New Geometric Approaches to the Singularity Analysis of Parallel Platforms

New Geometric Approaches to the Singularity Analysis of Parallel Platforms
New Geometric Approaches to the Singularity Analysis
of Parallel Platforms
Júlia Borràs, Federico Thomas and Carme Torras
Abstract— In general, rearranging the legs of a StewartGough platform, i.e., changing the locations of its leg attachments, modifies the platform singularity locus in a rather
unexpected way. Nevertheless, some leg rearrangements have
been recently found to leave singularities invariant. In this
work, a summary of the some of such singularity-invariant leg
rearrangements are presented, and their practical consequences
are illustrated with several examples including well-known
architectures.
I. INTRODUCTION
The Stewart-Gough platform triggered the research on
parallel manipulators, and it has remained one of the most
widely studied because, despite its geometric simplicity, its
analysis translates into challenging mathematical problems
[19], [12]. One important part of this analysis corresponds
to the characterization of its singularities.
The singularities of a Stewart-Gough platform are those
poses for which the manipulator loses stiffness. Characterizing such unstable poses has revealed as a challenging
problem during the last decades, resulting in an extensive
literature in the scientific kinematic world [15], [23], [14],
[3], [13].
The Stewart-Gough platform is defined as a 6-DoF parallel
mechanism with six identical SPS legs. The geometric and
topological characterization of its singularity locus in its sixdimensional configuration space is, in general, a huge task
which has only been completely solved for some specializations —i.e., designs in which some spherical joints in
the platform, the base, or both, coalesce to form multiple
spherical joints [2], [1].
The kinematics group at the Institut de Robòtica i Informàtica Industrial at Barcelona studies new approaches
to the singularity analysis of parallel platforms. This work
presents one of their indirect approaches: even when there
is no known solution to a given mathematical problem, it is
always possible to try to find the set of transformations to the
problem that leave its solution invariant. Although this does
not solve the problem itself, it provides a lot of insight into
its nature. This way of thinking is the one applied herein for
the characterization of the singularity loci of Stewart-Gough
platforms. In this context, this approach means finding leg
rearrangements in a given Stewart-Gough platform that leave
its singularity locus invariant.
Such singularity-invariant leg rearrangements are useful
for two main reasons: (a)
Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Barcelona,
Spain {jborras, fthomas, ctorras}@iri.upc.edu
1) If the singularity locus of the platform at hand has
already been characterized, it could be interesting to
modify the location of its legs to optimize some other
platform characteristics without altering such locus.
2) If the singularity locus of the analyzed platform has
not been characterized yet, it could be of interest
to simplify the platform’s geometry by changing the
location of its legs, thus easing the task of obtaining
this characterization.
In [5] it is shown how, for a leg rearrangement to be
singularity-invariant, it is necessary and sufficient that the
linear actuators’ velocities, before and after the rearrangement, are linearly related. It is important to realize that,
if this condition is satisfied, a one-to-one correspondence
between the elements of the platform forward kinematics
solution sets, before and after the rearrangement, exists.
Actually, the invariance in the singularities and the assembly
modes of a parallel platform are two faces of the same coin.
These ideas are closely related to those that made possible
the development of kinematic substitutions [18]. They are
general in the sense that they can be applied to any kind of
mechanism, not only parallel platforms.
This paper shows how the application of singularityinvariant leg rearrangements to well-studied platforms leads
to interesting new results.
Section II introduces the notation used in the paper and
Section III defines a singularity-invariant leg rearrangement
in mathematical terms. Then, three case studies are presented
(Sections IV, V and VI), with particular numerical examples
showing interesting results and the development and implementation of two prototypes based in them.
II. NOTATION
A general Stewart-Gough platform is a 6-SPS platform. In
other words, it has six actuated prismatic legs with lengths li ,
i = 1, . . . , 6, connecting two spherical passive joints centered
at ai = (xi , yi , zi )T and b̃i = (ri , si , ti )T , given in base
and platform reference frames, respectively (see Fig. 1). The
pose of the platform is defined by a position vector p =
(px , py , pz )T and a rotation matrix R


ix jx kx
R = (i, j, k) = iy jy ky  ,
iz jz kz
so that the platform attachments can be written in the base
reference frame as bi = p + Rb̃i , for i = 1, . . . , 6 (Fig. 1).
To simplify the notation, the same name will be used to
denote a point and its position vector.





P=




−r1
−r2
−r3
−r4
−r5
−r6
−r
−s1
−s2
−s3
−s4
−s5
−s6
−s
−t1
−t2
−t3
−t4
−t5
−t6
−t
x1
x2
x3
x4
x5
x6
x
y1
y2
y3
y4
y5
y6
y
z1
z2
z3
z4
z5
z6
z
r1 x1
r2 x2
r3 x3
r4 x4
r5 x5
r6 x6
rx
r1 y1
r2 y2
r3 y3
r4 y4
r5 y5
r6 y6
ry
r1 z1
r2 z2
r3 z3
r4 z4
r5 z5
r6 z6
rz
R
O
b
′
b̃i
bj
p
O
d
li
bi
ai
lj
a
aj
Fig. 1. A general Stewart-Gough platform with base attachments ai and
platform attachments at bi , i = 1, ..., 6. A single leg rearrangement consist
in the substitution of one of the legs by a new one, in gray in the drawing.
There are three types of parameters that fully define a
Stewart-Gough platform
Pose parameters:
X = (px , py , pz , ix , jx , kx , iy , jy , ky , iz , jz , kz )
Geometric parameters:
G = (x1 , y1 , z1 , r1 , s1 , t1 , . . . , x6 , y6 , z6 , r6 , s6 , t6 )
Joint parameters:
Θ = (l1 , . . . , l6 )
Finally, it will be useful to introduce a 6-dimensional space
defined by the coordinates (x, y, z, r, s, t), called the space
of leg attachments. Each point of this space defines a leg
that goes from base attachment a = (x, y, z)T to platform
attachment b̃ = (r, s, t)T .
III. SINGULARITY-INVARIANT LEG
REARRANGEMENTS
A leg rearrangement consists in a relocation of the attachments of the manipulator, without modifying the pose of the
platform, and thus, leading to new leg lengths d1 , d2 , . . . , d6
(Fig. 1). In general, such rearrangement completely modifies
the kinematics of the manipulator and also the location
of its singularities, because the forward kinematics of the
rearranged platform must be solved again, which leads to a
different number of assembly modes and to a different set
of singularities.
s1 x1
s2 x2
s3 x3
s4 x4
s5 x5
s6 x6
sx
s1 y1
s2 y2
s3 y3
s4 y4
s5 y5
s6 y6
sy
s1 z1
s2 z2
s3 z3
s4 z4
s5 z5
s6 z6
sz
t1 x1
t2 x2
t3 x3
t4 x4
t5 x5
t6 x6
tx
t1 y 1
t2 y 2
t3 y 3
t4 y 4
t5 y 5
t6 y 6
ty
t 1 z1
t 2 z2
t 3 z3
t 4 z4
t 5 z5
t 6 z6
tz
1
1
1
1
1
1
1





.




(1)
Despite of this, recently, we have been able to identify leg
rearrangements that do not modify the singularity locus of
the platform, nor the solution of its forward kinematics. In
other words, for the rearranged platform, the location of the
singularity poses within the workspace of the manipulator
remain at the same position. This kind of rearrangement are
called singularity-invariant leg rearrangements, and where
characterized in detail in [5].
In Fig. 1 we show the rearrangement of the leg j. In other
words, we relocate the attachments aj and b̃j to the new
coordinates a = (x, y, z)T and b̃ = (r, s, t)T . In [5], it was
shown how such leg rearrangement is singularity invariant if,
and only if, the coordinates (x, y, z, r, s, t) make the matrix
P in (1) to be rank defective.
Note that the first 6 rows of P contain only geometric
parameters of the manipulator, while the last row depends
on the coordinates of the new location of the rearranged
leg. The 6 first rows of P where used in [11], [21] to
characterize architectural singularities. With this additional
row, we are able to characterize any singularity-invariant leg
rearrangement by studying the rank of P.
Gaussian Elimination uses elementary row operations to
reduce a given matrix into a rank-equivalent one, with an
upper triangular shape. After it is applied to a matrix, rank
deficiency occurs when all the elements of the last row
are zero. Matrix P is 7 × 16 and, if we apply Gaussian
Elimination, the last row of the resulting matrix can be
expressed as:
¡
¢
0 0 0 0 0 0 0 P1 . . . P10 ,
(2)
where Pi , for i = 1, . . . , 10, are polynomials in the unknowns (x, y, z, r, s, t), and we can state that P is rank defective if, and only if, the 10 polynomials are simultaneously
vanished.
In conclusion, if any of the legs is relocated to the new
attachments a = (x, y, z)T and b = (r, s, t), the resulting leg
rearrangement is singularity-invariant if, and only if, {P1 =
0, . . . , P10 = 0}.
This is an overdetermined system that has no solution for
a generic case. We need to impose at least 5 more scalar
equations to obtain a 1-dimensional set of solutions. Next
we will see several cases for which matrix P is simplified
and solutions of dimension 1 and 2 are obtained.
IV. CASE STUDY I: DOUBLY-PLANAR
STEWART-GOUGH PLATFORMS
For any doubly planar Stewart-Gough platform, the coordinates of the base and platform attachments can be written,
without loss of generality, as ai = (xi , yi , 0) and b̃i =
(zi , ti , 0). In this case, a leg rearrangement with coordinates
(x, y, z, t) stands for the substitution of any of the legs by
another one going from the base attachment located at a =
(x, y, 0)T to the platform attachment at b = p + R(z, t, 0)T .
In this case, matrix P can be simplified to


−z1 −t1 x1 y1 x1 z1 y1 z1 x1 t1 y1 t1 1
 −z2 −t2 x2 y2 x2 z2 y2 z2 x2 t2 y2 t2 1 


 −z3 −t3 x3 y3 x3 z3 y3 z3 x3 t3 y3 t3 1 



P=
 −z4 −t4 x4 y4 x4 z4 y4 z4 x4 t4 y4 t4 1  .
 −z5 −t5 x5 y5 x5 z5 y5 z5 x5 t5 y5 t5 1 


 −z6 −t6 x6 y6 x6 z6 y6 z6 x6 t6 y6 t6 1 
−z −t x y xz
yz
xt
yt 1
(3)
Consider the example with attachment local coordinates
appearing in Table I.
Fig. 2. A general singularity-invariant leg rearrangement for a doublyplanar Stewart-Gough platform
b2
b3
t
TABLE I
b1
ATTACHMENT COORDINATES (ai = (xi , yi , 0)T , b̃i = (zi , ti , 0)T ).
b4
b6
b5
z
i
1
2
3
4
5
6
xi
3
7
8
12
5
9
yi
5
9
9
5
2
2
zi
5
7
9
9
6
9
a2
ti
6
8
8
6
4
5
a3
y
a1
a4
a5
a6
x
To check rank deficiency, Gaussian Elimination is applied
on P with the corresponding numerical values substituted.
In this case, the last row of the resulting matrix has only 3
nonzero terms dependent on x, y, z and t. Different but
equivalent equations arise depending on the order of the
columns. For example, Gaussian Elimination on matrix P
as it appears in equation (3) leads to a matrix whose last
row is
¢
1 ¡
0 0 0 0 0 0 P89 P79 P78 ,
P789
where Pij is the determinant of the submatrix obtained from
P after deleting columns i and j, and Pijk the determinant
of the submatrix formed by the first 6 rows of P after
deleting columns i, j and k. With the corresponding numerical values, P789 = −12180 and the singularity-invariant
leg rearrangements are defined by the condition defined by
{P89 = P79 = P78 = 0}, which reads as

1096
22713
338
xz + xt + 3706
− 609

3045 yz + 1015 x − 1015 y


19302

z
+
t
=
0
− 27743

3045
1015






470
10519
13274
61662
− 609 xz + 3045 yz + yt + 1015 x − 1015 y
51343
87557
− 3045 z + 1015 t = 0 






17
38
67
194

xz
−
yz
−
x
+
y

609
609
203
203


247
192
+ 609 z − 203 t + 1 = 0
(4)
Note that any equation consisting of a submatrix determinant Pij equated to zero will be bilinear in the unknowns,
Fig. 3. The base and the platform curves of the doubly-planar StewartGough platform depicted in Fig. 2.
but with different monomials. As the system is linear, both
in (x, y) and in (z, t), it can be rewritten in matrix form as
   
0
z
(5)
Sb  t  = 0 ,
0
1
where Sb is
 −27743
3045
+

 10519
 3045 y −

17
x
609
3706
y
3045
−
338
x
609
x+
19302
1015
87557
3045
−
470
x
609
y+
51343
1015
38
y
609
−
+
247
609
22713
y
1015
61662
y
1015
−1 +
−192
203
−
−
67
x
203
1096
x
1015
13274
x
1015
−
194
y
203





which only depends on x and y (b refers to base, as x and y
are the coordinates of the base attachments). The other way
round, the system can also be written as
   
x
0
Sp y  = 0 ,
(6)
1
0
where Sp is





t+
1096
1015
13274
1015
−67
203
−
−
+
338
z
609
470
z
609
17
z
609
−22713
1015
−61662
1015
+
−38
z
609
+
3706
z
3045
10519
z
3045
+
194
203
+t
27743
z
3045
−
19302
t
1015
87557
z
3045
−
51343
t
1015
−247
z
609
+
192
t
203
−1





Λp1
Λp2
b1
b2 Λp3
Λp1
b3
b4
b6
Λp2
b1 = b2 Λp3
b3 = b4
b5 = b6
b5
Λb1
a2
a4
Λb3
a3
Λb1
a4 = a5
Λb3
a5
a2 = a3
a1
Λb2
Λb2
a6
a6 = a1
Fig. 4. Griffis-Duffy type I platform with the attachment coordinates given in in Table II (left), and its equivalent octahedral manipulator after applying
a leg rearrangement (right).
that only depends on z and t (p refers to platform, as z and
t are the coordinates of the platform attachments).
From equation (5) it is clear that the system has a solution
for (z, t) only for those (x, y) that satisfy det(Sb ) = 0,
and this solution is unique (assuming that Sb has rank 2).
In the same way, there exists a solution for (x, y) only for
those (z, t) that make det(Sp ) = 0. Both determinants define
cubic curves on the base and platform planes, respectively. In
other words, system (4) defines a one-to-one correspondence
between generic points on two cubic curves. However, the
correspondence may be not one-to-one for special points on
the cubics for non-generic examples (see details in [10]).
For this particular example, the equations of the cubic on
the base is
253 2 142 3 1061 2
16 3 293 2
x −
x y+
xy −
y +
x +
145
609
1015
609
3045
4343
2313 2 17888
26032
261691
xy +
y −
x−
y+
=0
1015
1015
1015
1015
3045
and on the platform
representation of these manipulators can be found in Fig. 4(left).
396 2
293 2 192 3 282 2 1877
9 3
z −
z t+
zt −
t +
z +
zt
145
1015
1015
203
203
1015
98097
32922
2229 2 17799
t −
z−
t+
=0
+
145
1015
1015
145
which have been plotted in Fig. 3. The curves attached to
the manipulator base and platform are shown in Fig. 2.
Depending on the placement of the attachments, these
curves can be generic curves of degree 3, or a line and a
conic, or even 3 lines crossing 2 by 2. In the next example,
one of these degenerate cases is analyzed.

2t
−
y
+
yz
+
xt
=
0

√
√
(
3z
+
t
−
3)y
=
0
√
√
√
√

−2 3z + 4t + 3x − y + 3xz + 3yz − 2 3 = 0
(7)
The resolution of this system gives correspondences between
base and platform attachments that leave the singularities
invariant. The base and platform cubic curves, in this case,
factorize into the 3 lines:
TABLE II
C OORDINATES OF THE ATTACHMENTS ai = (xi , yi , 0) AND
bi = p + R(zi , ti , 0)T FOR THE ANALYZED ROBOTS
xi
1
2
3
4
5
6
1
2
2/3
−2
−2/3
0
yi
√
3
0
0
0√
(4/3)
√ 3
2 3
zi
ti
1
1/2
−1
−1/2
0
1/2
0
0
√0
3/2
√
√ 3
3/2
In this case, the system obtained by applying Gaussian
elimination on the corresponding matrix P results in :
√ √
√
√
( 3z − t + 3)( 3z + t − 3)t = 0,
A. An octahedral manipulator implementation
In 1993, Griffis and Duffy patented a manipulators named
thereafter Griffis-Duffy platform [17]. The platform have
his attachments distributed on triangles, three attachments
on the vertexes and three on the midpoints of the edges,
and platform is formed by joining the attachments on the
midpoints on the base to the vertexes on the platform, as the
example with attachment coordinates given in Table II. A
i
and
(−3x +
√
√
3y − 6)(3x + 3y − 6)y = 0,
respectively.
Actually, it can be checked that system (7) has 6 sets of
P3
δ5
m4
m5
P6
δ6
P5
δ4
m6
m3
δ2
P4
m1
P1
m2
δ1
δ3
P2
Fig. 5. Contrary to what happens to the Stoughton-Arai approximation,
the proposed modification leads to a 6-6 platform kinematically equivalent
to the octahedral manipulator.
solutions
∆b1 = {(x, y, z, t) |
∆b2
√
√
x = λ, y = (λ1 + 2) 3, z = 0, t = 3; λ1 ∈ R},
= {(x, y, z, t) |
√
x = λ2 , y = (2 − λ2 ) 3, z = 1, t = 0; λ2 ∈ R},
∆b3 = {(x, y, z, t) |
x = λ3 , y = 0, z = −1, t = 0; λ3 ∈ R},
∆p1 = {(x, y, z, t) |
∆p2
∆p3
√
x = −2, y = 0, z = λ4 , t = 3(λ4 + 1); λ4 ∈ R},
= {(x, y, z, t) |
√
√
x = 0, y = 2 3, z = λ5 , t = 3(1 − λ5 ); λ5 ∈ R},
= {(x, y, z, t) |
x = 2, y = 0, z = λ6 , t = 0; λ6 ∈ R}.
In other words, these are 6 point-line correspondences, that
is, to each vertex of the base (platform) triangle corresponds
a line on the platform (base) triangle. This means that, for the
Griffis-Duffy type manipulator, we can fix the attachments
at the vertexes of the platform (base), and then rearrange
the opposite attachments along a line in the base (platform)
without modifying the kinematics of the platform.
As a result, by moving the six midpoint attachments along
their supporting lines, the manipulator can be rearranged into
the manipulator depicted in Fig. 4-(right), which is the widely
known octahedral manipulator. This is an interesting result,
because we can avoid the use of multiple spherical joints
(that is, spherical joints sharing the same center) without
loosing the properties of the celebrated octahedral architecture [14]. A manipulator has been constructed following the
design in Fig. 5 in the Laboratory of Parallel Robots, at the
Institut de Robòtica i Informàtica Industrial [22] (Fig. 6).
Its advantage is that it is a 6-6 manipulator with the same
Fig. 6. This platform consists of six extensible legs connecting a moving
platform to a fixed base. We avoid the use of multiple spherical joints (that
is, spherical joints sharing the same center) without loosing the properties
of the celebrated octahedral architecture.
kinematics and singularities to the widely studied octahedral
manipulator (see more details in [20], [6]).
V. CASE STUDY II: A DECOUPLED
STEWART-GOUGH PLATFORM
Consider the manipulator in Fig. 7. It contains a tripod and
3 more legs, with all the base attachments coplanar. Thus,
without loss of generality, we can write the coordinates of
the attachments as ai = (xi , yi , 0)T and b̃i = (ri , si , ti )T .
This manipulator is said to be decoupled because the three
legs forming the tripod give the position of the platform,
while the three remaining ones orient it. When the tripod is
rigid, i. e., fixed at a position, this manipulator is also known
as spherical [4], [16].
Consider the example with numeric coordinates appearing
in Table III. After performing Gaussian Elimination on
TABLE III
ATTACHMENT COORDINATES ai = (xi , yi , 0) AND
bi = p + R(ri , si , ti )T
i
1
2
3
4
5
6
xi
2
5
-1
7
2
-3
yi
-1
4
4
-2
7
-2
ri
2
2
2
5
2
-1
si
2
2
2
0
5
0
ti
0
0
0
1
1
1
the corresponding matrix P, only six non-zero elements
b6
b5
b5
b5
b6
b6
b3
b4
b2
b1
b4
b4
a2a5
a2a5
a6a3
a5
a6a3
a1 a4
a3
a1 a4
Fig. 8. Singularity-invariant leg rearrangements from the example in Fig. 7
a2
a6
VI. CASE STUDY III: PENTAPODS
a1
a4
Fig. 7. A decoupled manipulator with non-planar platform. In blue, its
singularity-invariant leg rearrangement lines.
remain at the last row. That is, a leg rearrangement will be
singularity-invariant if it fulfills the following 6 conditions
−2xr + yr + 4x − 2y + 6r − 6s + 18t = 0,
−4xr/3 + xs + 2x/3 + 6r − 6s + 12t = 0,
1/5(17xr + ys − 34x − 10y − 34r + 34s − 207t) = 0,
5xr/3 + xt − 10x/3 − 5r + 5s − 17t = 0,
9xr/5 + yt − 18x/5 − 18r/5 + 18s/5 − 89t/5 = 0,
−1xr/2 + x + r − 3s/2 + 9t/2 + 1 = 0
This system of equations has 4 sets of solutions:
T = {(x, y), (r, s, t) |
x = λ, y = µ; r = 2, s = 2, t = 0, λ, µ ∈ R},
∆1 = {(x, y), (r, s, t) |
x = 2, y = 7; r = 2, s = 2 + 3λ, t = λ, λ ∈ R},
∆2 = {(x, y), (r, s, t) | x = 7, y = −2;
r = 5 − 3λ/2, s = λ, t = 1 − λ/2, λ ∈ R},
∆3 = {(x, y), (r, s, t) | x = −3, y = −2;
r = 2 − 3λ, s = 2 − 2λ, t = λ, λ ∈ R}.
The first one corresponds to the tripod component and it
means that base attachments can be rearranged to any point
of the base plane as long as its corresponding platform
attachment is the vertex of the tripod. The other 3 sets
correspond to point-line correspondences as before, depicted
as red lines in Fig. 7. This means that b4 , b5 and b6 can
be relocated to any other point of the red lines (as long as
their corresponding base attachment remains the same).
In Fig. 8 we show two possible singularity-invariant leg
rearrangements of the manipulator at hand. For all of them,
the decoupling properties remain the same as they are all
equivalent manipulators.
A pentapod is usually defined as a 5-degree-of-freedom
fully-parallel manipulator with an axial spindle as moving
platform. This kind of manipulators have revealed as an
interesting alternative to serial robots handling axisymmetric
tools. The moving platform can freely rotate around the axis
defined by the five aligned revolute joints, but if this rotation
axis is made coincident with the symmetry axis of the tool,
the uncontrolled motion becomes irrelevant in most cases.
Their particular geometry permits that, in one tool axis,
large inclination angles are possible thus overcoming the
orientation limits of the classical Stewart-Gough platform.
A pentapod involves only 5 of the 6 legs of the StewartGough platform, with the platform attachments collinear.
This 5 legs form a rigid component by itself that can be
studied separately. In addition to the platform attachments
collinearity, if we consider all the base attachments coplanar,
then we can write the coordinates of the attachments as
ai = (xi , yi , 0)T and b̃i = (zi , 0, 0)T for i = 1..5 and the
corresponding matrix P after some simplifications reads as


z1 x1 y1 x1 z1 y1 z1 1
z2 x2 y2 x2 z2 y2 z2 1


z3 x3 y3 x3 z3 y3 z3 1

(8)
P=
z4 x4 y4 x4 z4 y4 z4 1 .


z5 x5 y5 x5 z5 y5 z5 1
z x y
xz
yz 1
In this case, P is a square matrix, so its rank deficiency is
characterized only by the equation det(P) = 0. In [9] it was
shown that such condition defines a one-to-one correspondence between the platform attachments and the lines of a
pencil attached at the base. The center of this pencil, called
B-point in [9], [7], plays an important role in the geometric
characterization of the manipulator singularities.
Consider the example with numerical coordinates appearing in table VI.
After substituting the numerical values in P, we get that
the condition for the singularity invariance is
det(P) = x − z = 0.
(9)
This means that any leg can be rearranged to a leg going
from the base attachment a = (λ, y, 0)T to b̃ = (λ, 0, 0)T
Fig. 9. Pentapod analyzed in Section VI. Note that it is in an upside-down
configuration, so that the platform is located under the base.
TABLE IV
ATTACHMENTS ai = (xi , yi , 0) AND b̃i = (zi , 0, 0)
i
1
2
3
4
5
xi
−2
−1
0
1
2
yi
2
−2
3
−2
2
zi
−2
−1
0
1
2
without modifying the singularity locus (where for a fixed
λ, the y coordinate can take any value). This corresponds
to the rearrangements plotted in Fig. 9, that is, a one-to-one
correspondence between the attachments at the platform and
a pencil of parallel lines attached at the base. In this case,
the center of the pencil lies at infinity.
This particular architecture was proved to be quadratically
solvable in [8], [9], that is, its forward kinematics can be
solved by solving only 2 quadratic polynomials. If we fix
the attachments of the platform, the corresponding base
attachments can be relocated to any point of the red lines
plotted in Fig.9. Taking advantage of that idea, at Laboratory
of Parallel Robots at IRI we have developed a reconfigurable manipulator prototype based on this structure. Its
base attachments can be reconfigured along actuated guides,
without modifying the nature of its forward kinematics nor
the singularities of the manipulator, and thus increasing the
versatility of the manipulator, as for each task, the legs can
be reconfigured to equally distribute the forces among its
legs (Fig. 10).
VII. CONCLUSIONS
The present work shows how the application of
singularity-invariant leg rearrangements provide a new geometric approach to the study of Stewart-Gough platform
singularities. Indeed, we have presented three case studies
that illustrate several new results.
Fig. 10. Prototype of the reconfigurable quadratically-solvable pentapod
and its joint implementations.
We have presented a tool to detect equivalences between
manipulators, which means that we can use previous known
geometric interpretations of singularities to new architectures. That is the case of the Griffis-Duffy platform at Section
IV. The 6-6 Stewart-Gough platform prototype shown in
Fig. 6 has the same kinematic properties than the octahedral
manipulator, that is, the same geometric interpretation for its
singularities applies, as well as all other kinematic properties studied in the extensive literature about the octahedral
manipulator.
We have also shown how decoupled manipulators can
be rearranged to equivalent and apparently non-decoupled
manipulators, with different configurations of their spherical
joints that might be easy to construct.
Also, the hidden geometric structure reveled by these
curves of singularity-invariant leg rearrangements can help
in the simplification of the forward kinematics resolution.
For example, in the case study III, we show a manipulator
that is quadratically solvable.
Finally, new geometric interpretation of singularities have
been found thanks to singularity-invariant leg rearrangements. For example, for pentapods with planar bases, the
identified pencil of lines at the base of the manipulator
reveals to be crucial for the geometric interpretation of its
singularities. Similar interpretations represent a challenge for
the future work.
In conclusion, this indirect approach to the analysis of
Stewart-Gough platform singularities has succeed in finding
new results in a topic with an extensive previous literature.
VIII. ACKNOWLEDGMENTS
The authors gratefully acknowledge
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