Instant screw axis point synthesis of the RRSS mechanism

Instant screw axis point synthesis of the RRSS mechanism

Kevin Russell a, Raj S. Sodhi b,*

a Close Combat Armaments Center, US Army Research, Development and Engineering Center, Picatinny Arsenal,

NJ 07806-5000, USAb Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA

Received 2 January 2002; accepted 29 April 2002

Abstract

This paper presents a precision point synthesis of the RRSS motion generator, by specifying a set ofsuccessive points to the instantaneous screw axis. The method involves synthesizing RRSS mechanisms toachieve prescribed crank and coupler displacement angles by incorporating instant screw axis (ISA) pointsin the fixed axode point polynomial and calculating the R–R and S–S link parameters of this mechanism.The synthesis is facilitated by specific geometry of the RRSS mechanism, where the fixed axode is calculatedas intersection of the R–R member plane and the S–S member axis. The RRSS fixed axode point poly-nomial was developed using the Cosine law approach introduced by M€uuller [Kansas State UniversitySpecial Report No. 21, June 1962]. Complete expansion of the developed RRSS fixed axode point poly-nomial reveals that it is of order 56.� 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction

There have been a number of examples of using spatial RRSS mechanisms as motion gener-ators [2,3]. To facilitate the synthesis of these RRSS mechanisms, it can be noted that there is aclose link between the axodes and both a motion on one hand and a mechanism on the other. Forthis reason, ‘‘In synthesis, the mechanism for a prescribed motion can be found using the relationsbetween the axodes and the motion, and the axodes and the mechanism’’. This quote by Skreiner[4] will be fulfilled in this work since ISA points from the fixed axodes of the RRSS mechanismwill be used to calculate the R–R and S–S link variables of the RRSS mechanism needed toapproximate prescribed crank and coupler link displacement angles.

*Corresponding author. Tel.: +1-973-596-3333; fax: +1-973-642-4282.

E-mail address: [email protected] (R.S. Sodhi).

0094-114X/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0094-114X(02)00047-2

Mechanism and Machine Theory 37 (2002) 1117–1126www.elsevier.com/locate/mechmt

mail to: [email protected]

Several authors have made significant contributions in the area of axode-based spatial mech-anism synthesis. Hunt [5,6] has shown how the existence of many over-constrained linkages can beexplained, and even predicted, by applying simple theorems directly related to linear complexes,congruencies and ruled quadratic surfaces. Sodhi and Shoup [7,8] presented a general analyticalmethod for synthesizing four-revolute spherical mechanism based upon the fixed axode. Synthesisequations were developed which include a description of the linkage geometry and the axodegeometry. They also presented the relationships between the axodes and the geometric configu-ration of the spherical 4R mechanism. Fu and Chiang [9] presented a method to construct aspherical four-bar linkage so that the motion of its coupler matches a given spherical motion up toa certain order. Tong and Chinag [10] derived some basic equations and constructed compatibleequations to synthesize planar and spherical path generators. These equations are based on thegeometrical relations between the pole of the coupler and the joints of a mechanism.

This paper presents a method for synthesizing RRSS mechanisms to achieve prescribed crankand coupler displacement angles by incorporating instant screw axis (ISA) points in the fixed axodepoint polynomial and calculating the R–R and S–S link parameters of this mechanism. The RRSSfixed axode point polynomial is also developed for specific geometry of the RRSS mechanism.

2. The RRSS mechanism

2.1. General displacement equation

Displacement analysis of the RRSS mechanism is based on the constant length condition (Eq.(1)) of the output link ðb0 � b1Þ in Fig. 1. The RRSS mechanism displacement Eqs. (1)–(12) wereintroduced by Suh and Radcliffe [11].

Fig. 1. The RRSS mechanism.

1118 K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126

ðb� b0ÞTðb� b0Þ ¼ ðb1 � b0ÞTðb1 � b0Þ ð1Þwhere

a ¼ ½R#;u0�ða1 � a0Þ þ a0 ð2Þb ¼ ½Ra;ua�ðb01 � aÞ þ a ð3Þb01 ¼ ½R#;u0�ðb1 � a0Þ þ a0 ð4Þua ¼ ½R#;u0�ua1 ð5Þ

and

Ru;u

� �¼

u2x 1� cosuð Þ þ cosu uxuy 1� cosuð Þ � uz sinu uxuz 1� cosuð Þ þ uy sinuuxuy 1� cosuð Þ þ uz sinu u2y 1� cosuð Þ þ cosu uyuz 1� cosuð Þ � ux sinuuxuz 1� cosuð Þ � uy sinu uyuz 1� cosuð Þ þ ux sinu u2z 1� cosuð Þ þ cosu

24

35

ð6ÞFor each crank angle value #, the coupler angle a has two solutions.

a1;2 ¼ 2 tan�1 �F ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 þ F 2 � G2

p

G� Eð7Þ

where

E ¼ ða� b0ÞTf½I � Qua�ðb01 � aÞg ð8Þ

F ¼ ða� b0ÞTf½Pua�ðb01 � aÞg ð9Þ

G ¼ ða� b0ÞTf½Qua�ðb01 � aÞg þ 1

2fðb01 � aÞTðb01 � aÞ þ ða� b0ÞTða� b0Þ � ðb1 � b0ÞTðb1 � b0Þg

ð10Þand

Pu½ � ¼0 �uz uyuz 0 �ux�uy ux 0

24

35 ð11Þ

Qu½ � ¼u2x uxuy uxuzuxuy u2y uyuzuxuz uyuz u2z

24

35 ð12Þ

2.2. Fixed instant screw axis point equation

Geometrically speaking, a point on the fixed ISA of the RRSS mechanism is the point of in-tersection between a line that passes through b0 and b1 and a plane that passes through a0, a1 andjoint axis ua0 (see Fig. 2). Since the location of this fixed ISA point changes with the crank angle,a complete crank rotation would result in a locus of fixed ISA points from the fixed axode ofthe RRSS mechanism. In Fig. 2, the lengths R, S and C are the distances between pointP (the ISA point) and joint a1, point P and joint b1, and joint a1 and joint b1 respectively. The

K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126 1119

lengths A, B and D are the distances between joints a1 and a0, joints b1 and b0, and joints b0 and a0respectively.

M€uuller [1] derived the centrode polynomials for planar four bar mechanisms using the Cosinelaw. Using the notations given in Fig. 2, the Cosine law for the RRSS mechanism becomes Eq. (13)

2 cosðbÞ ¼ Aþ Rj j2 þ Bþ Sj j2 � Dj j2

Aþ Rj j Bþ Sj j ¼ Rj j2 þ Sj j2 � Cj j2

Rj j Sj j ð13Þ

where

P ¼ ½x; y; z�T ð14ÞAþ R ¼ ½x; y; z�T � a0 ð15ÞBþ S ¼ ½x; y; z�T � b0 ð16ÞD ¼ b0 � a0 ð17ÞR ¼ ½x; y; z�T � a ð18Þ

S ¼ ½x; y; z�T � b ð19ÞC ¼ b� a ð20Þ

Fig. 2. RRSS mechanism with fixed ISA point P.


Terms a and b in Eqs. (18)–(20) are the same as those in Eqs. (2) and (3) respectively. To in-corporate fixed ISA points in Eqs. (2) and (3), the cosð#Þ and sinð#Þ terms in the spatial angular

rotation matrix (6) will be replaced with either x=ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

por y=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pdepending on the

quadrant the ISA point P lies within.Depending upon the position of the crank link of the RRSS mechanism, Eq. (13) can be ex-

pressed as Eq. (21). To accommodate virtually every crank position of the RRSS mechanism, thesum and difference of the two fractions in Eq. (21) are multiplied. The resulting equation is givenin Eq. (22).

jAþ Rj2 þ jBþ Sj2 � jDj2

jAþ RjjBþ Sj jRj2 þ jSj2 � jCj2

jRjjSj ¼ 0 ð21Þ

jAþ Rj2 þ jBþ Sj2 � jDj2

jAþ RjjBþ Sj

" #2� jRj2 þ jSj2 � jCj2

jRjjSj

" #2¼ 0 ð22Þ

3. Example problem

Listed in Table 1 are seven prescribed crank and coupler displacement angles (and the corre-sponding ISA point coordinates) for the RRSS mechanism ð# and aÞ. The crank displacementangles were calculated by incorporating the x and y-coordinates of the ISA points in either

x=ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

por y=


p(depending on the quadrant the ISA point P lies within) and taking the

inverse sine or cosine of this ratio. The coupler displacement angles were calculated by incor-

porating the ISA-based cosð#Þ and sinð#Þ�x=


pand y=


p �in Eq. (7).

Generally, RRSS mechanism has 12 joint variables and 6 joint axis variables. These are the x, yand z-coordinates of a0, a1; b0, b1, ua0 and ua1. In this example problem a0, ua0 and ua1 are specified.

The value for a0 is (0, 0, 0), a1 is ð0; a1y; 0Þ, ua0 is (0,0, 1) and ua1 is 0:2; 0;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 0:22

ph iwhen the

crank lies along the positive y-axis. After the previous specifications are made, the remaining sevenunknowns are a1y , b0x, b0y, b0z, b1x, b1y and b1z.

Eq. (22) will be represented by F ðxj; yj; zjÞ ¼ 0 where j ¼ 1; 2; . . . ; 7. These results in a set ofseven simultaneous RRSS fixed ISA point equations for the seven unknown RRSS mechanismvariables. Given the following initial guesses:

Table 1

Prescribed ISA points, crank and coupler displacement angles for RRSS mechanism

Px Py Pz # (rad) a (rad)

0 50.21575 �7.41984 0 0

56.63391 �647.32860 104.12727 �0.08883 0.08727

7.66782 �43.48637 7.50617 �0.17548 0.17453

5.95617 �22.22872 4.06885 �0.26006 0.26180

5.34318 �14.68026 2.82147 �0.34268 0.34906

5.00945 �10.74280 2.14960 �0.42344 0.43633

4.78205 �8.28276 1.71277 �0.50244 0.52360


a1y ¼ 1; b0 ¼ ð3; 0; 0Þ; b1 ¼ ð2:7; 3:8;�0:5Þ

the set of seven RRSS fixed ISA point equations converges to

a1y ¼ 1:0200; b0 ¼ ð2:9440; 0:0483; 0:0003Þ; b1 ¼ ð2:7345; 3:6600;�0:3805Þ

using Newton’s method. This RRSS mechanism is illustrated in Fig. 3. Listed in Table 2 are theapproximated crank and coupler displacement angles (and the corresponding ISA point coordi-nates) for the RRSS mechanism.

Fig. 3. RRSS mechanism solution to example problem.

Table 2

ISA points, crank and coupler displacement angles for synthesized RRSS mechanism

Px Py Pz # (rad) a (rad)

0 50.80361 �5.35125 0 0

54.33382 �621.03844 73.42062 �0.08883 �0.08913

7.70892 �43.71944 5.67201 �0.17548 �0.17611

5.99954 �22.39061 3.13262 �0.26006 �0.26106

5.38547 �14.79645 2.20135 �0.34268 �0.34408

5.04981 �10.82935 1.69353 �0.42344 �0.42526

4.81973 �8.34802 1.35885 �0.50244 �0.50470


4. Expansion of the RRSS mechanism fixed axode point polynomial

In this section, the fixed ISA point polynomial for the RRSS mechanism will be expanded todetermine its order. Since it is anticipated that this polynomial will be too long to express in itsgeneral form, numerical values for the 18 variables of the RRSS mechanism will be incorporatedin Eq. (22). By doing this, the number of coefficients in the fixed ISA point polynomial will bereduced (making the equation more compact). The ISA point terms (x, y and z) will remain asthey are and will reveal the order of the fixed ISA point polynomial after Eq. (22) is fully ex-panded.

The variables of the RRSS mechanism illustrated in Fig. 4 will be used to expand the fixed ISApoint polynomial. These prescribed RRSS design variables are

a0 ¼ ½0; 3; 0�; a1 ¼ ½0; 4; 0�; b0 ¼ ½0; 0; 0�; b1 ¼ ½4; 0; 0�; ua0 ¼ ½0; 0; 1�; ua1 ¼ ½7=25; 0; 24=25�

cosð#Þ ¼ x=ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pand sinð#Þ ¼ y=


p:

Using the prescribed design variables for the RRSS mechanism, Eqs. (2), (3), (5),(8)–(10)become

a ¼

� yffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p ; 3þ xffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p ; 0

!T

ð23Þ

ua ¼ 7x

25ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p ;7y


p ;24

25

!T

ð24Þ

Fig. 4. RRSS mechanism for the fixed ISA point polynomial expansion.


b ¼

196x� 625y þ ð2304xþ 2500yÞ cosðaÞ þ 2400ðx� yÞ sinðaÞ625


p625xþ 196y þ 1875


pþ ð�2500xþ 2304yÞ cosðaÞ þ 2400ðxþ yÞ sinðaÞ


p� 28

625ð�24þ 24 cosðaÞ þ 25 sinðaÞÞ

0BBBBBBBB@

1CCCCCCCCA

ð25Þ

E ¼ � 4ð625x2 þ 625y2 þ 1875xffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p� 1728y


pÞ

625ðx2 þ y2Þ ð26Þ

F ¼ 96ðx2 þ y2 þ 3xffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pþ 3y


pÞ

25ðx2 þ y2Þ ð27Þ

and

G ¼ 8125x2 þ 8125y2 þ 1875xffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pþ 588y


p625ðx2 þ y2Þ ð28Þ

By multiplying Eq. (22) by the denominators in both of its fractions the form of the fixed ISApoint equation given in Eq. (29) was derived. In this form, the fixed ISA point equation expansioncan begin.

F ðx; y; zÞ ¼ jAh

þ Rj2 þ jBþ Sj2 � jDj2i2

jRjjSj½ �2 � jRj2h

þ jSj2 � jCj2i2

jA½ þ RjjBþ Sj�2 ¼ 0

ð29ÞAfter substituting Eq. (23) for a, Eq. (25) for b, and the prescribed values for a0 and b0 in Eq. (29),the form of the RRSS fixed ISA point equation given in Eq. (30) was derived.

F ðx; y; zÞ ¼ f1ðx; y; zÞ þ f2ðx; y; zÞ cosðaÞ þ f3ðx; y; zÞ cosðaÞ2 þ f4ðx; y; zÞ sinðaÞþ f5ðx; y; zÞ cosðaÞ sinðaÞ þ f6ðx; y; zÞ sinðaÞ2 ¼ 0 ð30Þ

In Eq. (30), the f ðx; y; zÞ terms represent the coefficients that exists when the terms cosðaÞ, cosðaÞ2,sinðaÞ, cosðaÞ sinðaÞ and sinðaÞ2 are grouped. The sine and cosine of Eq. (7) can also be expressedas Eqs. (31) and (32).

sinða1;2Þ ¼ð�Gþ EÞ

�F �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 þ F 2 � G2

p ��GE þ E2 þ F 2 � F


p ð31Þ

cosða1;2Þ ¼G2 � GE � F 2 F


p

�GE þ E2 þ F 2 � FffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 þ F 2 � G2

p ð32Þ

For the expansion of the fixed ISA point polynomial in this work, the coupler displacement angle(corresponding to counter-clockwise crank angular displacements) a1 was used. After Eqs. (31)and (32) are placed in Eq. (30), and the equation simplified, the fixed ISA point polynomial be-comes Eq. (33).


F ðx; y; zÞ ¼ ð�GE þ E2 þ F 2 � FffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 þ F 2 � G2

pÞ2f1ðx; y; zÞ

þ�G2 � GE � F 2 þ F


p �� GE þ E2 þ F 2

� FffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 þ F 2 � G2

p �f2ðx; y; zÞ þ

�G2 � GE � F 2 þ F


p �2f3ðx; y; zÞ

þ ð�Gþ EÞ�F �


p �� GE þ E2 þ F 2 � F


p �f4ðx; y; zÞ

þ�G2 � GE � F 2 þ F


p �ð�Gþ EÞ

�F �


p �f5ðx; y; zÞ

þ�ð�Gþ EÞðF �


pÞ�2f6ðx; y; zÞ ¼ 0 ð33Þ

After expanding Eq. (33), the square roots in this equation are eliminated by factoring them,placing them on one side of the equation (with the remaining terms on the other side) andsquaring both sides of Eq. (33). Eqs. (26)–(28) are then included in Eq. 33 as well as all of thef ðx; y; zÞ coefficients and this equation is expanded again. The remaining square roots in thisequation are eliminated as well. The fully expanded form of this equation representing the fixed-axode point polynomial of the RRSS mechanism illustrated in Fig. 4 comes out to be a 56th orderpolynomial and is very long for inclusion in this paper. This equation, which is about 20 pageslong or the MATHEMATICA notebook used in its derivation is available upon request from theauthors.

5. Discussion

The expanded fixed ISA point polynomial in this work was obtained using Mathematicasoftware. The coordinates of the fixed ISA points as well as the crank and coupler displacementangles in this work were calculated using Mathcad software. If Eq. (1) is used in place of Eq. (22)to calculate the parameters of RRSS mechanism, only the x- and y-coordinates of the fixed ISApoint is required. In the expanded fixed ISA point polynomial, replacing all of the numericalcoefficients with alphanumeric coefficients significantly reduces the overall size of the polynomial(from hundreds of pages to less than 20). However, since the purpose of expanding the RRSSfixed ISA point polynomial was to determine its order only, the numerical values of these coef-ficients are immaterial to the scope of this research.

6. Conclusion

A synthesis method for approximating crank and coupler displacement angles for RRSSmechanisms, given points from the fixed instant screw axes (ISAs) of their fixed axodes, is de-veloped and presented here. By incorporating ISA points in the fixed axode point polynomial andspecifying the initial guesses for the joint parameters of the RRSS mechanism, the actual jointparameters of this mechanism were obtained. The RRSS fixed axode point polynomial was


developed using the Cosine law approach introduced by M€uuller [1]. After expansion, the RRSSfixed axode point polynomial was shown to be a 56th order polynomial.

References

[1] R. M€uuller, Papers on geometrical theory of motion applied to approximate straight-line motion, pp. 217–219

(English Translation by D. Tesar, Kansas State University Special Report No. 21, June 1962).

[2] K. Russell, R.S. Sodhi, Kinematic synthesis of adjustable RRSS mechanisms for multi-phase motion generation,

Mechanism and Machine Theory 36 (2001) 939–952.

[3] K. Russell, R.S. Sodhi, Kinematic synthesis of adjustable RRSS mechanisms for multi-phase motion generation

with tolerances, Mechanism and Machine Theory 37 (2002) 279–294.

[4] M. Skreiner, A study of the geometry and the kinematics of instantaneous spatial motion, Journal of Mechanisms 1

(1966) 115–143.

[5] K.H. Hunt, Screw axes and mobility in spatial mechanisms via the linear complex, Mechanism and Machine

Theory 3 (1967) 307–327.

[6] K.H. Hunt, Kinematic Geometry of Mechanisms, Clarendon Press, 1978.

[7] R. Sodhi, T.E. Shoup, Axodes for the four-revolute spherical mechanism, Mechanism and Machine Theory 17 (3)

(1982) 173–178.

[8] R.S. Sodhi, T.E. Shoup, ISA synthesis of the four revolute spherical mechanism, International Symposium on

Design and Synthesis, Tokyo, Japan, 1984.

[9] T. Fu, C.H. Chiang, Simulating a given spherical motion by the Polode method, Mechanism and Machine Theory

29 (2) (1994) 237–249.

[10] S. Tong, C.H. Chiang, Synthesis of planar and spherical four-bar path generators by the Pole method, Mechanism

and Machine Theory 27 (2) (1992) 143–155.

[11] C.H. Suh, C.W. Radcliffe, Kinematics and Mechanisms Design, John Wiley and Sons, New York, 1978.


Instant screw axis point synthesis of the RRSS mechanism

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