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13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, 19-25 June, 2011 IMD-123

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,

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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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