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SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 1
On ∆-TransformsJulia Borras, Federico Thomas, Member, IEEE, Carme Torras, Member, IEEE
Abstract—Any set of two legs in a Gough-Stewart platformsharing an attachment is defined as a ∆ component. Thiscomponent links a point in the platform (base) to a line in thebase (platform). Thus, if the two legs involved in a ∆ componentare rearranged without altering the location of the line and thepoint in their base and platform local reference frames, thesingularity locus of the Gough-Stewart platform remains thesame, provided that no architectural singularities are introduced.Such leg rearrangements are defined as ∆-transforms, and theycan be applied sequentially and simultaneously. Although itmay seem counterintuitive at first glance, the rearrangementof legs using simultaneous ∆-transforms does not necessarilylead to leg configurations containing a ∆ component. As aconsequence, the application of ∆-transforms reveals itself asa simple and yet powerful technique for the kinematic analysisof large families of Gough-Stewart platforms. It is also shownthat these transforms shed new light on the characterization ofarchitectural singularities and their associated self-motions.
Index Terms—Gough-Stewart platform, kinematic compo-nents, pure condition, architectural singularities, self-motion.
I. INTRODUCTION
THE analysis of robot singularities is probably the mostactive research topic in kinematics nowadays. Peter
Donelan, from Victoria University of Wellington in NewZealand, maintains a database currently containing over 830publications on this topic [1]. Those devoted to the singularityanalysis of parallel robots represent a steadily increasingnumber but, while the pioneer works were of a general nature[2], [3], [4], [5], most of the recent works tackle the analysis ofparticular parallel designs. In this context, the development ofnew mathematical tools for the singularity analysis of parallelrobots, though difficult, is a must. This paper tries to contributeto this end by introducing the concept of ∆-transform.
In general, substituting one leg of a Gough-Stewart platformby another arbitrary leg modifies the platform singularity locusin a rather unexpected way. Nevertheless, in those cases inwhich the considered platform contains rigid subassemblies, orcomponents [6], legs can be rearranged so that the singularitylocus is modified in a controlled way provided that thekinematics of the components is not changed [7].
The simplest component consists of two legs sharing a legattachment, as shown in Fig. 1, which we will refer to as a∆ component for obvious reasons. A ∆ component is rigidin the sense that, given the lengths of the two involved legs,a point in the platform (base) is rigidly linked to a line onthe base (platform). A leg in a ∆ component can always besubstituted by another leg connecting the point and the line
The authors are with the Institut de Robotica i Informatica Industrial(CSIC-UPC), Llorens Artigas 4-6, 08028 Barcelona, Spain. E-mails: jborras,fthomas, [email protected]. This research has been partially supportedby the Catalan Research Commission through the Robotics group, and theSpanish Ministry of Science and Innovation under the I+D project DPI2007-60858.
PSfrag replacements
l1 l2
mnd
Fig. 1. A ∆ component consists of two legs sharing one attachment.
without altering the location of the point with respect to theline and hence the kinematics of this component. In otherwords, the transformed ∆ component spans the same set ofvelocities as the original one, i.e., all those parallel to the planeformed by the attachment points, and in addition there existsa one-to-one correspondence between both vectors fields. Thissimple observation is the basis for what is referred to as ∆-transform.
The classification of Gough-Stewart platforms on the basisof the components they contain was addressed in [6]. Eachclass consists of all the manipulators obtained by addingto a given component the remaining legs up to 6 in allpossible topological configurations. Note that the manipulatorsin a class have neither the same forward kinematics nor thesame singularity structure. The present work, on the contrary,tries to come up with a transformation that preserves theplatform’s singularities, thus opening up the possibility ofclassifying platforms in families sharing the same singularitystructure. One such family of mechanisms, whose commontripod component leads them to share parallel singularities,has been studied in the context of composite serial in-parallelrobots [9].
Two remarks are in place here. First, we are dealing onlywith parallel singularities, which are those relevant for the for-ward kinematic analysis of Gough-Stewart platforms. Hence,neither serial singularities nor constraint wrenches arising inrobots with less than 6 degrees-of-freedom are considered.Second, note that the leg rearrangements induced by a ∆-transform do modify the Jacobian matrix (although not thezeros of its determinant). Thus, each new design obtained maybehave differently near singularities.
The idea of using ∆-transforms for the kinematic analysisof parallel platforms was first proposed, in a rather intuitiveway, in [11] to generate the family of flagged parallel manipu-lators. Here we formalize and extend this idea introducing thepossibility of applying these transformations simultaneously.
When the two legs in a ∆ component are made coincidentby applying a ∆-transform, a trivial architectural singularity
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 2
is introduced [12]. In this paper, we show how applying aset of ∆-transforms simultaneously leads to the possibilityof introducing complex, non-trivial, architectural singularities.Obtaining architectural singular platforms in this way pro-vides, as a by-product, a simple way to characterize theirassociated self-motions.
The polynomial expressions of the self-motion curves forsome specializations of the Gough-Stewart platform have beensolved either analytically [13], [14] or numerically using con-tinuation [15], [16]. In contrast, here we obtain a parametriza-tion of such curves in configuration space, and they are plottedin the workspace by solving the forward kinematics of a non-architectural singular platform whose leg lengths depend on aparameter.
The paper is structured as follows. Section II briefly reviewsthe characterization of singularities for Gough-Stewart plat-forms and the associated concept of pure condition. SectionIII introduces the concept of ∆-transform and studies howits application modifies the pure condition of the analyzedplatform. To show the potential of this transform for parallelplatform kinematic analysis, Sections IV and V are devoted tothe detailed analysis of the Zhang-Song and Griffis-Duffy plat-forms, respectively, based on the sequential and simultaneousapplication of ∆-transforms. Section VI provides clues andhints to apply the presented transform to parallel platformsothers than those of the Gough-Stewart type. Section VIIsummarizes the main contributions presented in this paper.Finally, an appendix compiles basic facts on the concept ofpure condition.
II. THE SINGULARITIES OF GOUGH-STEWART PLATFORMS
Let us consider the general Gough-Stewart platform inFig. 2 whose six linear actuators’ lengths are given by l1, ..., l6.In an abuse of language, legs will be denoted by theirassociated length variable. The linear actuators’ velocities,l1, l2, . . . , l6, can be expressed in terms of the platform velocityvector (v,Ω) as follows:
l1l2...l6
= J
(
v
Ω
)
, (1)
with
J =
nT1 (n1 × a1)
T
nT2 (n2 × a2)
T
......
nT6 (n6 × a6)
T
, (2)
that is, the matrix of Plucker coordinates of the six leg lines(see [17] for details), where ai are the global coordinates of theattachment point of leg li to the base of the Gough-Stewartplatform, and ni is the unit free vector anchored at ai anddefining the orientation of leg li. Note that, in order to simplifynotation, the same symbol is used to represent a point and itsposition vector.
PSfrag replacements
bi
ai
ni
li
Fig. 2. The general Gough-Stewart platform. Each leg li, i = 1, . . . , 6, isattached to the base and the platform at the points with global coordinates ai
and bi, respectively, and ni is the unit free vector such that lini = bi −ai.
The matrix J can be factorized as follows:
J =
1/l1 0 · · · 00 1/l2 · · · 0...
.... . .
...0 0 · · · 1/l6
·
(b1 − a1)T ((b1 − a1) × a1)
T
(b2 − a2)T ((b2 − a2) × a2)
T
......
(b6 − a6)T ((b6 − a6) × a6)
T
= diag(1/l1, 1/l2, . . . , 1/l6)P.
Then, the singularities of the platform are those configurationsin which
det(J) =1
l1l2l3l4l5l6det(P) = 0. (3)
The dividing term is usually neglected because it is assumedthat, in practice, leg lengths cannot be null. Thus, only the termdet(P) is considered. This term can be expressed as productsand additions of 4×4 determinants, each of them involving thehomogenous coordinates of four different leg attachments [18].The resulting expression, which will be called pure expression,can be greatly simplified for most well-known special Gough-Stewart platforms. For example, for the basic flagged and theoctahedral Gough-Stewart platforms appearing in Fig. 3, thepure expression is:∣
∣
∣
∣
a1 a2 a3 b1
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
a1 a2 b1 b2
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
a2 b1 b2 b3
1 1 1 1
∣
∣
∣
∣
,
(4)and
∣
∣
∣
∣
b3 a2 a3 b2
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
b1 a1 a3 b3
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
b2 a1 a2 b1
1 1 1 1
∣
∣
∣
∣
−∣
∣
∣
∣
b3 a2 a3 b1
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
b1 a1 a3 b2
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
b2 a1 a2 b3
1 1 1 1
∣
∣
∣
∣
,
(5)
respectively [18]. These two expressions will be useful later,in Sections IV and V, respectively.
A pure expression, when equated to zero to characterizethe configurations in which a given platform is singular, is
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 3
known as a pure condition [46] (see the Appendix for a briefcompilation of basic facts on the concept of pure condition).
PSfrag replacements b1 b2
b3
a1 a2
a3
PSfrag replacementsb1
b2
b3
a1
a2
a3
b1 b2
b3
a1 a2
a3
Fig. 3. The basic flagged (left) and the octahedral (right) Gough-Stewartplatforms.
III. THE ∆-TRANSFORMPSfrag replacements
l1 l2l2
m nn r
d1d1 d2
a1 a2a2
b1b1
a3a3 a4
Fig. 4. A ∆ component formed by legs l1 and l2. Two ∆-transforms canbe applied to move the two attachments, a1 and a2, along the line.
Let us consider a Gough-Stewart platform containing the ∆component defined by legs l1 and l2 that appears in Fig. 4(left).Now, let us introduce leg d1, as shown in the same figure.Lengths l1, l2, and d1 are not independent and the relationbetween them can be straightforwardly obtained by realizingthat the volume of the tetrahedron defined by a1, a2, a3, andb1 is null, i.e., all four points are coplanar. Then, using Euler’sformula for the volume of a tetrahedron in terms of the squareof its edge lengths [19, Problem 68], we have:
f(l1, l2, d1) =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
0 1 1 1 11 0 (m + n)2 l21 m2
1 (m + n)2 0 l22 n2
1 l21 l22 0 d21
1 m2 n2 d21 0
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
= 0.
(6)The above determinant can be recognized as the Cayley-
Menger determinant of four points (see [20] for a reviewof these determinants and their generalizations). The Cayley-Menger determinant of n points is proportional to the squaredvolume of the simplex that these points define in R
n−1. Thiskind of determinants plays a fundamental role in the so-called Distance Geometry, a branch of Geometry devoted tothe characterization and study of sets of points on the basisof only their pairwise distances. Hence, Distance Geometryhas immediate relevance where distances between points aredetermined or considered. It started to receive a lot of attention
twenty years ago with the advent of molecular magneticresonance experiments to obtain distances between atoms inrigid molecules [21], and more recently in Robotics to obtainintrinsic formulations of different kinematics problems, thusavoiding the introduction of arbitrary reference frames [22],[23].
Now, equation (6) can be expressed as:
f(l1, l2, d1) = nl21 +ml22−(m+n)d21−mn(m+n) = 0. (7)
Thus, the time derivative of the leg length d1 can beexpressed as:
d1 = −∂f∂l1∂f∂d1
l1 −∂f∂l2∂f∂d1
l2 =l1d1
n
m + nl1 +
l2d1
m
m + nl2. (8)
Then, if we substitute leg l1 by leg d1, the platform leg lengthvelocities after this substitution are given by:
d1
l2...l6
=
− ∂f/∂l1∂f/∂d1
− ∂f/∂l2∂f/∂d1
. . . 0
0 1 . . . 0...
.... . .
...0 0 . . . 1
l1l2...l6
=
l1d1
nm+n
l2d1
mm+n . . . 0
0 1 . . . 0...
.... . .
...0 0 . . . 1
l1l2...l6
.
(9)
The determinant of the above matrix isl1d1
n
m + n. (10)
As a consequence, using (1) and (3), the pure expressionof the platform after this leg substitution, say det(P1), canbe expressed in terms of the pure expression of the originalplatform, say det(P0), as:
det(P1) =n
m + n· det(P0). (11)
Thus, the pure expression, after the substitution, is equalto the original one multiplied by the distance between thetwo attachments, divided by the same distance before thesubstitution, and affected by a sign change if the order ofthe attachments is permuted.
The above substitution is defined as a ∆-transform. In whatfollows, a ∆-transform will be denoted by ∆li,lj ,lk indicatingthat a ∆ component formed by li and lj is substituted by theone formed by li and lk.
By applying a ∆-transform to a ∆ component, we havemoved one of its two attachments on the line. We can movethe other by applying one more ∆-transform. For example,on the result of the above ∆-transform, we can substitute l2by d2, as shown in Fig. 4(right). Then, following the samereasoning as for the first ∆-transform, the pure expression ofthe resulting platform, say det(P2), can be expressed as:
det(P2) =n + r
n· det(P1) =
n + r
m + n· det(P0). (12)
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 4
PSfrag replacements
a1a2 a3 a4
a5
a6
b1 b2 b3 b4b5
b6
p1 p2p3
p4p5
p6
q1 = p1
q2 = p2
q3
q4
q5
q6
l1 = q1
l2l3 = q3
l4d1
d2
d3
d4 m1 n11 n12 r1
m2 n21 n22 r2
n1 = n11 + n12
n2 = n21 + n22
PSfrag replacements
a1 a2 a3 a4
a5
a6
b1 b2 b3 b4b5
b6
p1
p2
p3
p4
p5
p6
q1 = p1
q2 = p2
q3q4
q5
q6
l1 = q1
l2l3 = q3
l4d1
d2
d3
d4
m1
n11
n12
r1
m2
n21
n22
r2
n1 = n11 + n12
n2 = n21 + n22
(a) (b)
PSfrag replacements
a1 a2 a3 a4
a5
a6
b1 b2 b3 b4b5
b6
p1
p2
p3
p4
p5
p6
q1 = p1
q2 = p2
q3
q4
q5
q6
l1 = q1
l2
l3 = q3l4
d1
d2
d3
d4
m1
n11
n12
r1
m2
n21
n22
r2
n1 = n11 + n12
n2 = n21 + n22
PSfrag replacements
a1 a2 a3 a4
a5
a6
b1 b2 b3 b4b5
b6
p1
p2
p3
p4
p5
p6
q1 = p1
q2 = p2
q3
q4
q5
q6
l1 = q1
l2l3 = q3
l4
d1 d2
d3d4
m1
n11
n12
r1
m2
n21
n22
r2
n1 = n11 + n12
n2 = n21 + n22
(c) (d)
Fig. 5. The basic flagged platform in (a) can be transformed through a set of ∆-transforms into the Zhang-Song platform in (d).
Observe that we could apply the above two ∆-transformssimultaneously to move both attachments on the line at thesame time. In this case, the relationship between the leg lengthvelocities can be expressed as:
d1
d2
l3...l6
=
− ∂f1/∂l1∂f1/∂d1
− ∂f1/∂l2∂f1/∂d1
. . . 0
− ∂f2/∂l1∂f2/∂d2
− ∂f2/∂l2∂f2/∂d2
. . . 0...
.... . .
...0 0 . . . 1
l1l2l3...l6
. (13)
where
f1(l1, l2, d1) = nl21 + ml22 − (m + n)d21 − mn(m + n) = 0
f2(l1, l2, d2) = rl21 + (m + n)d22 − (m + n + r)l22
− r(m + n)(m + n + r) = 0.(14)
Computing the partial derivatives of these functions and sub-
stituting in (13), we obtain
d1
d2
l3...l6
=
l1d1
n(m+n)
l2d1
mm+n . . . 0
− l1d2
rm+n
l2d2
m+n+rm+n . . . 0
......
. . ....
0 0 . . . 1
l1l2l3...l6
. (15)
The determinant of the matrix in (15) is:
l1l2d1d2
n(m + n + r) + mr
(m + n)2,
which easily simplifies to
l1l2d1d2
· n + r
m + n. (16)
Then, using (1) and (3), we can conclude that the pureexpression of the platform after this double leg substitutioncan be expressed as in (12). Thus, the result is obviously thesame if we apply the two ∆-transforms either sequentially orsimultaneously. Nevertheless, there are some circumstances inwhich a set of ∆-transforms can only be applied simultane-ously. This will become evident in the next two sections.
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 5
In what follows, a set of ∆-transforms will be denoted
∆l1,l2,l3∆l4,l5,l6 . . . ,
when applied sequentially, and
∆l1,l2,l3
∆l4,l5,l6
...
,
when applied simultaneously.
IV. EXAMPLE I: THE ZHANG-SONG PLATFORM
Zhang and Song identified in [24] a whole family of specialGough-Stewart platforms with closed-form formulation fortheir forward kinematics. One member of this family appearsin Fig. 5(d) which is of interest because it has five alignedattachments both in the base and the platform. Next, we willshow that the pure condition of this platform is equal to that ofthe basic flagged platform in Fig. 3(a) (also known as 3-2-1)multiplied by a constant factor. To this end, let us first applythe sequence of ∆-transforms to the basic flagged platform inFig. 5(a)
∆p1,p3,q3∆p2,p4,q4
which leads to the platform in Fig. 5(b). Now, let us apply thesequence
∆q1,q2,l2∆q3,q4,l4
which leads to the platform in Fig. 5(c).Therefore, it can be checked that, using (11) four times, the
pure expression of the platform in Fig. 5(c) is that of the basicflagged platform in Fig. 5(a), i.e.,∣
∣
∣
∣
a1 a5 a6 b5
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
a1 a6 b2 b5
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
a6 b2 b5 b6
1 1 1 1
∣
∣
∣
∣
,
(17)multiplied by:
n21
n211
(m2 + n2 + r2)2
(m2 + n21)2. (18)
Now, let us apply the simultaneous set of ∆-transforms
∆l3l1d1
∆l1l2d2
∆l4l3d3
∆l2l4d4
This set of ∆-transforms is characterized by the followinglinear relation:
d1
d2
d3
d4
=
∂f1/∂l1∂f1/∂d1
0 ∂f1/∂l3∂f1/∂d1
0
∂f2/∂l1∂f2/∂d2
∂f2/∂l2∂f2/∂d2
0 0
0 0 ∂f3/∂l3∂f3/∂d3
∂f3/∂l4∂f3/∂d3
0 ∂f4/∂l2∂f4/∂d4
0 ∂f4/∂l4∂f4/∂d4
l1
l2
l3
l4
,
(19)
wheref1 =(m1 + n1)l
21 + n1d
21 + m1l
23 − m1n1(n1 + m1)
f2 =(n2 + r2)l21 + m2l
22 − (m2 + n2 + r2)d
22
− m2(n2 + r2)(m2 + n2 + r2)
f3 =(m2 + n2)l24 + r2l
23 − (m2 + n2 + r2)d
23
− r2(m2 + n2)(m2 + n2 + r2)
f4 =r1l22 + n1d
24 − (n1 + r1)l
24 − r1n1(r1 + n1)
(20)
The determinant of the matrix in (19), after computing thecorresponding derivatives, is:
(m1 + n1)(n1 + r1)m2r2 − (n2 + r2)(m2 + n2)r1m1
n21(m2 + n2 + r2)2
.
(21)As a consequence, the pure expression of the platform inFig. 5(d) is the product of factors (17), (18), and (21).
It is important to realize that factor (21) vanishes if, andonly if,
|a3 − a1||a2 − a4||a1 − a2||a3 − a4|
=|b3 − b1||b2 − b4||b1 − b2||b3 − b4|
, (22)
i.e., if the cross-ratio [25] of a1, a4, a3, and a2 equals that ofb1, b4, b3, and b2. When this happens, the pure expressionof the Zhang-Song platform in Fig. 5(d) is identically zero. Inthis case, the platform is said to be architecturally singular,i.e., it is always in a singularity independently of its leglengths. Alternatively, it is also said that the platform exhibitsa self-motion, i.e., it is movable while keeping its leg lengthsconstant. This architectural singularity corresponds to the line-line singular component studied in [26], and it also appearsas the fifth type of singularity in Theorem 1 of [27] (orequivalently type 8 in Theorem 3 of [28]). The cross-ratiosingularity condition was also found in [29] and [30] usingmuch more involved derivations than the one above.
TABLE ICOORDINATES OF THE ATTACHMENTS IN THEIR LOCAL REFERENCE
FRAMES FOR AN ARCHITECTURALLY SINGULAR ZHANG-SONGPLATFORM, AND LEG LENGTHS FOR WHICH ITS SELF-MOTION IS
ANALYZED (DRAWN IN FIG. 6)
i ai bi d2
i
1 (0, 0, 0) (0, 0, 0) 22
2 (2, 0, 0) (1, 0, 0) 15 −
√
2
3 (8, 0, 0) (4, 0, 0) 54 − 28√
2
4 (10, 0, 0) (5, 0, 0) 87 − 45√
2
5 (6,−1, 0) (3, 0, 0) 28 − 12√
2
6 (4, 0, 0) (2, 2, 0) 16 − (6 +√
3)√
2
A parameterization of the self-motions resulting from archi-tectural singularities introduced by a sequence of ∆-transformscan always be found by proceeding backwards. That is, byundoing the ∆-transforms and introducing parameters whenneeded. For example, for the Zhang-Song platform (Fig. 6)with the leg attachment coordinates appearing in Table I, thecross-ratio condition in (22) is satisfied. Hence, the platformis architecturally singular and, as a consequence, it willexhibit a self-motion. To obtain a parameterization of this
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 6
PSfrag replacements
x
y
z
x′
y′
z′
Fig. 6. The architecturally singular Zhang-Song platform given in Table Irepresented in one of its infinitely many possible configurations.
self-motion, observe that, for the Zhang-Song architecturallysingular platform, using (20), it can be concluded:
−4 0 1 08 2 0 00 0 2 80 1 0 −4
l21l22l23l24
=
12− 3d21
160 + 10d22
160 + 10d23
12− 3d24
. (23)
Given fixed values for d21, . . . , d
24, the above linear system
is underconstrained, as expected. In other words, there is aninfinite set of values for l21, . . . , l
24 compatible with a set of
values for d21, . . . , d
24. This set can be parameterized, for the
values of d2i in Table I, by taking one of the leg lengths as
parameter (l24 has been chosen here) yielding
l21 = −l24 + 101− 35√
2
l22 = 4l24 − 249 + 135√
2
l23 = −4l24 + 350− 140√
2
(24)
We can proceed to undo the four other ∆-transforms toobtain the leg lengths for the corresponding basic flaggedparallel platform in Fig. 5(a). The result is:
p21 = 101− 35
√2 − l24
p22 = −63 + 33
√2 + l24
p23 = −3l24 + 265− 105
√2
p24 = −l24 + 101− 45
√2
p25 = 28− 12
√2
p26 = 16− (6 +
√3)√
2
(25)
The basic flagged parallel platform can have up to eightassembly modes which can be expressed in closed-form interms of its leg lengths [11]. Thus, by sweeping l24 in therange (0,∞), eight curves in the configuration space of theplatform are traced. This provides a complete characterizationof the sought self-motion. Fig. 7 depicts the location of thebarycenter of the triangular moving platform for the obtainedself-motion. A Maple worksheet is attached as a multimediafile to the paper for the interested reader to be able to followthe computational steps in detail.
Each value of the parameter l24 defines a unique pointin joint space (a set of leg lengths), which leads to eight
3
3.5
4
4.5
5-1
-0.5
0
0.5
1
1.5-3
-2
-1
0
1
2
3
PSfrag replacements
x y
z
Fig. 7. Curve traced by the barycenter of the triangular platform in Fig. 6when it is moved along its self-motion. Each color corresponds to one branchof the forward kinematics generated as the parameter is swept.
Fig. 8. All eigenvalues for all assembly modes, excluding the one thatis always zero, of J
TJ, where J is the Jacobian matrix of the analyzed
architecturally-singular Zhang-Song platform, as a function of l24
.
solutions of the forward kinematics (one for each assemblymode) in configuration space. Note that these solutions maybe real only for some ranges of the parameter. In the example,the parameter in the interval (22.35, 31.34) yields the realsolutions plotted in Fig. 7. The extremes of this intervalcorrespond to transition points between different assemblymodes (marked with a change of color in the figure). Nev-ertheless, such transition points do not correspond to higher-order singularities of the architecturally-singular Zhang-Songplatform. Indeed, if the eigenvalues of JJ
T , where J is theJacobian matrix of the architecturally-singular Zhang-Songplatform, are computed along its self-motion, five of them arealways different from zero. Fig. 8 plots in logarithmic scaleall eigenvalues, excluding the one that is always zero, for allassembly modes as a function of l24. Note that none of them.
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 7
PSfrag replacementsa1
a2
a3
a4
a5
a6
b1
b2
b3
b4
b5
b6
l1
l2
l3
l4l5
l6
d1
d2
d3
d4
d5
d6
m1
m2
m3
m4
m5
m6
n1
n2
n3
n4
n5
n6
PSfrag replacements
a1
a2
a3
a4
a5
a6
b1
b2
b3
b4
b5
b6
l1l2l3l4l5l6
d1
d2
d3
d4
d5
d6
m1
m2
m3
m4
m5
m6
n1
n2
n3
n4
n5
n6
Fig. 9. The simultaneous application of 6 ∆-transforms allows us totransform an octahedral platform (top) into a Griffis-Duffy platform (bottom).
V. EXAMPLE II: THE GRIFFIS-DUFFY PLATFORM
Let us consider the octahedral parallel platform inFig. 9(top), also known as 2-2-2 platform, whose pure ex-pression is:
∣
∣
∣
∣
b5 a4 a6 b3
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
b1 a2 a6 b5
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
b3 a2 a4 b1
1 1 1 1
∣
∣
∣
∣
−∣
∣
∣
∣
b5 a4 a6 b1
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
b1 a2 a6 b3
1 1 1 1
∣
∣
∣
∣
∣
∣
∣
∣
b3 a2 a4 b5
1 1 1 1
∣
∣
∣
∣
.
(26)
Now, let us apply the following set of ∆-transforms to it
∆l2,l1,d1
∆l3,l2,d2
∆l4,l3,d3
∆l5,l4,d4
∆l6,l5,d5
∆l1,l6,d6
(27)
The resulting platform is the Griffis-Duffy platform [31]appearing in Fig. 9(bottom) (for a complete analysis of thisspecial Gough-Stewart platform, see [32]).
The transformation matrix between the leg lengths velocitiesassociated with the set of ∆-transforms in (27) is:0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
∂f1/∂l1∂f1/∂d1
∂f1/∂l2∂f1/∂d1
0 0 0 0
0 ∂f2/∂l2∂f2/∂d2
∂f2/∂l3∂f2/∂d2
0 0 0
0 0 ∂f3/∂l3∂f3/∂d3
∂f3/∂l4∂f3/∂d3
0 0
0 0 0 ∂f4/∂l4∂f4/∂d4
∂f4/∂l5∂f4/∂d4
0
0 0 0 0 ∂f5/∂l5∂f5/∂d5
∂f5/∂l6∂f5/∂d5
∂f6/∂l1∂f6/∂d6
0 0 0 0 ∂f6/∂l6∂f6/∂d6
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
(28)where
f1 = m1l22 + m2l
21 − (m1 + m2)d
21 − m1m2(m1 + m2)
f2 = n2l22 + n1l
23 − (n1 + n2)d
22 − n1n2(n1 + n2)
f3 = m3l24 + m4l
23 − (m3 + m4)d
23 − m3m4(m3 + m4)
f4 = n4l24 + n3l
25 − (n3 + n4)d
24 − n3n4(n3 + n4)
f5 = m5l26 + m6l
25 − (m5 + m6)d
25 − m5m6(m5 + m6)
f6 = n6l26 + n5l
21 − (n6 + n5)d
26 − n6n5(n5 + n6).
(29)After computing the corresponding derivatives, the determi-nant of the matrix in (28) is:
m2m4m6n2n4n6 − m1m3m5n1n3n5
(m1 + m2)(m3 + m4)(m5 + m6)(n1 + n2)(n3 + n4)(n5 + n6)(30)
Therefore, the pure expression of the platform inFig. 9(bottom) is that of Fig. 9(top) multiplied by factor(30). Note that this factor is constant: it only depends onarchitectural parameters.
It is easy to check that factor (30) vanishes if, and only if,|a2, a6, a3||a2, a6, a4|
|a1, a5, a4||a1, a5, a3|
=|b2,b6,b3||b2,b6,b4|
|b1,b5,b4||b1,b5,b3|
, (31)
where |ai, aj , ak| is the area of the triangle defined by pointsai, aj and ak. It is worth noting that these cross-ratios betweenareas are projective invariants whose role for coplanar pointsis similar to that of the cross-ratios between distances forcollinear points [33].
Using rather more complicated arguments, the algebraiccondition derived from factor (30) to detect architecturalsingularities in Griffis-Duffy platforms was already found in[32]. In this latter reference, the reader can also find analternative geometric interpretation which, from our point ofview, is not as elegant as the one given above in terms ofcross-ratios between areas. Another interpretation of the same
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 8
factor (30) can be found in [34]. In [35] the same manipulatorwas used as a particular case example of a general theoremon architectural singularities.
An important consequence of this result is that, if factor(30) is different from zero, the singularity locus of a Griffis-Duffy platform is the same as that of an octahedral platform.If it is zero, the platform is architecturally singular.
TABLE IICOORDINATES OF THE ATTACHMENTS IN THEIR LOCAL REFERENCE
FRAMES FOR AN ARCHITECTURALLY SINGULAR GRIFFIS-DUFFYPLATFORM, AND LEG LENGTHS FOR WHICH ITS SELF-MOTION IS
ANALYZED (DRAWN IN FIG. 10).
i ai bi d2
i
1 (0, 0, 0) (0,√
3, 0) 3 −
√
3
2 (2, 0, 0) (−1/2,√
3/2, 0) 2
3 (1,√
3, 0) (−1, 0, 0) 5 −
√
3
4 (0, 2√
3, 0) (0, 0, 0) 15 − 4√
3
5 (−1,√
3, 0) (1, 0, 0) 11 − 5√
3
6 (−2, 0, 0) (1/2,√
3/2, 0) 11 − 3√
3
PSfrag replacements x
y
z
x′
y′z′
Fig. 10. The architecturally singular Griffis-Duffy platform given in TableII represented in one of its infinitely many possible configurations.
As an example of architecturally singular Griffis-Duffyplatform, consider the platform with the leg attachment co-ordinates appearing in Table II (Fig. 10). In this case factor(30) is identically zero. Hence, the platform is architecturallysingular and, as a consequence, it will exhibit a self-motion. Toobtain a characterization of this self-motion, we will proceedas in the previous section. First, observe that, for the Griffis-Duffy architecturally singular platform, using the system ofequations in (29), it can be concluded that:
2 2 0 0 0 00 1 1 0 0 00 0 2 2 0 00 0 0 1 1 00 0 0 0 2 21 0 0 0 0 1
l21l22l23l24l25l26
=
16 + 4d21
2 + 2d22
16 + 4d23
2 + 2d24
16 + 4d25
2 + 2d26
. (32)
Since the above matrix is rank defective, there is an infinitenumber of values for l21, . . . l
66 compatible with d2
1, . . . , d26.
Substituting the values of d21, . . . , d
26 in Table II and solving
the above system taking l26 as parameter, we get:
l21 = −l26 + 24− 6√
3
l22 = l26 − 10 + 4√
3
l23 = −l26 + 16− 4√
3
l24 = l26 + 2 + 2√
3
l25 = −l26 + 30− 10√
3
(33)
The octahedral platform can have up to 16 assembly modeswhich can be obtained as the roots of an eight-degree poly-nomial in the square of the unknown [36, p. 161]. Thus,no algebraic formula exists for the forward kinematics ofthe octahedral platform. Nevertheless, the self-motion canbe characterized by sweeping l26 in the range (0,∞) andobtaining the roots of the bioctic polynomial numerically. The16 solutions obtained for each value of l26 form eight pairs ofmanipulator postures, one being the mirror image of anotherabout the base plane.
0.40.5
0.60.7
0.80.9
0.40.50.60.70.80.911.1-1.5
-1
-0.5
0
0.5
1
PSfrag replacements
xy
z
Fig. 11. Curve traced by the barycenter of the triangular platform in Fig. 10when it is moved along its self-motion. The color corresponds to the value ofthe parameter swept from its lower value (pure red) to its upper value (pureblue). It consists of two symmetric disjoint components with respect to theplane z = 0 (the base plane). Note that for the same color different pointsare obtained, corresponding to different assembly modes.
Fig. 11 shows the location of the barycenter of the triangularmoving platform for the obtained self-motion. The color ofthe plotted points encodes the parameter sweep from itslower value (pure red) to its upper value (pure blue). It isinteresting to note that four assembly modes are real forl26 ∈ (3.89, 6.025) and eight for l26 ∈ (6.025, 8.9). The plat-form cannot be assembled for l26 outside the range (3.89, 8.9).As in the example of the previous section, the configurations
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 9
Fig. 12. All eigenvalues for all assembly modes, excluding the one thatis always zero, of JT J, where J is the Jacobian matrix of the analyzedarchitecturally-singular Griffis-Duffy platform, as a function of l2
6.
obtained for the value of the parameter that lead to changes inthe number of assembly modes do not correspond to higher-order singularities of the architecturally-singular platform.Indeed, if the eigenvalues of JJ
T , where J is the Jacobianmatrix of the architecturally-singular Griffis-Duffy platform,are computed along the self-motion, five of them are alwaysdifferent from zero. Fig. 12 plots in logarithmic scale alleigenvalues, excluding the one that is always zero, for allassembly modes as a function of the chosen parameter. Notethat none of them vanishes.
x
y
z
Fig. 13. The self-motion discretization of an architecturally-singular Griffis-Duffy platform obtained using CUIK is plotted in light transparent blue, andthe samples obtained with the presented parametrization in dark blue.
In order to validate the obtained self-motions, we havecompared the results obtained using the proposed parameteri-zation technique and those obtained using the CUIK softwarepackage. CUIK uses linear relaxation techniques to discretizethe space of all the configurations that a multiloop linkage can
adopt [37]. The obtained results for the attachment coordinatesand the leg lengths of the Griffis-Duffy manipulator in [37](also available online at [38]) appear in Fig. 13. Details onthe obtained parametrization for this example are providedin a Maple Worksheet included in the attached multimediamaterial. The solution obtained using CUIK consists of a listof boxes approximating the self-motion with a resolution of10−3. It can be checked that the sampled points obtained usingthe proposed technique are all included in these boxes.
VI. FURTHER APPLICATION CONTEXT
The aim of this paper has been to define the concept of∆-transform, as well as to explain the details of its usage,showing how sets of such transforms can be applied sequen-tially and/or simultaneously.
In order to illustrate the potential of the approach, twoexamples have been worked out in detail, both unravelingthe singularity-wise equivalence of two well-known parallelplatforms (flagged and Zhang-Song, on the one hand, andoctahedral and Griffis-Duffy, on the other), thus permittingto simplify designs by removing multiple spherical joints.
Now let us mention that the ∆-transform has proven usefulin several other contexts. Aside from the extension of the corefamily of flagged manipulators [11] with some new designs[30], the interesting family of partially-flagged platforms hasnow been derived [39], which includes the celebrated 3-2-1Hunt-Primrose manipulator as one of its members. Of practicalrelevance is the synthesis of a singularity-free redundantdesign, combining two members of this family, that is alsofree of multiple spherical joints.
Another promising application is in the context of parallelmanipulators including a 5-dof line-plane component [40],where fruitful kinematic equivalences are currently beingexplored.
And, needless to say, the application of the ∆-transform to3-dof planar parallel manipulators is straightforward.
But the proposed methodology goes well beyond the fully-parallel Gough-Stewart platforms mentioned up to now, as itapplies to any parallel robot having a ∆ component or anykinematically equivalent serial chain. For instance, several 2-dof serial chains considered in [41] are kinematically equiv-alent to ∆ components, and we have exploited this fact inprevious works to analyze the singularity structure of, forinstance, a 3-legged manipulator with PRPS chains as legs(see Fig. 12 in [11]).
VII. CONCLUSIONS
The idea of using ∆-transforms for the kinematic analysisof parallel platforms put forth in this paper was first proposedin [11] to generate the family of flagged parallel manipulators.In this paper, we have formalized and extended this idea intro-ducing the possibility of applying sets of these transformationssimultaneously. This extension permits:
1) classifying platforms in families sharing the same sin-gularity structure,
2) modifying platforms, without modifying the location oftheir singularities, to satisfy extra design criteria (this
SUBMITTED TO THE IEEE TRANSACTIONS ON ROBOTICS 10
includes the possibility of eliminating multiple sphericaljoints, whose implementation always introduces somecomplications), and
3) characterizing self-motions associated with non-trivialarchitectural singularities.
Concerning the last point, note that the characterization ofself-motions has important practical applications in redundantparallel robots and the emerging area of reconfigurable parallelrobots. For example, a redundant parallel robot, in which itsbase attachments can be moved along guides [42], shouldbe controlled in such a way that it remains far from thiskind of singularities. Alternatively, one might be interestedin approaching them so that the robot stiffness is reduced in agiven direction to ease some tasks requiring accommodation.
A ∆-transform essentially refers to a leg rearrangement ina ∆ component, a rigid subassembly involving a line and apoint. Then, it is reasonable to wonder if other particular legrearrangements exist for point-plane, line-line, or line-planecomponents that leave singularities invariant. The point-planecase can be easily shown to reduce to the application of two∆-transforms [11], while the line-line one has been shown inSection IV (Figure 5) of the current paper to also boil down tothe application of four serial followed by four simultaneous ∆-transforms. The case of the line-plane component, and possiblyalso that of the general plane-plane one, remain as a promisingavenue of future research.
ACKNOWLEDGEMENT
The authors would like to thank J.M. Porta for his help inusing the CUIK software package for the validation of theproposed method to parameterize self-motions.
APPENDIX
Singularities of Gough-Stewart platforms correspond tozeros of det(P) in equation (3). It is known that the rowsof matrix P are the Plucker coordinates of the leg-lines [3],[43]. Thus, identifying singularities amounts to finding thoseconfigurations in which the leg-lines vectors become linearlydependent. These kind of linear dependencies were first treatedin [44], [45].
White showed in [46] that the condition det(P) = 0could be expressed as the sum of terms involving the productof three determinants, termed brackets, the four columns ofwhich correspond to the coordinates of four leg attachmentswritten in homogeneous coordinates. Thus, this expressionbecoming zero, known as the pure condition, is equivalentto the determinant of P vanishing.
There exist several equivalent expressions of the pure con-dition associated with the general Gough-Stewart platform.
Downing et al. [18] used the following 16-term expression:[b1a1b4b5][b2a2a4b6][b3a3a5a6] − [b4a4b1b2][b5a5a1b3][b6a6a2a3]
−[b1a1b4b5][b2a2a4a6][b3a3a5b6] + [b4a4b1b2][b5a5a1a3][b6a6a2b3]
−[b1a1b4a5][b2a2a4b6][b3a3b5a6] + [b4a4b1a2][b5a5a1b3][b6a6b2a3]
+[b1a1b4a5][b2a2a4a6][b3a3b5b6] − [b4a4b1a2][b5a5a1a3][b6a6b2b3]
−[b1a1a4b5][b2a2b4b6][b3a3a5a6] + [b4a4a1b2][b5a5b1b3][b6a6a2a3]
+[b1a1a4b5][b2a2b4a6][b3a3a5b6] − [b4a4a1b2][b5a5b1a3][b6a6a2b3]
+[b1a1a4a5][b2a2b4b6][b3a3b5a6] − [b4a4a1a2][b5a5b1b3][b6a6b2a3]
−[b1a1a4a5][b2a2b4a6][b3a3b5b6] + [b4a4a1a2][b5a5b1a3][b6a6b2b3]
= 0, (34)
while Ben-Horin and Shoham [47] used the following 24-termexpression instead:
[b1a1b2a2][b3a3b4b5][a4a5b6a6] − [b1a1b2a2][b3a3a4b5][b4a5b6a6]
−[b1a1b2a2][b3a3b4a5][a4b5b6a6] + [b1a1b2a2][b3a3a4a5][b4b5b6a6]
−[b1a1b2b3][a2a3b4a4][b5a5b6a6] + [b1a1a2b3][b2a3b4a4][b5a5b6a6]
−[b1a1a2a3][b2b3b4a4][b5a5b6a6] + [b1a1b2a3][a2b3b4a4][b5a5b6a6]
−[b1a1b2b3][a2b4a4b5][a3a5b6a6] + [b1a1a2b3][b2b4a4b5][a3a5b6a6]
−[b1a1a2a3][b2b4a4b5][b3a5b6a6] + [b1a1b2a3][a2b4a4b5][b3a5b6a6]
+[b1a1b2b3][a2b4a4a5][a3b5b6a6] − [b1a1a2b3][b2b4a4a5][a3b5b6a6]
+[b1a1a2a3][b2b4a4a5][b3b5b6a6] − [b1a1b2a3][a2b4a4a5][b3b5b6a6]
+[b1a1b2b4][a2b3a3b5][a4a5b6a6] − [b1a1b2a4][a2b3a3b5][b4a5b6a6]
−[b1a1a2b4][b2b3a3b5][a4a5b6a6] + [b1a1a2a4][b2b3a3b5][b4a5b6a6]
−[b1a1b2b4][a2b3a3a5][a4b5b6a6] + [b1a1b2a4][a2b3a3a5][b4b5b6a6]
+[b1a1a2b4][b2b3a3a5][a4b5b6a6] − [b1a1a2a4][b2b3a3a5][b4b5b6a6]
= 0 (35)
where ai and bi are the homogeneous coordinates of theattachments ai and bi, respectively, for i = 1 . . . , 6. In otherwords, ai = (aT
i , 1)T and bi = (bTi , 1)T (see Fig. 2).
The advantage of using the above expressions for represent-ing the dependence of leg-lines is seen when investigating sim-plified forms of the general Stewart-Gough platform in which,for example, some legs share attachments. In general, placingconstraints on the geometrical structure of the platform reducesthe number of bracket terms to a manageable level, thusoffering the opportunity for simple geometrical interpretationsof the singularities. For example, since a bracket involvingthe same point at least twice is zero, the pure condition ofa platform in which several attachments coincide is greatlysimplified. Although using either (34) or (35) to obtain thesimplified forms of the pure condition is fully equivalent, theresult might be more compact in one case than in the otherthus avoiding the need of further algebraic manipulations forits simplification. For instance, for the octahedral manipulatorin Fig. 3-right, equation (34) reduces directly to the purecondition in equation (5), and for the manipulator in Fig. 3-left, equation (35) simplifies directly to the pure condition inequation (4).
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