Motion control of cam mechanisms

Motion control of cam mechanisms

Mechanism and Machine Theory 35 (2000) 593±607 www.elsevier.com/locate/mechmt Motion control of cam mechanisms Yan-an Yao a,*, Ce Zhang a, Hong-Sen ...
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Mechanism and Machine Theory 35 (2000) 593±607

www.elsevier.com/locate/mechmt

Motion control of cam mechanisms Yan-an Yao a,*, Ce Zhang a, Hong-Sen Yan b a

School of Mechanical Engineering, Tianjin University, Tianjin 300072, China Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan China

b

Received 3 August 1998; received in revised form 14 December 1998

Abstract In this paper, optimal control theory is applied to create a theoretical frame of `Active Control of Cam Mechanisms.' The investigation presented here deals with the problem of `Motion Control' which is the foundation of `Active Control.' It is shown that, from a kinematics point of view, the motion characteristics of the follower can be improved by applying an optimal control to the cam speed. Some examples are given to demonstrate the procedure. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction The pursuit of high productivity in industry is the essential force driving investigations to improve mechanisms. High-speed, high-precision automatic machines are becoming increasingly important. E€ective mechanism design methods are required in order to develop superior machines. Analysis and synthesis of mechanisms have obtained many important results. Usually, the conventional methods assume that the input speed of a mechanism is constant. The assumption is acceptable for most motor-driven mechanisms. The variation of input speed, for instance, the so called `speed ¯uctuation' due to inconsistencies in characteristics of the motor and the load, is usually considered to have negative e€ects on the behavior of mechanisms and should be eliminated or reduced. However, if the variation of the input speed can be actively controlled according to the designer's will, the motion of the mechanism will be more free

* Corresponding author. Tel.: +886-21-62932671; fax: +886-21-6293203. E-mail address: [email protected] (Y. Yao). 0094-114X/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 9 9 ) 0 0 0 2 5 - 7

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Nomenclature _ y,  y_ y,y, s,v,a,j f…y† t th yh ,h T _ _ Y,  Y Y,Y, S,V,A,J F…Y† P w1 ,w2 ,w3 X U m H m, n

displacement, velocity, acceleration, and jerk of the cam displacement, velocity, acceleration, and jerk of the follower displacement function of the follower actual time total rise time of the cam total displacement of cam and follower during rise time dimensionless time dimensionless displacement, velocity, acceleration, and jerk of the cam dimensionless displacement, velocity, acceleration, and jerk of the follower dimensionless displacement function of the follower optimal performance index weight factors state vector control vector co-state vector Hamilton function constants

compared with that in the constant-input-speed case. Hence, the designer will have a wider design space to improve properties of mechanisms. Combined mechanisms ful®ll the designer's will to some extent. In a combined mechanism, the input of the `driven mechanism' is the output of the `driving mechanism' whose speed is usually non-constant. However, the form of input speed variation is very limited. The use of combined mechanisms brought about the idea of controlling the input speed of a mechanism. In past decades, the servomotor and its control system have developed rapidly. Using a microcomputer-controlled servomotor as an actuator of a mechanism in place of an ordinary motor is becoming more frequent in high-precision automatic machines. The technology a€ords huge freedom to control variations of input speeds of mechanisms. The design of such servo-integrated mechanism systems presents a new problem for researchers. The idea of varying the input speed of cam mechanisms may be traced to the cams designed by Rothbart [1] whose input is the output of a Withworth quick-return mechanism. Yan [2,3] has proved both theoretically and experimentally that the motion characteristics of a cam follower system, from a kinematics point of view, can be improved by using an input speed function realized by a microcomputer-controlled dc servomotor. Chew [4] demonstrated that controlling the input command to a dc servomotor could minimize the residual vibrations in a high-speed electromechanical bonding machine. The idea of applying microcomputer-controlled servo systems as actuators to improve the motion characteristics of mechanisms may be called `Active Control of Mechanisms.' The

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`Active Control' here means that the input of a mechanism could be actively controlled according to the designer's will. The objective of the technology is to develop superior machines, which take advantage of both the ¯exibility of servo systems and the rigidity of mechanisms. This paper deals with the problem of `Motion Control' whose objective is to improve the kinematic characteristics of cam-follower systems. Obviously, `Motion Control' is the foundation of `Active Control.' Another paper to be published will deal with the problem of `Vibration Control' whose objective is to minimize the residual vibration of the follower.

2. Dimensionless parameters For a rigid cam-follower system, the input motion is displacement y of the cam and the _ acceleration y output motion is displacement s of the follower. The displacement y, velocity y, _ and jerk y of the cam are expressed as y ˆ y…t†

…1a†

dy y_ ˆ dt

…1b†

d2 y y ˆ 2 dt

…1c†

d3 y y_ ˆ 3 dt

…1d†

The displacement s, velocity v, acceleration a and jerk j of the follower are expressed as s ˆ s…t† ˆ f…y…t††

…2a†

v ˆ v…t† ˆ

ds ˆ f 0 …y †y_ dt

…2b†

a ˆ a…t† ˆ

d2 s 2 ˆ f}…y †y_ ‡ f 0 …y †y dt2

…2c†

j ˆ j…t† ˆ

3 d3 s ˆ f 000 …y †y_ ‡ 3f 00 …y †y_ y ‡ f 0 …y † y_ 3 dt

…2d†

where f 0 …y † ˆ

df dy

…3a†

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f 0 …y † ˆ

d2 f dy2

…3b†

f}0 …y † ˆ

d3 f dy3

…3c†

The dimensionless parameters are de®ned as Tˆ

t th

…4†

Yˆ

y yh

…5a†

_ ˆ dy Y dT

…5b†

2  ˆdY Y dT 2

…5c†

3 _ ˆ d Y Y dT 3

…5d†

S ˆ S…T † ˆ F…Y…T †† ˆ

s h

…6a†

V ˆ V…T † ˆ

dS _ ˆ F 0 …Y †Y dT

…6b†

A ˆ A…T † ˆ

d2 S _ 2 ‡ F 0 …Y †Y  ˆ F 00 …Y †Y dT 2

…6c†

J ˆ J…T † ˆ

d3 S _ _ 3 ‡ 3F 00 …Y †Y _Y  ‡ F 0 …Y †Y ˆ F 000 …Y †Y dT 3

…6d†

where F0 ˆ

dF dY

…7a†

F0 ˆ

d2 F dY2

…7b†

Y. Yao et al. / Mechanism and Machine Theory 35 (2000) 593±607

F 000 ˆ

d3 F dY3

597

…7c†

The relationship between the actual and dimensionless parameters can be found as y ˆ yh  Y

…8a†

yh _ y_ ˆ Y th

…8b†

yh  y ˆ 2 Y th

…8c†

yh _ Y t3h

…8d†

sˆhS

…9a†

vˆ

h V th

…9b†

aˆ

h A t2h

…9c†

jˆ

h J t3h

…9d†

yˆ

3. Optimal control strategy Yan [2] has presented a method for ®nding an appropriate cam input speed trajectory by application of an optimum program. Optimum control theory is used to deal with the problem in this study. 3.1. Formulation of the optimal criterion The motion characteristics of the follower of a cam mechanism should ful®ll various requirements presented by practical processes. This is the essential consideration when selecting an optimal performance index. For example, Minimization of Vmax :

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Pˆ

…1 0

V…T †

2

dT;

Minimization of Amax : …1 2 A…T † dT, Pˆ 0

whereVmax and Amax are the peak values of velocity and acceleration of the follower. Meanwhile, considering the limited tracing capability of the servo system, the following performance index may be used to control the variation of cam speed: …1 2 _ ÿ 1 dT Y Pˆ 0

Hence, the performance index can be de®ned as …1 …1 …1 2 2 2 _ ÿ 1 dT V…T † dT ‡ w2 A…T † dT ‡ w3 P ˆ w1 Y 0

0

0

…10†

where the weighting factors w1 ,w2 and w3 can be adjusted according to di€erent objectives and limitations. 3.2. Boundary conditions The dimensionless displacement of the cam should satisfy the following boundary conditions: Y…0 † ˆ 0,

Y…1 † ˆ 1

…11†

3.3. Subjects Without loss of generality, the velocity of the cam is assumed to be kept positive that: _ …T † > 0, 0RTR1 Y

…12†

The subject can be ful®lled by adjusting the weighting factors in the performance index shown in Eq. (11).

4. Examples 4.1. Minimization of Vmax The performance index is de®ned as

Y. Yao et al. / Mechanism and Machine Theory 35 (2000) 593±607

P ˆ w1

599

…1 …1 2 2 _ ÿ 1 dT Y V…T † dT ‡ w2 0

…13†

0

It is rewritten as …1 …1 2 2 F 0 …X †U dT ‡ w2 ‰U ÿ 1 Š dT P ˆ w1 0

…14†

0

where the state vector is XˆY

…15†

and the control vector is _ UˆY

…16†

The state equation is Ç ˆU X

…17†

The boundary conditions are X…0 † ˆ 0,

X…1 † ˆ 1

…18†

The Hamilton's method is used to solve this problem [5]. The Hamilton function is de®ned as  2 2 mU …19† H ˆ w1 F 0 …X †U ‡w2 ‰U ÿ 1 Š ‡m The control equation is @H ˆ0 @U

…20†

Table 1 Peak values of motion characteristics of the followerwith constant-cam and variable-cam speed (Example 1) Peak values

Vmax Amax Jmax

Constant -cam speed

1.88 25:77 +60.0, ÿ30.0

Variable-cam speed w1

w2

w1

w2

1.0

1.0

1.0

10.0

1.32 +8.79, ÿ8.85 +203.10, ÿ74.11

1.81 +6.24, ÿ6.26 +48.40, ÿ25.62

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The co-state equation is mÇ ˆ ÿ

@H @X

…21†

Solving Eq. (19), yields Uˆ

2w2 ÿ mÇ  2 2w1 F 0 …X † ‡2w2

…22†

Eq. (21) yields mÇ ˆ ÿ2w1 F 0 …X †F 00 …X †U2

…23†

Substituting the control vector U from Eq. (22) to Eqs. (17) and (23), a two-point boundaryvalue problem is hence obtained with the boundary conditions expressed in Eq. (18). An improved shoot-target method [6] is adopted to solve the problem. Example 1 is given to illustrate the procedure. The cam pro®le is the standard 3±4±5 polynomial. Two groups of weight factors are adopted that, w1 ˆ 1:0,w2 ˆ 1:0 and w1 ˆ 1:0,w2 ˆ 10:0: The peak values of motion characteristics of the follower are shown in Table 1. Figs. 1 and 2 show the corresponding motion curves of the cam and the follower.

Fig. 1. Motion curves of the cam (Example 1): (a) displacement; (b) velocity; (c) acceleration; (d) jerk (Q: w1 ˆ 1:0;w2 ˆ 1:0, W: w1 ˆ 1:0;w2 ˆ 10:0).

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Fig. 2. Motion curves of the follower (Example 1): (a) displacement; (b) velocity; (c) acceleration; (d) jerk (R: constant-cam speed, Q: w1 ˆ 1:0;w2 ˆ 1:0, W:w1 ˆ 1:0;w2 ˆ 10:0).

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It can be seen that the peak value of velocity of the follower can be obviously decreased by controlling the cam speed. At the same time, we note that greater variations of cam speed produce greater reductions in the peak velocity of the follower. The application of optimal control theory has been shown to be capable of obtaining a satisfactory trade-o€ between the characteristics of the output motion (i.e. the follower motion) and the input motion (i.e. the cam motion) by adjusting the weighting factors in the performance index. A trial and error procedure is required for ®nding the appropriate weighting factors. 4.2. Minimization of Amax The performance index is de®ned as …1 …1 2 2 _ ÿ 1 dT A…T † ÿ A …T † dT ‡ w2 P ˆ w1 Y 0

…24†

0

 † is a function which can be adjusted in order to gain a desired reduction of the where A…T objective. The performance index is rewritten as …1 …1 2 00 2 0  …25† F …X1 †X 2 ‡ F …X1 †U ÿ A…T † dT ‡ w2 ‰X2 ÿ 1Š2 dT P ˆ w1 0

0

where the state vector is _ XT ˆ ‰X1 ,X2 Š ˆ bY,Yc

…26†

and the control vector is  UˆY

…27†

The state equation are written in the state-space form: Ç ˆ AX ‡ bU X where



0 Aˆ 0

 1 , b

…28†   0 bˆ 1

The boundary conditions are: X1 …0 † ˆ 0,

X1 … 1 † ˆ 1

The Hamilton's method [5] is used to solve the problem. The Hamilton function is  2 H ˆ w1 F 00 …X1 †X 22 ‡ F 0 …X1 †U ÿ A …T † ‡w2 ‰X2 ÿ 1Š2 ‡mT ‰X2 ,UŠT where the co-state vector is

…29†

…30†

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m T ˆ ‰m1 ,m2 Š

603

…31†

The control equation is @H ˆ0 @U

…32†

The co-state equation is mÇ ˆ ÿ

@H @X

…33†

The transversality conditions are m2 …0 † ˆ 0,

m2 …1 † ˆ 0

…34†

Solving Eq. (32), yields Uˆÿ

2w1 F 0 …X †bF 00 …X1 †X 22 ÿ A …T †c ‡ m2 2  2w1 F 0 …X1 † ‡2w2

…35†

Eq. (33) yields  mÇ ˆ

m_ 1 m_ 2



  3 ÿ2w1 F 000 …X1 †X 22 ‡ F 0 …X1 †U ÿ A …T † F 000 …X1 †X 22 ‡ F 00 …X1 †U 5 ˆ4    ÿ2w1 F 000 …X1 †X 22 ‡ F 0 …X1 †U ÿ A …T † 2F 000 …X1 †X2 ÿ m1 2

…36†

Substituting the control vector U from Eq. (35) to Eqs. (28) and (36), a two-point boundaryvalue problem is thus obtained with the boundary conditions expressed in Eqs. (29) and (34).  † is An improved shoot-target method [6] is adopted to solve the problem.The function A…T de®ned as shown in Fig. 3. The mathematical expression is

 †: Fig. 3. Diagram ofA…T

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Table 2 Peak values of motion characteristics of the followerwith constant-cam and variable-cam speed (Example 2) Peak values

Vmax Amax Jmax

Constant-cam speed

1.88 25:77 +60.0, ÿ30.0

n T m n n n A …T † ˆ f ÿ T ‡ m 2m ÿn n n Tÿ m m

Variable-cam speed w1

w2

w1

w2

1.0

1.0

1.0

0.1

1.83 +5.59, ÿ5.63 +70.03, ÿ33.07

1.95 +4.91, ÿ4.92 +77.96, ÿ53.18

0RT < m mRT < 0:5 ÿ m 0:5 ÿ mRT < 0:5 ‡ m

…37†

0:5 ‡ mRT < 1:0 ÿ m 1:0 ÿ mRT < 1:0

where m, n are variables that can be adjusted. Other functions may be selected according to the di€erent performance index as well as the corresponding optimal results. Example 2 is given to illustrate the procedure. The cam pro®le is the standard 3±4±5  † is de®ned where m = 1/12, n = 4.3. polynomial motion. The function A…T Two groups of weighting factors are adopted that, w1 ˆ 1:0,w2 ˆ 1:0 and w1 ˆ 1:0,w2 ˆ 0:1: The peak values of motion characteristics of the follower are shown in Table 2. Figs. 4 and 5 show the corresponding motion curves of the cam and the follower. It can be seen that the peak value of acceleration of the follower can be obviously decreased by controlling the cam speed. At the same time, we note that greater variations of cam speed produce greater reductions in the peak acceleration of the follower. The application of optimal control theory has been shown to be capable of obtaining a satisfactory trade-o€ between the characteristics of the output motion (i.e. the follower motion) and the input motion (i.e. the cam motion) by adjusting the weighting factors in the performance index. A trial and error procedure is required for ®nding the appropriate weighting factors.

5. Conclusions The concept of `Active Control of Cam Mechanisms' is presented in this paper. `Motion Control' is studied as the foundation of `Active Control'. It is shown that, from a kinematics point of view, the motion characteristics of the follower can be improved by controlling the cam input speed. Optimal control theory has been shown to be an e€ective tool which can obtain a satisfactory trade-o€ between the improvement of the output motion of the follower and the limitation of capability of the servo control systems.

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Fig. 4. Motion curves of the cam (Example 2): (a) displacement; (b) velocity; (c) acceleration; (d) jerk (Q: w1 ˆ 1:0;w2 ˆ 1:0, W: w1 ˆ 1:0,w2 ˆ 0:1).

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Fig. 5. Motion curves of the follower (Example 2): (a) displacement; (b) velocity; (c) acceleration; (d) jerk (R: constant-cam speed, Q: w1 ˆ 1:0;w2 ˆ 1:0, W: w1 ˆ 1:0;w2 ˆ 0:1).

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The technology of `Active Control of Cam Mechanisms' has great potentiality in highprecision automatic machines. The idea presented in this paper could also be extended to linkages or other mechanisms. References [1] H.A. Rothbart, Cams: Design, Dynamics, and Accuracy, Wiley, New York, 1956. [2] Hong-Sen Yan, Mi-Ching Tsai, Meng-Hui Hsu, A variable-speed method for improving motion characteristics of cam-follower systems, Transactions of the ASME 118 (1996). [3] H.S. Yan, M.C. Tsai, M.H. Hsu, An experimental study of the e€ects of cam speeds on cam-follower systems, Mech. Mach. Theory 31 (4) (1996) 397±412. [4] M. Chew, M. Plan, Application of learning control theory to mechanisms. Part I: Inverse kinematics and parametric error compensation; Part II: Reduction of residual vibrations in high-speed electro-mechanical bonding machines, in: Proc. of the 23rd ASME Mechanisms Conference, Minneapolis, USA, 1994. [5] Pei-Yu Liu, Applied Optimal Control (in Chinese), The Press of Dalian University of Technology, Dalian, 1990. [6] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, The Art of Scienti®c Computing, 2nd edition, Cambridge University Press, Cambridge, 1992.