2014 14th International Conference on Control, Automation and Systems (ICCAS 2014) Oct. 22-25, 2014 in KINTEX, Gyeonggi-do, Korea

High-Speed Cam Design for Quick-Action Linkage System 1

Jing Wu , Rui-Jun Yan1, Chao Yuan1, Yong Seok Lee2, Kyoo-Sik Shin3, and Chang-Soo Han3* 1

Department of Mechatronics Engineering, Hanyang University, Republic of Korea ([email protected]; [email protected]; [email protected]) 2 Department of Mechanical Engineering, Hanyang University, Republic of Korea ([email protected]) 3 Department of Robot Engineering, Hanyang University, Republic of Korea ([email protected]; [email protected]* ) * Corresponding author

Abstract: As a specially designed mechanism, cam is usually be used in many high-speed structures and mechanisms. Diverse geometric cam counters support for diverse motion applications, including the displacement, the velocity and the acceleration. This paper proposes a design of a high-speed cam which can be applied in a quick-action-based four bar linkage system. This variable high-speed cam is design by using the graphical combination of 5th and 7th polynomial cam curves. In addition, the kinematic model of a linkage system based on our proposed cam is constructed. Finally, the simulation results of the cam counter, velocity and acceleration of the designed cam are shown to validate the appropriate application of a quick-action linkage system. Keywords: Polynomial high-speed cam, quick action system, linkage system, kinematic model. high speed motions and robots such as electric systems or hydraulic systems. Nonetheless, compare to the cam-based linkage mechanical system, energy-intensive and complicated control would become the big problems of electric system. For hydraulic system, complicated structural, bulky size and high cost would be some non-negligible significant predicament which would weaken the performance in quick action movement [11]. For a linkage mechanism, the motion of high-speed could consist of three patterns, 1) Fast-forward (FF); 2) Fast-reverse (FR); 3) Fast-forward-reverse (FFR) [11]. From this classification, the definition of quick-action motion could be settled affiliate to the high speed motivation but just emphasis on short-time performance. Therefore, these two structures are similar in the analysis of kinematic and dynamic models. In this paper, the contribution of the designed cam is that the cam counter offers a non-constant velocity actuates the four-bar linkage to produce a quick-action motivation. In the following sections, we first introduced the underling calculation of the cam-follower system, outline the designed cam profile, present its velocity and acceleration curves. Then the mechanism design of the cam-based linkage system will be given in section2. Section 3 shows the kinematic model and the calculated equations. Finally, section 4 concludes the paper and offers some perspectives.

1. INTRODUCTION Conventional linkages are usually composed of closed-loop single degree of freedom kinematic chains. They can operate at high speed with heavy loads and high output precisions, but they cannot provide any versatility such as quick action motion that is often required in many automated instruments and industrial applications[1]. In recent years, linkage systems are extensively applied in high speed structures and parallel manipulators. The linkage structures especially four-bar linkage systems are frequently used in high speed parallel robot, such as the Japanese Quattro parallel robot or high speed wire-driven robots [2, 3]. In this paper, a cam-based four-bar linkage system is applied to allow for a very high speed and adjustment the motivation in quick-action movement. Cam-follower mechanisms are widely used in many types of modern machines not only because of arbitrarily complicated motions have been achieved by the brief simple-compact cam-based construction [4], but also based on their excellent properties for operation speed, motion accuracy, structural rigidity, and production cost [5, 6]. However, in many cased, machines require cam to provide a quick follower movement or to work at high-operating speed, which always required the special designed cam. For instance, a high speed cam has been calculated for a down-swing paper transferring mechanism in a printing press, a quick-action cam has been diagrammatized to control the switching action for a vacuum circuit breaker, and a floating cam has been improved to satisfy the motion of a jumping robot [7-9] . Traditionally, several smooth profiles, such as trigonometric functions, s-curves, and polynomial profiles are used to produce high speed quick-action cams [6, 10]. However, as the requirements in high-speed conditions, linkage structure and cam-follower system are not the alternative mechanism, multitudinous of techniques are proposed in many aspect to achieve the

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2. HIGH-SPEED CAM SYSTEM DESIGN In the classification of cam follower motion, there are two basic curves: dwell-rise-dwell (DRD), rise-return-rise (RRR) [10]. For the application of our system, we compound the follower motion curve as rise-dwell-return-dwell (RDRD). In Fig1, picture (a) shows the displacement curve of cam follower motion and the boundary conditions that will apply in the follow polynomial equations.

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Table 2 Boundary conditions in RDRD motion.

Displacement curve 60 50

Displacement

40

30

10

0

0

50

100

150

200 Rotation

250

300

350

Dwell Angle

0° ∼ 90°

90° ∼ 180°

Return Angle

Dwell Angle

(far-off)

(close back)

180° ∼ 270°

270° ∼ 360°

Although the characteristics of 5th, 6th and 7th polynomial are similar, there still are some different in their velocity and acceleration curves. To adapt the application of quick action preferable, we use 7th polynomial curve in the rise phase and 5th polynomial curve in the return phase. The 5th polynomial equation is given as:

20

-10

Rise Angle

400

s(θ ) = a + bθ + cθ 2 + dθ 3 + eθ 4 + f θ 5

Fig. 1. RDRD displacement curve of cam follower.

v (θ ) = b + 2cθ + 3dθ 2 + 4eθ 3 + 5 f θ 4 a (θ ) = 2c + 6dθ + 12eθ + 20 f θ 2

2.1 Cam –follower profile design. Referring to the require application in speed and load, there are numerous cam follower motion curves such as: constant velocity curve, constant acceleration curve, polynomial curve, cosine acceleration curve, sine acceleration curve and modified sine acceleration curve. The comparison of these curves is given in Table 1:

(1)

3

The 7th polynomial equation is given as: s(θ ) = a + bθ + cθ 2 + dθ 3 + eθ 4 + f θ 5 + gθ 6 + mθ 7 v (θ ) = b + 2cθ + 3dθ 2 + 4eθ 3 + 5 f θ 4 + 6 gθ 5 + 7mθ 6

(2)

a (θ ) = 2c + 6dθ + 12eθ 2 + 20 f θ 3 + 30 gθ 4 + 42mθ 5 j (θ ) = 6d + 24eθ 2 + 60 f θ 3 + 120 gθ 4 + 210mθ 5

Table 1 Comparison of cam follower motion curve

After these calculations, the compounded velocity curve and the acceleration curve are shown in Figure 2.

Motion curve

Impact

Vmax

amax

Speed

Load

Constant velocity

Rigidness

1.0

∞

Low

Light

1.5

Constant acceleration

Flexibility

2.0

4.0

Medium

Light

1

5th-6th -7th polynomial

No

1.88

5.77

High

Medium

Cosine acceleration

Flexibility

1.57

4.93

Medium

Medium

Sine acceleration

No

2.0

6.28

High

Light

Modified Sine acceleration

No

1.76

5.53

High

Heavy

Velocity curve

Velocity

0.5

0

-0.5

-1

-1.5 0

50

100

150

200 Rotation

250

300

350

400

300

350

400

Acceleration curve

Depend on the comparison that shows in Table1, the characteristics of polynomial curve could be an ideal curve for its application in high-speed and medium-load conditions. Meanwhile, the characteristic of no impact is also fit for our quick-action system for impact would produce an infinite jerk phenomenon and should be avoid especially in high-speed mechanisms.

0.05 0.04 0.03

Acceleration

0.02

2.2 Polynomial cam profiles design. A polynomial cam profile allows for versatility and customization through the selection of boundary conditions based on desired motion characteristics. Based on the RDRD motion characteristic, cam follower motion displacement curve is separate into four intervals among one rotational periodicity of cam contour and that is showing in Table 2 as the boundary condition of polynomial equations.

0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 0

50

100

150

200 Rotation

250

Fig. 2. Velocity curve and acceleration curve of the compounded curve of cam follower.

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Simultaneously, cam counter curve can be established base on the calculated profile curve of cam follower by combining the previous equations with cam counter calculations in Equation (3). The final profile curve is shown in Figure 3. X = E ∗ cos( J ) + ( R 2 − E 2 + S ) ∗ sin( J )

3.2 Kinematic model of cam-based quick-action linkage system.

As shown in Figure 4, the mechanism can be analyzed as a closed-chain linkage. This system is separated into two loops at point D, which are loop ABCD and loop GD. In these two loops, both prismatic joint and relative joint exist in the link CD. The nine coordinates are built for the kinematics analysis, shown in Figure 5.

(3)

Y = ( R 2 − E 2 + S ) ∗ cos( J ) − E ∗ sin( J ) Cam profile curve 40

The meaning of these parameters is explained as follow:

20

ϕC : Angle from Z0 to ZC along X0 .

0

LC : Distanse from Z0 to ZC along X0 .

-20 y

DC : Distanse from X0 to XC along Z0 .

-40

φC : Angle from X0 to XC along Z0 .

-60

CC : Distanse from cam center to roller center.

θC : Angle of cam roller. θr : Angle between cam roller and link1. θ5 : Angle between link2 and link3.

-80

-100 -60

-40

-20

0

20 x

40

60

80

100

d : lengh of CD.(d < l3 )

Fig. 3. Profile curve of high-speed cam.

ϕ g : Angle from Z0 to Z g along X0 . Lg : Distanse from Z0 to Z g along X0 .

3. DESIGH OF CAM-BASED QUICK ACTION MANIOULATOR

Dg : Distanse from X0 to X g along Z0 .

3.1 Build the cam-based quick-action linkage system

φg : Angle from X0 to X g along Z0 .

Normally, four-bar linkage need at least more than one actuator to drive the system. In our system, the system is actuated by the contact force between cam and its follower which link is belong to one of the four-bar linkage. The mechanism structure diagram is given in Figure 4.

θ7 : Angle between link3 and link4. θ8 : Rotation angle of link4. Finally, the coordinate calculations of the end points of this quick-action linkage are done by using DH-Table , shown in equation (4) and (5): 0 8

D

X = LC + (CC + l1 ) cφC + (l2 + d ) c c + r −5−φC + (l3 - d ) sc + r −5−φC

l3

C

= Lg + l7 cφg + (l3 - d ) cφG +7

d 0 8

Y = − DC sϕC + (CC + l1 ) cϕC sφC +

l2

(l2 + d ) cϕC s c + r −5−φC + (l3 - d ) cϕC c c + r −5−φC

B

= − DC sϕ g − (l3 - d ) cϕ g sφG +7 n

0 8

l1

A

(l2 + d ) sϕC s c + r −5−φC + (l3 - d )sϕC c c + r −5−φC

E

B

Z = DC cϕC + (CC + l1 ) sϕC sφC + = Dg cϕ g + l7 sϕ g sφg − (l3 - d )sϕ g sφG +7

α

ϕ1 G

ωc

Fig. 4. High-speed cam-based quick-action linkage.

496

(4)

2014 14th International Conference on Control, Automation and Systems (ICCAS 2014) Oct. 22-25, 2014 in KINTEX, Gyeonggi-do, Korea xe

x6

ye z6

d

xe

y7

y5

ye

l3

x7

x5

C x4 l2

B

y4

n

l1

x3

y3

yg

x2

y2

G

B A

α

ϕ1 xc

G

Z0

xg

yc X0

ωc

Y0

Fig. 5. Coordinate diagram of the separated Two loops. Here,

Therefore, the velocity in the end point is calculated in equation 8 as:

sin(θ c + θ r − θ 5 − φC ) = s c + r − 5−φC

−1

u = ⎡⎣G1 ⎤⎦ ⎡⎣G2 ⎤⎦ φa = ⎡⎣ r Gφu ⎤⎦ r φa ,

cos(θ c + θ r − θ5 − φC ) = c c + r − 5−φC cos ϕ C = cϕC ;sin ϕ C = sϕC cos φC = cφC ;sin φC = sφC

Depends on the two independent inputs, the redundancy cam-cased linkage system could be adapted to many applications.

(5)

sin(φG + θ 7 ) = sφG + 7 cos(φG + θ 7 ) = cφG + 7

4. CONCLUSION

cos ϕ G = cϕG ;sin ϕ G = sϕG

In this paper, a special high-speed cam was designed for a quick-action linkage system. The cam counter is calculated based on a synthetic 5th and 7th polynomial curve. The motion of linkage endpoint was given and the kinematic model was also shows in this paper. Based on the characteristic of this mechanism, it could also form the propulsion system of a jumping robot or be used for self-deployment of sensors. This proposed mechanism can be combined with a manipulator having many more degrees of freedom, which can be developed as a biomimetic arm or used as a special industry manipulator with many applications.

cos φG = cφG ;sin φG = sφG To control the system more preferably, there are two independent input torques located in the high-speed cam and point G. Therefore, after the calculation, the forward kinematic equation connecting input vector φa with output vector φ p is built in equation (6) as follows:

⎡⎣G1 ⎤⎦ φp = ⎡⎣G 2 ⎤⎦ φa ,

(6)

Here,

φ p = ⎡⎣θ r θ5 d θ7 ⎤⎦ φa = ⎡⎣Cc θC θ8 ⎤⎦

(8)

ACKNOWLEDGEMENT

T

This research was funded by Building-façade Maintenance Robot Research Center, supported by Korea Institute of Construction and Transportation Technology Evaluation and Planning under the Ministry of Land, Transport, and Maritime Affairs (MLTM).

(7)

In equation (6), G1 and G 2 are the jacobia matrix of loop ABCD and loop GD.

497

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