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Overconstraint analysis on spatial 6-link loops
Mechanism and Machine Theory 37 (2002) 267±278
www.elsevier.com/locate/mechmt
Overconstraint analysis on spatial 6-link loops Qiong Jin a, Tingli Yang a
b,*
Department of mechanical Engineering, Southeast University, Nanjing 210096, PR China b Jinling Petrochemical Corporation, Nanjing 210037, PR China Received 5 July 2000; accepted 25 September 2001
Abstract In order to reduce the complexity of overconstraint analysis on spatial 6-link loops, we present a complete closed-form analytical method using only four basic equations combined with a ``turn technique'', in which the closure conditions can be obtained and input±output (I±O) equations can be further deduced. Based on this method, we can verify the overconstraint of the Bricard trihedral mechanism and derive its closure condition easily from its overconstraint condition. Also, the closure condition and I±O equations of a known 4R2P overconstrained mechanism are deduced for the ®rst time. Ó 2002 Published by Elsevier Science Ltd. Keywords: Overconstraint criterion; Existence condition; I±O relationship; 6-Link loop
1. Introduction Among all types of overconstrained mechanisms, the 6-link mechanism is one of the most dicultly analyzed mechanisms. After Baker [1] listed all known 6R overconstrained mechanisms in 1984, Wohlhart [2] proposed a 6R symmetric linkage in 1987, and presented a 6R overconstrained mechanism from merging two general Goldberg 5R overconstrained mechanisms in 1991 [3]; Mavroidis and Roth [4] developed a 6R overconstrained mechanism with two Bennett joints that have no common axis in 1995. There are also some 6-link overconstrained mechanisms containing prismatic joints, such as Waldron's [5] hybrid overconstrained mechanisms. In literature [6], a 4R2P mechanism with two pairs of parallel revolute joints and a 5R1P mechanism with one planar joint and one pair of parallel revolute joints are presented.
*
Corresponding author. Tel.: +86-25-509-8935. E-mail address: [email protected] (T. Yang).
0094-114X/02/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 7 2 - 6
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In general, 6-link overconstrained mechanisms can be classi®ed into three categories: (a) Bennett based mechanisms; (b) symmetric mechanisms; and (c) variant mechanisms of (a) or (b). For the ®rst category, Wohlhart [3], Mavroidis and Roth [4], Waldron [5], Goldberg [7], Baker [8], and Huang and Sun [9], etc. have studied it. For the second category, Bricard [10], Waldron [11], Baker [12,13], Mavroidis and Roth [4] analyzed it in detail. As for the third category, a typical example is Schatz mechanism [14] derived from a special plane-symmetric Bricard trihedral loop [10], which was further investigated and developed for practical use by many researchers [15±18]. Although many 6-link overconstrained mechanisms have been proposed, few of them possess a wide application like Schatz mechanism because these so-called overconstrained mechanisms have not been understood thoroughly [19]. Overconstraint analysis of mechanisms should include three parts: overconstraint criterion, overconstraint existence conditions, and input±output (I±O) equations. The existence conditions of overconstrained mechanisms include the overconstraint condition and the closure condition. The former, i.e., linear complex condition in screw theory, is very important for achieving overconstraint, and the latter makes sure that the kinematic chain will be closed. There are following three methods to analyze the overconstraint mechanisms: (i) Geometrical method. Using geometric intuition or synthetic reasoning, most of the early known overconstrained mechanisms were discovered by geometricians such as Goldberg [7], Bricard [10], Bennett [20], and Myard [21] mechanisms. However, the identical mode cannot be obtained using this method because the deductions of existence conditions are dierent for various overconstrained mechanisms. (ii) Screw theory. Screw theory had been used to analyze overconstraint for a long time by many researchers such as Hunt [22], Dizioglu [23] and Yang [24,25]. However, the closure conditions are dicult to be obtained in some degree using this method. (iii) Analytical method. In recent decades, analytical method has become more important. Its main idea is that: if a univariate polynomial derived from the matrix-loop equation is identically equal to zero, i.e., the coecient of each power of the joint variable is equated to zero, then the corresponding loop is an overconstrained mechanism. Equating the coecients of the polynomial to zero gives a set of equations in the constant geometric parameters, and these equations are the existence conditions for the loop to be mobile. Hence, the I±O equations for the mechanism could be easily obtained. Baker [1], Dimentberg [26], Waldron [27], Lee and Yan [28], and Mavroidis and Roth [4,29], etc. found some new mechanisms or obtained valuable results using this method. Since Lee and Liang [30] derived the univariate polynomial of 16th degree for the displacement analysis of general 7R loop in 1988, it has become clear that the inverse kinematics problem of serial-chain manipulators can be reduced to the univariate polynomials from 14 original equations. Based on this consideration, Mavroidis and Roth [29] pointed out that the problem of overconstrained mechanisms could also be analyzed using the 16-solution equation set for general 6R manipulators. Theoretically, it is no doubt a systematical idea. In fact, due to the great calculation work and the limitation of calculating capability of computers nowadays, only a numerical 16th degree univariate polynomial equation can be obtained. Therefore only numerical analysis can be operated in such mode, and I±O equations can be obtained numerically but in closed-form. In order to discover new overconstrained mechanisms, researchers usually assume some overconstraint conditions, and further deduce the perfect existence conditions and the closure
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conditions using the criterion of overconstrained mechanisms. The key step here is to establish a kind of simple and eective criterion. Hence in this paper, a new criterion, which can eectively distinguish the ®nite and in®nite solutions of displacement analysis, is presented for the 6-link overconstrained mechanism. According to this criterion, analytical I±O equations of overconstrained mechanisms can be deduced ®nally. After using this criterion to verify the known overconstrained mechanisms, some drawbacks, such as the unperfectness of existence conditions and the existing of some rigid joints in the loop, can be easily identi®ed.
2. Principle of overconstraint analysis 2.1. Basic equations It is certain that the solution number for displacement analysis of a general 6R loop is ®nite. The main goal in displacement analysis is to achieve the minimum degree of a univariate polynomial. Hence in this case, the original equations should be built as many as possible so that extraneous roots could be circumvented, for examples, Lee and Reinholtz [31], and Raghavan and Roth [32] derived 16 solutions from 14 basic equations. However, for overconstrained mechanisms, to determinate overconstraint is virtually to distinguish the ®nite solutions and in®nite solutions of displacement equations since there always exists correlatability among basic equations in which all variables have in®nite solutions. In order to diminish the calculating scale for studying overconstraint analytically, it would be better to directly analyze the independence of the original equations than to seek all coecients of a univariate polynomial equation being zero. Therefore, the original equations needed should be as few as possible under condition that the equation set possesses ®nite solutions in general. Fig. 1 shows a general 6-link loop, in which zi and xi represent joint-axis-direction and linkdirection unit vectors, respectively. And the parameters di ; ai ; aij and hi are de®ned so that: di is the oset along joint axis zi ; ai is the normal distance along xi1 ; aij is the twist angle between the joint axes zi and zi1 , and hi is the rotation angle about joint axis zi . In the following sections, sin hi , cos hi , sin aij and cos aij are abbreviated as si ; ci ; sij and cij , respectively. When the 6-link loop is
Fig. 1. A general spatial 6-link single loop.
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broken at two kinematic joints it becomes two single-opened-chains (SOCs): SOCA and SOCB . Generally, from the viewpoint of kinematics compatibility of two SOCs, we can establish an equation set with four equations including four kinematic variables and possessing ®nite solutions. If the loop is broken at joints z1 and z3 , the corresponding four equations can be expressed as follows: eq1 z1 z3 A z1 z3 B ;
1a
eq2 R z1 z3 A R z1 z3 B ;
1b
eq3 R z1 A R z1 B ;
1c
eq4 R z3 A R z3 B ;
1d
where the SOC vectors are: RA a1 x2 d2 z2 a2 x3 ; RB
d3 z3 a3 x4 d4 z4 a4 x5 d5 z5 a5 x6 d6 z6 a6 x1 d1 z1 ;
RA RB : Equation set (1a)±(1d) re¯ecting the restrictive relationship in spatial single loop, possesses geometrical meaning clearly. Under the ane coordinate system consisting of unit vectors z1 ; z3 and z1 z3 , Eqs. (1b), (1c) and (1d), corresponding to the three projections of vector R, are independent each other. As to Eq. (1a), it is the scalar product of unit vectors z1 and z3 , and has no relationship with the three components of vector R. Therefore the four Eqs. (1a)±(1d) are independent. It should be noted that the loop must be broken at two appropriate joints in order to make equation set (1a)±(1d) contain only four variables. 2.2. Overconstraint criterion If the geometrical parameters of the loop are given specially so that the number of dependent equations Nequ and the number of dependent variables Nvar of equation set (1a)±(1d) satisfy the following relationship Nequ < Nvar ;
2
and if expression (2) is always true when all terms' subscripts are taken turns, then all variables of the 6-link loop cannot be solved ®nitely. Such a loop is called the overconstrained mechanism. This is the criterion of overconstrained mechanism. The turn operation is virtually to change the broken positions of the loop till all six variables in the loop are analyzed. Inequality (2) indicates that at least one dependent equation exists in equation set (1a)±(1d), which can be linearly (see Section 4(a) and (c)) or nonlinearly derived from other independent equations in equation set (1a)±(1d). The nonlinear derivation manner can be normally implemented in two steps: ®rstly eliminating some unknowns to derive a polynomial equation, and secondly proving the polynomial equation to be identical equation. One kind of identical equations is a polynomial identically equated to zero in which the coecient of each power of the joint variable can be proved to be zero (see Section 4(b)). Another kind of identical equations contains at least one common factor being zero (see Section 3).
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2.3. Closure condition The conditions, on the linkage's structural parameters that ful®ll the overconstraint criterion, will be called existence conditions of the overconstrained mechanism. Sometimes, only the overconstraint condition but not the closure condition can be given using the screw theory. While using analytical method presented in this paper, when overconstraint conditions are substituted into the above ``turned'' equation sets, some constant equations can be derived, which are in fact closure conditions. The closure conditions are often outcomes of overconstraint determination. 2.4. Main steps (i) Assuming some overconstraint conditions. In order to assume the overconstraint conditions, three aspects should be considered: symmetry, dependency possibly existing in the simpler hand side of equation set (1a)±(1d), and topological structures of overconstrained mechanisms. (ii) Determinating overconstraint and perfecting existence conditions. When the assumed part of overconstraint conditions is substituted into equation set (1a)±(1d), the overconstraint of the loop could be determined using the criterion. It should be noted that the turn operation is needed before the ®nal conclusion is given. Meanwhile overconstraint conditions and closure condition can be supplemented. (iii) Deducing the I±O equations. Synthesizing the results obtained by the turn operation, ®ve simplest equations can be selected as I±O relationships from all independent equations. 3. Bricard trihedral mechanism In 1927, Bricard found some symmetric 6R overconstrained mechanisms using geometrical method [10]. Here we only discuss the trihedral mechanism. (i) The given existence conditions are: a12 a34 a56 p=2;
a23 a45 a61 3p=2;
a21
a26 ;
a23
a25
a22
a24
di 0 i 1±6;
3a 3b
where condition (3a) is the overconstraint condition with exacting symmetry, and condition (3b) is a complicated closure condition. The following step shows that condition (3b) can be derived from condition (3a) easily. (ii) Determinating overconstraint and deducing the closure condition. After breaking the loop at joints z1 and z3 , the remainder variables are h2 ; h4 ; h5 and h6 . The four basic equations built in this case are the same as those indicated in equation set (1a)±(1d). To simplify equation set (1a)±(1d) condition (3a) can be substituted into it. The result is eq1 c2
s4 s6 c5 c4 c6 ;
eq2 0 s5 a3 s6
4a
a6 s4 a4 c4 s6
a5 c6 s4 ;
4b
eq3 a2 s2 a4 c5 s6 a3 s4 c6 a3 c4 c5 s6 a5 s6 ;
4c
eq4
4d
a1 s 2
a6 c 4 s 6
a4 s4
a6 s4 c6 c5
a5 s 4 c 5 :
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If h5 is a variable, the following result can be deduced from Eq. (4b) easily a3 s6
a6 s4 a4 c4 s6
a5 c6 s4 0:
5
Hence the I±O relationship between variables h4 and h6 can be obtained from Eq. (5). Assuming that the variables of h4 and h6 are given, from Eqs. (4c) and (4d) we can derive: s2 a4 a6 c4 s26 a24 s4 s6 a3 a6 c24 s26 s4 s6 c4 a3 a4
a5 a6 s4 s6 c6
a25 s4 s6
a3 a6 s24 c26
a3 a5 s24 c6 =D; c5 a4 a2 s4 a6 a2 c4 s6
6 a1 a3 s4 c6
a1 a5 s6 =D;
7
where D a1 a3 s6 c4 a1 a4 s6 a5 a2 s4 a6 a2 s4 c6 : Using Eq. (5), Eq. (6) can be further simpli®ed as s2 s4 s6 a26 a24
a25
a23 =D:
8
Substituting c5 of Eq. (7) into Eq. (4a) and simplifying it using Eq. (5), we can obtain c2 a1 a5 s4
a4 a2 s6
a2 a3 c4 s6 a1 a6 c6 s4 =D:
9
It can be realized that the original equation set 4a,(4b)±(4d) can be expressed as Eqs. (5), (7), (8) and (9). Considering the distribution of variables in these four equations, if there exist dependent equations, only Eqs. (8) and (9) are possibly dependent. If Eqs. (8) and (9) satisfy the Pythagorean theorem s22 c22 1, then Eq. (9) can be derived from Eq. (8), i.e., only one of these two equations is dependent. Therefore, the condition that satis®es the relationship s22 c22 1 is just the supplemental existence condition or closure condition. Based on the above consideration, substituting Eqs. (8) and (9) into the following expression: t2 s22 c22
1;
10
and simplifying it using Eq. (5), we get t2 s24 s26 AB=D2 ;
11
where A a26 a24 a25 a23 and B a26 a24 a25 a23 a21 a22 . If t2 0, then A 0 or B 0. Due to the symmetry of the linkage, when subscripts of equation set (4a)±(4d) are taken turns, Eq. (10) always contains factor B but not factor A, indicating that the precondition of ti 0 is just B 0, i.e., a26 a24
a25
a23
a21 a22 0:
12
Eq. (12) is none other but the closure condition given in the known condition (3a) and (3b). Hence, only Eqs. (4b)±(4d) among the four basic equations are independent, i.e., Nequ 3 < Nvar 4: This indicates that none of the variables h2 ; h4 ; h5 and h6 can be solved ®nitely. For another two variables h1 and h3 , they can be analyzed by breaking the loop at joints z2 and z4 . Due to the symmetry of the loop, the four equations in this case are identical to those in equation set (4a)±(4d) which results in in®nite solutions for both variables h1 and h3 . Therefore this linkage is an overconstrained mechanism.
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This example shows that even if the overconstraint existence conditions are not perfect the complementary condition can be always found out during overconstraint determination. A univariate polynomial is not needed in the overconstraint analysis of this linkage, which makes the overconstraint determination simple. (iii) Deducing the I±O equations. Eqs. (5), (7) and (8) can be regarded as three independent equations. Using symmetry and turn operation, another two relationships similar to Eq. (5) can be derived as s3 a6 a1 c1 s1 a3 a2 c3 ;
13
s5 a2 a3 c3 s3 a5 a4 c5 :
14
These ®ve equations constitute the I±O relationships. The I±O relationships may be expressed in various forms. For example, we can give another group I±O equations for this linkage, four of which are obtained by turning the subscripts of Eq. (5) and the ®fth equation is Eq. (4a). 4. 4R2P mechanism In 1968, Waldron [5] found a group of hybrid overconstrained mechanisms from screw theory. One of these mechanisms is a RRR±PHP linkage formed from a spherical linkage and a screwdouble slider linkage. Now we introduce a special case of this 6-link loop with the zero-pitch screw joints. The existence conditions and I±O relations of such a 4R2P linkage have not been discussed explicitly till now. Here we let both prismatic joints (P4 and P5 ) be perpendicular to and intersect with the revolute joint (R5 ) (see Fig. 2). (i) According to the screw theory, the overconstraint condition of this linkage is a1 a2 d2 a4 a5 0;
a45 a56 p=2:
15
(ii) Determinating overconstraint and perfecting existence conditions. In order to ensure that the equation set has only four variables, the broken positions should not be selected at the prismatic joints. (a) Breaking at joints z5 and z1 . In this case variables are h2 ; h3 ; d4 and d6 . Turning the subscripts of equation set (1a)±(1d) we get
Fig. 2. A 4R2P overconstrained mechanism.
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eq1 z5 z1 A z5 z1 B ; eq2 R z5 z1 A R z5 z1 B ; eq3 R z5 A R z5 B ; eq4 R z1 A R z1 B ;
16
where RA a5 x6 d6 z6 a6 x1 ; RB
d1 z1 a1 x2 d2 z2 a2 x3 d3 z3 a3 x4 d4 z4 a4 x5 d5 z5 ;
RA RB : Substituting overconstraint condition (15) into equation set (16), we obtain a large equation set (17a)±(17d). For the convenience of description, the right-hand side of equation set (17a)±(17d) is expressed by function fi i 1; 2; 3; 4. Terms in each bracket are power products of function fi , and ``1'' expresses the constant term. eq1
s61 c6 f1 s2 s3 ; s2 c3 ; c2 s3 ; c2 c3 ; c2 ; s3 ; c3 ; 1;
eq2 a6 c6 c61
17a
s61 s6 d6
f2 s2 s3 d4 ; s2 c3 d4 ; c2 s3 d4 ; c2 c3 d4 ; s2 s3 ; s2 c3 ; c2 s3 ; c2 c3 ; c2 d4 ; s3 d4 ; c3 d4 ; c2 ; s3 ; c3 ; d4 ; 1;
17b
eq3 a6 s6 f3 s2 s3 ; s2 c3 ; c2 s3 ; c2 c3 ; c2 ; s3 ; c3 ; 1;
17c
eq4 d6 c61 f4 s2 s3 d4 ; c2 c3 d4 ; s2 s3 ; s2 c3 ; c2 s3 ; c2 c3 ; c2 d4 ; c3 d4 ; c2 ; s3 ; c3 ; d4 ; 1:
17d
Eqs. (17a) and (17c) have the same power products and there is a linear relationship between them f3
d5
a3 s4 d3 s34 c4
d1 f1 :
Substituting Eq. (17a) into Eq. (17c), we have a6 s6
d1 c6 s61 a3 s4
d3 c4 s34 d5 0:
18
Eq. (18), a constant equation free from unknown variables, is the closure condition of the linkage. Eq. (17a) shows the relationship between variables h2 and h3 . From simultaneous Eqs. (17b) and (17d), we can express variables d4 and d6 in terms of h2 and h3 . Since Eqs. (17a) and (17c) are equivalent, only three of equation set (17a)±(17d) are independent, i.e., Nequ 3 < Nvar 4: Therefore, all variables possess in®nite solutions when joints of z5 and z1 are broken. (b) Breaking at joints z1 and z3 . In this case, the variables discussed here are h2 ; d4 ; h5 and d6 . Substituting overconstraint condition (15) into equation set (1a)±(1d), we ®gure out that eq1 c23 c12
s23 c2 s12 g1 s5 ; c5 ; 1;
19a
eq2 0 g2 d4 s5 ; d4 c5 ; d6 s5 ; d6 c5 ; d4 ; d6 ; s5 ; c5 ; 1;
19b
eq3 0 g3 d4 s5 ; d4 c5 ; d6 ; s5 ; c5 ; 1;
19c
eq4 0 g4 d6 s5 ; d6 c5 ; d4 ; s5 ; c5 ; 1;
19d
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where Eq. (19a) can be expanded as c23 c12 s23 c2 s12 c34 c5 c61 c34 s5 s61 s6 s34 s4 s5 c61 s34 s4 s6 c5 s61 s34 c4 s61 c6 :
275
20
Solving the simultaneous Eqs. (19c) and (19d) for variables d4 and d6 in terms of h5 , we get d4 s34 s4 s5 d1 d5 s34 c4 c61 d1 c34 c5 c261 c234 c25 d3 c61 a6 s34 c4 s6 c61 a6 s34 s4 c6 c5 c61 c234 c5 d3 s5 s61 s6 c34 c25 a3 c4 s6 s61 c34 c25 d3 s34 s4 s6 s61 c34 c5 d1 c34 c5 a3 c4 s5 c61 s34 s4 s5 d5 s61 c6 s34 s24 s5 a3 c6 s61
s234 s24 s5 d3 s6 c5 s61
s234 s4 s5 d3 c4 s61 c6
s34 s4 s5 a3 c4 c5 s6 s61 2c34 c5 d3 s34 s4 s5 c61 a6 c34 s5 c6 c61 c34 c5 d3 s34 c4 s61 c6 s234 s24 s25 d3 c61
c34 c5 d5 s61 c6
s34 s4 s25 d3 c34 s61 s6
d1 c34 s5 s61 s6 c61 = c25 c61 c34
c5 c61 s34 s4 s5
d6 d3 s34 c4 s61 c6 c34
c34 c5 a3 s4 c6 s61 d1 s34 s4 s5 c261 d1 s34 s4 s6 c5 s61 c61 s34 s4 s25 a3 c4 c61 d1 s34 c4 s61 c6 c61 d3 c61
s5 s61 s6 c34 c5 s25 s61 s6 s34 s4
a3 s4 c6 s61 c34
c25 c261 d1 c34
c34 c61 ;
21
d1 c34
s25 s61 s6 a6 c34 c6
s25 s61 s6 d1 s34 s4 c61
s5 s61 s6 d5 s34 c4
d3 c234 s5 s61 s6
s5 s61 s6 d3
s25 s261 s26 d1 c34
s5 s61 s6 a6 s34 s4 c6 c5 a3 c4 s5 c61 c34 c5 c261 d1 s34 s4 s5 s5 s61 s26 a6 s34 c4
d5 s61 c6 c34
c5 c61 d5 s34 c4 c5 c61 d1 s34 c4 s61 c6 c5 c61 a6 c34 s5 c6 c25 c61 a6 s34 s4 c6 s5 s261 s26 d1 s34 s4 c5 a3 c4 c5 s6 s61 c34 2c5 c61 d1 c34 s5 s61 s6
d3 s34 s4 s6 c5 s61 c34 d3 s34 s4 s5 c61 c34 = c25 c61 c34
s5 s261 s6 d1 s34 c4 c5
d3 c234 c5 c61
c5 c61 a6 s34 c4 s6 c25 c61 d1 s34 s4 s6 s61 c5 c61 d3
c5 c61 s34 s4 s5
s5 s61 s6 c34 c5 s25 s61 s6 s34 s4
c34 c61 :
22
A trigonometric-function equation containing only variable h5 can be obtained by substituting Eqs. (21) and (22) into Eq. (19b). It can be changed into a univariate polynomial equation after half-angle replacement x5 tan h5 =2. Using Eq. (18), each coecient of the polynomial equation can be proved to be zero. Hence Eq. (19b) can be ®nally simpli®ed as type ``0 0'', indicating that this equation is only dependent on Eqs. (19c) and (19d) so that Nequ 3 < Nvar 4: Therefore, all variables possess in®nite solutions. (c) Breaking at joints z3 and z5 . In this case the variables discussed are h1 ; h2 ; d4 and d6 . Turning the subscripts of equation set (1a)±(1d), we get the following basic equation set: eq1 z3 z5 A z3 z5 B ; eq2 R z3 z5 A R z3 z5 B ; 23 eq3 R z3 A R z3 B ; eq4 R z5 A R z5 B ; where RA a3 x4 d4 z4 a4 x5 ; RB
d5 z5 a5 x6 d6 z6 a6 x1 d1 z1 a1 x2 d2 z2 a2 x3 d3 z3 ;
RA RB :
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Considering overconstraint condition (15), equation set (23) can be simpli®ed as: eq1
s34 c4 g1 s1 s2 ; s1 c2 ; c1 s2 ; c1 c2 ; s1 ; c1 ; c2 ; 1;
eq2 a3 c4 c34
24a
s4 d4 s34
g2 s1 s2 d6 ; s1 c2 d6 ; c1 s2 d6 ; c1 c2 d6 ; s1 s2 ; s1 c2 ; c1 s2 ; c1 c2 ; s1 d6 ; c1 d6 ; c2 d6 ; s1 ; c1 ; c2 ; d6 ; 1;
24b
eq3 d4 c34 g3 s1 s2 d6 ; c1 c2 d6 ; s1 s2 ; s1 c2 ; c1 s2 ; c1 c2 ; c1 d6 ; c2 d6 ; s1 ; c1 ; c2 ; d6 ; 1;
24c
eq4 a3 s4 g4 s1 s2 ; s1 c2 ; c1 s2 ; c1 c2 ; s1 ; c1 ; c2 ; 1:
24d
Similar to case (a), Eq. (24a) has the same power products as those in Eq. (24d) and there is also a linear transformation between them: g4
a6 s6
d5 d1 c6 s61
d3 g1 :
So we can also derive Eq. (18) from Eqs. (24a) and (24d). Hence equation set (24a)±(24d) has only three independent equations and Nequ 3 < Nvar 4: Based on the above results of (a), (b) and (c), such a 4R2P linkage can be con®rmed as an overconstrained mechanism with the following existence conditions: a1 a2 d2 a4 a5 0; a6 s6
d1 c6 s61 a3 s4
a45 a56 p=2;
25a
d3 c4 s34 d5 0;
25b
where Eq. (25a) is the overconstraint condition and Eq. (25b) the closure condition. (iii) Deducing the I±O equations. Eqs. (20)±(22), (17a) and (24a) constitute I±O relationships, where Eq. (17a) can be expanded as s61 c6 c12 s23 s3 s4
c12 s23 c3 c4 c34
c12 c23 s34 c4 s12 s2 s4 c3 s12 c2 s3 c23 s4 s12 c2 s34 s23 c4
s12 c2 c3 c4 c23 c34 s12 s2 c34 s3 c4 ; and Eq. (24a) can be expanded as s34 c4 c23 s12 s6 s1 s23 s6 s2 c1 s23 c6 s1 c61 s2
c61 s12 c6 c1 c23 s61 c6 s12 s23 c2
c12 c23 s61 c6
s23 c6 c1 c2 c61 c12 s23 s6 c12 s1 c2 : 5. Conclusions In this paper we are interested in establishing a practical and general analytical method for analyzing overconstriant of spatial 6-link loop. The main characteristics of this method are: · Only four basic equations are needed to analyze overconstraint of mechanisms by using overconstraint criterion and turn technique. · Using algebraic operation, we can ®nd the dependency among the four basic equations, namely, one of which can be changed into a ``0 0'' type equation. The exact operation may be in linear or nonlinear mode. The nonlinear mode may be completed through eliminating variables.
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· The purpose of turn subscripts of all parameters is to avoid some rigid joints existing in the linkage and to ®nd I±O relationships completely. · The method described in this paper can be used to determinate overconstraint of known 6-link overconstrained mechanisms and perfect their existence conditions. This approach can also be applied for 5-link and 4-link loops. For the 5-link loop only three basic equations are needed, and for the 4-link loop just two needed. As to the assumption of overconstraint conditions for new mechanisms, the further research is required. · The theory of overconstraint analysis of linkages presented in this paper also provides an important theoretical foundation for ®nding new types of serial and parallel robots.
Acknowledgements The ®nancial support of the National Science Foundation of PR China (Grant NSFF 59875084) is acknowledged and appreciated. References [1] J.E. Baker, On 5-revolute linkages with parallel adjacent joint axes, Mechanism and Machine Theory 19 (6) (1984) 467±475. [2] K. Wohlhart, A new 6R space mechanism, in: Proceedings of the Seventh World Congress on IFToMM, Sevilla, vol. 1, 1987, pp. 193±198. [3] K. Wohlhart, Merging two general goldberg 5R linkages to obtain a new 6R space mechanism, Mechanism and Machine Theory 26 (1991) 659±668. [4] C. Mavroidis, B. Roth, New and revised overconstrained mechanisms, ASME Journal of Mechanical Design 117 (1995) 75±82. [5] K.J. Waldron, Hybrid overconstrained linkages, Journal of Mechanisms 3 (1968) 73±78. [6] C. Mavroidis, B. Roth, Analysis and synthesis of overconstrained mechanisms, in: G.R. Pennock (Ed.), Mechanism Synthesis and Analysis, vol. DE-70, ASME, New York, 1994, pp. 115±133. [7] M. Goldberg, New ®ve-bar and six-bar linkages in three dimensions, Transaction at the ASME 65 (1943) 649±661. [8] J.E. Baker, A comparative survey of the Bennett-based 6-revolute kinematic loops, Mechanism and Machine Theory 28 (1) (1993) 83±96. [9] C. Huang, C.C. Sun, An investigation of screw systems in the ®nite displacements of Bennett-based 6R linkages, in: Proceedings of the 1998 ASME Design Engineering Technical Conference, Atlanta, Georgia, DETC98/MECH5905, 1998. [10] R. Bricard, Lecons de cinematique, Bd. II, Paris (1927) 7±12. [11] K.J. Waldron, Symmetric overconstrained linkages, ASME Journal of Engineering for Industry 91 (1969) 158±162. [12] J.E. Baker, An analysis of the Bricard linkages, Mechanism and Machine Theory 15 (1980) 267±286. [13] J.E. Baker, On seeking new plane-symmetric six-bars, Mechanism and Machine Theory 21 (3) (1986) 243±252. [14] P. Schatz, Rhytthmusforschung und Technik, Freies Geistesleken, Stuttgart (1975) C.C. [15] V. Brat, A six-link spatial mechanism, Journal of Mechanisms 4 (4) (1969) 325±336. [16] H.C. Yu, Geometrical investigation of a general octahedral linkage and the turbula, Mechanism and Machine Theory 15 (1980) 463±478. [17] J.E. Baker, T. Duclong, P.S.H. Khoo, On attempting to reduce undesirable inertial characteristics of the Schatz mechanism, ASME Journal of Mechanical Design 104 (1982) 192±205. [18] C.C. Lee, A geometric±algebraic exploration of two kinds of Schatz 6R linkages, in: Proceedings of the 1998 ASME Design Engineering Technical Conference, Atlanta, Georgia, DETC98/MECH-5903, 1998.
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[19] J. Phillips, Freedom in Machinery, Vol. 2, Screw Theory Exempli®ed, Cambridge University Press, Cambridge, 1990. [20] G.T. Bennett, A new mechanism, Engineering 76 (1903) 777±778. [21] F.E. Myard, Contribution a la geometrie des systemes articules, Bulletin of the Society of Mathematics, France 59 (1931) 183±210. [22] K.H. Hunt, Prismatic Pairs in spatial linkages, Journal of Mechanisms 2 (1967) 213±230. [23] B. Dizioglu, Theory and practice of spatial mechanisms with special positions of the axes, Mechanism and Machine Theory 13 (1978) 139±153. [24] T.L. Yang, Structural analysis and number synthesis of spatial mechanisms, in: Proceedings of the Sixth World Congress or the T.M.M., vol. 1, 1983, pp. 280±283. [25] T.L. Yang, Kinematic structural analysis and synthesis of overconstrained spatial single-opened-chains, in: Proceedings of the 19th Biennial Mechanisms Conference, ASME paper 86-DET-189, 1986. [26] F.M. Dimentberg, A general method for the investigation of ®nite displacements of spatial mechanisms and certain cases of passive joints, Akad. Nauk. SSSR, Trudi Sem. Teorii Mash. Mekh 5 (1948) 5±39. [27] K.J. Waldron, Overconstrained linkages, Environment and Planning 6 (1979) 393±402. [28] Lee, H.S. Yan, Movable spatial 6R mechanisms with three adjacent parallel axes, ASME Journal of Mechanical Design 115 (1993) 522±529. [29] C. Mavroidis, B. Roth, Analysis of overconstrained mechanisms, ASME Journal of Mechanical Design 117 (1995) 69±74. [30] H.Y. Lee, C.G. Liang, Displacement analysis of the general spatial 7-link 7R mechanism, Mechanism and Machine Theory 23 (3) (1988) 209±217. [31] H.Y. Lee, C.F. Reinholtz, Inverse kinematics of serial-chain manipulators, ASME Journal of Mechanical Design 118 (1996) 396±404. [32] M. Raghavan, B. Roth, Inverse kinematics of the general 6R manipulator and related linkages, ASME Journal of Mechanical Design 105 (1993) 502±508.
www.elsevier.com/locate/mechmt
Overconstraint analysis on spatial 6-link loops Qiong Jin a, Tingli Yang a
b,*
Department of mechanical Engineering, Southeast University, Nanjing 210096, PR China b Jinling Petrochemical Corporation, Nanjing 210037, PR China Received 5 July 2000; accepted 25 September 2001
Abstract In order to reduce the complexity of overconstraint analysis on spatial 6-link loops, we present a complete closed-form analytical method using only four basic equations combined with a ``turn technique'', in which the closure conditions can be obtained and input±output (I±O) equations can be further deduced. Based on this method, we can verify the overconstraint of the Bricard trihedral mechanism and derive its closure condition easily from its overconstraint condition. Also, the closure condition and I±O equations of a known 4R2P overconstrained mechanism are deduced for the ®rst time. Ó 2002 Published by Elsevier Science Ltd. Keywords: Overconstraint criterion; Existence condition; I±O relationship; 6-Link loop
1. Introduction Among all types of overconstrained mechanisms, the 6-link mechanism is one of the most dicultly analyzed mechanisms. After Baker [1] listed all known 6R overconstrained mechanisms in 1984, Wohlhart [2] proposed a 6R symmetric linkage in 1987, and presented a 6R overconstrained mechanism from merging two general Goldberg 5R overconstrained mechanisms in 1991 [3]; Mavroidis and Roth [4] developed a 6R overconstrained mechanism with two Bennett joints that have no common axis in 1995. There are also some 6-link overconstrained mechanisms containing prismatic joints, such as Waldron's [5] hybrid overconstrained mechanisms. In literature [6], a 4R2P mechanism with two pairs of parallel revolute joints and a 5R1P mechanism with one planar joint and one pair of parallel revolute joints are presented.
*
Corresponding author. Tel.: +86-25-509-8935. E-mail address: [email protected] (T. Yang).
0094-114X/02/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 0 9 4 - 1 1 4 X ( 0 1 ) 0 0 0 7 2 - 6
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In general, 6-link overconstrained mechanisms can be classi®ed into three categories: (a) Bennett based mechanisms; (b) symmetric mechanisms; and (c) variant mechanisms of (a) or (b). For the ®rst category, Wohlhart [3], Mavroidis and Roth [4], Waldron [5], Goldberg [7], Baker [8], and Huang and Sun [9], etc. have studied it. For the second category, Bricard [10], Waldron [11], Baker [12,13], Mavroidis and Roth [4] analyzed it in detail. As for the third category, a typical example is Schatz mechanism [14] derived from a special plane-symmetric Bricard trihedral loop [10], which was further investigated and developed for practical use by many researchers [15±18]. Although many 6-link overconstrained mechanisms have been proposed, few of them possess a wide application like Schatz mechanism because these so-called overconstrained mechanisms have not been understood thoroughly [19]. Overconstraint analysis of mechanisms should include three parts: overconstraint criterion, overconstraint existence conditions, and input±output (I±O) equations. The existence conditions of overconstrained mechanisms include the overconstraint condition and the closure condition. The former, i.e., linear complex condition in screw theory, is very important for achieving overconstraint, and the latter makes sure that the kinematic chain will be closed. There are following three methods to analyze the overconstraint mechanisms: (i) Geometrical method. Using geometric intuition or synthetic reasoning, most of the early known overconstrained mechanisms were discovered by geometricians such as Goldberg [7], Bricard [10], Bennett [20], and Myard [21] mechanisms. However, the identical mode cannot be obtained using this method because the deductions of existence conditions are dierent for various overconstrained mechanisms. (ii) Screw theory. Screw theory had been used to analyze overconstraint for a long time by many researchers such as Hunt [22], Dizioglu [23] and Yang [24,25]. However, the closure conditions are dicult to be obtained in some degree using this method. (iii) Analytical method. In recent decades, analytical method has become more important. Its main idea is that: if a univariate polynomial derived from the matrix-loop equation is identically equal to zero, i.e., the coecient of each power of the joint variable is equated to zero, then the corresponding loop is an overconstrained mechanism. Equating the coecients of the polynomial to zero gives a set of equations in the constant geometric parameters, and these equations are the existence conditions for the loop to be mobile. Hence, the I±O equations for the mechanism could be easily obtained. Baker [1], Dimentberg [26], Waldron [27], Lee and Yan [28], and Mavroidis and Roth [4,29], etc. found some new mechanisms or obtained valuable results using this method. Since Lee and Liang [30] derived the univariate polynomial of 16th degree for the displacement analysis of general 7R loop in 1988, it has become clear that the inverse kinematics problem of serial-chain manipulators can be reduced to the univariate polynomials from 14 original equations. Based on this consideration, Mavroidis and Roth [29] pointed out that the problem of overconstrained mechanisms could also be analyzed using the 16-solution equation set for general 6R manipulators. Theoretically, it is no doubt a systematical idea. In fact, due to the great calculation work and the limitation of calculating capability of computers nowadays, only a numerical 16th degree univariate polynomial equation can be obtained. Therefore only numerical analysis can be operated in such mode, and I±O equations can be obtained numerically but in closed-form. In order to discover new overconstrained mechanisms, researchers usually assume some overconstraint conditions, and further deduce the perfect existence conditions and the closure
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269
conditions using the criterion of overconstrained mechanisms. The key step here is to establish a kind of simple and eective criterion. Hence in this paper, a new criterion, which can eectively distinguish the ®nite and in®nite solutions of displacement analysis, is presented for the 6-link overconstrained mechanism. According to this criterion, analytical I±O equations of overconstrained mechanisms can be deduced ®nally. After using this criterion to verify the known overconstrained mechanisms, some drawbacks, such as the unperfectness of existence conditions and the existing of some rigid joints in the loop, can be easily identi®ed.
2. Principle of overconstraint analysis 2.1. Basic equations It is certain that the solution number for displacement analysis of a general 6R loop is ®nite. The main goal in displacement analysis is to achieve the minimum degree of a univariate polynomial. Hence in this case, the original equations should be built as many as possible so that extraneous roots could be circumvented, for examples, Lee and Reinholtz [31], and Raghavan and Roth [32] derived 16 solutions from 14 basic equations. However, for overconstrained mechanisms, to determinate overconstraint is virtually to distinguish the ®nite solutions and in®nite solutions of displacement equations since there always exists correlatability among basic equations in which all variables have in®nite solutions. In order to diminish the calculating scale for studying overconstraint analytically, it would be better to directly analyze the independence of the original equations than to seek all coecients of a univariate polynomial equation being zero. Therefore, the original equations needed should be as few as possible under condition that the equation set possesses ®nite solutions in general. Fig. 1 shows a general 6-link loop, in which zi and xi represent joint-axis-direction and linkdirection unit vectors, respectively. And the parameters di ; ai ; aij and hi are de®ned so that: di is the oset along joint axis zi ; ai is the normal distance along xi1 ; aij is the twist angle between the joint axes zi and zi1 , and hi is the rotation angle about joint axis zi . In the following sections, sin hi , cos hi , sin aij and cos aij are abbreviated as si ; ci ; sij and cij , respectively. When the 6-link loop is
Fig. 1. A general spatial 6-link single loop.
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broken at two kinematic joints it becomes two single-opened-chains (SOCs): SOCA and SOCB . Generally, from the viewpoint of kinematics compatibility of two SOCs, we can establish an equation set with four equations including four kinematic variables and possessing ®nite solutions. If the loop is broken at joints z1 and z3 , the corresponding four equations can be expressed as follows: eq1 z1 z3 A z1 z3 B ;
1a
eq2 R z1 z3 A R z1 z3 B ;
1b
eq3 R z1 A R z1 B ;
1c
eq4 R z3 A R z3 B ;
1d
where the SOC vectors are: RA a1 x2 d2 z2 a2 x3 ; RB
d3 z3 a3 x4 d4 z4 a4 x5 d5 z5 a5 x6 d6 z6 a6 x1 d1 z1 ;
RA RB : Equation set (1a)±(1d) re¯ecting the restrictive relationship in spatial single loop, possesses geometrical meaning clearly. Under the ane coordinate system consisting of unit vectors z1 ; z3 and z1 z3 , Eqs. (1b), (1c) and (1d), corresponding to the three projections of vector R, are independent each other. As to Eq. (1a), it is the scalar product of unit vectors z1 and z3 , and has no relationship with the three components of vector R. Therefore the four Eqs. (1a)±(1d) are independent. It should be noted that the loop must be broken at two appropriate joints in order to make equation set (1a)±(1d) contain only four variables. 2.2. Overconstraint criterion If the geometrical parameters of the loop are given specially so that the number of dependent equations Nequ and the number of dependent variables Nvar of equation set (1a)±(1d) satisfy the following relationship Nequ < Nvar ;
2
and if expression (2) is always true when all terms' subscripts are taken turns, then all variables of the 6-link loop cannot be solved ®nitely. Such a loop is called the overconstrained mechanism. This is the criterion of overconstrained mechanism. The turn operation is virtually to change the broken positions of the loop till all six variables in the loop are analyzed. Inequality (2) indicates that at least one dependent equation exists in equation set (1a)±(1d), which can be linearly (see Section 4(a) and (c)) or nonlinearly derived from other independent equations in equation set (1a)±(1d). The nonlinear derivation manner can be normally implemented in two steps: ®rstly eliminating some unknowns to derive a polynomial equation, and secondly proving the polynomial equation to be identical equation. One kind of identical equations is a polynomial identically equated to zero in which the coecient of each power of the joint variable can be proved to be zero (see Section 4(b)). Another kind of identical equations contains at least one common factor being zero (see Section 3).
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2.3. Closure condition The conditions, on the linkage's structural parameters that ful®ll the overconstraint criterion, will be called existence conditions of the overconstrained mechanism. Sometimes, only the overconstraint condition but not the closure condition can be given using the screw theory. While using analytical method presented in this paper, when overconstraint conditions are substituted into the above ``turned'' equation sets, some constant equations can be derived, which are in fact closure conditions. The closure conditions are often outcomes of overconstraint determination. 2.4. Main steps (i) Assuming some overconstraint conditions. In order to assume the overconstraint conditions, three aspects should be considered: symmetry, dependency possibly existing in the simpler hand side of equation set (1a)±(1d), and topological structures of overconstrained mechanisms. (ii) Determinating overconstraint and perfecting existence conditions. When the assumed part of overconstraint conditions is substituted into equation set (1a)±(1d), the overconstraint of the loop could be determined using the criterion. It should be noted that the turn operation is needed before the ®nal conclusion is given. Meanwhile overconstraint conditions and closure condition can be supplemented. (iii) Deducing the I±O equations. Synthesizing the results obtained by the turn operation, ®ve simplest equations can be selected as I±O relationships from all independent equations. 3. Bricard trihedral mechanism In 1927, Bricard found some symmetric 6R overconstrained mechanisms using geometrical method [10]. Here we only discuss the trihedral mechanism. (i) The given existence conditions are: a12 a34 a56 p=2;
a23 a45 a61 3p=2;
a21
a26 ;
a23
a25
a22
a24
di 0 i 1±6;
3a 3b
where condition (3a) is the overconstraint condition with exacting symmetry, and condition (3b) is a complicated closure condition. The following step shows that condition (3b) can be derived from condition (3a) easily. (ii) Determinating overconstraint and deducing the closure condition. After breaking the loop at joints z1 and z3 , the remainder variables are h2 ; h4 ; h5 and h6 . The four basic equations built in this case are the same as those indicated in equation set (1a)±(1d). To simplify equation set (1a)±(1d) condition (3a) can be substituted into it. The result is eq1 c2
s4 s6 c5 c4 c6 ;
eq2 0 s5 a3 s6
4a
a6 s4 a4 c4 s6
a5 c6 s4 ;
4b
eq3 a2 s2 a4 c5 s6 a3 s4 c6 a3 c4 c5 s6 a5 s6 ;
4c
eq4
4d
a1 s 2
a6 c 4 s 6
a4 s4
a6 s4 c6 c5
a5 s 4 c 5 :
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If h5 is a variable, the following result can be deduced from Eq. (4b) easily a3 s6
a6 s4 a4 c4 s6
a5 c6 s4 0:
5
Hence the I±O relationship between variables h4 and h6 can be obtained from Eq. (5). Assuming that the variables of h4 and h6 are given, from Eqs. (4c) and (4d) we can derive: s2 a4 a6 c4 s26 a24 s4 s6 a3 a6 c24 s26 s4 s6 c4 a3 a4
a5 a6 s4 s6 c6
a25 s4 s6
a3 a6 s24 c26
a3 a5 s24 c6 =D; c5 a4 a2 s4 a6 a2 c4 s6
6 a1 a3 s4 c6
a1 a5 s6 =D;
7
where D a1 a3 s6 c4 a1 a4 s6 a5 a2 s4 a6 a2 s4 c6 : Using Eq. (5), Eq. (6) can be further simpli®ed as s2 s4 s6 a26 a24
a25
a23 =D:
8
Substituting c5 of Eq. (7) into Eq. (4a) and simplifying it using Eq. (5), we can obtain c2 a1 a5 s4
a4 a2 s6
a2 a3 c4 s6 a1 a6 c6 s4 =D:
9
It can be realized that the original equation set 4a,(4b)±(4d) can be expressed as Eqs. (5), (7), (8) and (9). Considering the distribution of variables in these four equations, if there exist dependent equations, only Eqs. (8) and (9) are possibly dependent. If Eqs. (8) and (9) satisfy the Pythagorean theorem s22 c22 1, then Eq. (9) can be derived from Eq. (8), i.e., only one of these two equations is dependent. Therefore, the condition that satis®es the relationship s22 c22 1 is just the supplemental existence condition or closure condition. Based on the above consideration, substituting Eqs. (8) and (9) into the following expression: t2 s22 c22
1;
10
and simplifying it using Eq. (5), we get t2 s24 s26 AB=D2 ;
11
where A a26 a24 a25 a23 and B a26 a24 a25 a23 a21 a22 . If t2 0, then A 0 or B 0. Due to the symmetry of the linkage, when subscripts of equation set (4a)±(4d) are taken turns, Eq. (10) always contains factor B but not factor A, indicating that the precondition of ti 0 is just B 0, i.e., a26 a24
a25
a23
a21 a22 0:
12
Eq. (12) is none other but the closure condition given in the known condition (3a) and (3b). Hence, only Eqs. (4b)±(4d) among the four basic equations are independent, i.e., Nequ 3 < Nvar 4: This indicates that none of the variables h2 ; h4 ; h5 and h6 can be solved ®nitely. For another two variables h1 and h3 , they can be analyzed by breaking the loop at joints z2 and z4 . Due to the symmetry of the loop, the four equations in this case are identical to those in equation set (4a)±(4d) which results in in®nite solutions for both variables h1 and h3 . Therefore this linkage is an overconstrained mechanism.
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273
This example shows that even if the overconstraint existence conditions are not perfect the complementary condition can be always found out during overconstraint determination. A univariate polynomial is not needed in the overconstraint analysis of this linkage, which makes the overconstraint determination simple. (iii) Deducing the I±O equations. Eqs. (5), (7) and (8) can be regarded as three independent equations. Using symmetry and turn operation, another two relationships similar to Eq. (5) can be derived as s3 a6 a1 c1 s1 a3 a2 c3 ;
13
s5 a2 a3 c3 s3 a5 a4 c5 :
14
These ®ve equations constitute the I±O relationships. The I±O relationships may be expressed in various forms. For example, we can give another group I±O equations for this linkage, four of which are obtained by turning the subscripts of Eq. (5) and the ®fth equation is Eq. (4a). 4. 4R2P mechanism In 1968, Waldron [5] found a group of hybrid overconstrained mechanisms from screw theory. One of these mechanisms is a RRR±PHP linkage formed from a spherical linkage and a screwdouble slider linkage. Now we introduce a special case of this 6-link loop with the zero-pitch screw joints. The existence conditions and I±O relations of such a 4R2P linkage have not been discussed explicitly till now. Here we let both prismatic joints (P4 and P5 ) be perpendicular to and intersect with the revolute joint (R5 ) (see Fig. 2). (i) According to the screw theory, the overconstraint condition of this linkage is a1 a2 d2 a4 a5 0;
a45 a56 p=2:
15
(ii) Determinating overconstraint and perfecting existence conditions. In order to ensure that the equation set has only four variables, the broken positions should not be selected at the prismatic joints. (a) Breaking at joints z5 and z1 . In this case variables are h2 ; h3 ; d4 and d6 . Turning the subscripts of equation set (1a)±(1d) we get
Fig. 2. A 4R2P overconstrained mechanism.
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eq1 z5 z1 A z5 z1 B ; eq2 R z5 z1 A R z5 z1 B ; eq3 R z5 A R z5 B ; eq4 R z1 A R z1 B ;
16
where RA a5 x6 d6 z6 a6 x1 ; RB
d1 z1 a1 x2 d2 z2 a2 x3 d3 z3 a3 x4 d4 z4 a4 x5 d5 z5 ;
RA RB : Substituting overconstraint condition (15) into equation set (16), we obtain a large equation set (17a)±(17d). For the convenience of description, the right-hand side of equation set (17a)±(17d) is expressed by function fi i 1; 2; 3; 4. Terms in each bracket are power products of function fi , and ``1'' expresses the constant term. eq1
s61 c6 f1 s2 s3 ; s2 c3 ; c2 s3 ; c2 c3 ; c2 ; s3 ; c3 ; 1;
eq2 a6 c6 c61
17a
s61 s6 d6
f2 s2 s3 d4 ; s2 c3 d4 ; c2 s3 d4 ; c2 c3 d4 ; s2 s3 ; s2 c3 ; c2 s3 ; c2 c3 ; c2 d4 ; s3 d4 ; c3 d4 ; c2 ; s3 ; c3 ; d4 ; 1;
17b
eq3 a6 s6 f3 s2 s3 ; s2 c3 ; c2 s3 ; c2 c3 ; c2 ; s3 ; c3 ; 1;
17c
eq4 d6 c61 f4 s2 s3 d4 ; c2 c3 d4 ; s2 s3 ; s2 c3 ; c2 s3 ; c2 c3 ; c2 d4 ; c3 d4 ; c2 ; s3 ; c3 ; d4 ; 1:
17d
Eqs. (17a) and (17c) have the same power products and there is a linear relationship between them f3
d5
a3 s4 d3 s34 c4
d1 f1 :
Substituting Eq. (17a) into Eq. (17c), we have a6 s6
d1 c6 s61 a3 s4
d3 c4 s34 d5 0:
18
Eq. (18), a constant equation free from unknown variables, is the closure condition of the linkage. Eq. (17a) shows the relationship between variables h2 and h3 . From simultaneous Eqs. (17b) and (17d), we can express variables d4 and d6 in terms of h2 and h3 . Since Eqs. (17a) and (17c) are equivalent, only three of equation set (17a)±(17d) are independent, i.e., Nequ 3 < Nvar 4: Therefore, all variables possess in®nite solutions when joints of z5 and z1 are broken. (b) Breaking at joints z1 and z3 . In this case, the variables discussed here are h2 ; d4 ; h5 and d6 . Substituting overconstraint condition (15) into equation set (1a)±(1d), we ®gure out that eq1 c23 c12
s23 c2 s12 g1 s5 ; c5 ; 1;
19a
eq2 0 g2 d4 s5 ; d4 c5 ; d6 s5 ; d6 c5 ; d4 ; d6 ; s5 ; c5 ; 1;
19b
eq3 0 g3 d4 s5 ; d4 c5 ; d6 ; s5 ; c5 ; 1;
19c
eq4 0 g4 d6 s5 ; d6 c5 ; d4 ; s5 ; c5 ; 1;
19d
Q. Jin, T. Yang / Mechanism and Machine Theory 37 (2002) 267±278
where Eq. (19a) can be expanded as c23 c12 s23 c2 s12 c34 c5 c61 c34 s5 s61 s6 s34 s4 s5 c61 s34 s4 s6 c5 s61 s34 c4 s61 c6 :
275
20
Solving the simultaneous Eqs. (19c) and (19d) for variables d4 and d6 in terms of h5 , we get d4 s34 s4 s5 d1 d5 s34 c4 c61 d1 c34 c5 c261 c234 c25 d3 c61 a6 s34 c4 s6 c61 a6 s34 s4 c6 c5 c61 c234 c5 d3 s5 s61 s6 c34 c25 a3 c4 s6 s61 c34 c25 d3 s34 s4 s6 s61 c34 c5 d1 c34 c5 a3 c4 s5 c61 s34 s4 s5 d5 s61 c6 s34 s24 s5 a3 c6 s61
s234 s24 s5 d3 s6 c5 s61
s234 s4 s5 d3 c4 s61 c6
s34 s4 s5 a3 c4 c5 s6 s61 2c34 c5 d3 s34 s4 s5 c61 a6 c34 s5 c6 c61 c34 c5 d3 s34 c4 s61 c6 s234 s24 s25 d3 c61
c34 c5 d5 s61 c6
s34 s4 s25 d3 c34 s61 s6
d1 c34 s5 s61 s6 c61 = c25 c61 c34
c5 c61 s34 s4 s5
d6 d3 s34 c4 s61 c6 c34
c34 c5 a3 s4 c6 s61 d1 s34 s4 s5 c261 d1 s34 s4 s6 c5 s61 c61 s34 s4 s25 a3 c4 c61 d1 s34 c4 s61 c6 c61 d3 c61
s5 s61 s6 c34 c5 s25 s61 s6 s34 s4
a3 s4 c6 s61 c34
c25 c261 d1 c34
c34 c61 ;
21
d1 c34
s25 s61 s6 a6 c34 c6
s25 s61 s6 d1 s34 s4 c61
s5 s61 s6 d5 s34 c4
d3 c234 s5 s61 s6
s5 s61 s6 d3
s25 s261 s26 d1 c34
s5 s61 s6 a6 s34 s4 c6 c5 a3 c4 s5 c61 c34 c5 c261 d1 s34 s4 s5 s5 s61 s26 a6 s34 c4
d5 s61 c6 c34
c5 c61 d5 s34 c4 c5 c61 d1 s34 c4 s61 c6 c5 c61 a6 c34 s5 c6 c25 c61 a6 s34 s4 c6 s5 s261 s26 d1 s34 s4 c5 a3 c4 c5 s6 s61 c34 2c5 c61 d1 c34 s5 s61 s6
d3 s34 s4 s6 c5 s61 c34 d3 s34 s4 s5 c61 c34 = c25 c61 c34
s5 s261 s6 d1 s34 c4 c5
d3 c234 c5 c61
c5 c61 a6 s34 c4 s6 c25 c61 d1 s34 s4 s6 s61 c5 c61 d3
c5 c61 s34 s4 s5
s5 s61 s6 c34 c5 s25 s61 s6 s34 s4
c34 c61 :
22
A trigonometric-function equation containing only variable h5 can be obtained by substituting Eqs. (21) and (22) into Eq. (19b). It can be changed into a univariate polynomial equation after half-angle replacement x5 tan h5 =2. Using Eq. (18), each coecient of the polynomial equation can be proved to be zero. Hence Eq. (19b) can be ®nally simpli®ed as type ``0 0'', indicating that this equation is only dependent on Eqs. (19c) and (19d) so that Nequ 3 < Nvar 4: Therefore, all variables possess in®nite solutions. (c) Breaking at joints z3 and z5 . In this case the variables discussed are h1 ; h2 ; d4 and d6 . Turning the subscripts of equation set (1a)±(1d), we get the following basic equation set: eq1 z3 z5 A z3 z5 B ; eq2 R z3 z5 A R z3 z5 B ; 23 eq3 R z3 A R z3 B ; eq4 R z5 A R z5 B ; where RA a3 x4 d4 z4 a4 x5 ; RB
d5 z5 a5 x6 d6 z6 a6 x1 d1 z1 a1 x2 d2 z2 a2 x3 d3 z3 ;
RA RB :
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Considering overconstraint condition (15), equation set (23) can be simpli®ed as: eq1
s34 c4 g1 s1 s2 ; s1 c2 ; c1 s2 ; c1 c2 ; s1 ; c1 ; c2 ; 1;
eq2 a3 c4 c34
24a
s4 d4 s34
g2 s1 s2 d6 ; s1 c2 d6 ; c1 s2 d6 ; c1 c2 d6 ; s1 s2 ; s1 c2 ; c1 s2 ; c1 c2 ; s1 d6 ; c1 d6 ; c2 d6 ; s1 ; c1 ; c2 ; d6 ; 1;
24b
eq3 d4 c34 g3 s1 s2 d6 ; c1 c2 d6 ; s1 s2 ; s1 c2 ; c1 s2 ; c1 c2 ; c1 d6 ; c2 d6 ; s1 ; c1 ; c2 ; d6 ; 1;
24c
eq4 a3 s4 g4 s1 s2 ; s1 c2 ; c1 s2 ; c1 c2 ; s1 ; c1 ; c2 ; 1:
24d
Similar to case (a), Eq. (24a) has the same power products as those in Eq. (24d) and there is also a linear transformation between them: g4
a6 s6
d5 d1 c6 s61
d3 g1 :
So we can also derive Eq. (18) from Eqs. (24a) and (24d). Hence equation set (24a)±(24d) has only three independent equations and Nequ 3 < Nvar 4: Based on the above results of (a), (b) and (c), such a 4R2P linkage can be con®rmed as an overconstrained mechanism with the following existence conditions: a1 a2 d2 a4 a5 0; a6 s6
d1 c6 s61 a3 s4
a45 a56 p=2;
25a
d3 c4 s34 d5 0;
25b
where Eq. (25a) is the overconstraint condition and Eq. (25b) the closure condition. (iii) Deducing the I±O equations. Eqs. (20)±(22), (17a) and (24a) constitute I±O relationships, where Eq. (17a) can be expanded as s61 c6 c12 s23 s3 s4
c12 s23 c3 c4 c34
c12 c23 s34 c4 s12 s2 s4 c3 s12 c2 s3 c23 s4 s12 c2 s34 s23 c4
s12 c2 c3 c4 c23 c34 s12 s2 c34 s3 c4 ; and Eq. (24a) can be expanded as s34 c4 c23 s12 s6 s1 s23 s6 s2 c1 s23 c6 s1 c61 s2
c61 s12 c6 c1 c23 s61 c6 s12 s23 c2
c12 c23 s61 c6
s23 c6 c1 c2 c61 c12 s23 s6 c12 s1 c2 : 5. Conclusions In this paper we are interested in establishing a practical and general analytical method for analyzing overconstriant of spatial 6-link loop. The main characteristics of this method are: · Only four basic equations are needed to analyze overconstraint of mechanisms by using overconstraint criterion and turn technique. · Using algebraic operation, we can ®nd the dependency among the four basic equations, namely, one of which can be changed into a ``0 0'' type equation. The exact operation may be in linear or nonlinear mode. The nonlinear mode may be completed through eliminating variables.
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· The purpose of turn subscripts of all parameters is to avoid some rigid joints existing in the linkage and to ®nd I±O relationships completely. · The method described in this paper can be used to determinate overconstraint of known 6-link overconstrained mechanisms and perfect their existence conditions. This approach can also be applied for 5-link and 4-link loops. For the 5-link loop only three basic equations are needed, and for the 4-link loop just two needed. As to the assumption of overconstraint conditions for new mechanisms, the further research is required. · The theory of overconstraint analysis of linkages presented in this paper also provides an important theoretical foundation for ®nding new types of serial and parallel robots.
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