Mech. Mach. Theory Vol. 23, No. 4, pp. 287-293, 1988 0094-114X/88 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1988 Pergamon Press pie

T H E S T A T I O N A R Y C O N F I G U R A T I O N S O F P L A N A R SIX-BAR K I N E M A T I C C H A I N S

H O N G - S E N Y A N and L O N G - I O N G W U Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.

(Received in revised form 16 August 1987)

Abstract--Based on the concepts of instant centers and kinematic inversion, a method is proposed to determine the stationary configurations of planar linkage mechanisms. All stationary configurations of six-bar kinematic chains are determined and listed. Quadruple stationary configurations are found in the six-bar chains, and one of them is applied as a nose landing gear.

1. INTRODUCTION

During the course of motion of a mechanism, some of the pair-freedoms may be temporarily inactive, and these configurations are called stationary configurations [1, 2]. The determination of these con- figurations will provide a foundation for an under- standing of the range of motion of a mechanism. AI~ stationary configurations, the mechanism may be locked there provided that an actuator, which is located along or about the inactive pair, is adopted to drive the mechanism. On the other hand, the absence of such configurations indicates that continuous motion with respect to the relevant pair- freedom is possible. Thus, it is convenient to use such a pair to drive the mechanism.

Several methods are available for determining stationary configurations. Some of them are listed in Refs. [1-5]. These methods, originally intended for spatial linkages, are based on the handling of mathematical equations. Our purpose is to provide a simple graphical approach, based on the concepts of instant centers and kinematic inversion, to find out the stationary configurations of planar linkage mechanisms.

2. STATIONARY CONFIGURATIONS AND INSTANT CENTERS

Figure 1 shows two stationary configurations of a planar four-bar linkage mechanism, in which joint D in Fig. 1 (a) and joint C in Fig. l(b) are temporarily inactive. In Fig. l(a), links 1 and 4 are joined together at instant center I14. At this configuration, links 1 and 4 have temporarily equivalent angular velocity, and do not have relative motion with respect to each other. Similarly, links 3 and 4 in Fig. l(b) are joined together at instant center 134, and also do not have relative motion with respect to each other. In other words, two adjacent links joined together by an in- active revolute joint can be regarded as a sole link [6]. In fact, it can be shown that if and only if the inactive

287

joint and its adjacent links constitute a (temporary) rigid chain, these two links should have identical instant centers with respect to all the remaining links in the mechanism. Here, a rigid chain is a kinematic chain with zero degrees of freedom [7]. Therefore, we conclude that the determination of the stationary configurations of a mechanism is equivalent to find out states such that two adjacent links of a specified joint have identical instant centers with respect to all the remaining links.

It is obvious that both configurations shown in Figs l(a) and (b) can be derived from their corresponding kinematic chain, as shown in Fig. 2.

c 13~

1 (a) 1

C~I3~ 3 /~

(b) Fig. I. Stationary configurations of a four-bar linkage

mechanism.

lij

Iil Ijk Ijt - I lkt k - l i k

Fig. 2. Stationary configuration of the four-bar chain.

288 HONG-SEN YAN and LONG-lONG WU

Furthermore, it is well known that the relative motion between any two links in a linkage mechanism is not affected by the process of inversion. Therefore, it would be more systematic to identify inactive joints from kinematic chains than from their corresponding mechanisms separately.

Based on the above reasoning, we propose the following steps for the determination and identi- fication of stationary configuration of a specified planar kinematic chain (or mechanism):

1. Specify the joint which is to be inactive, and let its adjacent links be links i and j.

2. Find the configuration of the kinematic chain (or mechanism) such that those incident joints corresponding to these two adjacent links have coincident instant centers with respect to a third link, say link k, by applying Kennedy's theorem. In other words, instant center Iik should coincide with Ijk in the determined configuration.

3. For the configuration obtained in step 2, check if the two adjacent links have identical instant centers with respect to all the remaining links in the kinematic chain (or mechanism).

Here, we call a kinematic chain (or mechanism), which has single or multiple loops and possesses one inactive joint, a simple stationary configuration. A

C~ah3

I14 I L ~ ~

(a) ~,136

A E

114 I~ 116 (b)

Fig. 3. The six-bar kinematic chains.

I~4 (a) 146(116)

/

1~4 (b) I46, I16

Fig. 4. A simple stationary configuration of the Watt six-bar chain.

kinematic chain (or mechanism) possesses more than one inactive joint is called a multiple stationary con- figuration. Multiple stationary configurations can be divided into double, triple, quadruple . . . . stationary configurations corresponding to the numbers of their inactive joints. In the following, we synthesize the stationary configurations of planar six-bar kinematic chains.

3. STATIONARY CONFIGURATIONS OF SIX-BAR CHAINS

Two planar six-bar kinematic chains are available, that is, the Watt chain and the Stephenson chain as shown in Figs 3(a) and (b), respectively. Owing to their topological structures, joints A, C, E and G, as well as joints B and F of the Watt chain are symmetric, respectively; joints A, B, C and D, as well as joints E and G of the Stephenson chain are symmetric, respectively. Furthermore, all instant centers of these two chains are located in their corresponding figures.

Example 1

Joint A of the Watt chain shown in Fig. 3(a) is specified to be inactive.

1. Since joint A is inactive and links I and 4 are its adjacent links, links 1, 4 and joint A should constitute a rigid chain. Therefore, I24 should coincide with Ii2, Ij3 should coincide with I34, and I16 should coicide with I46.

2. According to Kennedy's theorem, instant center I23 must be on the line of I12 I13, and also on the line of I24 I34. Therefore, I23 must be on the line of Ii2 I34, as shown in Fig. 4(a).

Stationary configurations of planar six-bar kinematic chains 289

A ~ 5 6 , 1 3 6 Fig. 5. A double stationary configuration of the Watt

six-bar chain.

3. Locate all the instant centers of links 1 and 4 with respect to all the remaining links, i.e. Ijs, I16, I45 and I~, as shown in Fig. 4(b). Since It5 coincides with 145, and Il6 coincides with I46, Fig. 4(a) is a simple stationary configuration.

Example 2

Joints A and E of the Watt chain shown in Fig. 3(a) are specified to be inactive.

1. Since joints A and E are inactive and links 1 and 4 as well as links 3 and 5 are their corresponding adjacent links, links 1, 4 and joint A, as well as links 3, 5, and joint E should constitute two rigid chains, respectively.

2. Applying the results of Example 1 and note that joints A and E are symmetric, it is reasonable to infer that the configuration with link 2 in-line with 123 I34 and link 6 in-line with 134 146 simul- taneously will cause both joints A and E to be inactive, Fig. 5.

3. Locate all the instant centers of this configuration. Since 1,2 coincides with I24 , It3 coincides with It,, I15 coincides with I45, and 1,6 coincides with I46, joint A is inactive. Since I13 coincides with I15, I:3 coincides with I:5, 134 coincides with Ls, and I36 coincides with I56, joint E is also inactive. Therefore, Fig. 5. is a double stationary configuration.

Example 3

Joints A, B, C and D of the Stephenson chain shown in Fig. 3(b) are specified to be inactive.

1. Since joints A, B, C and D are inactive, and links 1, 2, 3, and 4 are their adjacent links; links 1, 2, 3 and 4 as well as joints A, B, C and D should constitute a rigid chain. Therefore, 115, I35 and 145 should coincide with 125; and I16, 1:6 and Is6 should coincide with 146.

2. According to Kennedy's theorem, instant center I25,156 and I:6 should be colinear. Therefore, link 5 should be in-line with link 6, Fig. 6(a).

3. Locate all the instant centers of links 1, 2, 3 and 4 with respect to all the remaining links. Since 115,125 , I35 and 145; as well as I16, I26 , I36 and 146 are coincident, respectively, Fig. 6(a) is a quadruple stationary configuration.

II~

s, h5

:! ~Is6

" ~ 6 , h6 (a)

112 12s, I~5

(b)

Fig. 6. A quadruple stationary configuration of the Stephenson six-bar chain.

Furthermore, this quadruple stationery configur- ation will be a nested one provided that link 3 is also in-line with I,, 134, as shown in Fig. 6(b). This additional constraint causes joint A to be inactive, due to the fact that links I and 2 have identical instant centers with respect to links 3, 4, 5 and 6. Therefore, links l, 2 and joint A constitute a sub-rigid chain of the former larger one.

Since either Watt chain or Stephenson chain has 7 joints, the number of all the possible combinations of inactive joint(s) for each chain is |27. After eliminating those cases, which are symmetric, trivial, or impossible to exist, the following results are obtained. For the Watt six-bar chain, we find three simple stationary configurations as shown in Figs 7,

123 (a)

3t,

N (b) (c) Fig. 7. Simple stationary configurations of Watt six-bar chain.

M,M.T. 23/~C

290 HONG-SEN YAN and LONG-IONG WU

123

(a)

A,

(b) (c)

(d) (el

Fig. 8. Double stationary configurations of Watt six-bar chain.

five double stationary configurations as shown in Figs 8, zero triples stationary configurations, and three quadruple stationary configurations as shown in Figs 9. Among these quadruple ones, those two shown in Figs 9(b) and (c) are nested cases because they also include simple stationary configurations. In

I14 Ca)

12~ ~ ~ 6 (b) (c)

Fig. 10. Simple stationary configurations of the Stephenson six-bar chain.

addition, for the Stephenson six-bar chain, we find three simple stationary configurations as shown in Figs 10, five double stationary configurations as shown in Figs 11, one triple stationary configuration as shown in Figs 12, and two quadruple stationary

(a)

B2C•5 (a)

(b) (c) Fig. 9. Quadruple stationary configurations of Watt six-bar

chain.

(b)

3 5

11

(c)

Cd) Ce) Fig. 11. Double stationary configurations of the Stephenson

six-bar chain.

Stationary configurations of planar six-bar kinematic chains 291

14 5 bF

Fig. 12. Triple stationary configuration of the Stephenson six-bar chain.

1 B

Fig.

h~ (a) (b) 13. Quadruple stationary configurations of

Stephenson six-bar chain. the

configurations as shown in Figs 13 in which the one shown in Fig. 13(b) is a nested one.

Tables 1 and 2 list the constraints of s tat ionary configurations for Wat t and Stephenson fix-bar kinematic chains.

4. APPLICATION

When landing wheels of aircraft are either fully retracted or extended, they must be positively locked in those respective positions. Thus, aircraft landing gear retraction systems usually involve four-bar linkages which are locked in dead-center posit ions to isolate forces that might tend to fold or unfold the linkages during landing or flight [8, 9]. In fact,

it will be more reliable provided that a nest-type stat ionary configuration is adopted. Fo r example, the mechanism derived from the nested quadruple s tat ionary configuration in Fig. 9(b), with link 1 being selected as the frame member, has been adopted as the nose landing gear for an aircraft. At the down locking position, Fig. 14(a), the wheel link (link 4) is doubly locked there because either link 5 is in-line with link 6, or I23I~ and link 2 are colinear alone is sufficient to secure it. This doubly locked configuration is adopted due to the necessity that we always need a quite reliable state of landing gear system at the down locking posit ion during landing or take-off. To fold this mechanism, on the other hand, it is necessary to unlock the larger rigid chain before the sub-rigid one. That is, this mechanism can

Table 1. The stationary configurations of Watt six-bar chain

Inactive Case joints Constraints Figure

1 A Link 2 in-line with I23134 Fig. 7(a) 2 B Ii4134 in-line with I23I~4 Fig. 7(b) 3 D Link 1 in-line with link 2 and link 5 in-line with link 6 Fig. 7(c) 4 A and E Link 2 in-line with I2~I34 and link 6 in-line with I~I4~ Fig. 8(a) 5 A and F Link 2 in-line with 123134 and I24, I35 and I4e are colinear Fig. 8(b) 6 A and G Link 2 in-line with I23134 and link 5 in-line with I34135 Fig. 8(c) 7 B and E Link 6 in-line with I3414~ and It4 , 123 and 134 are colinear Fig. 8(d) 8 B and F It4, I23 and I~ are colinear and I34, I35 and I46 are colinear Fig. 8(e) 9 A, B, C and D Link 5 in-line with link 6 Fig. 9(a)

10 AT, B C and D Link 5 in-line with link 6 and link 2 in-line with I23134 Fig. 9(b) 1 1 A, Bl', C and D Link 5 in-line with link 6 and I23 , 124 and I14 are colinear Fig. 9(c)

i" Nested inactive joints.

Table 2. The stationary configurations of Stephenson six-bar chain

Inactive Case joints Constraints Figure

1 A Link 3 in-line with I14134 Fig. 10(a) 2 E I24 in-line with link 6 Fig. 10(b) 3 F I24, I25 and I~ are colinear Fig. 10(c) 4 A and E Link 3 in-line with It4134 and link 6 in-line with Ii4I~ Fig. l l(a) 5 A and F Link 3 in-line with Ii4134 and It4, I2s and I46 are ¢olinear Fig. l l(b) 6 A and G Link 3 in-line with It4134 and I14 in-line with link 5 Fig. ll(c) 7 E and F L6 in-line with links 1 and 3 Fig. 1 l(d) 8 E and G I~ coincides with I24 Fig. 1 l(e) 9 A, F and G Link 3 in-line with Ii4124 and I25 coincides with It4 Fig. 12

10 A, B, C and D Link 5 in-line with link 6 Fig. 13(a) 11 AT, B, C and D Link 5 in-line with link 6 and link 3 in-line with Ii4134 Fig. 13(b)

T Nested inactive joint.

292 HONG-SEN YAN and LONG-lONG WU

1 /'

~L3. (a)

(b) Fig. 14. An aircraft nose landing gear mechanism.

be folded only if external forces are applied on link 5 or 6 at this configuration. Alternatively, Fig. 14(b) shows the mechanism at the up locking position. This configuration, in which only links 5 and 6 are colinear, is a quadruple stationary configuration. Therefore, the mechanism is simply locked there. In addition, it occupies a small space, and this is usually a desirable characteristic of landing gears.

(a)

(b) Fig. 15. Degenerate Watt six-bar chain.

Fig. 16. A

135,113 5 I56,I16

'146

1

configuration with stationary link without inactive joints.

5. DISCUSSION

It is assumed that both inner in-line and outer in-line positions are included while colinear links are cited. In the results shown in Figs. 7-13, however, only one of their proper types is listed for simplicity. The only exception is the simple stationary configuration with inactive joint D of the Watt chain, as shown in Fig. 7(c). In this case, the existence of two inner in-line positions, Fig. 15(a), or two outer in-line positions, Fig. 15(b), will cause this kinematic chain to degenerate into a structure. To identify a degenerate chain, a configuration alone is generally not sufficient. When this is the case, a rough additive motion analysis will be helpful.

It is frequently true that a stationary configuration of a mechanism also possesses a stationary link with respect to the frame, for example, Fig. l(a). It is also true that a stationary configuration of a mechanism need not have a stationary link with respect to the frame, for example, Fig. l(b). On the other hand, a special configuration without any inactive joint may possess a stationary link [10]. The configuration shown in Fig. 16 is the simplest case. At this configuration, link 5 is stationary with respect to the frame because there are more than one stationary point on this link, in spite of all the joints are active [11].

The design steps for obtaining the stationary configurations of planar six-bar kinematic chains are carried out by listing all possible combinations of inactive joints, then deleting those cases which are useless or impossible to exist. Therefore, we believe that the results is complete. However, for kinematic chains with more than six links, the design steps are suggested to be computerized to obtain all possible stationary configurations efficiently.

6. CONCLUSION

In conclusion, we propose a method, based on the concepts of instant centers and kinematic inversion, to determine the configurations of planar linkages with inactive joints. This method is straightforward and can avoid handling any mathematical equations.

Stationary configurations of planar six-bar kinematic chains 293

All stationary configurations of six-bar kinematic chains are found and listed. These results may be of interest to the mechanism designers in their synthesis process, either to determine them or to back away from them. A practical application of nest-type quadruple stationary configurations as an aircraft landing gear is illustrated. Furthermore, this proposed method can also be applied to find stationary configurations of planar kinematic chains with more than six links.

REFERENCES

1. K. H. Hunt, Kinematic Geometry of Mechanisms, pp. 37-40 and pp. 384-390. O.U.P., Oxford (1978).

2. K. Sugimoto, J. Duffy and K. H. Hunt, Special configurations of spatial mechanisms and robot arms. Mech. Mach. Theory 17, 119-132 (1982).

3. A. T. Yang and F. Freudenstein, Application of dual- number quaternion algebra to the analysis of spatial mechanisms, J. Appl, Mech, 31. Also Trans. ASME 86, (E), 300-308 (1964).

4. J. E. Baker and K, J. Waldron, Limit positions of

spatial linkages via screw system theory. ASME Paper No. 74-DET-107.

5. F. L. Litvin, P. Fanghella, J. Tan and Y. Zhang, Singularities in motion and displacement functions of spatial linkages. Trans. ASME JI Mech. Transmiss. Automat. Design 108, 516-523 (1986).

6. J. E. Baker, Limit positions of spatial linkages via connectivity sum reduction. J. Mech. Design Trans. ASME 101, 504-508 (1979).

7. H. S. Yan and W. M. Hwang, Atlas of basic rigid chain. 9th Applied Mechanisms Conf. Proc. Session IV.B, Oct. 28-30 Kansas City, Miss. (1985),

8. D. W. Peterson, Design of dead-center linkages for airplane landing gear retraction systems. Proc. 2nd OSU Applied Mechanisms Conf. Paper No. 2, Stillwater, Okla. (1971).

9. I. L Artobolevsky, Mechanisms in Modern Engineering Design--Aircraft Landing Gear Mechanisms, Vol. II, Part I, pp. 416-444 (English Translation). Mir, Moscow (1976).

10. J. E. Baker, On the investigation of extrema in linkage analysis, using screw system algebra. Mech. Mach. Theory 13, 333-343 (1978).

11. H. S. Yan and L. I. Wu, On the dead-center positions of planar linkage mechanisms. ASME 20th Mechanisms Conf. Proc., Sept. 25-28, Orlando, Fla. 0988).

D I E S T A T I O N ~ R E N K O N F I G U R A T I O N E N S E C H S G L I E D R I G E R K I N E M A T I S C H E R K E T T E N

Zt~mamenf~ng--Ein Verfahren wird vorgeschlagen, die station~ren konfigurationen ebener Gelenkge- triebe zu ermitteln. Diesem Verfahren liegen die Gendanken von den Momentanpolen und kinematischen Umkehrungen zugrunde. S~mtliche stationiiren konfigurationen yon sechsgliedrigen kinematischen Ketten werden ermittelt und registriert. Es wird gezeigt, dass vierfach stationClre konfignrationen in den sechsgliedrigen Ketten existieren, und zwar ist eine davon ais Vorderfahrgestell eines Luftfahrzenges angewendet worden ist. Es wird auch gefunden, dass eine spezielle konfiguration ohne irgendwelche unwirksame Gelenke ein stationiires Glied besitzen kann.