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Instant screw axis point synthesis of the RRSS mechanism
Mechanism and Machine Theory 37 (2002) 1117–1126 www.elsevier.com/locate/mechmt
Instant screw axis point synthesis of the RRSS mechanism Kevin Russell a, Raj S. Sodhi a
b,*
Close Combat Armaments Center, US Army Research, Development and Engineering Center, Picatinny Arsenal, NJ 07806-5000, USA b 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 of successive points to the instantaneous screw axis. The method involves synthesizing RRSS mechanisms to achieve prescribed crank and coupler displacement angles by incorporating instant screw axis (ISA) points in 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 calculated as intersection of the R–R member plane and the S–S member axis. The RRSS fixed axode point polynomial was developed using the Cosine law approach introduced by M€ uller [Kansas State University Special Report No. 21, June 1962]. Complete expansion of the developed RRSS fixed axode point polynomial reveals that it is of order 56. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction There have been a number of examples of using spatial RRSS mechanisms as motion generators [2,3]. To facilitate the synthesis of these RRSS mechanisms, it can be noted that there is a close link between the axodes and both a motion on one hand and a mechanism on the other. For this reason, ‘‘In synthesis, the mechanism for a prescribed motion can be found using the relations between 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 mechanism will be used to calculate the R–R and S–S link variables of the RRSS mechanism needed to approximate 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 Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 0 4 7 - 2
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K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126
Several authors have made significant contributions in the area of axode-based spatial mechanism synthesis. Hunt [5,6] has shown how the existence of many over-constrained linkages can be explained, 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 analytical method for synthesizing four-revolute spherical mechanism based upon the fixed axode. Synthesis equations were developed which include a description of the linkage geometry and the axode geometry. They also presented the relationships between the axodes and the geometric configuration of the spherical 4R mechanism. Fu and Chiang [9] presented a method to construct a spherical four-bar linkage so that the motion of its coupler matches a given spherical motion up to a certain order. Tong and Chinag [10] derived some basic equations and constructed compatible equations to synthesize planar and spherical path generators. These equations are based on the geometrical 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 crank and coupler displacement angles by incorporating instant screw axis (ISA) points in the fixed axode point polynomial and calculating the R–R and S–S link parameters of this mechanism. The RRSS fixed 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 b1 Þ in Fig. 1. The RRSS mechanism displacement Eqs. (1)–(12) were introduced by Suh and Radcliffe [11].
Fig. 1. The RRSS mechanism.
K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126
ðb b0 ÞT ðb b0 Þ ¼ ðb1 b0 ÞT ðb1 b0 Þ
1119
ð1Þ
where a ¼ ½R#;u0 ða1 a0 Þ þ a0 b¼
and
½Ra;ua ðb01
aÞ þ a
ð2Þ ð3Þ
b01 ¼ ½R#;u0 ðb1 a0 Þ þ a0
ð4Þ
ua ¼ ½R#;u0 ua1
ð5Þ
2 2 3 ux uy ð1 cos uÞ uz sin u ux uz ð1 cos uÞ þ uy sin u ux ð1 cos uÞ þ cos u u2y ð1 cos uÞ þ cos u uy uz ð1 cos uÞ ux sin u 5 Ru;u ¼ 4 ux uy ð1 cos uÞ þ uz sin u ux uz ð1 cos uÞ uy sin u uy uz ð1 cos uÞ þ ux sin u u2z ð1 cos uÞ þ cos u ð6Þ
For each crank angle value #, the coupler angle a has two solutions. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2 þ F 2 G2 1 F a1;2 ¼ 2 tan GE where T
ð7Þ
E ¼ ða b0 Þ f½I Qua ðb01 aÞg
ð8Þ
F ¼ ða b0 ÞT f½Pua ðb01 aÞg
ð9Þ
1 G ¼ ða b0 ÞT f½Qua ðb01 aÞg þ fðb01 aÞT ðb01 aÞ þ ða b0 ÞT ða b0 Þ ðb1 b0 ÞT ðb1 b0 Þg 2 ð10Þ
and
2
3 0 uz uy 0 ux 5 ½Pu ¼ 4 uz uy ux 0 2 2 3 ux ux uy ux uz uy uz 5 ½Qu ¼ 4 ux uy u2y ux uz uy uz u2z
ð11Þ
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 intersection between a line that passes through b0 and b1 and a plane that passes through a0 , a1 and joint 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 of the RRSS mechanism. In Fig. 2, the lengths R, S and C are the distances between point P (the ISA point) and joint a1 , point P and joint b1 , and joint a1 and joint b1 respectively. The
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Fig. 2. RRSS mechanism with fixed ISA point P.
lengths A, B and D are the distances between joints a1 and a0 , joints b1 and b0 , and joints b0 and a0 respectively. M€ uller [1] derived the centrode polynomials for planar four bar mechanisms using the Cosine law. Using the notations given in Fig. 2, the Cosine law for the RRSS mechanism becomes Eq. (13) 2 cosðbÞ ¼
jA þ Rj2 þ jB þ Sj2 jDj2 jRj2 þ jSj2 jCj2 ¼ jA þ RjjB þ Sj jRjjSj
ð13Þ
where P ¼ ½x; y; zT
ð14Þ T
ð15Þ
T
B þ S ¼ ½x; y; z b0
ð16Þ
D ¼ b0 a0
ð17Þ
T
R ¼ ½x; y; z a
ð18Þ
S ¼ ½x; y; zT b
ð19Þ
C¼ba
ð20Þ
A þ R ¼ ½x; y; z a0
K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126
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Terms a and b in Eqs. (18)–(20) are the same as those in Eqs. (2) and (3) respectively. To incorporate fixed ISA points in Eqs. (2) and (3), the cosð#Þ and sinð#Þp terms in the spatial angular pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi rotation matrix (6) will be replaced with either x= x2 þ y 2 or y= x2 þ y 2 depending 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 expressed as Eq. (21). To accommodate virtually every crank position of the RRSS mechanism, the sum and difference of the two fractions in Eq. (21) are multiplied. The resulting equation is given in Eq. (22). jA þ Rj2 þ jB þ Sj2 jDj2 jRj2 þ jSj2 jCj2 ¼0 jA þ RjjB þ Sj jRjjSj " # " # 2 2 2 2 2 2 2 2 jA þ Rj þ jB þ Sj jDj jRj þ jSj jCj ¼0 jA þ RjjB þ Sj jRjjSj
ð21Þ
ð22Þ
3. Example problem Listed in Table 1 are seven prescribed crank and coupler displacement angles (and the corresponding ISA point coordinates) for the RRSS mechanism ð# and aÞ. The crank displacement angles were calculated by incorporating the x and y-coordinates of the ISA points in either pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 x= x þ y or y= x þ y 2 (depending on the quadrant the ISA point P lies within) and taking the inverse sine or cosine of this ratio. The coupler displacement p angles pffiffiffiffiffiffiffiffiffiffiffiffiffiffi calculated by incorffiffiffiffiffiffiffiffiffiffiffiffiffiffiwere 2 2 2 2 porating the ISA-based cosð#Þ and sinð#Þ x= x þ y and y= x þ y in Eq. (7). Generally, RRSS mechanism has 12 joint variables and 6 joint axis variables. These are the x, y specified. and z-coordinates of a0 , a1 ; b0 , b1 , ua0 and ua1 . In this example problemha0 , ua0 and ua1 are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 2 The value for a0 is (0, 0, 0), a1 is ð0; a1y ; 0Þ, ua0 is (0,0, 1) and ua1 is 0:2; 0; 1 0:2 when the crank lies along the positive y-axis. After the previous specifications are made, the remaining seven unknowns are a1y , b0x , b0y , b0z , b1x , b1y and b1z . Eq. (22) will be represented by F ðxj ; yj ; zj Þ ¼ 0 where j ¼ 1; 2; . . . ; 7. These results in a set of seven simultaneous RRSS fixed ISA point equations for the seven unknown RRSS mechanism variables. 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 56.63391 7.66782 5.95617 5.34318 5.00945 4.78205
50.21575 647.32860 43.48637 22.22872 14.68026 10.74280 8.28276
7.41984 104.12727 7.50617 4.06885 2.82147 2.14960 1.71277
0 0.08883 0.17548 0.26006 0.34268 0.42344 0.50244
0 0.08727 0.17453 0.26180 0.34906 0.43633 0.52360
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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 the approximated crank and coupler displacement angles (and the corresponding ISA point coordinates) 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 54.33382 7.70892 5.99954 5.38547 5.04981 4.81973
50.80361 621.03844 43.71944 22.39061 14.79645 10.82935 8.34802
5.35125 73.42062 5.67201 3.13262 2.20135 1.69353 1.35885
0 0.08883 0.17548 0.26006 0.34268 0.42344 0.50244
0 0.08913 0.17611 0.26106 0.34408 0.42526 0.50470
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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 to determine its order. Since it is anticipated that this polynomial will be too long to express in its general form, numerical values for the 18 variables of the RRSS mechanism will be incorporated in Eq. (22). By doing this, the number of coefficients in the fixed ISA point polynomial will be reduced (making the equation more compact). The ISA point terms (x, y and z) will remain as they are and will reveal the order of the fixed ISA point polynomial after Eq. (22) is fully expanded. The variables of the RRSS mechanism illustrated in Fig. 4 will be used to expand the fixed ISA point 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 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð#Þ ¼ x= x2 þ y 2 and sinð#Þ ¼ y= x2 þ y 2 : Using the prescribed design variables for the RRSS mechanism, Eqs. (2), (3), (5),(8)–(10) become !T y x ð23Þ a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 3 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 0 x2 þ y 2 x2 þ y 2 !T 7x 7y 24 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð24Þ ua ¼ 25 x2 þ y 2 25 x2 þ y 2 25
Fig. 4. RRSS mechanism for the fixed ISA point polynomial expansion.
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1 196x 625y þ ð2304x þ 2500yÞ cosðaÞ þ 2400ðx yÞ sinðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B 625 x2 þ y 2 C B C B C B 625x þ 196y þ 1875pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 x þ y þ ð2500x þ 2304yÞ cosðaÞ þ 2400ðx þ yÞ sinðaÞ C b¼B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B 2 þ y2 C B 625 x C B A @ 28 ð24 þ 24 cosðaÞ þ 25 sinðaÞÞ 625 0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ð625x2 þ 625y 2 þ 1875x x2 þ y 2 1728y x2 þ y 2 Þ E¼ 625ðx2 þ y 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 96ðx2 þ y 2 þ 3x x2 þ y 2 þ 3y x2 þ y 2 Þ F ¼ 25ðx2 þ y 2 Þ and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8125x2 þ 8125y 2 þ 1875x x2 þ y 2 þ 588y x2 þ y 2 G¼ 625ðx2 þ y 2 Þ
ð25Þ ð26Þ ð27Þ
ð28Þ
By multiplying Eq. (22) by the denominators in both of its fractions the form of the fixed ISA point equation given in Eq. (29) was derived. In this form, the fixed ISA point equation expansion can begin. h i2 h i2 2 2 2 2 2 2 2 2 F ðx; y; zÞ ¼ jA þ Rj þ jB þ Sj jDj ½jRjjSj jRj þ jSj jCj ½jA þ RjjB þ Sj ¼ 0 ð29Þ
After 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 f1 ðx; y; zÞ þ f2 ðx; y; zÞ cosðaÞ þ f3 ðx; y; zÞ cosðaÞ2 þ f4 ðx; y; zÞ sinðaÞ þ f5 ðx; y; zÞ cosðaÞ sinðaÞ þ f6 ðx; 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 expressed as Eqs. (31) and (32). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðG þ EÞ F E2 þ F 2 G2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð31Þ sinða1;2 Þ ¼ GE þ E2 þ F 2 F E2 þ F 2 G2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 GE F 2 F E2 þ F 2 G2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð32Þ cosða1;2 Þ ¼ GE þ E2 þ F 2 F E2 þ F 2 G2 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 becomes Eq. (33).
K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126
1125
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 F ðx; y; zÞ ¼ ðGE þ E2 þ F 2 F E2 þ F 2 G2 Þ f1 ðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ G2 GE F 2 þ F E2 þ F 2 G2 GE þ E2 þ F 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 F E2 þ F 2 G2 f2 ðx; y; zÞ þ G2 GE F 2 þ F E2 þ F 2 G2 f3 ðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðG þ EÞ F E2 þ F 2 G2 GE þ E2 þ F 2 F E2 þ F 2 G2 f4 ðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ G2 GE F 2 þ F E2 þ F 2 G2 ðG þ EÞ F E2 þ F 2 G2 f5 ðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ ðG þ EÞðF E2 þ F 2 G2 Þ f6 ðx; 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) and squaring both sides of Eq. (33). Eqs. (26)–(28) are then included in Eq. 33 as well as all of the f ðx; y; zÞ coefficients and this equation is expanded again. The remaining square roots in this equation are eliminated as well. The fully expanded form of this equation representing the fixedaxode point polynomial of the RRSS mechanism illustrated in Fig. 4 comes out to be a 56th order polynomial and is very long for inclusion in this paper. This equation, which is about 20 pages long or the MATHEMATICA notebook used in its derivation is available upon request from the authors.
5. Discussion The expanded fixed ISA point polynomial in this work was obtained using Mathematica software. The coordinates of the fixed ISA points as well as the crank and coupler displacement angles 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 ISA point is required. In the expanded fixed ISA point polynomial, replacing all of the numerical coefficients 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 RRSS fixed ISA point polynomial was to determine its order only, the numerical values of these coefficients are immaterial to the scope of this research.
6. Conclusion A synthesis method for approximating crank and coupler displacement angles for RRSS mechanisms, given points from the fixed instant screw axes (ISAs) of their fixed axodes, is developed and presented here. By incorporating ISA points in the fixed axode point polynomial and specifying the initial guesses for the joint parameters of the RRSS mechanism, the actual joint parameters of this mechanism were obtained. The RRSS fixed axode point polynomial was
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K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126
developed using the Cosine law approach introduced by M€ uller [1]. After expansion, the RRSS fixed axode point polynomial was shown to be a 56th order polynomial.
References [1] R. M€ uller, 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 Kevin Russell a, Raj S. Sodhi a
b,*
Close Combat Armaments Center, US Army Research, Development and Engineering Center, Picatinny Arsenal, NJ 07806-5000, USA b 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 of successive points to the instantaneous screw axis. The method involves synthesizing RRSS mechanisms to achieve prescribed crank and coupler displacement angles by incorporating instant screw axis (ISA) points in 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 calculated as intersection of the R–R member plane and the S–S member axis. The RRSS fixed axode point polynomial was developed using the Cosine law approach introduced by M€ uller [Kansas State University Special Report No. 21, June 1962]. Complete expansion of the developed RRSS fixed axode point polynomial reveals that it is of order 56. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction There have been a number of examples of using spatial RRSS mechanisms as motion generators [2,3]. To facilitate the synthesis of these RRSS mechanisms, it can be noted that there is a close link between the axodes and both a motion on one hand and a mechanism on the other. For this reason, ‘‘In synthesis, the mechanism for a prescribed motion can be found using the relations between 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 mechanism will be used to calculate the R–R and S–S link variables of the RRSS mechanism needed to approximate 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 Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 0 2 ) 0 0 0 4 7 - 2
1118
K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126
Several authors have made significant contributions in the area of axode-based spatial mechanism synthesis. Hunt [5,6] has shown how the existence of many over-constrained linkages can be explained, 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 analytical method for synthesizing four-revolute spherical mechanism based upon the fixed axode. Synthesis equations were developed which include a description of the linkage geometry and the axode geometry. They also presented the relationships between the axodes and the geometric configuration of the spherical 4R mechanism. Fu and Chiang [9] presented a method to construct a spherical four-bar linkage so that the motion of its coupler matches a given spherical motion up to a certain order. Tong and Chinag [10] derived some basic equations and constructed compatible equations to synthesize planar and spherical path generators. These equations are based on the geometrical 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 crank and coupler displacement angles by incorporating instant screw axis (ISA) points in the fixed axode point polynomial and calculating the R–R and S–S link parameters of this mechanism. The RRSS fixed 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 b1 Þ in Fig. 1. The RRSS mechanism displacement Eqs. (1)–(12) were introduced by Suh and Radcliffe [11].
Fig. 1. The RRSS mechanism.
K. Russell, R.S. Sodhi / Mechanism and Machine Theory 37 (2002) 1117–1126
ðb b0 ÞT ðb b0 Þ ¼ ðb1 b0 ÞT ðb1 b0 Þ
1119
ð1Þ
where a ¼ ½R#;u0 ða1 a0 Þ þ a0 b¼
and
½Ra;ua ðb01
aÞ þ a
ð2Þ ð3Þ
b01 ¼ ½R#;u0 ðb1 a0 Þ þ a0
ð4Þ
ua ¼ ½R#;u0 ua1
ð5Þ
2 2 3 ux uy ð1 cos uÞ uz sin u ux uz ð1 cos uÞ þ uy sin u ux ð1 cos uÞ þ cos u u2y ð1 cos uÞ þ cos u uy uz ð1 cos uÞ ux sin u 5 Ru;u ¼ 4 ux uy ð1 cos uÞ þ uz sin u ux uz ð1 cos uÞ uy sin u uy uz ð1 cos uÞ þ ux sin u u2z ð1 cos uÞ þ cos u ð6Þ
For each crank angle value #, the coupler angle a has two solutions. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 2 þ F 2 G2 1 F a1;2 ¼ 2 tan GE where T
ð7Þ
E ¼ ða b0 Þ f½I Qua ðb01 aÞg
ð8Þ
F ¼ ða b0 ÞT f½Pua ðb01 aÞg
ð9Þ
1 G ¼ ða b0 ÞT f½Qua ðb01 aÞg þ fðb01 aÞT ðb01 aÞ þ ða b0 ÞT ða b0 Þ ðb1 b0 ÞT ðb1 b0 Þg 2 ð10Þ
and
2
3 0 uz uy 0 ux 5 ½Pu ¼ 4 uz uy ux 0 2 2 3 ux ux uy ux uz uy uz 5 ½Qu ¼ 4 ux uy u2y ux uz uy uz u2z
ð11Þ
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 intersection between a line that passes through b0 and b1 and a plane that passes through a0 , a1 and joint 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 of the RRSS mechanism. In Fig. 2, the lengths R, S and C are the distances between point P (the ISA point) and joint a1 , point P and joint b1 , and joint a1 and joint b1 respectively. The
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Fig. 2. RRSS mechanism with fixed ISA point P.
lengths A, B and D are the distances between joints a1 and a0 , joints b1 and b0 , and joints b0 and a0 respectively. M€ uller [1] derived the centrode polynomials for planar four bar mechanisms using the Cosine law. Using the notations given in Fig. 2, the Cosine law for the RRSS mechanism becomes Eq. (13) 2 cosðbÞ ¼
jA þ Rj2 þ jB þ Sj2 jDj2 jRj2 þ jSj2 jCj2 ¼ jA þ RjjB þ Sj jRjjSj
ð13Þ
where P ¼ ½x; y; zT
ð14Þ T
ð15Þ
T
B þ S ¼ ½x; y; z b0
ð16Þ
D ¼ b0 a0
ð17Þ
T
R ¼ ½x; y; z a
ð18Þ
S ¼ ½x; y; zT b
ð19Þ
C¼ba
ð20Þ
A þ R ¼ ½x; y; z a0
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Terms a and b in Eqs. (18)–(20) are the same as those in Eqs. (2) and (3) respectively. To incorporate fixed ISA points in Eqs. (2) and (3), the cosð#Þ and sinð#Þp terms in the spatial angular pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi rotation matrix (6) will be replaced with either x= x2 þ y 2 or y= x2 þ y 2 depending 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 expressed as Eq. (21). To accommodate virtually every crank position of the RRSS mechanism, the sum and difference of the two fractions in Eq. (21) are multiplied. The resulting equation is given in Eq. (22). jA þ Rj2 þ jB þ Sj2 jDj2 jRj2 þ jSj2 jCj2 ¼0 jA þ RjjB þ Sj jRjjSj " # " # 2 2 2 2 2 2 2 2 jA þ Rj þ jB þ Sj jDj jRj þ jSj jCj ¼0 jA þ RjjB þ Sj jRjjSj
ð21Þ
ð22Þ
3. Example problem Listed in Table 1 are seven prescribed crank and coupler displacement angles (and the corresponding ISA point coordinates) for the RRSS mechanism ð# and aÞ. The crank displacement angles were calculated by incorporating the x and y-coordinates of the ISA points in either pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 x= x þ y or y= x þ y 2 (depending on the quadrant the ISA point P lies within) and taking the inverse sine or cosine of this ratio. The coupler displacement p angles pffiffiffiffiffiffiffiffiffiffiffiffiffiffi calculated by incorffiffiffiffiffiffiffiffiffiffiffiffiffiffiwere 2 2 2 2 porating the ISA-based cosð#Þ and sinð#Þ x= x þ y and y= x þ y in Eq. (7). Generally, RRSS mechanism has 12 joint variables and 6 joint axis variables. These are the x, y specified. and z-coordinates of a0 , a1 ; b0 , b1 , ua0 and ua1 . In this example problemha0 , ua0 and ua1 are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 2 The value for a0 is (0, 0, 0), a1 is ð0; a1y ; 0Þ, ua0 is (0,0, 1) and ua1 is 0:2; 0; 1 0:2 when the crank lies along the positive y-axis. After the previous specifications are made, the remaining seven unknowns are a1y , b0x , b0y , b0z , b1x , b1y and b1z . Eq. (22) will be represented by F ðxj ; yj ; zj Þ ¼ 0 where j ¼ 1; 2; . . . ; 7. These results in a set of seven simultaneous RRSS fixed ISA point equations for the seven unknown RRSS mechanism variables. 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 56.63391 7.66782 5.95617 5.34318 5.00945 4.78205
50.21575 647.32860 43.48637 22.22872 14.68026 10.74280 8.28276
7.41984 104.12727 7.50617 4.06885 2.82147 2.14960 1.71277
0 0.08883 0.17548 0.26006 0.34268 0.42344 0.50244
0 0.08727 0.17453 0.26180 0.34906 0.43633 0.52360
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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 the approximated crank and coupler displacement angles (and the corresponding ISA point coordinates) 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 54.33382 7.70892 5.99954 5.38547 5.04981 4.81973
50.80361 621.03844 43.71944 22.39061 14.79645 10.82935 8.34802
5.35125 73.42062 5.67201 3.13262 2.20135 1.69353 1.35885
0 0.08883 0.17548 0.26006 0.34268 0.42344 0.50244
0 0.08913 0.17611 0.26106 0.34408 0.42526 0.50470
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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 to determine its order. Since it is anticipated that this polynomial will be too long to express in its general form, numerical values for the 18 variables of the RRSS mechanism will be incorporated in Eq. (22). By doing this, the number of coefficients in the fixed ISA point polynomial will be reduced (making the equation more compact). The ISA point terms (x, y and z) will remain as they are and will reveal the order of the fixed ISA point polynomial after Eq. (22) is fully expanded. The variables of the RRSS mechanism illustrated in Fig. 4 will be used to expand the fixed ISA point 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 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð#Þ ¼ x= x2 þ y 2 and sinð#Þ ¼ y= x2 þ y 2 : Using the prescribed design variables for the RRSS mechanism, Eqs. (2), (3), (5),(8)–(10) become !T y x ð23Þ a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 3 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 0 x2 þ y 2 x2 þ y 2 !T 7x 7y 24 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð24Þ ua ¼ 25 x2 þ y 2 25 x2 þ y 2 25
Fig. 4. RRSS mechanism for the fixed ISA point polynomial expansion.
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1 196x 625y þ ð2304x þ 2500yÞ cosðaÞ þ 2400ðx yÞ sinðaÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B 625 x2 þ y 2 C B C B C B 625x þ 196y þ 1875pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 x þ y þ ð2500x þ 2304yÞ cosðaÞ þ 2400ðx þ yÞ sinðaÞ C b¼B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B 2 þ y2 C B 625 x C B A @ 28 ð24 þ 24 cosðaÞ þ 25 sinðaÞÞ 625 0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ð625x2 þ 625y 2 þ 1875x x2 þ y 2 1728y x2 þ y 2 Þ E¼ 625ðx2 þ y 2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 96ðx2 þ y 2 þ 3x x2 þ y 2 þ 3y x2 þ y 2 Þ F ¼ 25ðx2 þ y 2 Þ and
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8125x2 þ 8125y 2 þ 1875x x2 þ y 2 þ 588y x2 þ y 2 G¼ 625ðx2 þ y 2 Þ
ð25Þ ð26Þ ð27Þ
ð28Þ
By multiplying Eq. (22) by the denominators in both of its fractions the form of the fixed ISA point equation given in Eq. (29) was derived. In this form, the fixed ISA point equation expansion can begin. h i2 h i2 2 2 2 2 2 2 2 2 F ðx; y; zÞ ¼ jA þ Rj þ jB þ Sj jDj ½jRjjSj jRj þ jSj jCj ½jA þ RjjB þ Sj ¼ 0 ð29Þ
After 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 f1 ðx; y; zÞ þ f2 ðx; y; zÞ cosðaÞ þ f3 ðx; y; zÞ cosðaÞ2 þ f4 ðx; y; zÞ sinðaÞ þ f5 ðx; y; zÞ cosðaÞ sinðaÞ þ f6 ðx; 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 expressed as Eqs. (31) and (32). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðG þ EÞ F E2 þ F 2 G2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð31Þ sinða1;2 Þ ¼ GE þ E2 þ F 2 F E2 þ F 2 G2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G2 GE F 2 F E2 þ F 2 G2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð32Þ cosða1;2 Þ ¼ GE þ E2 þ F 2 F E2 þ F 2 G2 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 becomes Eq. (33).
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1125
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 F ðx; y; zÞ ¼ ðGE þ E2 þ F 2 F E2 þ F 2 G2 Þ f1 ðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ G2 GE F 2 þ F E2 þ F 2 G2 GE þ E2 þ F 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 F E2 þ F 2 G2 f2 ðx; y; zÞ þ G2 GE F 2 þ F E2 þ F 2 G2 f3 ðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ðG þ EÞ F E2 þ F 2 G2 GE þ E2 þ F 2 F E2 þ F 2 G2 f4 ðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ G2 GE F 2 þ F E2 þ F 2 G2 ðG þ EÞ F E2 þ F 2 G2 f5 ðx; y; zÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ ðG þ EÞðF E2 þ F 2 G2 Þ f6 ðx; 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) and squaring both sides of Eq. (33). Eqs. (26)–(28) are then included in Eq. 33 as well as all of the f ðx; y; zÞ coefficients and this equation is expanded again. The remaining square roots in this equation are eliminated as well. The fully expanded form of this equation representing the fixedaxode point polynomial of the RRSS mechanism illustrated in Fig. 4 comes out to be a 56th order polynomial and is very long for inclusion in this paper. This equation, which is about 20 pages long or the MATHEMATICA notebook used in its derivation is available upon request from the authors.
5. Discussion The expanded fixed ISA point polynomial in this work was obtained using Mathematica software. The coordinates of the fixed ISA points as well as the crank and coupler displacement angles 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 ISA point is required. In the expanded fixed ISA point polynomial, replacing all of the numerical coefficients 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 RRSS fixed ISA point polynomial was to determine its order only, the numerical values of these coefficients are immaterial to the scope of this research.
6. Conclusion A synthesis method for approximating crank and coupler displacement angles for RRSS mechanisms, given points from the fixed instant screw axes (ISAs) of their fixed axodes, is developed and presented here. By incorporating ISA points in the fixed axode point polynomial and specifying the initial guesses for the joint parameters of the RRSS mechanism, the actual joint parameters of this mechanism were obtained. The RRSS fixed axode point polynomial was
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developed using the Cosine law approach introduced by M€ uller [1]. After expansion, the RRSS fixed axode point polynomial was shown to be a 56th order polynomial.
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