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Kinematic Analysis of the TSAI-3UPU Parallel Manipulator
using Algebraic Methods
D. R. Walter M. L. Husty
University of Innsbruck University of Innsbruck
Innsbruck, Austria Innsbruck, Austria
Abstract This paper discusses in detail the TSAI 3-
UPU parallel manipulator. This very special parallel ma-
nipulator was presented by Lung-Wen Tsai in 1996. Up to
the knowledge of the authors practically all published work
about this manipulator deals with the property that it ex-
hibits pure translational motion if it is properly assembled.
Quite evidently the question arises which type of motion oc-curs, when the manipulator is not properly assembled for
translational motion. Here it will be explained, how the
manipulator can be described by a set of algebraic equa-
tions. This set is used to analyze its motion capabilities ex-
haustively using methods from algebraic geometry. It turns
out that the manipulator has theoretically up to 78 solutions
of the direct kinematics (including complex ones), and that
is has in addition to the translational mode four other op-
eration modes. It is shown that there exist poses where a
transition from one assembly mode to another is possible.
All necessary conditions on the leg lengths are determined
which lead to changes of the operation mode.
Keywords: TSAI-3UPU manipulator, direct kinematics, assembly
mode, constraint equations, linear implicitization algorithm, opera-
tion mode, primary decomposition.
I. IntroductionThe TSAI 3-UPU parallel manipulator was presented by
Tsai in 1996 [1] as a new 3-dof manipulator to generate
pure translational motion. This mechanism and its general-
izations were discussed with respect to kinematic properties
in e.g. [2], [3], [4], and [5]. It is interesting to note that al-
most all of these papers focus on the translational operation
mode. On the other hand it is well known that the manipula-
tor also exhibits some other operation modes. Especially in
[2] it is pointed out nicely that the manipulator only showstranslational motion if it is properly assembled.
Due to the fact that in [6] it was possible to find all different
operation modes of the SNU 3-UPU manipulator, one could
ask if this can also be achieved for the TSAI version of this
manipulator, especially because both manipulators are very
similar.
Using an algebraic description of the manipulator and
methods from algebraic geometry a complete description
of the manipulators operation modes can be given. Addi-
tionally conditions are presented which lead to assemblies,
in which a change of operation modes is possible.
This paper is organized as follows: A description of the ma-
nipulators design is given in Section II. The deduction of
the constraint equations follows in Section III. These equa-
tions are used to discuss possible assembly modes and op-
eration modes in Sections IV and V. Finally the possibility
of changing the operation mode is discussed exhaustivelyin VI.
II. The manipulators design
First of all an exact description of the manipulators de-
sign is necessary. Due to the fact that the TSAI 3-UPU can
be obtained from the SNU 3-UPU by simply rotating each
limb by 90 degrees about the axis of the prismatic joint, the
following is almost identical to the description of the design
in [6].
x
y
z
x
y
z
A1
B1
A2
B2
A3
B3
0
1
d1
d2
d3
2
1
4
3
h1
h2
Fig. 1. The numbers at the limb fromA1 to B1describe the order of the
rotational axes of the U-joints.
In the base there are three points A1, A2 and A3 form-
ing an equilateral triangle with circumradius h1 (Fig.1).The frame0 is fixed in the base such that its origin liesin the circumcenter of the triangle, its yz-plane coincides
with the plane formed by the triangle and its z-axis passes
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through A3. The same situation is established in the plat-
form. There we have an equilateral triangle with vertices
B1, B2, B3, circumradiush2, and the platforms frame isdenoted by1. The parametersh1 and h2 are the two first
design parameters, which are always different, if not men-tioned explicitly.
Now, each pair of corresponding pointsAi,Biis connected
by a limb with U-joints at each end. The length of each limb
is denoted bydi and adjusted by a prismatic joint. The firstand the fourth axis are embedded in the base resp. platform
such that each of them is tangent to the corresponding cir-
cumcircle (see Fig. 1). The second and the third axes of this
link-combination are parallel to each other and perpendic-
ular to the axis of the limb and its first and fourth axis. All
together we need five parameters to describe the design of
the TSAI 3-UPU mechanism: d1,d2,d3,h1 and h2, wherethe first three of them are used to actuate the manipulator,
the latter are fixed design parameters.
III. Deduction of constraint equationsIn the following equations are derived which describe the
position of1 with respect to0, and with it the positionof the platform with respect to the fixed base.
First of all for each limbLi a chain of4 4transformationmatrices can be given, according to the Denavit-Hartenberg
convention, such that the position of1 with respect to0via limbLi can be described by the matrix product
Ti = F1iM1i1M2i2iM3i3M4iF2i (1)
where the matrices Fji are fixed transformations, matri-
ces Mji are responsible for the rotations about the axes
of the U-joints, depending on rotation angles uji , and the-matrices manage the transformations from one rotational
axis to the next one. Due to the special arrangement of the
axes in each limb no separate transformation matrix was in-
troduced for the active prismatic joints, their parameter diappears in the matrix 2i. A clear and brief introduction of
the Denavit-Hartenberg convention and the systematic de-
duction of the forward transformation matrices Ti can befound e.g. in [7] and [8]. The DH parameters are nearly the
same for all three legsLi and read as follows:
a d 1 0 0 /22i di 0 03 0 0 /2
TABLE I. DH parameters occurring in matrices 1,2i, and 3
It has to be noted explicitly that in Table I thedhas nothingto do withdi.Next, the so called Study parameters are introduced, which
are a very convenient way to parametrize spatial displace-
ments, see e.g. [8] for a concise review of that concept.
The most important fact here is that there is a one-to-one
correspondence between all spatial displacements and the
projective points[x0: x1: x2: x3: y0: y1: y2: y3]of theso called Study-quadricS P7 which is a semi-algebraicset described by
x0y0+x1y1+x2y2+x3y3 = 0 (2)
x20+x21+x
22+x
23 = 0 (3)
The relation between points ofSand the corresponding dis-placements is established by the4 4matrix operator
x20+x21+x
22+x
23 0
MT MR
, (4)
where the expressions for MT and MR are as follows
2 (x0y1+x1y0 x2y3+x3y2)2 (
x0y2+x1y3+x2y0
x3y1)
2 (x0y3 x1y2+x2y1+x3y0)
,
0@x20
+ x21 x2
2 x2
3 2 (x1x2 x0x3) 2 (x1x3+ x0x2)
2 (x1 x2 + x0x3) x2
0 x2
1+ x2
2 x2
3 2 (x2x3 x0x1)
2 (x1x3 x0x2) 2 (x2x3+ x0x1) x2
0 x2
1 x2
2+ x2
3
1A .
For the inverse operation, namely to obtain the Study pa-
rameters from a given displacement matrix, there exists a
quite nice method introduced by Study himself. Due to
space limitations this method, although used in the paper,
is not explained here, see e.g. [8] for further information.
Due to the fact that the limbs do not allow free motion
of the platform, there have to be some constraints on the
Study parameters describing the platforms possible poses,i.e. only a subset of the Study-quadric describes all possi-
ble poses of the manipulators endeffector frame. In the fol-
lowing equations depending onx0, x1, x2, x3, y0, y1, y2, y3and describing this subset ofSare deduced from all threetransformation matrices T1, T2, T3.
First of all half-tangent substitutions for all uji are per-formed to get rid of the trigonometric functions. This re-
sults in displacement matrices Ti containing the new pa-
rameterstji . Then from each matrix Ti the Study parame-ters are computed using Studys method, leading to expres-
sions
x0 = f1i(t1i, t2i, t3i, t4i)... (5)
y3 = f8i(t1i, t2i, t3i, t4i) i= 1, . . . , 3
From each matrix Tiwe obtain a parametrization of a sub-
set ofS depending on four parameters. This subset essen-tially describes the motion capability of one limb. The fi-
nal step is then to eliminate from each parametrization the
corresponding four parameters to obtain equations which
contain only the Study parameters and, of course, the de-
sign and motion parametersd1,d2,d3,h1andh2. This canbe achieved easily by using the linear implicitization algo-
rithm (LIA) published in [9]. This algorithm is a method to
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find all equations of given degree which are fulfilled by a
parametrization like (5).
Because the kinematic chain of each limb has four degrees
of freedom, represented by the four parameters in (5), the
elimination will result in two constraint equations and theconstraint equationS. Each of the three eliminations yieldsequation (2) and two other quadratic equations, i.e. all to-
gether there are seven equations in P7 whose solutions de-
scribe all possible poses of the manipulator, given that (3) is
fulfilled. This result was checked by performing the elimi-
nation with other methods and it can be said that definitely
not more than these seven equations are necessary to de-
scribe the motion capabilities of the TSAI manipulator. Af-
ter some simplifications to remove
3 from at least someequations the following equations are obtained:
g1 : x0y0 + x1y1 + x2y2 + x3y3 = 0 (6)
g2: (h1 h2) x0x2+ (h1+h2) x1x3 x2y3 x3y2= 0 (7)
g3: (h1 h2) x0x3 (h1+h2) x1x2 4 x1y1 3 x2y2 x3y3= 0 (8)
g4: (h1 h2) x0x3 (h1+h2) x1x2+
+ 2 x1y1+ 2 x3y3= 0 (9)
g5: (h21 2 h1h2 +h22 d21) x20 + 2
3 (h1 h2) x0y2
2 (h1 h2) x0y3+ (h21+ 2 h1h2+h22 d21) x21 2 (h1+h2) x1y2 2
3 (h1+h2) x1y3+
+ (h21 h1h2+h22 d21) x22+ 2
3 h1h2x2x3 2
3 (h1 h2) x2y0+ 2 (h1+h2) x2y1+
+ (h21+h1h2+h22 d21) x23+ 2 (h1 h2) x3y0+
+ 2
3 (h1+h2) x3y1+ 4 (y20+ y
21+ y
22+ y
23) = 0
(10)
g6: (h21 2 h1h2 +h22 d22) x20 2
3 (h1 h2) x0y2
2 (h1 h2) x0y3+ (h21+ 2 h1h2+h22 d22) x21 2 (h1+h2) x1y2+ 2
3 (h1+h2) x1y3+
+ (h21 h1h2+h22 d22) x22 2
3 h1h2x2x3+
+ 2
3 (h1 h2) x2y0+ 2 (h1+h2) x2y1++ (h21+h1h2+h
22 d22) x23+ 2 (h1 h2) x3y0
2
3 (h1+h2) x3y1+ 4 (y20+ y
21+ y
22+ y
23) = 0
(11)
g7: (h21 2 h1h2+h22 d23) x20+ 4 (h1 h2) x0y3+
+ (h21+ 2 h1h2+h22 d23) x21+ 4 (h1+h2) x1y2+
+ (h21+ 2 h1h2+h22 d23) x22 4 (h1+h2) x2y1+
+ (h21 2 h1h2+h22 d23) x23 4 (h1 h2) x3y0++ 4 (y20+y
21+ y
22+ y
23) = 0 (12)
This system of algebraic equations describes the mecha-
nism. Fixing the motion parametersdi it can be asked nowfor all projective points in P7 which fulfill all these seven
equations, under the condition thatx20+ x21+ x
22+ x
23= 0.
These points represent then all possible poses of the plat-
form which are the solution of the direct kinematics of the
TSAI 3-UPU manipulator. Because it is more convenient
to do all computations in affine space, without loss of gen-
erality, the following normalization equation is added:
g8: x20+x
21+x
22+x
23 1 = 0 (13)
Now it is guaranteed that no solution of this final system lies
in the forbidden variety described by x20 +x21 +x
22 +x
23= 0.
The downside of the normalization is that for each projec-
tive solution point two affine representatives as solutions for
(6)-(13) are obtained. This has to be taken in account when
different solutions are counted.
IV. Solving the system
Now the system of equations (6)-(13) is studied. In the
following this system of equations is always written as apolynomial ideal. Therefore, the ideal that has to be dealt
with is
I=g1, g2, g3, g4, g5, g6, g7, g8,
where eachgi here denotes the polynomial on the left handside of the corresponding constraint equation. First of all
the following ideal is inspected, which obviously is inde-
pendent of the joint parametersd1,d2, andd3
J =g1, g2, g3, g4
Allthough that ideal isnt that complicated, it is tried tomake it simpler. The computation of the primary decom-
position ofJ shows that it indeed can be written in a verysimple way:
J =6
i=1
Ji
with
J1=y0, x1, x2, x3,J2=x0, y1, x2, x3,
J3=y0, y1, x2, x3,J4=x0, x1, y2, y3,
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J5=(h1 h2) x0x2 +(h1 +h2) x1x3 x2y3 x3y2,(h1 h2) x0x3 (h1+h2) x1x2 x2y2+x3y3,
2 x1y1+x2y2+x3y3, x0y0 x1y1,
(h1 h2)2
x20+ (h1+h2)
2
x21 y
22 y
23 ,
(h1+h2) x32y0 3 (h1 h2) x22x3y1 2 x22y0y1
3 (h1 h2) x2x23y0+ (h1 h2) x33y1 2 x23y0y1J6=x0, x1, x2, x3.
Actually all these ideals are prime ideals and there are
no embedded components. The primary decomposition
was computed over Q[x0, x1, x2, x3, y0, y1, y2, y3, h1, h2]to find possible changes of dimension, which could occur
for special values ofh1 andh2. Fortunately idealJ5 hasalways the same dimension, i.e. dimension 3. For the zero
set or vanishing set
V(
J)of
J it follows that
V(J) =6
i=1
V(Ji).
Now the remaining equations are added and by writing
Ki :=Ji g5, g6, g7, g8 the vanishing set of the wholesystemIcan be written as
V(I) =6
i=1
V(Ki).
So, instead of studying the system as a whole, each of the
smaller systemsKi can be inspected separately. Then thesolution of systemIis the union of the solutions of the sub-systems.
It can easily be seen that the last set V(K6) is empty becauseK6contains equations {x0, x1, x2, x3, x20+x21+x22+x231}which cannot vanish simultaneously. Therefore it is only
necessary to study systems K1, . . . ,K5.Furthermore, it has to be noted, that for each assembly
mode described by a solution, there exists another solution
where the manipulator can be assembled mirrored with re-
spect to the base.
A. Solutions for arbitrary design parameters
In the following subsection all computations are per-formed under the assumption that the five design parame-
ters are arbitrary i.e. generic. To find out the Hilbert di-
mension of each ideal Ki the necessary Groebner bases arecomputed for general parameters, except the basis forK5,where randomly chosen parameters h1,h2 are substitutedbefore the computation of the basis. So, for arbitrary de-
sign parameters the result is that
dim(Ki) = 0, i= 1, . . . , 5which means that all sub-systems have finitely many solu-
tions. Reusing the computed bases from above the number
of solutions can be determined for each systemKi. Due to
the fact that always two solutions of a system describe the
same position of the platform, each number has to be halved
(see paragraph below (13)). In the following only these es-
sentially different solutions are considered. The number of
solutions for each system Ki in the generic case is|V(K1)|=|V(K2)|= 2,|V(K3)|= 4,
|V(K4)|= 6,|V(K5)|= 64So all together there are 78 essentially different solutions
for generically given parameters h1,h2,d1,d2, and d3, i.e.78 possible poses of the platform, theoretically. It is clear
that for arbitrarily chosen parameters all these solutions will
be complex. Mechanically this means that the manipula-
tor cannot be assembled because of e.g. too different limb
lengths. But of course, when design and joint parameters
are chosen thoughtfully, many of the solution poses alsocan be real. In Section V design parameters will be given
such that for almost allKi all solutions of the correspond-ing system are real. Using reasonably chosen parameters,
attempts were made to maximize the total number of real
solutions. Surprisingly this number never exceeded 28. A
strict proof for 28 to be an upper bound for real solutions is
missing.
B. Solutions for equal limb lengths
In the following subsection it is assumed that all limbs
are of equal length. Because of the structure of the equa-
tions this is a non-generic case and has to be treated sepa-rately.
d1:= d d2:= d d3:= d
Now the same computations can be performed which were
done in the previous subsection to obtain the Hilbert dimen-
sion of each ideal. Due to the fact that there are less parame-
ters all Groebner bases can be computed without specifying
any parameter. For the dimension the result is the same as
in the previous case.
dim(Ki) = 0, i= 1, . . . , 5.
When the number of solutions is computed for each systemand halved afterwards the following results are obtained.
|V(K1)|=|V(K2)|=|V(K3)|= 2,
|V(K4)|= 6,|V(K5)|= 60Here there exist theoretically 72 solutions for the platforms
position, six less than before. Concerning the question
where they could have gone, one should not forget that we
have that forbidden subvariety on the Study-quadric. It is
possible that for special design parameters a solution lies
on this subvariety. In the previous section it was mentioned
that for special design parameters 28 real solutions can be
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obtained. Actually it was a set of parameters with equal
limb lengths. The exact values for the parameters were
h1= 12, h2= 7, d1= d2= d3= 181
13 13.923
A very important difference to the SNU 3-UPU manipula-
tor which was discussed in [6] is the fact that here in general
all 72 solutions have multiplicity 1, i.e. particularly the so-
lution which corresponds to the so called home position
has multiplicity 1, and not 4, which is the case for the SNU
3-UPU. One could conclude that the TSAI should show a
better behavior in the home position, not being that shaky
like the SNU. The home position can be seen in Fig. 1 and
is described by the following values
x0= 1, x1= 0, x2= 0, x3= 0
y0= 0, y1=
d2 (h1 h2)2/2, y2= 0, y3= 0.The following section will deal with the most interest-
ing interpretation of the systemsK1, . . . , K5. So far theywere simply seen as a decomposition of the original system
of equations, that made solving the system at least a little
easier.
V. The manipulators operation modesUntil nowd1,d2 and d3 were treated as fixed design pa-
rameters. In this section they will be treated as parameters
which are allowed to change, i.e. the behavior of this mech-
anism will be studied, when the prismatic joints are actu-
ated. Computation of the Hilbert dimension of each ideal
Kiwithd1, d2, d3 used as unknowns shows that
dim(Ki) = 3, i= 1, . . . , 5
where dim denotes the dimension over C[h1, h2], incontrast to dim which denotes the dimension overC[h1, h2, d1, d2, d3]as in the previous sections. It followsthat in general the 3-UPU manipulator has 3 DOFs.
As it was shown in [6] for the SNU 3-UPU manipulator
each subsystem Kiof a mechanisms set of equations corre-sponds to a specific operation mode of the manipulator. Inthe following each systemKi will be discussed separately,particularly with regard to the type of motion and possible
singular poses. It has to be mentioned explicitly that the
singularities, where a change of operation mode can occur,
are not the subject here. They will be discussed in the next
section.
The following algorithm is applied in the next paragraphs:
Each system Jiis solved and the solution is substituted intothe transformation matrix (4). From the obtained results
properties of the solutions of the sub systemsKi can bededuced and from these follow properties of the motion of
the platform. It is absolutely not necessary to use equations
(10)-(12) for this inspection, because they describe only the
limb lengths which are now free parameters. Equation (13)
is used to simplify the matrix entries, if possible, i.e. if three
unknowns of (13) are 0, the remaining unknown is set to 1.
Furthermore the position of the platform is described by a
series of simpler transformations, starting from the planarhome position.
System K1: Translational Mode
{y0= 0, x1= 0, x2= 0, x3= 0}
1 0 0 02 y1 1 0 02 y2 0 1 02 y3 0 0 1
This is the operation mode which was discussed in almost
all articles about this manipulator. The transformation ma-trix was simplified by substitutingx0 = 1. From the ma-trix it can easily be seen that it can be parameterized using
y1, y2, y3. It is possible to compute the necessary conditionfor singular positions by making use of the determinant of
the Jacobian matrix of the equations ofK1. The result is anequation of degree 4 ind1, d2, d3 and looks as follows:
d41+d42+d
43 d21d22 d21d23 d22d23 (14)
3 (h1 h2)2 (d21+d22+d23) + 9 (h1 h2)4 = 0
Further inspection shows that for given values for h1 and
h2describes a Kummer surface in the joint parameter spaced1, d2, d3. The Kummer surface is a very famose algebraicdegree four surface, possessing the maximum number of
16 double points (see e.g. [11]). The part of the surface
in the first octant is the interesting part. If the parameters
d1, d2, d3are chosen inside resp. outside the pipe (Fig.2),both solutions of this system are real resp. complex. All
solutions are real for e.g. the parameters
h1= 12, h2= 7, d1= 670
21 , d2=
243
7 , d3=
611
21
On the other hand, if the parameters of the limbs are chosen
such that the condition (14) is fulfilled, it is not difficultto show that both solutions coincide and the corresponding
positions are described by
1 0 0 00 1 0 0
p2 0 1 0p3 0 0 1
i.e. the singular positions of the translational mode are
those, where base and platform lie in the same plane.
System K2: Twisted translational Mode
{x0= 0, y1= 0, x2= 0, x3= 0}
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Fig. 2. The singularity surface of the translational mode.
1 0 0 02 y0 1 0 02 y3 0 1 0
2 y2 0 0 1
This is a mode which was already mentioned in [10]. Here
x1 = 1was used to simplify. Each solution of systemK2corresponds to a rotation of the platform about its normal
axisNby 180 degrees and a subsequent translation. It fol-lows that the described operation mode is basically a pure
translation. To parameterize it one could usey0, y2, y3 asparameters. Once again it is possible to compute the neces-
sary condition for singular positions. The result is again an
equation of degree 4 ind1, d2, d3:
d41+d42+d
43 d21d22 d21d23 d22d23 (15)
3 (h1+h2)2 (d21+d22+d23) + 9 (h1+h2)4 = 0It it is again a Kummer surface with the difference that the
pipe (Fig.3) has a larger diameter. Again if the limb pa-
rameters are chosen inside resp. outside the pipe, both
solutions of this system are real resp. complex. All solu-
tions are real for e.g. the parameters
h1= 12, h2= 7, d1= 121
3 , d2=
965
21 , d3=
62
3
If the condition (15) fulfilled, both solutions coincide and
the corresponding positions are described by
1 0 0 00 1 0 0
p2 0 1 0p3 0 0 1
i.e. the singular positions of the twisted translational mode
are once more those, where base and platform lie in the
same plane.
System K3: Planar Mode
{y0= 0, y1= 0, x2= 0, x3= 0}
Fig. 3. The singularity surface of the twisted translational mode.
1 0 0 00 1 0 0
2 (x0y2+x1y3) 0 x20 x21 2 x0x12 (x0y3 x1y2) 0 2 x0x1 x20 x21
Solutions ofK3 correspond to poses of the platform whereit is coplanar to the base. To parameterize this planar op-
eration mode one could use x0, y2, y3 in connection withx20+ x
21= 1, wherex0is responsible for the rotation of the
platform about its normal axisNandy2, y3 for the transla-
tion in the base-plane, i.e free planar motion is available inthis mode. Once more it is possible to compute the neces-
sary condition for singular positions. The result is an equa-
tion of degree 12 which can be factorized into
F1F2(d1+ d2 d3) (d1+ d3 d2) (d2+ d3 d1) F3= 0where F1 and F2 are exactly the singularity-conditionsfrom the previous modes andF3 is the factord1+d2+d3which does never vanish. The union of the corresponding
varieties of these five factors separates the joint parame-
ter space into several zones where zero, two, or four real
solutions are obtained. All solutions are real for e.g. the
parameters
h1= 12, h2= 7, d1= 125
7 , d2=
493
21 , d3=
66
7
Due to the fact that the condition is a product of essentially
five factors there are more than one representatives for a
singular position. First of all both matrices from the previ-
ous modes, describing their singular positions. Furthermore
from the small factors one obtains
1 0 0 00 1 0 0
p2 0 v33 v32
p3 0 v32 v33
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with
v232+v233= 1, p
22+p
23= h
21+h
22 2 h1h2v33
i.e. amongst others a singular position is obtained when the
origin of the platform frame lies on a circle in the base plane
whose radius is determined by the rotation of the frame
about its x-axis.
Fig. 4. The singularity surface of the planar mode. It contains the surfaces
from Fig. 2 and Fig. 3 .
System K4: Upside-down planar mode{x0= 0, x1= 0, y2= 0, y3= 0}
1 0 0 00 1 0 0
2 (x2y0 x3y1) 0 x22 x23 2 x2x32 (x2y1+x3y0) 0 2 x2x3 x22+x23
Solutions ofK4 correspond to positions of the platformwhere it is turned upside down and coplanar to the base.This can be achieved by starting in the planar home position
and turning the platform about its y-axis, while the limbs
are always attached. To parameterize the upside-down pla-
nar operation mode one could usex3, y0, y1 in connectionwithx22+ x
23 = 1, wherex3 is responsible for the rotation
of the platform about its normal axis N andy0, y1 for thetranslation in the base-plane.
Computation of the singularity condition was rather hard in
this case. The result is a very lengthy equation of degree
24 which cannot be factorized over Q. Due to space limita-
tions it is not displayed here. Several plots of it were made
for given h1, h2. The variety is again tube-like but it has
also self-intersections which again separate the parameter
space in different zones. It was possible to find parameters
where all six solutions are real, which are e.g.
h1= 12, h2= 7, d1= 1257
, d2= 49321
, d3= 667
Up to now it was not possible to deduce all singular posi-
tions from the condition above due to its complexity.
Fig. 5. The singularity surface of the upside-down planar mode.
System K5: General Mode
This mode is the most difficult one, because of the com-
plexity of the equations in idealJ5. What definitely canbe said is that is has in general for given limb length 64
solutions and that the system has 24 real solutions if the
parameters are
h1= 12, h2= 7, d1= d2= d3= 181
13
Although the equations ofJ5can be solved linearly fory1,y2,y3, and y4, due to the complexity of the equations there is
no possibility to find a neat description of this mode, neitherthe condition for singularities nor a description of them.
All together there are five essential operation modes of
the SNU 3-UPU manipulator. It is quite obvious that this
mechanism is more complex than the SNU 3-UPU whose
essentially seven operation modes were quite simple.
VI. Changing operation modesAs already mentioned there exist poses where the mech-
anism can change from one mode into another mode. One
of them is e.g. the planar home position where the mech-
anism can bifurcate into the planar mode or in the trans-
lational mode. In the following an overview is presented
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of those the poses where a mode-change is possible. This
is done by inspecting each pair of ideals{Ki, Kj} withrespect to common real solutions. For each pair the di-
mension of the intersection of the corresponding varieties
is computed and the following results are obtained.
K1 K2 K3 K4 K5K1 3 1 2 1 2K2 1 3 2 1 2K3 2 2 3 1 2K4 1 1 1 3 2K5 2 2 2 2 3
TABLE II. All values ofdim(Ki Kj)
The numbers in Table VI correspond to the dimension
of the intersection variety. As it can easily be seen there
are four possible combinations of operation modes which
have no pose in common, to change from one to the other
a detour has to be made, which is always possible via the
general mode, corresponding toK5. It has to be noted thatmode changing poses of the manipulator are singular, be-
cause each of them lies in the intersection of two varieties.
In the following all possible changes are discussed with re-
spect to necessary conditions and a description of related
poses.
Translational mode planar mode
The condition for this change is exactly the singularity con-
dition from the translational mode, i.e. all singularities of
this mode coincide with the intersection singularities with
the planar mode. Hence the possible mode change poses
have already been given in Section V and the correspond-
ing singularity surface can be seen in Fig. 2.
Translational mode general mode
For this change a new condition appears, that is
d41+ d42+ d
43 d21d22 d21d23 d22d23 36 (h1 h2)4 = 0
and for the corresponding positions of the platform the fol-
lowing description can be deduced
1 0 0 0p1 1 0 0p2 0 1 0p3 0 0 1
with
p22+p23= 4 (h1 h2)2
i.e. platform and base are in the same orientation and the
origin of the platforms frame has to lie on a cylinder with
radius2 (h1 h2). This is a result which verifies statementsabout singularity loci from [10].
Fig. 6. The singularity surface for changes from translational to general
mode.
Twisted translational mode planar mode
The condition for this change is exactly the singularity con-
dition from the twisted translational mode, i.e. all singular-
ities of this mode coincide with the intersection singulari-
ties with the planar mode. Hence the possible mode change
poses have already been mentioned and the corresponding
singularity surface can be seen in Fig. 3.
Twisted translational mode general mode
The condition for this mode change is as follows:
d41+ d42+ d
43 d21d22 d21d23 d22d23 36 (h1+ h2)4 = 0
and for the corresponding poses of the platform the follow-
ing description can be deduced
1 0 0 0
p1 1 0 0p2 0
1 0
p3 0 0 1
with
p22+p23= 4 (h1+h2)
2
i.e. once more platform nd base have the same orientation
and the origin of the platforms frame has to lie on a cylin-
der, but this time with radius2 (h1+h2). The same resultcan again be found in [10], p. 598.
Planar mode general mode
The condition which has to be fulfilled for this case reads
7 (d41+d42+d43) 11 (d21d22 d21d23 d22d23) = 0
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Fig. 7. The singularity surface for changes from twisted translational to
general mode.
The corresponding poses of the platform are given by
1 0 0 00 1 0 0
p2 0 v33 v32p3 0 v32 v33
with
v232+v233= 1, p
22+p
23= 4 (h
21+h
22 2 h1h2v33)
It is noticeable that these positions are very similar to the
singular poses of the planar mode itself.
Fig. 8. The singularity surface for changes from planar to general mode.
Upside-down planar mode general mode
The condition for this case can be computed but it is rather
complicated. It is an equation of degree 24 and it is not
equal to the singularity condition of the upside-down planarmode itself. As a result of the complexity up to now also no
description of the platforms poses could be found.
Fig. 9. The singularity surface for changes from upside-down planar to
general mode.
All other combinations which are
{K1,
K2
},
{K1,
K4
},
{K2, K4}, and{K3, K4} do not allow any operation modeswap.
VII. Conclusions
Like in [6] methods from algebraic geometry have
proven to be very useful to analyze a mechanism like the
TSAI 3-UPU manipulator. In particular primary decompo-
sitions can be used to inspect a manipulator with respect to
possible different operation modes.
It could be shown that the direct kinematics of the TSAI 3-
UPU has up to 78 solutions. The maximum number of real
solutions is 28 so far. Furthermore the mechanism seems to
be less special than the SNU 3-UPU. Maybe a precisely
manufactured model of it would not be that unstable, at
least in a region around the home position. Nevertheless
several regions could be found where one has to expect sin-
gular positions, not to mention possible singularities of the
general mode, which could not be found until now.
References
[1] Tsai L.-W., Kinematics of a Three-DOF Platform with Three Exten-sible Limbs, Recent Advances in Robot Kinematics, J. Lenarcic andV. Parenti-Castelli (eds.), Kluwer Academic Publishers, 401410,1996.
[2] Di Gregorio R., Parenti-Castelli V., A Translational 3-DOF ParallelManipulator, Advances in Robot Kinematics: Analysis and Control ,J. Lenarcic and V. Parenti-Castelli (eds.), Kluwer Academic Publish-
ers, 4958, 1998.
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[3] Di Gregorio R., Parenti-Castelli V., Mobility Analysis of the 3-UPU Parallel Mechanism Assembled for a Pure Translational Mo-tion, IEEE-ASME International Conference on Advanced Intelligent
Mechatronics, Atlanta, 520525, 1999.[4] Parenti-Castelli V., Di Gregorio R., Bubani F., Workspace and Opti-
mal Design of a Pure Translation Parallel Manipulator, Meccanica,Vol. 35, No. 3, 203214, 2000.[5] Tsai L.-W., Joshi S., Kinematics and Optimization of a Spatial 3-
UPU Parallel Manipulator, ASME Journal of Mechanical Design,Vol. 122, 439446, 2000.
[6] Walter D. R., Husty M. L., Pfurner M., A Complete Kinematic Anal-ysis of the SNU 3-UPU Parallel Robot, Interactions of Classicaland Numerical Algebraic Geometry, Vol. Contemporary Mathemat-ics 496, 331346, 2009.
[7] Husty M. L., Karger A., Steinhilper W., Kinematik und Robotik,Springer-Verlag, Berlin, Heidelberg, New York, 1997.
[8] Pfurner M., Analysis of spatial serial manipulators using kinematicmapping, Doctoral Thesis, University of Innsbruck, 2006, (availableat http://repository.uibk.ac.at).
[9] Walter D. R., Husty M. L., On Implicitization of Kinematic Con-straint Equations,Machine Design & Research (CCMMS 2010) , Vol.26, 218226, ISSN 1006-2343, Shanghai, 2010.
[10] Chebbi A. H., Parenti-Castelli V., Geometric and Manufactur-ing Issues of the 3-UPU Pure Translational Manipulator, NewTrends in Mechanism Science, D. L. Pisla et al. (eds.), Springer Sci-ence+Business Media B.V., 595603, 2010.
[11] http://en.wikipedia.org/wiki/Kummer surface
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