s h i p function. We b u i l d seven i n p u t m e m b e r s h i p functions in the universe s h o w n in Fig. 4(a), or b u i l d bell-shape
I f f x is PB then Vx is PB
(8a)
I f f x is P M then Vx is P M
(8b)
I f f x is PS then Vx is PS
(8c)
I f f x is Z O then Vx is Z O
(8d)
I f f x is N S then Vx is N S
(8e)
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
272
ENAUgLE~
r ~ DOWNCOUNT R,OHTLM DECODER
SINGLE ONE SHOT L"
-~
~..~,~LM~E LINEARS-• L'~
V
R~
tt
OIRCUIT
[POWER
PC
> ~ RELAY
i - -
INTERFACE
"~AC220V
~ ~ ;
& ~~z~
FORC~E SENSOR
~
CONTROLLER
DRIVER
FORCE SENSOR
J
Fig. 5. Block diagram of the hardware.
Iffx is NM then Vx is NM
(80
If efz is ZO then Vz is ZO
(lOd)
Iffx is NB then Vx is NB
(8g)
If efz is NS then Vz is NS
(lOe)
If efz is NM then Vz is NM
(lOf) (lOg)
Y axis direction:
Iffy is PB then Vy is PB
(9a)
If efz is NB then Vz is NB
Iffy is PM then Vy is PM
(9b)
Iffy is PS then Vy is PS
(9c)
Iffy is ZO then Vy is ZO
(9d)
Iffy is NS then Vy is NS
(9e)
in which PB (positive big), PM (positive medium), PS (positive small), ZO (zero), NS (negative small), NM (negative medium), NB (negative big); efz = f z d - f z ; f z d : desired force in the Z-axis direction; fx, fy, fz: contact force in the three axis directions measured with the force/torque sensor. There are twenty-one rules in total.
Iffy is NM then Vy is NM
(9f)
Iffy is NB then Vy is NB
(9g)
Z axis direction: If efz is PB then Vz is PB
(10a)
If efz is PM then Vz is PM
(lOb)
If efz is PS then Vz is PS
(lOc)
3.2. Force and variation offorce feedback in a fuzzy controller Because signals from the force/torque sensor vary largely in the Z-axis direction, we add a fuzzy variable for variation of contact force feedback to improve the system response. The control rules in each axis are shown in Tables 1-3, in which Afx, Afy, Aefz: variation offx, fy, efz, respectively.
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
273
Fig. 6. The main experimental equipments.
Table 4 Hole character hole diameter
chamfer type
A
10.30
cl
B
10.025 + 0.0025 -
C
0
10.0075 + 0.0025 -- 0
cl cl
4. Experiment and results The block diagram of the system is illustrated in Fig. 5 and the main experimental equipment is shown in Fig. 6. It can be divided into several parts as follows:
(1) Robot: cartesian manipulator, repeat precision is 0.05 mm. (2) Linear scale: 0.01 mm/count (3) Force/torque sensor: Lord 30/100 type. (4) work parts (unit:mm), peg: diameter 10.002, hole: There are three sizes as shown in Table 4. (5) Controller: PC-AT. The software program is prepared in the C language. The purpose of this work was to attain the greatest precision inserting with desired contact force under various conditions of clearances. In order to compare controller performances, we performed the following tests and experiments. With only the input and output of fuzzy controller, the characteristic relation between single input and output is shown in Fig. 7. The effects are insensitive to triangular and bell-shaped membership function. If there is another input to the fuzzy controller, i.e., force and variation of force, the
274
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
0.|11
o. 44
Triangle shapc
. /
0.t2 0.40
0.01 0.0G m
o.o4-
I~l 0 . 0 2 0.00-
"~
-0.02
-
> -0.04 *0.08
-0.0| -0.t0 *0. t2 -0.t4
-0.11
i
I
!
I
I
I
I
I
-+4o-~o-+oo-,o-.-,to-2o unLt
|
!
o 8o force
I
I
4o
ao
I
I
,o ~oo
I
!
t20
t40
Fig. 7. Comparison of two different shapes.
Table 5 The figures of force feedback only in Fuzzy controller Hole
Position
Force
A B C
Fig. 8(a) Fig. 9(a) Fig. 10(a)
Fig. 8(b) Fig. 9(b) Fig. 10(b)
Table 6 Modified control rules in the Z-axis direction efz
PB PM PS ZO NS NM NB
Aefz PB
PM
PS
ZO
NS
NM
NB
NB NB NM NM ZO ZO NS
NB NB NM ZO ZO ZO ZO
NB NM NM ZO PS PM PB
NB NM NS ZO PM PB PB
NM NM NS ZO PM PM PB
NS ZO ZO PS PM PB PB
ZO ZO PS PM PM PB PB
differences are also very small. Therefore, we use the function of bell-shape because they are easier to implement. Although the sensor element of the force/torque sensor is of a strain gauge type and it is an analog device. We can read only digital data that have been converted by a force/torque sensor controller. Its unit force of resolution is ¼ oz near 7.8 g. Because in c o m p u t e r p r o g r a m integer calculation is faster than that in floating point, we calculate with unit force in the experiment. F o r the force feedback in a fuzzy controller as mentioned in Section 3.1, the desired force is - 2 0 0 unit force, F M A X = 100 unit force, V M A X = 0.15 voltage. The peg was inserted into A, B and C, three holes with diameters shown in Table 4. The results of position and force responses are shown in the figures and presented in Table 5. F o r the position responses shown in Figs. 8(a), 9(a) and 10(a), the unit of the vertical axis of the coordinate is 0.01 m m and the position is relative to the peg's final position. The unit of vertical axis is near 7.8 g in response in Figs. 8(b), 9(b) and 10(b).
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
I-x
275
...... y
@0
Z
40
E E 0
c;
20
"%,_
q
_
-
Q-
-_-
-~0 O
.
I
I
I
I
I
I
I
I
2
4
G
9
lo
12
14
"~
t
I lg
I 20
(sec)
X
50-
0
-~5o-
0 4~
-lOO
r-
It It
-zoo -
0
I 2
I 4
I £
I 2
I 10
+.
I 12
I 14
! lg
I 111
! 20
(see)
Fig. 8. (a) Position response of the peg inserted to hole A; (b) force response of the peg inserted to hole A.
For the force and variation of force feedback in the fuzzy controller mentioned in Section 3.2, the desired force is also - 2 0 0 unit force and e F M A X = 100. By performing the experiment, we
found that the response in the Z axis is unsati@factory. Through trial and error, we found the final control rules in Table 6. We insert peg into the A, B and C holes again; the results of position and
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
276
413
Y
30,
E .¢..,.
10
C3
-
:-
-..
~_. _,-
D-
..-.
~...
~..:
/
"~..._
"~_
•
-10
-~'0
fl
I
I
I
I
I
I
I
l
i
2
4
8
~
1O
1:~
14
1G
19
l 2fl
t (see)
Q)
I
:t
x
L, 0 <4-100 r-
-1513 IF, III Ill
-z13o
lilt ~'" P 4
0
i 2
i 4
I G
'
i 9
"'i '" 10
t
i 12
i 14
I
1G
I
i
19
20
(SAC)
Fig. 9. (a) Position response of the peg inserted to hole B; (b) force response of the peg inserted to hole B.
force response are shown in the figures and presented in Table 7. Comparing Tables 3-6, we see that they differ appreciably. Hence the control rules, which are
designed according to operational experience, must be revised again (o become suitable to implement. In the similar conditions, such as FMAX, VMAX, the response improved on adding the
/
277
P.-C. Tung, Y.-P. Hsu Fuzzy Sets and Systems 66 (1994) 267-281
I -- X 1o -
E
(:3 c~
~,_ ~.,
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".%
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Fig. 10. (a) Position response of the peg inserted to hole C; (b) force response of the peg inserted to hole C.
variation of force feedback in the fuzzy controller. The experimental results reveal that the minimal fitting clearance is 6.8 IJm. This value is obtained from the average diameter of several measured pegs
subtracting the hole diameter that is the average of two critical pin gauges. F r o m Figs. 8(a) and 8(b) we observed that the P I D controller switched to the fuzzy controller at
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
278
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time near 0.2 s. At that time the peg did not contact the chamfer, but each axis produced the bias force. Because the clearance between holes and peg was 0 . 3 m m in this example, without chamfering the
hole we inserted the peg successfully. O n l y a slight c o r r e c t i o n was r e q u i r e d when t = 6.5, t = 7, a n d t = 12.3. W h e n t = 16s, the peg t o u c h e d the bottom; each axis m a k e s a little oscillation. The steady
279
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281
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state of the X and Y-axis force decays to 0, and Z-axis inclines to - 2 0 0 unit force. Except for hole A with the clearance 0.3 mm, the other holes need to be chamfered more or less.
According to Figs. 9(a), 9(b), the peg position is regulated by contact force. In Figs. 10(a), 10(b), the clearance for hole C is 6.8lam. After t---10s, Z-axis force increased
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994.) 267-281
280
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Fig. 13. (a) Position response of the peg inserted to hole C; (b) force response of the peg inserted to hole C.
gradually. The peg had a tendency to jam. When the peg reached the bottom, the force produced a large oscillation, and the response was poor.
Figs. 13(a) and 13(b) show the responses of inserting peg into holes C using force and variation of force feedback in the fuzzy controller. The response was better than that of simple force feedback.
P.-C. Tung, Y.-P. Hsu / Fuzzy Sets and Systems 66 (1994) 267-281 Table 7 The figures of force and variation of force in Fuzzy controller Hole
Position
Force
A B C
Fig. ll(a) Fig. 12(a) Fig. 13(a)
Fig. ll(b) Fig. 12(b) Fig. 13(b)
During all insertion processes, the maximum position error in the X and Y-directions that relates to the final position is 0.1 mm. However, as the measured clearance is just 6.8 pm, it means that the real position of peg differs from the value measured by the position sensor. In these circumstances, if we perform only by the position controller, even if the manipulators have good compliance or if we use RCC device, some system becomes damaged.
5. Conclusion
In the control strategy of precision insertion, force feedback is necessary especially in tiny clearances. Using the control method of this work, we need no precise position control and we can achieve successfully precision insertion. The experimental results reveal that by using the proposed control strategy we can insert a peg into a hole with fitting clearance less than 0.01 mm.
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