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Kinematic analysis of the Generalized Clemens Coupling and its variant mechanisms
Pergamon
Mech. Mach. Theory Vol. 32, No. 6, pp. 699-714, 1997 © 1997 ElsevierScienceLtd. All rights reserved Printed in Great Britain PII: S0094-114X(96)00062-6 0094-114x/97 $17.00+ 0.00
KINEMATIC ANALYSIS OF THE GENERALIZED CLEMENS C O U P L I N G A N D ITS V A R I A N T M E C H A N I S M S Y. B. ZHOUI" and R. BUCHAL Department of Mechanical Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9
F. G. FENTON Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M 1S 1A4 (Received 6 June 1995)
Abstract--The Generalized Clemens Coupling Ro-RSRR and its variant mechanisms Ro-SRRR, Ro-RRSR and Ro-RRRS are kinematically analysed by using the vector algebraic method. The closed-form input-output displacement equations of those mechanisms are derived as fourth-order polynomials. As compared to previous related works, the new approach presented in this paper is characterized by its standard analysis steps, compact expressions and simplicity. © 1997 Elsevier Science Ltd
I. I N T R O D U C T I O N
The Generalized Clemens Coupling Ro-RSRR and its variant mechanisms Ro-SRRR, Ro-RRSR and Ro-RRRS are very useful 5-1ink spatial mechanisms. They have already had much practical application in engineering, especially in agricultural and textile machines. The kinematic analysis of these mechanisms has been studied in the past. Using the method of generated surfaces, Torfason and Sharma [I] derived an 8th-order polynomial displacement equation for the Generalized Clemens Coupling; though they were aware of the fact that at least four of the eight roots were spurious solutions. Wallace and Freudenstein [2, 3] were probably the first who successfully obtained a 4th-order polynomial for the Generalized Clemens Coupling, using the geometric configuration method, by first disassembling the linkage into two configurations and then reassembling them under appropriate geometric constraints. In 1980, Zhang [4] obtained an 8th-order polynomial for mechanism Ro-RRSR using direction cosine matrices, Xie et al. [5] studied the same mechanism Ro-RRSR using rotation transformation tensors and the resulting polynomial was also of 8th-order. In the same year Duffy [6] (pp. 384-404) derived 4th-order polynomials for the Ro-SRRR, Ro-RSRR and Ro-RRSR mechanisms, using
spherical trigonometry. In 1987, Youm and Huang [7] analysed the Ro-SRRR, Ro-RSRR and Ro-RRSR mechanisms using a numerical scheme based on the direction cosine matrix. The analysis starts by converting the 5-1ink mechanism into a 4-1ink mechanism, so that the results of the displacement analysis of the corresponding 4-1ink mechanism can be applied. In the same year, Alizade et al. [8] presented mathematical models for analysis and synthesis of the Ro-SRRR and Ro-RRSR mechanisms. Each of the models is composed of a set of 12 simultaneous equations, governing the motion of the mechanism. However, the procedure and the feasibility for deriving a closed-form solution was neither presented nor mentioned. In 1988, Yih and Youm [9] introduced a numerical method, based on the spherical-Euler transformation, for the analysis of the Ro-SRRR, Ro-RSRR and Ro-RRSR mechanisms. Reference [9] is perhaps the latest related work. The method of generated surfaces [1] and the geometric configuration method [2, 3] can be classified as geometric methods (or graphical methods). They can solve the symmetric Ro-RSRR mechanism, however, they do not work for the unsymmetric cases: Ro-SRRR, Ro-RRSR and
Ro-RRRS. tPrescnt address: Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. 699
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The approaches using direction cosine matrices [4] and rotation transformation tensors [5] are actually quite similar. They exhibit the same problem: though they can yield 4th-order polynomials, the derivation of algebraic manipulation is not easy. Both of the two methods fall into the category of matrix methods, however, the approaches of Refs [4] and [5] have shown more flexibility in using matrices as compared to the traditional approach of the D - H matrix method [10]. For using the latter, one usually performs a series of matrix multiplications (or matrix homogeneous transformations) and obtain a set of 12 simultaneous algebraic equations by equating the corresponding elements of the two resulting matrices. However, only six or less than six of these equations are useful for deriving the I/O displacement equation; therefore, the calculation for the unused equations constitutes unavoidable extraneous work. Moreover, using the D - H approach, the spherical pair S as shown in Fig. l(a) cannot be dealt with directly unless it is simulated by the (RRR) pair as shown in Fig. l(b). Though the approaches of [4] and [5] are devoid of both the extraneous work and the necessity of simulating the S pair, they are more complex than the vectorial approach presented in this paper. Both the numerical scheme [7] and the numerical method [9] do not yield closed-form solutions, namely, polynomials. An algebraic approach, as opposed to a numerical approach, "can shed light on the basic aspects of linkage behavior, such as, for example, proportions for overclosure, criteria for rotability, determination of type, number of branches, transmission characteristics, number of ways of assembly, etc." ([6] p. 1). The idea of converting an n-link mechanism into an (n - 1)-link mechanism for the purpose of analysis [7] is novel. It is nevertheless by no means easy to apply and its applicability is limited. Of all the aforementioned related works, Duffy's approach [6] is probably the best for the following reasons: (i) it rendered the correct order of the displacement polynomials; (ii) it is efficient for being applicable to the analyses of all the variants of the Ro-3R-S mechanisms; (iii) it is devoid of the extraneous work encountered by the D - H method[10]. In fact, Duffy's spherical trigonometry method [6] has long been considered one of the most powerful methods for the kinematics of mechanisms and robots. However, its algebraic symbolic system, being unique but complicated, cannot be explained clearly in a couple of pages. This fact hinders the comprehension of the physical meaning of the algebraic equations. In this paper the Generalized Clemens Coupling Ro-RSRR and its variant mechanisms Ro-SRRR, Ro-RRSR and Ro-RRRS are kinematically analysed by using the vector algebraic
(a)
(b)
Fig. 1. (a) Spherical pair. (b) The (RRR) pair.
The generalized Clemens coupling
701
method [11]. The closed-form input-output displacement equations of these mechanisms are obtained as fourth order polynomials. The proposed approach is characterized by its efficiency, uniformity and simplicity. It is free of any of the problems encountered by all the approaches discussed above. Following is a brief summary of the analysis steps of the new approach: (a) develop the vector loop equation and the direction equations, based on the structure of the mechanism; (b) derive the first equation relating the input, output and auxiliary angles by squaring both sides of the vector loop equation; (c) derive the second equation relating the input, output and auxiliary angles from the scalar product of the end-axis vector and the vector loop equation; (d) eliminate the auxiliary angle from the two equations derived in steps (b) and (c) to obtain the desired input-output displacement equation. 2. A N A L Y S I S OF THE G E N E R A L I Z E D C L E M E N S C O U P L I N G Ro-RSRR
2.1. Displacement analysis
Mechanism Ro-RSRR is shown in Fig. 2. Let the input angle be 0~, then the basic variables of the mechanism are {02, 03, 04, 05}, where 03 is the single rotary variable corresponding to the triple of unconstrained rotary freedoms of the spherical pair. When the input angle 01 is given, unknowns to be determined are {02, 03, 04, 05}. Let 05 = O5 be the output angle, the first rotary variable to be determined; Let 02 = ~'2 be the auxiliary angle, the rotary variable to be eliminated from the simultaneous equations, as shown in Fig. 2(a). The vector loop equation of the mechanism can now be written as: K+L=F
f
(1)
K = J + I
(2)
L = p2q2 J = p4q4 - - $3a3 I = ($2a2 + p.ql) + I0 10 = Sial + p s q s - $4a4 F = -paq3
(1) (3) (4) (5) (6)
(la)
I is defined as the input vector of the mechanism. It is the sum of those vectors in the loop of the mechanism which are given or known at the beginning. J is the output vector of the mechanism. It is the sum of those constant-magnitude vectors in the loop of the mechanism that can be expressed as a function of the output angle (04 in this case). L is the auxiliary vector. It is the sum of those constant-magnitude vectors in the loop of the mechanism that can be expressed as a function of the auxiliary angle (~2 in this case). After the input, output and auxiliary angles are specified, there are two (unspecified) angles left, i.e. 03 and 04, which set the location of the "ends" of the floating vector F. Cutting off the loop at the two ends of the floating vector, we get two separate chains, where one part is fixed to the ground and the other part is floating. This is how the floating vector got its name. The following set of vector equations is called direction equations, which specifies the relative direction of any individual (unit) vector with its two adjacent (unit) vectors on the vector loop. Those who are not familiar with these equations are referred to equations (A2) and (A3) of the Appendix. q2 = C~k2ql + s~2a2 × ql a2 = Cgl2al + S~t2q~ X a~ ql = c01q5 + s 0 1 a l x q5 q3 = c04q4 - sO4a3 × q4
(1) (2) (3)
a3
(5)
----
C~34a4 - - S~X34q4 X Ih
q4 = cOsq~ -- sesa4 x q5
(4)
(6)
(Ib)
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/¥2
81
\ /
(b)
83
1
...
/
P4q
/ Psqs
Fig. 2. (a) The Ro-RSRR mechanism. (b) The Ro-RSRR mechanism.
(i) Derivation of the first equation relating 0,, O5 and ~2 Squaring both sides of equation (1) yields 2(K. L) = - K 2 + (F 2 - L 2)
(2)
Substituting (la-1) into (2) yields U • q2 --- V
(3)
The generalized Clemens coupling
-K 2+ (~ -p~)
703
(3a)
Substituting (lb-1) into (3) yields A cos 42 + B sin 42 = C U ' a 2 x q~ V
(4) (4a)
Since 4//2 is the auxiliary angle, the variable to be eliminated from the simultaneous equations, thus we transform (2) into the standard form (3), where only q2 is a function of qs2. Equation (3) is a transition from (2) to (4).
(iO Derivation of the second equation relating 01, 6)s and 42 An end-axis vector is a constrained rotary axial vector at the end of the floating vector. We can see from Fig. 2 that a3 is an end-axis vector of F. The dot product of a3 with both sides of (1) yields a3'
L = -a3" K
(5)
Substituting (la-1) into (5) yields U " q 2 = V'
(6)
U ' = p2a3
V' = - (a3" K)
(6a)
Substituting (lb-1) into (6) yields A' cos ~02 + B' sin ~k2 = C'
(7)
A' = U' • ql
t
B'=U'
as x q 2
(7a)
C'=V"
(iiO Derivation of the input-output displacement equation relating 6), Since {A, B, C} and {A', B', C'} of (4) and (7) contain only the input and output angles, therefore, eliminating the auxiliary angle ~02 from the two equations, we obtain the required input-output displacement equations. Solving (4) and (7) yields cos ~k2 = - Q2/Q3 sin 42 = QI/Q3
(8)
Q, = (AC' - A'C) Q2 ( B C ' - B'C)
(8a)
Q,
(aB" - A'B)
From (8) and the identity cos: 42 + sin: ~b2 = 1 we get
Qf + Qg = Q~
(9)
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Substituting (4a) and (7a) into (9a) yields Q2 Q3
(v'u(U × U'
(10)
a2 × ql
Substituting (3a) and (6a) into (10) we get {Q, = -2pz(a3 • K)(q, • K) - p 2 ( - K 2 + p~ - p~)(a3 • ql) -2p2[a3 .(I + J)] [q,. (I + J)] - p 2 [ - ( I + j)2 + p~ _ p~](a3 "q,) 2p2(q, x I). (a3 x J) + pfffl,q, - 2/t2I) • a3 + (2p2p4S3q,)'q4 + 2p2S3f12 3
(ll)
2p~a2" (K x a3) = 2p:2[(a2 × l ) ' a 3 - a2' (a3 × J)]
where f
/ ~1
=
2
2
2
2
(F + P2 - P3 + P4 -- $3)
S~ + p~ + 2S2[S~c~,2 + pssO~s~2 - $4(c~41ccq2 - s~,~s~,2cO~)]
+ 2p~(p~cO, - S4s~41sO,) -[- ( S 2 -~-p~ + SJ - 2SiS4co~41)
flz
(lla)
(q, • I) = (p, + psCOl -- S , SO~,ISOI)
Replacing tl~ by (a2 x q~) in QI of (11), we obtain an expression for Q2. Since (a3 × J), a3 and q4 of (1 1) are linear in cos O5 and sin 05, hence Q~, Q: and Q3 are also linear in cos O5 and sin 05. This implies that we can derive a fourth-order polynomial displacement equation relating O5 from (9). Substituting (la-3) and (lb-6) into (11) we obtain Q, = ( w , "q4 + z i ) Q2 (w2 q4 -I- z2) 03 (w3 q4 q- z3)
f
(12)
where "Wl = H(ql) W2 = H(a2 x ql) W3 = 2p2[sct34(a2 × I) x a4 -p4cot34a2 x a,] Z, = h(q,)
(12a)
Z2 = h(a2 x ql) Z3 = 2p22[c~34(a2 × |)" a4 + p4scta4a2" a4]
H(x) = 2p2p, co%4(x × I) × a4 + p2s0t34(fllX -- 2fl2I) × a, + 2p2p4S3x h ( x ) = 2p2p4sot34(x × I ) . a4 + p2c~34(fllx - 2flJ)" a, + 2p2S3(I • x)
(12b)
Substituting (12) into (9) yields ( W l . q , y + (w~. q , y - ( w , . q , y + 2 [ z , ( w , • q,) + z~(w~ • q,) - z , ( w , , q,)] + (Z? + Z~ - Z~) = O
(13)
Substituting (1b-6) into (13) yields /~l cos 2 05 + g: sin 20~ +/,t3 cos O~ sin O~ + #4 cos 05 + #5 sin 05 + #b = 0
(14)
The generalized Clemens coupling
705
//i = ( W , . qs) 2 + (W2. qs) 2 - - (W3" qs) 2 //2 = (W, "q5 x a4) 2 + (W2 "q5 x a4) 2 - - ( W 3 " q5 x a4) 2 //3 = 2(W, • qs)(W, 'q5 x a4) + 2(W2' qs)(W2 • q5 × a4) - - 2(W3. qs)(W3 • q5 x a4) //4 :
(14a)
2(Wl " q5)Zl + 2(W2" as)Z2 -- 2(W3" qs)Z3
//5 : 2(W~ • q5 x R4)Z~ --1-2 ( W 2 " q5 x a4)Z2 - - 2 ( W 3 " q5 X a4)Z3 //6 :
Z~ + Z 2 - - Z 2
Let y = tan(O5/2), then cos 05 = (1 - y2)/(1 + y2), sin O4
=
2y/(1 + y2). F r o m (14) we get
4
2 viy4-i : 0
(15)
/=0
f V0 = //1 -- //4 + [.26 Vl = --2l/3 -+- 2//5 V2 = --2//1 + 4//2 + 2//6
(15a)
V3 = 2//3 + 2//5 V4 =//1 +//4 +//6 Solving (15) we obtain y, then 05 = 2 tan -~ y; from (8) we get ~b2; now the vectors {K, L} can be determined from (la) and (lb). F r o m (1) and (laqS) we get q3 = - - ( K + L)/p3 = c o s 04q4 - - s i n 04a3 × q4
(16)
Using q4 and a5 × a4, respectively, to dot product both sides of (16), we obtain 04: cos 04 = - ( K + L ) . q4/p4 sin 04 (K + L ) ' a3 × q4/p4
(17)
:
The angle 03 can be determined from cos 03 = q2 • q3, namely, 03 = c o s - l ( q 2 • q3). At this point, the theoretical displacement analysis for the Generalized Clemens Coupling is complete. If we want to calculate numerical values for a specific mechanism, we need to scalarize (14a). Since there are quite a few dot products of W~ (i = 1-3) with q5 and q5 x a5 in (14a), respectively, it is convenient for the calculation to express W~ as W t = w,lq5 + w~2q5 x a4 + w~3a4 (i = 1-3)
(18)
Hence we have
{W, q5
Wil
W ~ • q5 × a4 =
(i = 1-3)
(19)
w,-2
As an example, the scalarized Z~ and the coefficients of W~ are as follows Z1 = - 2 p 2 p 4 s ~ ( c O l p 2 + C0~41S01pl) + 2p:,S3f12 + p2C~34(flIS~41S01 -- 2/~2p3)
(20)
Wlt = 2p2p4cot34(SOt41sOlpl -- c01p3) + p2s~34(fllC~41sOl + 2f12p2) + 2p2p4S3c01 W12 =" 2p2p4cOt34(SOt4tsOlp2 + COI41sOIp3) "at-p2sot:~(fllc01 -- 2f12pl) - 2p2p4S3c~41sOt
(21)
f
w~3 = 2p2p4cot34(sot41sOlp3 -- sot4tcOlp3) + 2p2p4S3sct41sOi
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I
I=p,qs+p2qsxa4+p~a4
pl =S~sOIs~2+p,c01 +p5 p2
(21a)
= 52s0~41c~12 - - plC~41S01-~-51S~41
p3 = 52(c~41c~12--s~41s~12COl)-~-pls~41s01-Jt- 51c~41-- S4
2.2. Velocity and acceleration analysis Differentiating both sides of (14) yields
~5 =
~1 COS2 05 "~ /12sin 2 O5 -F/13 cos O5 sin O5 +/i4 cos O5 +/is sin O5 +/16 (#1 -- #2)sin(2Os) --/h cos(205) + #4 sin O3 -- #5 cos O5
(22)
{/1,} (i = 1-6) of (22) can be calculated from (14a). From d2/dt ~ (14) we get O5. Other velocity variables {02, 03, 04} and the corresponding acceleration variables {0"2,/)'3,/)~} can also be easily determined. Here we only displayed the expression of 05, for it is the most important one among all the velocity and acceleration variables. 3. D I S P L A C E M E N T A N A L Y S I S OF THE Ro--SRRR M E C H A N I S M Mechanism Ro-SRRR is shown in Fig. 3. When the input angle 0, is given, the unknowns to be determined are {02, 03, 04, 05}. If the first displacement equation we want to derive isf(0,, 05) = 0, we can let 05 = 05. In this case, the auxiliary angle can be chosen from 03 and 04. If we choose 04 as the auxiliary angle, the procedure for deriving f(01, 05) = 0 will be identical to that of Section 2. If we choose 03 as the auxiliary angle, the procedure for deriving f(0], 05) = 0 will be somewhat different. Here we let 05 = 05, 03 = if3, then, the vector loop equation of the mechanism can be written as follows,
(23)
K=F K=I-t-J
(1)
J=p4q4-S3a3
(2) (3) (4)
I=plq~ +(Sial + p 3 a s - $4a4) F= -(p2q2-S2a2+p3q3)
(23a)
jil3 P3,
Plql P4~
,¢sq~
Fig. 3. The Ro-RSRR mechanism.
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The generalized Clemens coupling
(1) q2 = c@2q3- s*2a2 x q3 (2)
qI = c&q5 + s&a, x q5
a2 = ca23a3- sa23q3x a3 (3) q3 = d&q4 - s&a3 x q4 (4)
Wb)
a3 = ca34a4 - sa34q4x a4 (5) q4 = d&q5 - s&a4 x qs (6) 3.1. Derivation of the first equation relating Or, Oj and ti3 Squaring both sides of (23) yields K2 = F2
(24)
Substituting (23a-4) into (24) yields COS $3 =
[K2 - (p: f s: + fJ:)]/(@psp3)
(25)
3.2. Derivation of the second equation relating 8,, O5 and ~5, We can see from Fig. 2 that a3 is an end-axis vector of the floating vector F. Dot product a3 with both sides of (23) yields K.a3=F.a3
(26)
Substituting (23a-4) into (26) we get sin $3 =
[-(K.a3) +
(27)
S2ca231/(p2sa23)
3.3. Derivation of the input-output displacement equation relating OS From (25) (27) and the identity cos2 II/3+ sin2 Ic/3= 1 we obtain Q: + Q: = Q:
(28)
I = sa23K2 - sad& + Si + pi) E2= -2p3(K * a3) + 2p&ca23
(284
Q3 = 2p2pm23
QI = (WI .q4) + Z, = (WI * q5)cos 05 + (W, . q5 x a)sin O5 + Z, Q2 = (W2 * 44) + Z2 = (W2 *qs)cos 05 + (w, . q5 x a)sin Q, + z2
GW
WI = 2sa23(p41- S3sa3aa4x I) W2 = 2p3sa34a4x I ZI = -2S3sa23ca34(I * a4) + sa23(12+ J2 - pi - S: - pi) z2
=
-@3casr(I
* %)+
&'3(s3
+
(28~)
s2Ca23)
Substituting (28b) into (28) yields PI C0S2
OS
+
p2
sin2 6% + p3 cos OS sin OS + j4 cos Q5 + cc5sin OS + pa = 0
PI = (W,
.q5)*
p2
.
=
WI
P3 = WI pl = WI
+
(W2
(29)
*q5j2
q5 x a4)2+ (W2 . q5 x a4)*
* q5wI
*q5
x
lh)
+
2(w*
,q5)(W2.
* q5)Z + 2(W2 * q5)Z2
P5 = 2(W, * 95 x a4)ZI + 2oy2 * q, /46 = z: + z; - Q:
x adz*
q, x a4)
(294
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Let y = tan(Os/2), then cos O5 = (1 - y2)/(1 + y2), sin O5 = 2y/(1 + y2). From (29) we obtain 4
Z v~y4-i__ - 0 i=0
(30)
1)0 =/.~1 - /~4 + f16 Vl = -2/.t3 q- 2/z5
f
(30a)
vz = -2/11 + 4fl2 "+- 2fl6 1)5 2~3 + 2/~5 I)4 = /[~l "Jl-~4 "31-/~6
Solving (30) we get y, then 05 = 2 tan -~ y; From (25) and (27) we obtain ~3, namely 05; using q4 and a3 × q4, respectively, to dot product both sides of (23) yields -(P2q2-S2a2+p3q3)'q4 (I + J) a3 x q4 = -(Pzq2 - $2a2 + p 3 q 3 ) " a3 × q4
(31)
(I. q4) "]- p4 = --p2[C03 COS 04 -- C~23S03 sin 04] + $2 + s~23 sin 04 - p3 cos 04 (l a3 × q4) = p2[c03 sin 04 + c0~23s03 cos 04] ---[-82s~23 cos 04 + p3 sin 04
(31a)
f - - (p2c03 + p3)cos 04 + (p2co~23s03 q- S2s~23)sin 04 - (I. q4) + p4 (pzc~23s03 + S2s~23)cos 04 + (p2c03 + p3)sin 04 = ~ " a3 × q4)
(31b)
(I+J).q,=
i.e.
i.e.
Solving (31b) yields 04: ICOS 04
--
(p2c0~23S03q- $2s~23)(I • a3 × q4) - (p2c03 + p3)(I • q4 + p4) (p2c03 q-p3) 2 --t- (p2c~23S03 -~- 82S0f23) 2
(32) [.sin 04 = (p2c~23s03 + $2s~23)(I • q, + p4) + (p2c03 + p3)(I • a3 × q4) (p2c03 --]-p3) 2 "l- (p2c~23S03 "3!-S2S0~23)2 The angle 02 can be determined from 02 = cos-l(ql • q2). At this point, the theoretical displacement analysis for the R o - S R R R mechanism is complete.
4. D I S P L A C E M E N T
ANALYSIS
O F T H E Ro-RRRS M E C H A N I S M
Mechanism R o - R R R S is shown in Fig. 4. This mechanism has never been analysed before. When the input angle 01 is given, the unknowns to be determined are {02, 03, 04, 05}. Let f(01, 02) = 0 be the first displacement equation to be derived. Let 02 = 02, 04 = @4. The vector loop equation of the mechanism can be written as K=F
=i+j =(p2q2-S3a3) =(-S2a2+plq,)+(Sta,
fi
= --p3q3"}-S4a4--p4q4
(33)
+psas)
(1) (2) (3) (4)
The generalized Clemens coupling
709
?
II1
Ptqt
~/~4
Fig 4. The
Ro-RRRS mechanism.
"q4 -- c~4q3 -q- s~b4a4 x q3
(1)
a4 ---- c~34a3 + s~34q3 × 83
(2)
q3 = c03q2 d- s03a3 x q2
(3)
a3 = c~23a2 d- s~23q2 x a:
(4)
q2 = cO2ql "t- st92a2 x qj
(5)
a2 = c~2a~ + s~2ql x a~
(6)
ql = cOiq5 + sOlal x q5
(7)
(33b)
4.1. Der&ation of the first equation relating 01, O: and ~4 Squaring both sides o f (33) yields K2 : F2
(34)
(p2 + $24 q_ p24)]/(2p3p4 )
(35)
Substituting (33a--4) into (34) yields COS ~4 = [K 2 -
4.2. Derivation of the second equation relating 01, 02 and ~k4 We can see from Fig. 3 that a3 is an axial vector between L and K. D o t product a3 with both sides of (33) yields K • a3
=
F • a3
(36)
Substituting (33a--4) into (36) yields sin ~k, = [ - ( K " a3) + S4c~34]/(p4s~34)
(37)
4.3. Derivation of the input-output displacement equation relating 02 F r o m (35), (37) and the identity cos 2 ~k2 + sin: ~k2 = 1 we obtain +
=
(38)
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{
a~ = s~t34K2 - sct34(p2 + S 2 + P[) Q2 -2p3((K. 43) + 2p3S4cot34 Q3 2p3p4s~34
(38a)
{Qt = (Wt ' q2) + Z~ = (W~ • q0cos 02 + (W~ • a2 × qt)sin 02 + Zt Q2 (W2 q2) + Z2 (W2 q,)cos 02 + (WE 42 × q,)sin 02 + Z2
(38b)
Wl = 2S~34(p2I -- $3S~23a2 × I)
I
W2
--2p3s~34a2 x I
ZI
--2S3c0~23s~34(I"82) "31-SOt34(I2 "t- j2
Z2
--2p3c~23(I " a2) + 2p3($3 + $4c~t34)
_ p~ _ $42 _ p2)
(38c)
Substituting (38b) into (38) yields /2~ cos 2 02 +/22 sin 2 O2 +/23 COS [92 sin 02 +/24 cos O2 +/23 sin O2 +/26 = 0
(39)
/2i ~---(Wt" q,)2 + (WE" ql) 2 /22 (W, " 42 × ql) 2 -t- (W2" 42 x q,)2 2(W,- q,)(Wt • a2 × q,) + 2(W2. q0(W2 • a2 × q,) = 2(W,. q,)Z, + 2(W2" qt)Z2 /25 /26
L
(39a)
2(W, a2 x q,)Z, + 2(W2. a2 x q~)Z2 Z~ "-t-Z2 -- a 2
Let y = tan(Od2), then cos O4 = (1 - y2)/(1 + y2), sin O4 = 2y/(1 + fl). From (39) we obtain 4 E v,Y4 - ' = 0 i=0
(40)
v0 =/21 --/24 +/26 v~ = --2/23 + 2/25 v2 = --2/2t + 4/22 + 2/26
(40a)
v3 = 2/23 + 2/2s v4 =/21 +/24 +/26 Solving (40) we gety, then 02 = tan -~ y; from (35) and (37) we obtain 04; using q2 and (43 x q2), respectively, to dot product both sides of (33) yields 03:
f COS 03 =
(P3 + p4c04)(I" q2 + p2) + (S4s~34 "31-p4s04c~3,)(I " a3 X q2) (t73 q- p4C04) 2 d- (84s~34 "Jr-p4s04c~34) 2
(41) sin 03
($4s~3, + p, s04cu34)(I, q2 + p2) - (1o3 31- p4cO,)(I" a3 × q2) (t93 4-/94C04) 2 4" ($4s~t34 + p4sO4cot34) 2
The theoretical displacement analysis for the R o - R R R S mechanism is now complete. 5. A N A L Y S I S OF T H E Ro-RRSR M E C H A N I S M
Mechanism R o - R R S R is shown in Fig. 5. The main purpose of this section is to show the flexibility of the vector algebraic method in the analysis of spatial mechanisms, therefore, details are not given. (1) Letf(0t, 02) = 0 be the first displacement equation to be derived; let 02 = 02, then the auxiliary angle can be either 03 or 05.
The generalized Clemens coupling
711
Case 1.1. Let 02 = 02, 03 = 43. The vector loop equation o f the mechanism can be written as follows, K + L = F
(42)
L = p3q3 K=I+J J = p2q2 - $3a3 I = ( - $ 2 a 2 + p,ql) + I0
(42a)
Io = (Sial + psq5 -- $4a4) F = --p4q4 Case 1.2. Let 02 =
~)2, 05 :
4/]5.
The vector loop equation is K + L = F
(43)
" L = p4q4 K=I+J J = p2q2 S383 I = ( - $ 2 a 2 +plq~) + Io - -
I0 = (Sial + psq5
-
(43a)
$4a4)
F = -p3q3
(2) Let f(01, 03) = 0 be the first displacement equation to be derived; let 03 = 03, then the auxiliary angle can be either 02 or 05. Case 2.1. Let 03 = O3, 02 = ~'2; then I + L = F
(44)
{ ~ = p2q2 $3a3 + p3q3 L ( - S 2 a 2 + p ~ q ~ ) + Io -
(S~a~ + psq5 - $4a4)
(44a)
-p4q4
jr as
03 a2
at
/Ptql
P4¢
\ Psqs
Fig. 5. The Ro-RRSR mechanism. MMT 32/6~U
Y. B. Zhou et al.
712 Case 2.2. Let 03 = 03, 05 = ffs; then
(45)
I+L=F L = p4q4 I = -($2a2 + p~q~) + (S~al + psq5 F : --(P2q2 -- 83a3 Jr- p3q3)
-
-
$4a4)
(45a)
(3) Let the first I / 0 displacement equation we want to derive bef(O~, 05) = O; let 05 = 05, then the auxiliary angle can be either 02 or 03. Case 3.1. Let 05 = 05, 02 = ~2; then K + L = F
(46)
L = p2q2 - $3a3 K=I+J
J = p4q4 I = (-$2a2 + p~q~) + Io
(46a)
Io = (& aj + psq5 - $4a4) F = -p3q3 Case 3.2. Let 05 = 05, 03 = ~3; then K = F
(47)
K=I+J
f
J = p4q4 I = (-$2a2 + plql) + Io
Io
=
(47a)
(Sial + psq5 -- $4a4)
F = -(p2q2 - $3a3 + p3q3)
The analysis procedures for Cases 1.1, 1.2, 2.1, 2.2 and 3.1 are identical to that of Section 2, whereas the analysis for Case 3.2 is identical to that of Section 3. 6. CONCLUSION Since all the variants of the Ro-3R-S mechanisms can be analysed, the efficiency of the proposed approach warrants no additional comment. As for the uniformity and simplicity of the proposed approach, let us take a close look at the analysis procedures. Though the analysis steps for each of the four variants of the Ro-3R-S mechanisms are the same as outlined at the end of the Introduction, there are two slightly different versions of the analysis procedures, as represented in Sections 2 and 3. Let us first examine Section 2. In equations (3) and (6) everything unrelated to q2, a function of the auxiliary angle ~k2, are included in [U, V} and {U', V'}, respectively. Similarly, in equations (4) and (7) those terms unrelated to ~2 are also collected into {A, B, C} and {A', B', C'}. Unlike all the matrix related approaches, we do not need to expand (i.e. scalarize) everything at the beginning, we manipulate only those vectors we need, i.e. q2 and ~02; thus we are able to obtain equations (4) and (7) by only one transformation and a few lines of derivation, respectively. In order to easily avoid rendering an 8th-order polynomial from equation (9), we should linearize {Or, Q2, Q3} of equation (8a) in terms of sin o5 and cos 05. This can be achieved by only one or two lines of derivation as shown in equation (11). The analysis procedure in Section 3 is similar to but simpler than that of Section 2. Comparing equations {(4), (7), (9), (14), (15), (15a)} of Section 2 with the equations {(25), (27), (28), (29), (30), (30a)} of Section 3, respectively, the uniformity of the proposed approach is not hard to perceive.
The generalized Clemens coupling
713
q
II2
v
111 Fig. A1.
It is obvious that in terms of simplicity no other mathematic tool can beat vectors. Therefore, the proposed approach is unlikely to be matched in simplicity by other possible approaches employing quaternion algebra, line-coordinates or screw theory, etc.
REFERENCES
1. Torfason, L. E. and Sharma, A. K., Analysis of spatial R - R ~ 3 - R - R mechanisms by the method of generated surfaces. Journal of Engineering for Industry, Trans. ASME, Series B, 1973, 95(3), 704-780. 2. Wallace, D. M., Displacement analysis of spatial mechanisms with more than four links. Doctoral dissertation, Columbia University, New York. No. 69, 13006 University Microfilms, Ann Arbor, Michigan, 1968. 3. Wallace, D. M. and Freudenstein, F., Displacement analysis of the Generalized Clemens Coupling, the R - R - S - R - R spatial linkage. Journal of Engineering for Industry, Trans. ASME, Series B, 1975, 97(2), 575-580. 4. Zhang, Q. X., Analysis and Synthesis of Spatial Mechanisms. South China University of Technology Press, Guangzhou, 1980; Mechanical Industry Press, Beijing, 1984, pp. 218-232. 5. Xie. C. X., Zheng, S. X. and Ou, H. H., Configuration analysis of spatial five-link mechanism RRRSR by means of the tensor rotational transformation method. Journal of South China University of Technology, 1980, 8(2). 6. Duffy, J., Analysis of Mechanisms and Robot Manipulators, Chap. 10. Arnold, Great Britain/Halstead Press, U.S.A., 1980, pp. 384-404. 7. Youm, Y. and Huang, T. C., Displacement analysis of 4RIG five-link spatial mechanisms using exact solution of four-link spatial mechanisms. Mechanism and Machine Theory, 1987, 22(3), 189-197. 8. Alizade, R. I., Duffy, J. and Azizov, A. A., Mathematical models for analysis and synthesis of spatial mechanisms--II. Mechanism and Machine Theory, 1987, 18(5), 309-315. 9. Yih, T. C. and Youm, T., Kinematics of spatial mechanisms in matrix notation. Trends and Developments in
Mechanisms, Machines, and Robotics--1988, Proceedings of The 20th ASME Biennial Mechanisms Conference, DE-VoI.15-1, Kissimmee, FL, September 1988, pp. 429-433. 10. Denavit, J. and Hartenberg, R. S., A kinematic notation for lower-pair mechanisms based on matrices. Trans. ASME, Journal of Applied Mechanisms, 1956, 22, 215-221. 11. Zhou, Y. B., Kinematics of spatial mechanisms. Ph.D. dissertation, The University of Western Ontario, 1994.
APPENDIX
Several Important Formulae (1) In Fig. A1, unit vector q is perpendicular to both unit vectors a, and a2; angle ~t2 is the right-hand-rotation angle from a~ to a2 to about q, then, we have a~ × a2 = q sin ct,2
(A1)
a2 = cos ~t,2a, ~ sin ~t12q x al
(A2)
a t ---- COS Ctt2a2 - - s i n ~ i 2 q
× a2
(A3)
(a-b)c
(A4)
(2) Let a, b, e and d be four arbitrary vectors, then a x (b × c) = ( a . c ) b -
(axb)
xe=(a.e)b-(b.e)a
(a x b)' (e x d) = ( a . e)(b" d) - (a' d)(b. e)
(A5) (A6)
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Y . B . Z h o u et al.
(3) Let e~, e2 and e3 be a positive triple of mutually perpendicular unit vectors, a and b are two arbitrary vectors, then a = ~ (a. e,)e~
(A7)
a . b = ~ (a. e,)(b • e,)
(A8)
a 2 = ~ (a. e,) 2
(A9)
(e. el) 2 -t.- (a. e2)2 = (a × e3)2
(A10)
Formulae (A1)-(A6) are frequently used in this paper. Their proofs can be easily verified or found in books on engineering mathematics.
Mech. Mach. Theory Vol. 32, No. 6, pp. 699-714, 1997 © 1997 ElsevierScienceLtd. All rights reserved Printed in Great Britain PII: S0094-114X(96)00062-6 0094-114x/97 $17.00+ 0.00
KINEMATIC ANALYSIS OF THE GENERALIZED CLEMENS C O U P L I N G A N D ITS V A R I A N T M E C H A N I S M S Y. B. ZHOUI" and R. BUCHAL Department of Mechanical Engineering, The University of Western Ontario, London, Ontario, Canada N6A 5B9
F. G. FENTON Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M 1S 1A4 (Received 6 June 1995)
Abstract--The Generalized Clemens Coupling Ro-RSRR and its variant mechanisms Ro-SRRR, Ro-RRSR and Ro-RRRS are kinematically analysed by using the vector algebraic method. The closed-form input-output displacement equations of those mechanisms are derived as fourth-order polynomials. As compared to previous related works, the new approach presented in this paper is characterized by its standard analysis steps, compact expressions and simplicity. © 1997 Elsevier Science Ltd
I. I N T R O D U C T I O N
The Generalized Clemens Coupling Ro-RSRR and its variant mechanisms Ro-SRRR, Ro-RRSR and Ro-RRRS are very useful 5-1ink spatial mechanisms. They have already had much practical application in engineering, especially in agricultural and textile machines. The kinematic analysis of these mechanisms has been studied in the past. Using the method of generated surfaces, Torfason and Sharma [I] derived an 8th-order polynomial displacement equation for the Generalized Clemens Coupling; though they were aware of the fact that at least four of the eight roots were spurious solutions. Wallace and Freudenstein [2, 3] were probably the first who successfully obtained a 4th-order polynomial for the Generalized Clemens Coupling, using the geometric configuration method, by first disassembling the linkage into two configurations and then reassembling them under appropriate geometric constraints. In 1980, Zhang [4] obtained an 8th-order polynomial for mechanism Ro-RRSR using direction cosine matrices, Xie et al. [5] studied the same mechanism Ro-RRSR using rotation transformation tensors and the resulting polynomial was also of 8th-order. In the same year Duffy [6] (pp. 384-404) derived 4th-order polynomials for the Ro-SRRR, Ro-RSRR and Ro-RRSR mechanisms, using
spherical trigonometry. In 1987, Youm and Huang [7] analysed the Ro-SRRR, Ro-RSRR and Ro-RRSR mechanisms using a numerical scheme based on the direction cosine matrix. The analysis starts by converting the 5-1ink mechanism into a 4-1ink mechanism, so that the results of the displacement analysis of the corresponding 4-1ink mechanism can be applied. In the same year, Alizade et al. [8] presented mathematical models for analysis and synthesis of the Ro-SRRR and Ro-RRSR mechanisms. Each of the models is composed of a set of 12 simultaneous equations, governing the motion of the mechanism. However, the procedure and the feasibility for deriving a closed-form solution was neither presented nor mentioned. In 1988, Yih and Youm [9] introduced a numerical method, based on the spherical-Euler transformation, for the analysis of the Ro-SRRR, Ro-RSRR and Ro-RRSR mechanisms. Reference [9] is perhaps the latest related work. The method of generated surfaces [1] and the geometric configuration method [2, 3] can be classified as geometric methods (or graphical methods). They can solve the symmetric Ro-RSRR mechanism, however, they do not work for the unsymmetric cases: Ro-SRRR, Ro-RRSR and
Ro-RRRS. tPrescnt address: Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. 699
700
Y.B. Zhou et al.
The approaches using direction cosine matrices [4] and rotation transformation tensors [5] are actually quite similar. They exhibit the same problem: though they can yield 4th-order polynomials, the derivation of algebraic manipulation is not easy. Both of the two methods fall into the category of matrix methods, however, the approaches of Refs [4] and [5] have shown more flexibility in using matrices as compared to the traditional approach of the D - H matrix method [10]. For using the latter, one usually performs a series of matrix multiplications (or matrix homogeneous transformations) and obtain a set of 12 simultaneous algebraic equations by equating the corresponding elements of the two resulting matrices. However, only six or less than six of these equations are useful for deriving the I/O displacement equation; therefore, the calculation for the unused equations constitutes unavoidable extraneous work. Moreover, using the D - H approach, the spherical pair S as shown in Fig. l(a) cannot be dealt with directly unless it is simulated by the (RRR) pair as shown in Fig. l(b). Though the approaches of [4] and [5] are devoid of both the extraneous work and the necessity of simulating the S pair, they are more complex than the vectorial approach presented in this paper. Both the numerical scheme [7] and the numerical method [9] do not yield closed-form solutions, namely, polynomials. An algebraic approach, as opposed to a numerical approach, "can shed light on the basic aspects of linkage behavior, such as, for example, proportions for overclosure, criteria for rotability, determination of type, number of branches, transmission characteristics, number of ways of assembly, etc." ([6] p. 1). The idea of converting an n-link mechanism into an (n - 1)-link mechanism for the purpose of analysis [7] is novel. It is nevertheless by no means easy to apply and its applicability is limited. Of all the aforementioned related works, Duffy's approach [6] is probably the best for the following reasons: (i) it rendered the correct order of the displacement polynomials; (ii) it is efficient for being applicable to the analyses of all the variants of the Ro-3R-S mechanisms; (iii) it is devoid of the extraneous work encountered by the D - H method[10]. In fact, Duffy's spherical trigonometry method [6] has long been considered one of the most powerful methods for the kinematics of mechanisms and robots. However, its algebraic symbolic system, being unique but complicated, cannot be explained clearly in a couple of pages. This fact hinders the comprehension of the physical meaning of the algebraic equations. In this paper the Generalized Clemens Coupling Ro-RSRR and its variant mechanisms Ro-SRRR, Ro-RRSR and Ro-RRRS are kinematically analysed by using the vector algebraic
(a)
(b)
Fig. 1. (a) Spherical pair. (b) The (RRR) pair.
The generalized Clemens coupling
701
method [11]. The closed-form input-output displacement equations of these mechanisms are obtained as fourth order polynomials. The proposed approach is characterized by its efficiency, uniformity and simplicity. It is free of any of the problems encountered by all the approaches discussed above. Following is a brief summary of the analysis steps of the new approach: (a) develop the vector loop equation and the direction equations, based on the structure of the mechanism; (b) derive the first equation relating the input, output and auxiliary angles by squaring both sides of the vector loop equation; (c) derive the second equation relating the input, output and auxiliary angles from the scalar product of the end-axis vector and the vector loop equation; (d) eliminate the auxiliary angle from the two equations derived in steps (b) and (c) to obtain the desired input-output displacement equation. 2. A N A L Y S I S OF THE G E N E R A L I Z E D C L E M E N S C O U P L I N G Ro-RSRR
2.1. Displacement analysis
Mechanism Ro-RSRR is shown in Fig. 2. Let the input angle be 0~, then the basic variables of the mechanism are {02, 03, 04, 05}, where 03 is the single rotary variable corresponding to the triple of unconstrained rotary freedoms of the spherical pair. When the input angle 01 is given, unknowns to be determined are {02, 03, 04, 05}. Let 05 = O5 be the output angle, the first rotary variable to be determined; Let 02 = ~'2 be the auxiliary angle, the rotary variable to be eliminated from the simultaneous equations, as shown in Fig. 2(a). The vector loop equation of the mechanism can now be written as: K+L=F
f
(1)
K = J + I
(2)
L = p2q2 J = p4q4 - - $3a3 I = ($2a2 + p.ql) + I0 10 = Sial + p s q s - $4a4 F = -paq3
(1) (3) (4) (5) (6)
(la)
I is defined as the input vector of the mechanism. It is the sum of those vectors in the loop of the mechanism which are given or known at the beginning. J is the output vector of the mechanism. It is the sum of those constant-magnitude vectors in the loop of the mechanism that can be expressed as a function of the output angle (04 in this case). L is the auxiliary vector. It is the sum of those constant-magnitude vectors in the loop of the mechanism that can be expressed as a function of the auxiliary angle (~2 in this case). After the input, output and auxiliary angles are specified, there are two (unspecified) angles left, i.e. 03 and 04, which set the location of the "ends" of the floating vector F. Cutting off the loop at the two ends of the floating vector, we get two separate chains, where one part is fixed to the ground and the other part is floating. This is how the floating vector got its name. The following set of vector equations is called direction equations, which specifies the relative direction of any individual (unit) vector with its two adjacent (unit) vectors on the vector loop. Those who are not familiar with these equations are referred to equations (A2) and (A3) of the Appendix. q2 = C~k2ql + s~2a2 × ql a2 = Cgl2al + S~t2q~ X a~ ql = c01q5 + s 0 1 a l x q5 q3 = c04q4 - sO4a3 × q4
(1) (2) (3)
a3
(5)
----
C~34a4 - - S~X34q4 X Ih
q4 = cOsq~ -- sesa4 x q5
(4)
(6)
(Ib)
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Y . B . Zhou et al. (a)
/¥2
81
\ /
(b)
83
1
...
/
P4q
/ Psqs
Fig. 2. (a) The Ro-RSRR mechanism. (b) The Ro-RSRR mechanism.
(i) Derivation of the first equation relating 0,, O5 and ~2 Squaring both sides of equation (1) yields 2(K. L) = - K 2 + (F 2 - L 2)
(2)
Substituting (la-1) into (2) yields U • q2 --- V
(3)
The generalized Clemens coupling
-K 2+ (~ -p~)
703
(3a)
Substituting (lb-1) into (3) yields A cos 42 + B sin 42 = C U ' a 2 x q~ V
(4) (4a)
Since 4//2 is the auxiliary angle, the variable to be eliminated from the simultaneous equations, thus we transform (2) into the standard form (3), where only q2 is a function of qs2. Equation (3) is a transition from (2) to (4).
(iO Derivation of the second equation relating 01, 6)s and 42 An end-axis vector is a constrained rotary axial vector at the end of the floating vector. We can see from Fig. 2 that a3 is an end-axis vector of F. The dot product of a3 with both sides of (1) yields a3'
L = -a3" K
(5)
Substituting (la-1) into (5) yields U " q 2 = V'
(6)
U ' = p2a3
V' = - (a3" K)
(6a)
Substituting (lb-1) into (6) yields A' cos ~02 + B' sin ~k2 = C'
(7)
A' = U' • ql
t
B'=U'
as x q 2
(7a)
C'=V"
(iiO Derivation of the input-output displacement equation relating 6), Since {A, B, C} and {A', B', C'} of (4) and (7) contain only the input and output angles, therefore, eliminating the auxiliary angle ~02 from the two equations, we obtain the required input-output displacement equations. Solving (4) and (7) yields cos ~k2 = - Q2/Q3 sin 42 = QI/Q3
(8)
Q, = (AC' - A'C) Q2 ( B C ' - B'C)
(8a)
Q,
(aB" - A'B)
From (8) and the identity cos: 42 + sin: ~b2 = 1 we get
Qf + Qg = Q~
(9)
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Y.B. Zhou
et al.
Substituting (4a) and (7a) into (9a) yields Q2 Q3
(v'u(U × U'
(10)
a2 × ql
Substituting (3a) and (6a) into (10) we get {Q, = -2pz(a3 • K)(q, • K) - p 2 ( - K 2 + p~ - p~)(a3 • ql) -2p2[a3 .(I + J)] [q,. (I + J)] - p 2 [ - ( I + j)2 + p~ _ p~](a3 "q,) 2p2(q, x I). (a3 x J) + pfffl,q, - 2/t2I) • a3 + (2p2p4S3q,)'q4 + 2p2S3f12 3
(ll)
2p~a2" (K x a3) = 2p:2[(a2 × l ) ' a 3 - a2' (a3 × J)]
where f
/ ~1
=
2
2
2
2
(F + P2 - P3 + P4 -- $3)
S~ + p~ + 2S2[S~c~,2 + pssO~s~2 - $4(c~41ccq2 - s~,~s~,2cO~)]
+ 2p~(p~cO, - S4s~41sO,) -[- ( S 2 -~-p~ + SJ - 2SiS4co~41)
flz
(lla)
(q, • I) = (p, + psCOl -- S , SO~,ISOI)
Replacing tl~ by (a2 x q~) in QI of (11), we obtain an expression for Q2. Since (a3 × J), a3 and q4 of (1 1) are linear in cos O5 and sin 05, hence Q~, Q: and Q3 are also linear in cos O5 and sin 05. This implies that we can derive a fourth-order polynomial displacement equation relating O5 from (9). Substituting (la-3) and (lb-6) into (11) we obtain Q, = ( w , "q4 + z i ) Q2 (w2 q4 -I- z2) 03 (w3 q4 q- z3)
f
(12)
where "Wl = H(ql) W2 = H(a2 x ql) W3 = 2p2[sct34(a2 × I) x a4 -p4cot34a2 x a,] Z, = h(q,)
(12a)
Z2 = h(a2 x ql) Z3 = 2p22[c~34(a2 × |)" a4 + p4scta4a2" a4]
H(x) = 2p2p, co%4(x × I) × a4 + p2s0t34(fllX -- 2fl2I) × a, + 2p2p4S3x h ( x ) = 2p2p4sot34(x × I ) . a4 + p2c~34(fllx - 2flJ)" a, + 2p2S3(I • x)
(12b)
Substituting (12) into (9) yields ( W l . q , y + (w~. q , y - ( w , . q , y + 2 [ z , ( w , • q,) + z~(w~ • q,) - z , ( w , , q,)] + (Z? + Z~ - Z~) = O
(13)
Substituting (1b-6) into (13) yields /~l cos 2 05 + g: sin 20~ +/,t3 cos O~ sin O~ + #4 cos 05 + #5 sin 05 + #b = 0
(14)
The generalized Clemens coupling
705
//i = ( W , . qs) 2 + (W2. qs) 2 - - (W3" qs) 2 //2 = (W, "q5 x a4) 2 + (W2 "q5 x a4) 2 - - ( W 3 " q5 x a4) 2 //3 = 2(W, • qs)(W, 'q5 x a4) + 2(W2' qs)(W2 • q5 × a4) - - 2(W3. qs)(W3 • q5 x a4) //4 :
(14a)
2(Wl " q5)Zl + 2(W2" as)Z2 -- 2(W3" qs)Z3
//5 : 2(W~ • q5 x R4)Z~ --1-2 ( W 2 " q5 x a4)Z2 - - 2 ( W 3 " q5 X a4)Z3 //6 :
Z~ + Z 2 - - Z 2
Let y = tan(O5/2), then cos 05 = (1 - y2)/(1 + y2), sin O4
=
2y/(1 + y2). F r o m (14) we get
4
2 viy4-i : 0
(15)
/=0
f V0 = //1 -- //4 + [.26 Vl = --2l/3 -+- 2//5 V2 = --2//1 + 4//2 + 2//6
(15a)
V3 = 2//3 + 2//5 V4 =//1 +//4 +//6 Solving (15) we obtain y, then 05 = 2 tan -~ y; from (8) we get ~b2; now the vectors {K, L} can be determined from (la) and (lb). F r o m (1) and (laqS) we get q3 = - - ( K + L)/p3 = c o s 04q4 - - s i n 04a3 × q4
(16)
Using q4 and a5 × a4, respectively, to dot product both sides of (16), we obtain 04: cos 04 = - ( K + L ) . q4/p4 sin 04 (K + L ) ' a3 × q4/p4
(17)
:
The angle 03 can be determined from cos 03 = q2 • q3, namely, 03 = c o s - l ( q 2 • q3). At this point, the theoretical displacement analysis for the Generalized Clemens Coupling is complete. If we want to calculate numerical values for a specific mechanism, we need to scalarize (14a). Since there are quite a few dot products of W~ (i = 1-3) with q5 and q5 x a5 in (14a), respectively, it is convenient for the calculation to express W~ as W t = w,lq5 + w~2q5 x a4 + w~3a4 (i = 1-3)
(18)
Hence we have
{W, q5
Wil
W ~ • q5 × a4 =
(i = 1-3)
(19)
w,-2
As an example, the scalarized Z~ and the coefficients of W~ are as follows Z1 = - 2 p 2 p 4 s ~ ( c O l p 2 + C0~41S01pl) + 2p:,S3f12 + p2C~34(flIS~41S01 -- 2/~2p3)
(20)
Wlt = 2p2p4cot34(SOt41sOlpl -- c01p3) + p2s~34(fllC~41sOl + 2f12p2) + 2p2p4S3c01 W12 =" 2p2p4cOt34(SOt4tsOlp2 + COI41sOIp3) "at-p2sot:~(fllc01 -- 2f12pl) - 2p2p4S3c~41sOt
(21)
f
w~3 = 2p2p4cot34(sot41sOlp3 -- sot4tcOlp3) + 2p2p4S3sct41sOi
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Y.B. Zhou et al.
I
I=p,qs+p2qsxa4+p~a4
pl =S~sOIs~2+p,c01 +p5 p2
(21a)
= 52s0~41c~12 - - plC~41S01-~-51S~41
p3 = 52(c~41c~12--s~41s~12COl)-~-pls~41s01-Jt- 51c~41-- S4
2.2. Velocity and acceleration analysis Differentiating both sides of (14) yields
~5 =
~1 COS2 05 "~ /12sin 2 O5 -F/13 cos O5 sin O5 +/i4 cos O5 +/is sin O5 +/16 (#1 -- #2)sin(2Os) --/h cos(205) + #4 sin O3 -- #5 cos O5
(22)
{/1,} (i = 1-6) of (22) can be calculated from (14a). From d2/dt ~ (14) we get O5. Other velocity variables {02, 03, 04} and the corresponding acceleration variables {0"2,/)'3,/)~} can also be easily determined. Here we only displayed the expression of 05, for it is the most important one among all the velocity and acceleration variables. 3. D I S P L A C E M E N T A N A L Y S I S OF THE Ro--SRRR M E C H A N I S M Mechanism Ro-SRRR is shown in Fig. 3. When the input angle 0, is given, the unknowns to be determined are {02, 03, 04, 05}. If the first displacement equation we want to derive isf(0,, 05) = 0, we can let 05 = 05. In this case, the auxiliary angle can be chosen from 03 and 04. If we choose 04 as the auxiliary angle, the procedure for deriving f(01, 05) = 0 will be identical to that of Section 2. If we choose 03 as the auxiliary angle, the procedure for deriving f(0], 05) = 0 will be somewhat different. Here we let 05 = 05, 03 = if3, then, the vector loop equation of the mechanism can be written as follows,
(23)
K=F K=I-t-J
(1)
J=p4q4-S3a3
(2) (3) (4)
I=plq~ +(Sial + p 3 a s - $4a4) F= -(p2q2-S2a2+p3q3)
(23a)
jil3 P3,
Plql P4~
,¢sq~
Fig. 3. The Ro-RSRR mechanism.
107
The generalized Clemens coupling
(1) q2 = c@2q3- s*2a2 x q3 (2)
qI = c&q5 + s&a, x q5
a2 = ca23a3- sa23q3x a3 (3) q3 = d&q4 - s&a3 x q4 (4)
Wb)
a3 = ca34a4 - sa34q4x a4 (5) q4 = d&q5 - s&a4 x qs (6) 3.1. Derivation of the first equation relating Or, Oj and ti3 Squaring both sides of (23) yields K2 = F2
(24)
Substituting (23a-4) into (24) yields COS $3 =
[K2 - (p: f s: + fJ:)]/(@psp3)
(25)
3.2. Derivation of the second equation relating 8,, O5 and ~5, We can see from Fig. 2 that a3 is an end-axis vector of the floating vector F. Dot product a3 with both sides of (23) yields K.a3=F.a3
(26)
Substituting (23a-4) into (26) we get sin $3 =
[-(K.a3) +
(27)
S2ca231/(p2sa23)
3.3. Derivation of the input-output displacement equation relating OS From (25) (27) and the identity cos2 II/3+ sin2 Ic/3= 1 we obtain Q: + Q: = Q:
(28)
I = sa23K2 - sad& + Si + pi) E2= -2p3(K * a3) + 2p&ca23
(284
Q3 = 2p2pm23
QI = (WI .q4) + Z, = (WI * q5)cos 05 + (W, . q5 x a)sin O5 + Z, Q2 = (W2 * 44) + Z2 = (W2 *qs)cos 05 + (w, . q5 x a)sin Q, + z2
GW
WI = 2sa23(p41- S3sa3aa4x I) W2 = 2p3sa34a4x I ZI = -2S3sa23ca34(I * a4) + sa23(12+ J2 - pi - S: - pi) z2
=
-@3casr(I
* %)+
&'3(s3
+
(28~)
s2Ca23)
Substituting (28b) into (28) yields PI C0S2
OS
+
p2
sin2 6% + p3 cos OS sin OS + j4 cos Q5 + cc5sin OS + pa = 0
PI = (W,
.q5)*
p2
.
=
WI
P3 = WI pl = WI
+
(W2
(29)
*q5j2
q5 x a4)2+ (W2 . q5 x a4)*
* q5wI
*q5
x
lh)
+
2(w*
,q5)(W2.
* q5)Z + 2(W2 * q5)Z2
P5 = 2(W, * 95 x a4)ZI + 2oy2 * q, /46 = z: + z; - Q:
x adz*
q, x a4)
(294
708
Y.B. Zhou et al.
Let y = tan(Os/2), then cos O5 = (1 - y2)/(1 + y2), sin O5 = 2y/(1 + y2). From (29) we obtain 4
Z v~y4-i__ - 0 i=0
(30)
1)0 =/.~1 - /~4 + f16 Vl = -2/.t3 q- 2/z5
f
(30a)
vz = -2/11 + 4fl2 "+- 2fl6 1)5 2~3 + 2/~5 I)4 = /[~l "Jl-~4 "31-/~6
Solving (30) we get y, then 05 = 2 tan -~ y; From (25) and (27) we obtain ~3, namely 05; using q4 and a3 × q4, respectively, to dot product both sides of (23) yields -(P2q2-S2a2+p3q3)'q4 (I + J) a3 x q4 = -(Pzq2 - $2a2 + p 3 q 3 ) " a3 × q4
(31)
(I. q4) "]- p4 = --p2[C03 COS 04 -- C~23S03 sin 04] + $2 + s~23 sin 04 - p3 cos 04 (l a3 × q4) = p2[c03 sin 04 + c0~23s03 cos 04] ---[-82s~23 cos 04 + p3 sin 04
(31a)
f - - (p2c03 + p3)cos 04 + (p2co~23s03 q- S2s~23)sin 04 - (I. q4) + p4 (pzc~23s03 + S2s~23)cos 04 + (p2c03 + p3)sin 04 = ~ " a3 × q4)
(31b)
(I+J).q,=
i.e.
i.e.
Solving (31b) yields 04: ICOS 04
--
(p2c0~23S03q- $2s~23)(I • a3 × q4) - (p2c03 + p3)(I • q4 + p4) (p2c03 q-p3) 2 --t- (p2c~23S03 -~- 82S0f23) 2
(32) [.sin 04 = (p2c~23s03 + $2s~23)(I • q, + p4) + (p2c03 + p3)(I • a3 × q4) (p2c03 --]-p3) 2 "l- (p2c~23S03 "3!-S2S0~23)2 The angle 02 can be determined from 02 = cos-l(ql • q2). At this point, the theoretical displacement analysis for the R o - S R R R mechanism is complete.
4. D I S P L A C E M E N T
ANALYSIS
O F T H E Ro-RRRS M E C H A N I S M
Mechanism R o - R R R S is shown in Fig. 4. This mechanism has never been analysed before. When the input angle 01 is given, the unknowns to be determined are {02, 03, 04, 05}. Let f(01, 02) = 0 be the first displacement equation to be derived. Let 02 = 02, 04 = @4. The vector loop equation of the mechanism can be written as K=F
=i+j =(p2q2-S3a3) =(-S2a2+plq,)+(Sta,
fi
= --p3q3"}-S4a4--p4q4
(33)
+psas)
(1) (2) (3) (4)
The generalized Clemens coupling
709
?
II1
Ptqt
~/~4
Fig 4. The
Ro-RRRS mechanism.
"q4 -- c~4q3 -q- s~b4a4 x q3
(1)
a4 ---- c~34a3 + s~34q3 × 83
(2)
q3 = c03q2 d- s03a3 x q2
(3)
a3 = c~23a2 d- s~23q2 x a:
(4)
q2 = cO2ql "t- st92a2 x qj
(5)
a2 = c~2a~ + s~2ql x a~
(6)
ql = cOiq5 + sOlal x q5
(7)
(33b)
4.1. Der&ation of the first equation relating 01, O: and ~4 Squaring both sides o f (33) yields K2 : F2
(34)
(p2 + $24 q_ p24)]/(2p3p4 )
(35)
Substituting (33a--4) into (34) yields COS ~4 = [K 2 -
4.2. Derivation of the second equation relating 01, 02 and ~k4 We can see from Fig. 3 that a3 is an axial vector between L and K. D o t product a3 with both sides of (33) yields K • a3
=
F • a3
(36)
Substituting (33a--4) into (36) yields sin ~k, = [ - ( K " a3) + S4c~34]/(p4s~34)
(37)
4.3. Derivation of the input-output displacement equation relating 02 F r o m (35), (37) and the identity cos 2 ~k2 + sin: ~k2 = 1 we obtain +
=
(38)
710
Y. B. Zhou et al.
{
a~ = s~t34K2 - sct34(p2 + S 2 + P[) Q2 -2p3((K. 43) + 2p3S4cot34 Q3 2p3p4s~34
(38a)
{Qt = (Wt ' q2) + Z~ = (W~ • q0cos 02 + (W~ • a2 × qt)sin 02 + Zt Q2 (W2 q2) + Z2 (W2 q,)cos 02 + (WE 42 × q,)sin 02 + Z2
(38b)
Wl = 2S~34(p2I -- $3S~23a2 × I)
I
W2
--2p3s~34a2 x I
ZI
--2S3c0~23s~34(I"82) "31-SOt34(I2 "t- j2
Z2
--2p3c~23(I " a2) + 2p3($3 + $4c~t34)
_ p~ _ $42 _ p2)
(38c)
Substituting (38b) into (38) yields /2~ cos 2 02 +/22 sin 2 O2 +/23 COS [92 sin 02 +/24 cos O2 +/23 sin O2 +/26 = 0
(39)
/2i ~---(Wt" q,)2 + (WE" ql) 2 /22 (W, " 42 × ql) 2 -t- (W2" 42 x q,)2 2(W,- q,)(Wt • a2 × q,) + 2(W2. q0(W2 • a2 × q,) = 2(W,. q,)Z, + 2(W2" qt)Z2 /25 /26
L
(39a)
2(W, a2 x q,)Z, + 2(W2. a2 x q~)Z2 Z~ "-t-Z2 -- a 2
Let y = tan(Od2), then cos O4 = (1 - y2)/(1 + y2), sin O4 = 2y/(1 + fl). From (39) we obtain 4 E v,Y4 - ' = 0 i=0
(40)
v0 =/21 --/24 +/26 v~ = --2/23 + 2/25 v2 = --2/2t + 4/22 + 2/26
(40a)
v3 = 2/23 + 2/2s v4 =/21 +/24 +/26 Solving (40) we gety, then 02 = tan -~ y; from (35) and (37) we obtain 04; using q2 and (43 x q2), respectively, to dot product both sides of (33) yields 03:
f COS 03 =
(P3 + p4c04)(I" q2 + p2) + (S4s~34 "31-p4s04c~3,)(I " a3 X q2) (t73 q- p4C04) 2 d- (84s~34 "Jr-p4s04c~34) 2
(41) sin 03
($4s~3, + p, s04cu34)(I, q2 + p2) - (1o3 31- p4cO,)(I" a3 × q2) (t93 4-/94C04) 2 4" ($4s~t34 + p4sO4cot34) 2
The theoretical displacement analysis for the R o - R R R S mechanism is now complete. 5. A N A L Y S I S OF T H E Ro-RRSR M E C H A N I S M
Mechanism R o - R R S R is shown in Fig. 5. The main purpose of this section is to show the flexibility of the vector algebraic method in the analysis of spatial mechanisms, therefore, details are not given. (1) Letf(0t, 02) = 0 be the first displacement equation to be derived; let 02 = 02, then the auxiliary angle can be either 03 or 05.
The generalized Clemens coupling
711
Case 1.1. Let 02 = 02, 03 = 43. The vector loop equation o f the mechanism can be written as follows, K + L = F
(42)
L = p3q3 K=I+J J = p2q2 - $3a3 I = ( - $ 2 a 2 + p,ql) + I0
(42a)
Io = (Sial + psq5 -- $4a4) F = --p4q4 Case 1.2. Let 02 =
~)2, 05 :
4/]5.
The vector loop equation is K + L = F
(43)
" L = p4q4 K=I+J J = p2q2 S383 I = ( - $ 2 a 2 +plq~) + Io - -
I0 = (Sial + psq5
-
(43a)
$4a4)
F = -p3q3
(2) Let f(01, 03) = 0 be the first displacement equation to be derived; let 03 = 03, then the auxiliary angle can be either 02 or 05. Case 2.1. Let 03 = O3, 02 = ~'2; then I + L = F
(44)
{ ~ = p2q2 $3a3 + p3q3 L ( - S 2 a 2 + p ~ q ~ ) + Io -
(S~a~ + psq5 - $4a4)
(44a)
-p4q4
jr as
03 a2
at
/Ptql
P4¢
\ Psqs
Fig. 5. The Ro-RRSR mechanism. MMT 32/6~U
Y. B. Zhou et al.
712 Case 2.2. Let 03 = 03, 05 = ffs; then
(45)
I+L=F L = p4q4 I = -($2a2 + p~q~) + (S~al + psq5 F : --(P2q2 -- 83a3 Jr- p3q3)
-
-
$4a4)
(45a)
(3) Let the first I / 0 displacement equation we want to derive bef(O~, 05) = O; let 05 = 05, then the auxiliary angle can be either 02 or 03. Case 3.1. Let 05 = 05, 02 = ~2; then K + L = F
(46)
L = p2q2 - $3a3 K=I+J
J = p4q4 I = (-$2a2 + p~q~) + Io
(46a)
Io = (& aj + psq5 - $4a4) F = -p3q3 Case 3.2. Let 05 = 05, 03 = ~3; then K = F
(47)
K=I+J
f
J = p4q4 I = (-$2a2 + plql) + Io
Io
=
(47a)
(Sial + psq5 -- $4a4)
F = -(p2q2 - $3a3 + p3q3)
The analysis procedures for Cases 1.1, 1.2, 2.1, 2.2 and 3.1 are identical to that of Section 2, whereas the analysis for Case 3.2 is identical to that of Section 3. 6. CONCLUSION Since all the variants of the Ro-3R-S mechanisms can be analysed, the efficiency of the proposed approach warrants no additional comment. As for the uniformity and simplicity of the proposed approach, let us take a close look at the analysis procedures. Though the analysis steps for each of the four variants of the Ro-3R-S mechanisms are the same as outlined at the end of the Introduction, there are two slightly different versions of the analysis procedures, as represented in Sections 2 and 3. Let us first examine Section 2. In equations (3) and (6) everything unrelated to q2, a function of the auxiliary angle ~k2, are included in [U, V} and {U', V'}, respectively. Similarly, in equations (4) and (7) those terms unrelated to ~2 are also collected into {A, B, C} and {A', B', C'}. Unlike all the matrix related approaches, we do not need to expand (i.e. scalarize) everything at the beginning, we manipulate only those vectors we need, i.e. q2 and ~02; thus we are able to obtain equations (4) and (7) by only one transformation and a few lines of derivation, respectively. In order to easily avoid rendering an 8th-order polynomial from equation (9), we should linearize {Or, Q2, Q3} of equation (8a) in terms of sin o5 and cos 05. This can be achieved by only one or two lines of derivation as shown in equation (11). The analysis procedure in Section 3 is similar to but simpler than that of Section 2. Comparing equations {(4), (7), (9), (14), (15), (15a)} of Section 2 with the equations {(25), (27), (28), (29), (30), (30a)} of Section 3, respectively, the uniformity of the proposed approach is not hard to perceive.
The generalized Clemens coupling
713
q
II2
v
111 Fig. A1.
It is obvious that in terms of simplicity no other mathematic tool can beat vectors. Therefore, the proposed approach is unlikely to be matched in simplicity by other possible approaches employing quaternion algebra, line-coordinates or screw theory, etc.
REFERENCES
1. Torfason, L. E. and Sharma, A. K., Analysis of spatial R - R ~ 3 - R - R mechanisms by the method of generated surfaces. Journal of Engineering for Industry, Trans. ASME, Series B, 1973, 95(3), 704-780. 2. Wallace, D. M., Displacement analysis of spatial mechanisms with more than four links. Doctoral dissertation, Columbia University, New York. No. 69, 13006 University Microfilms, Ann Arbor, Michigan, 1968. 3. Wallace, D. M. and Freudenstein, F., Displacement analysis of the Generalized Clemens Coupling, the R - R - S - R - R spatial linkage. Journal of Engineering for Industry, Trans. ASME, Series B, 1975, 97(2), 575-580. 4. Zhang, Q. X., Analysis and Synthesis of Spatial Mechanisms. South China University of Technology Press, Guangzhou, 1980; Mechanical Industry Press, Beijing, 1984, pp. 218-232. 5. Xie. C. X., Zheng, S. X. and Ou, H. H., Configuration analysis of spatial five-link mechanism RRRSR by means of the tensor rotational transformation method. Journal of South China University of Technology, 1980, 8(2). 6. Duffy, J., Analysis of Mechanisms and Robot Manipulators, Chap. 10. Arnold, Great Britain/Halstead Press, U.S.A., 1980, pp. 384-404. 7. Youm, Y. and Huang, T. C., Displacement analysis of 4RIG five-link spatial mechanisms using exact solution of four-link spatial mechanisms. Mechanism and Machine Theory, 1987, 22(3), 189-197. 8. Alizade, R. I., Duffy, J. and Azizov, A. A., Mathematical models for analysis and synthesis of spatial mechanisms--II. Mechanism and Machine Theory, 1987, 18(5), 309-315. 9. Yih, T. C. and Youm, T., Kinematics of spatial mechanisms in matrix notation. Trends and Developments in
Mechanisms, Machines, and Robotics--1988, Proceedings of The 20th ASME Biennial Mechanisms Conference, DE-VoI.15-1, Kissimmee, FL, September 1988, pp. 429-433. 10. Denavit, J. and Hartenberg, R. S., A kinematic notation for lower-pair mechanisms based on matrices. Trans. ASME, Journal of Applied Mechanisms, 1956, 22, 215-221. 11. Zhou, Y. B., Kinematics of spatial mechanisms. Ph.D. dissertation, The University of Western Ontario, 1994.
APPENDIX
Several Important Formulae (1) In Fig. A1, unit vector q is perpendicular to both unit vectors a, and a2; angle ~t2 is the right-hand-rotation angle from a~ to a2 to about q, then, we have a~ × a2 = q sin ct,2
(A1)
a2 = cos ~t,2a, ~ sin ~t12q x al
(A2)
a t ---- COS Ctt2a2 - - s i n ~ i 2 q
× a2
(A3)
(a-b)c
(A4)
(2) Let a, b, e and d be four arbitrary vectors, then a x (b × c) = ( a . c ) b -
(axb)
xe=(a.e)b-(b.e)a
(a x b)' (e x d) = ( a . e)(b" d) - (a' d)(b. e)
(A5) (A6)
714
Y . B . Z h o u et al.
(3) Let e~, e2 and e3 be a positive triple of mutually perpendicular unit vectors, a and b are two arbitrary vectors, then a = ~ (a. e,)e~
(A7)
a . b = ~ (a. e,)(b • e,)
(A8)
a 2 = ~ (a. e,) 2
(A9)
(e. el) 2 -t.- (a. e2)2 = (a × e3)2
(A10)
Formulae (A1)-(A6) are frequently used in this paper. Their proofs can be easily verified or found in books on engineering mathematics.