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The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015

DOI Number: 10.6567/IFToMM.14TH.WC.OS2.046

On Regular Kinematotropies

Andreas M¨

uller Samuli Piipponen

Institute of Robotics School of Information Sciences

Johannes Kepler University, Linz, Austria University of Tampere, Finland

a.mueller@jku.at samuli.piipponen@uta.ﬁ

Abstract— Kinematotropic linkage are fascinating as

they can switch between motion mode with different DOF.

Beside the mere curiosity, this phenomenon has also ap-

plication potential. From this perspective it is crucial to

consider the limitations that the actual motions impose on

the operation of such linkages. Such a limiting factor is

the smoothness of the transitory motion when switching be-

tween motion modes with different DOF. Given a particu-

lar kinematotropic mechanism, the feasible motions can be

analyzed by means of a local higher-order analysis of the

conﬁguration space. This is addressed in this paper. The

computational part of the method involves only Lie brack-

ets of the instantaneous joint screws. Two examples are

presented: a 7R linkage and the Wunderlich linkage. It is

shown that the latter possesses a regular kinematotropy, i.e.

it can smoothly move between its motion modes.

Keywords: Kinematotropy, singularities, overconstrained, Wun-

derlich linkage, higher-order kinematics, screws, Lie groups

I. Introduction

A remarkable property of linkages with special geom-

etry, which was recognized and investigated relatively re-

cently, is their ability to change the ﬁnite degree of free-

dom (DOF) without reassembling them. Such linkages are

now called kinematotropic [15]. Since Wohlhart introduced

the notion of kinematotropic linkages, the phenomenon has

attracted some attention. Initially, this phenomenon was

found in multi-loop linkages, e.g. [13], [15]. Later, Gal-

leti & Fangella [3] discussed kinematotropy of single-loop

kinematotropic linkages based on the intersection of mo-

tion space. This was continued by Lee & Herve in a series

of publications [5], [6], [7], [8]. Since then a number of

publications have addressed this problem. An interesting

recent contribution is [4] by Kong & Pfurner where kine-

matotropic single-loop linkages were analyzed by means of

computational algebraic geometry.

It should be remarked that usually a linkage is said to

be kinematotropic if it changes its DOF. This has indeed

accompanied by a change of its mobility, i.e. the way it

can actually move. In [3] the notion of kinematotropy was

relaxed so to include situations where the relative mobility

of members of the linkage (called connectivity) changes.

The study of kinematotropic linkages has basically been

focussed on speciﬁc examples, while the change of DOF

has been ensured by strategic selections of designated mo-

tion spaces. The actual kinematotropy, i.e. the linkage’s

motion leading to a change of DOF, have not been investi-

gated explicitly. Moreover, the latter is commonly not even

considered. However, whereas the DOF change is the pri-

mary focus, from an application point of view, it is impor-

tant to investigate the motion characteristics during tran-

sition between motion modes with different DOFs. Most

of the analyzed mechanisms can exhibit necessarily non-

smooth transitory motions. That is, some joints have dis-

continuous velocities and accelerations, which is also made

explicit by the adjective ’discontinuously’ in title of [5], [6],

[7], [8]. For any application it is crucial to avoid disconti-

nuities, however. Whether a given kinematotropic linkage

allows for smooth transitory motion is has not been a re-

search topic yet.

An important step towards a better understanding of

kinematotropies is the analysis of the transitory motions. To

this end a local analysis procedure is proposed in this paper.

This makes use of a local analysis method [9], [12] based

on the tangent cone to the conﬁguration space (c-space),

which is the best local approximation of the c-space and

thus of the transitory motions. It is shown for the Wunder-

lich linkage [16] that it possesses a smooth kinematotropy.

For a strategic design of kinematotropic linkages it will be

necessary to develop criteria that lead to smooth kinema-

totropies. The presented method to analyze a given linkage

is expected to provide a good starting point for this.

II. Linkage Kinematics

The members of a linkage are subjected to a system of

geometric constraints that can be written as

h(q) = 0(1)

where q∈Vnis the vector of joint variables. These con-

straints deﬁne the c-space of the linkage as

V:= h−1(0).(2)

This is a variety in the parameter space Vn. In this paper,

joint angles and relative displacements are used as variables

so that Vis an analytic variety in Vn. The velocity con-

straints to (1) are expressed as

J(q)˙

q=0(3)

with constraint Jacobian J. The latter is given in terms of

instantaneous joint screws (next section).

The differential (instantaneous) DOF of the linkage at

q∈Vis denoted δdiﬀ (q) := n−rank J(q). This is

the number of independent ﬁrst-order motions. The lo-

cal DOF at qis the local dimension of V, denoted as

δloc (q) := dimqV. This is the highest dimension of man-

ifolds passing through q. The c-space is locally a smooth

manifold at q∈Vif δdiﬀ is constant in a neighborhood of

qin V(the reverse is not necessarily true). A conﬁguration

qis a kinematic singularity iff there is no neighborhood

where δdiﬀ is constant.

The c-space consists of admissible conﬁgurations and its

geometry characterizes the admissible ﬁnite motions.

Deﬁnition 1: Consider a linkage with c-space variety V.

The assembly modes of a linkage are the connected sub-

varieties Ai⊆Vsuch that V=A1∪A2∪A3∪. . . and

Ai∩Aj=∅.

It should be remarked that this deﬁnition differs from an-

other concept [] according to which the assembly modes of

a mechanism are the different ways the passive links can be

assembled once the active links are positioned. According

to this deﬁnition, a linkage cannot transit from one assem-

bly mode to another (without disassembling it). As an ex-

ample, two assembly modes of a planar 4-bar linkage are

shown in Fig. 1. These are both 1-dimensional. The as-

sembly modes can possess different dimensions, however.

It is known for instance that a 6R linkage [2], with special

geometry, can be assembled to a structure (0 DOF) but also

to a linkage with mobility 1, namely the Bricard linkage.

Fig. 1. Two assembly modes of a planar 4-bar linkage.

III. Kinematotropy

The remarkable property of a kinematotropic linkage is

that it can change its local DOF without disassembling it.

Deﬁnition 2: A linkage is kinematotropic in the as-

sembly mode A⊆V, iff Acomprises submanifolds

M1, M2, . . . ⊆Awith different dimensions i. These sub-

manifolds Miare called the motion modes with DOF i.

A point q∈Vis a kinematotropic conﬁguration iff

q∈Mi∩Mj, where Mis the closure of Min V.

Kinematotropic points, where the linkage can transit

between different motion modes, are necessarily singular

points of the c-space V, i.e. there is no neighborhood of q

in Vwhere δdiﬀ is constant.

Consider the single-loop 7R linkage in Fig. 2 that was

discussed in [4]. Fig. 3 shows a conﬁguration in a motion

mode M2with 2 DOF, and Fig. 4 a conﬁguration in a mo-

tion mode M3with DOF 3. The linkage can pass between

the two modes via the singular conﬁguration in Fig. 2.

The conﬁgurations where two motion modes intersect

are singularities of the c-space V. This generally indi-

cates the possibility of non-smooth motions when passing

through these conﬁgurations. The mere fact that a linkage

can change its DOF does not mean that it must necessarily

exhibit non-smooth motions. Moreover, for the practical

exploitation of kinematotropic linkages, the possible mo-

tions when switching between different motion modes is

crucial. This has to do with the kinematotropy.

Deﬁnition 3: A curve q: [t1, t2]7→ Vis a kinema-

totropy between motion mode M1and M2, iff q(t1)∈M1

and q(t2)∈M2, and dim M16= dim M2. The kinema-

totropy q(t)between M1and M2is regular, iff it is a reg-

ular curve.

The remarkable property of a regular kinematotropy is

that the joint velocities evolve continuously when transiting

between different motion modes. Therefore, linkages with

regular kinematotropies are supposed to have some appli-

cation potential. However, yet there is no necessary and/or

sufﬁcient conditions on the geometry for the linkage to ex-

hibit regular kinematotropy that would eventually allow for

a purposeful design.

From a differential geometric perspective, the condition

on the c-space Vis that the motion modes (submanifolds)

of different DOF intersect non-transversally. Let q0∈

M1∩M2be an intersection point of the motion modes

M1and M2, i.e. q0is a singularity of V. A necessary con-

dition for the existence of a regular kinematotropy through

q0is that dim(Tq0M1∩Tq0M2)≥1, where Tq0M1and

Tq0M2is the respective tangent space [1].

IV. Local Analysis of a Kinematotropy

A. Tangent Cone of Single Loop Linkages

Analyzing the kinematotropy of a linkage boils down to

analyzing its mobility at a point where the linkage may

change its DOF. A global analysis is not possible in general,

and the considerations must resort to local characterization.

The best faithful approximation of the c-space variety at

any point is its tangent cone. The latter is the set of the tan-

gent vectors to curves in V, denoted CqV. Moreover it is

the algebraic variety of smallest degree that best approxi-

mates V.

The tangent cone reveals the ﬁnite motions through q,

and its dimension is the local DOF at q. The general deﬁ-

nition of CqVdoes not give rise to an explicit method for

its determination. However, a computational method can

be derived resorting to the physical meaning of the tangent

vectors x∈CqVas those velocities satisfying the veloc-

ity constraints for which there exist acceleration, jerks, and

so forth that satisfy the acceleration constraints, jerk con-

straints, and higher order constraints, respectively [9], [12].

The computational algorithm is based on the formulation

of geometric constraints in terms of joint screws. To this

end, the loop constraints for a kinematic loop are formu-

lated as

f(q) = I(4)

with Ibeing the 4×4identity matrix, and the constraint

mapping f:Vn→SE (3) given by

f(q) = exp(Y1q1)·. . . ·exp(Ynqn).(5)

Here Yi= (ei,si×ei+hiei)is the screw coordinate

vector of joint iexpressed in a global reference frame in

the zero reference conﬁguration q=0. The c-space is then

the analytic variety determined as

V:= h−1(I)⊂Vn.(6)

The corresponding velocity constraints at q∈Vare

0=X

i≤n

Si˙qi=J(q)˙q (7)

with

Si=AdgiYi(8)

where gi(q) = exp(Y1q1)·. . . ·exp(Yiqi)is the instan-

taneous screw coordinate vector of joint iexpressed in the

spatial reference frame. The matrix Adgitransforms the

screw coordinate vectors from the zero reference conﬁgu-

ration to the current conﬁguration according to q[14]. The

acceleration constraints are

0=X

i≤n

Si(q) ¨qi+X

j<i≤n

[Sj,Si] ˙qj˙qi.(9)

These follow from the fact that the partial derivative of the

instantaneous screw coordinate is given by the Lie bracket

as ∂

∂qjSi= [Sj,Si], j < i ≤n. (10)

This is an important relation since it allows expressing all

necessary objects in the higher-order constraints, i.e. the

time derivatives of the velocity constraints, in terms of al-

gebraic operations in the screw coordinates since (10) can

be applied recursively. The according explicit expressions

have been reported in [10], [11]. Moreover, in [12] a com-

putationally simple recursive formulation is reported that

does not require constructing the Lie brackets explicitly.

With the higher constraints, a cone of order ican be de-

ﬁned as

Ki

q:= {x|∃y,z, . . . ∈Rn:H(1)(q,x) = 0,

H(2)(q,x,y) = 0,

H(3)(q,x,y,z) = 0,

···

H(i)(q,x,y,z, . . .) = 0}

(11)

where at q∈V

H(1)(q,˙

q) := J(q)˙q

H(2)(q,˙

q,¨

q) := d

dtH(1)(q,˙

q)(12)

···

H(i)

(q,˙

q,...,q(i)) := di−1

dti−1H(1)(q,˙

q).

Ki

qis the space of velocities (not necessarily a vector space)

that satisfy the constraints up to order i. The tangent cone

is then the limit of these ith order cones

CqV=Kκ

q. . . ⊂K2

q⊂K1

q(13)

where Kκ

q=Kκ+1

q. It is important to notice that all poly-

nomials deﬁning these cones are available in terms of Lie

brackets of the instantaneous joint screws.

B. Tangent Cone of Multiple-Loop Linkages

Mutli-loop linkages are treated as a system of kinematic

loops. To this end a topological graph is introduced and

the constraints are formulated for each of the topologically

independent fundamental cycles [12].

A topologically graph, denoted Γ (B, J ), represents the

arrangement of bodies and joints in the linkage. A vertex

Ba∈Brepresent a body, and an edge Ji∈Jrepresents

a joint connecting two bodies. On Γcan be introduced a

system fundamental cycles (FC).The number of FCs is γ:=

|J|−|B|+ 1, where |J|is the of number joints, and |B|the

number of bodies. Assuming 1 DOF joint, it is |J|=n.

A spanning tree Gand cotree Hcan be introduced on

Γ. Each FC comprises exactly one cotree edge. De-

note with Λlthe FC associated with the cotree edge l.

Each FC has a direction, and thus induces an order of the

joints when traversing the loop. The latter is denoted by

i <lj <l. . . <llfor all edges in Λl. That is, edge lis re-

garded as the terminal body of the chain.

Also the joints, i.e. edges of Γ, have an orientation (but

Γis a non-directed graph). Introduce the indicator func-

tion so that σl(i)=1, if edge iorientated along the FC

Λl;σl(i) = −1, if edge iis oriented opposite to Λl; and

σl(i)=0, if Ji/∈Λl.

Now for each FC a loop constraints can be formulated

analogously to (5) as

fl(q)=exp(σl(i)Yiqi) exp(σl(j)Yjqj)·. . . ·exp(Ylql),

i <lj <l. . . <ll∈Λl(14)

that gives rise to the geometric loop constraints for FC Λl

fl(q) = I.(15)

The geometric constraints for all γFCs yields the overall

constraint system for a linkage. This deﬁnes the linkage’s

c-space V=V1∩V2∩ · ·· ∩ Vγ, with Vl:= f−1

l(I).

The velocity constraints are accordingly

X

i∈Λl

σl(i)Sl

i(q) ˙qi=0(16)

where the instantaneous joint screw coordinates are now

Sl

i=Adgl

iYi(17)

with gl

i(q) = exp(σl(j)Yjqj)·. . . ·exp(σl(i)Yiqi),

j <l. . . <li∈Λl. All higher-order constraints follow di-

rectly with (10) and (12). Hence the constraints of arbi-

trary order for multi-loop linkages are available in terms of

simple algebraic expressions. The ith order cones are thus

determined in an obvious manner as (11) and hence the tan-

gent cone.

In the following the local analysis based on the tangent

cone is applied to study the kinematotropy of linkages.

V. A non-regular Kinematotropy: 7R Linkage

In a recent paper [4] the single-loop kinematotropic 7R

linkage shown in Fig. 2 was analyzed. In that paper com-

putational algebraic geometry was used determine the kine-

matotropy explicitly. In particular the case with DH param-

eters α2=π/4, α6=π/6was considered.

In the reference conﬁguration q=0the joint screw co-

ordinates w.r.t. the shown reference frame are

Y1= (0,0,1,110,0,0),

Y2= (0,0,1,110,−40,0)

Y3= (−(1/2),0,√3/2,30√3,−20√3,30)

Y4= (0,0,1,0,−40,0)

Y5= (0,0,1,−b, a, 0),

Y6= (0,0,1,25,0,0)

Y7= (−(1/2),0,√3/2,30√3,0,30).

The ﬁrst-order constraints H(1) (0,x) = 0in (11) at q=0

are then given with the mapping

H(1) (0,x) = Y1x1+Y2x2+Y3x3+Y4x4

+Y5x5+Y6x6.

1

2

3

4

5

67

12

b

a

Fig. 2. 7R-linkage in a bifurcation conﬁguration.

The solution set is the vector space

K1

0=Vp1(x), p2(x), p3(x), p4(x), p5(x)∈R7,

where V(p1, . . . , pi)represents the algebraic solution va-

riety of the polynomials

p1(x) = x3+x7

p2(x) = 40x2+ 20√3x3+ 40x4−ax5

p3(x) = x1+x2+√3/2x3+x4+x5+x6+√3/2x7

p4(x) = 110 (x1+x2) + 30√3 (x3+x7)−bx5+ 25x6.

Hence at q=0there are 3-dimensional ﬁrst-order motions.

Evaluating the second-order constraints yields the second-

order cone

K2

0=K2,1

0∪K2,2

0

as the union of the two cones

K2,1

0={x∈R7|x1=1

680 ((200 + 5a+ 8b)x5−880x2),

x3= 0, x4=1

40 (ax5−40x2),

x6=1

340 (440x2−(440 + 11a+ 4)bx5),

x7= 0}(18)

K2,2

0={x∈R7|x1=1

200 (200 + 5a+ 8b)x5,

x2=−1

200 (200 + 5a+ 8b)x5,

x3=√3

150 (200 + 5a+ 8b)x5,

x4=−1

25 (25 + b)x5,

x6=b

25 x5,

x7=−√3

150 (200 + 5a+ 8b)x5}(19)

with dim K2,1

0= 1 and dim K2,2

0= 2. The third-order

cone turns out be identical to the second-order cone, so that

the tangent cone is C0V=K3

0=K2

0, and splits according

to C0V=C0V1∪C0V2, with C0V1=K2,1

0. The latter

is the tangent space to the 1 DOF 5-bar motion mode V1

where joint 3 and 7 are locked (Fig. 3), and C0V2=K2,2

0

is the tangent space to the 2 DOF motion mode (Fig. 4).

The conﬁguration q0is indeed a singularity, since

T0V6=C0V, where linkage can bifurcate between

the 1 DOF motion mode V1and the 2 DOF mode

V2. The local DOF is thus δloc (0) = dim C0V=

max(dim K2,1

0,dim K2,2

0)=2.

Fig. 3. A conﬁguration of the 7R-linkage in a 1 DOF mode.

Fig. 4. A conﬁguration of the 7R-linkage in a 2 DOF mode.

From the explicit parameterization of the tangent cone,

i.e. of K2,1

0and K2,2

0, at q=0, is obvious that there is no

regular kinematotropy.

VI. A regular Kinematotropy: Wunderlich’s Linkage

A kinematotropic multi-loop linkage was reported by

Wunderlich in [16]. It was shown that this mechanism

possesses ﬁve different motion modes: three with 2 DOF

(‘Hauptform’, ‘1. Nebenform’, ‘5. Nebenform’), and two

with 1 DOF (‘3. Nebenform’, ‘4. Nebenform’). The link-

age can change between these motion modes via singular

conﬁgurations. In the following the linkage is analyzed in

the conﬁguration in Fig. 5a).

The linkage consists of 12 bodies that are connected by

16 joints. Its topological graph is shown in Fig. 5b). The

spanning tree, cotree, and FCs are chosen as shown. It pos-

sesses γ= 5 independent FCs.

The geometric parameters (Fig. 5a)) are set to a= 3, b =

1, c =√5. Then, in the reference conﬁguration, the joint

screw coordinates are

Y1=Y11 = (0,0,1,0,−3,0)

Y2=Y4= (0,0,1,0,−4,0)

Y3=Y15 = (0,0,1,0,−1,0)

Y5=Y7= (0,0,1,2,−3,0)

Y6=Y8= (0,0,1,2,0,0)

Y9=Y13 = (0,0,1,2,−1,0)

Y10 =Y14 = (0,0,1,0,0,0)

Y12 =Y16 = (0,0,1,2,−4,0).

With the selected FCs, the ﬁrst-order constraints (14) at the

reference q=0for the 5 FCs are 0=H(1)

l(0,x)with

H(1)

12 (0,x) = Y12x12 −Y11 x11 +Y1x1+Y2x2

+Y4x4+Y5x5+Y8x8+Y9x9

H(1)

13 (0,x) = Y13x13 −Y10 x10 +Y1x1+Y2x2

+Y4x4+Y5x5+Y8x8+Y9x9

H(1)

14 (0,x) = Y14x14 −Y3x3−Y2x2−Y1x1

H(1)

15 (0,x) = Y15x15 +Y4x4+Y5x5+Y6x6

H(1)

16 (0,x) = Y16x16 −Y9x9−Y8x8+Y7x7

(20)

The solution of (20) is the 3-dimensional vector space

K1

0=V(p1(x), . . . , p12 (x)) ⊂R16 (21)

deﬁned by the linear equations

p1(x) = x1+ 2x14 + 6x15 −3x16,

p2(x) = x2−x14 −4x15 + 2x16,

p3(x) = x3−2x14 −2x15 +x16,

p4(x) = x4+x15,

p5(x) = x5−x15,

p6(x) = x6+x15,

p7(x) = x7−x14 −2x15 + 2x16,

p8(x) = x8+ 2x14 + 4x15 −x16,

p9(x) = x9−3x14 −6x15 + 2x16,

p10 (x) = x10 +x14 + 3x15 −x16,

p11 (x) = x11 +x14 + 3x15 −x16,

p12 (x) = x12 +x14 + 3x15 −x16,

p13 (x) = x13 +x14 + 3x15 −x16.

Here V(p1, . . . , pi)represents the algebraic solution vari-

ety of the polynomials p1, . . . , pi. Performing the higher-

order analysis shows that all higher-order cones are identi-

cal to the second-order cone, i.e. CqV=K2

q. The latter

is

K2

0=K2,1

0∪K2,2

0∪K2,3

0

where

K2,1

0={x∈R16|x1=s−2t, x2=−s+ 2t,

x3=s−2t, x4=−t, x5=t,

x6=−t, x7=−s, x8=−s,

x9=s, x10 =−t, x11 =−t,

x12 =−t, x13 =−t, x14 =s−2t,

x15 =t, x16 =s}(22)

K2,2

0={x∈R16|x1=−5s, x2= 5s, x3=−5s,

x4=−s, x5=−3s, x6=−s,

x7= 3s, x8= 3s, x9=−3s,

x10 =−s, x11 =−s, x12 =−s,

x13 =−s, x14 =−5s, x15 =s

x16 =s}(23)

B2

H

G

B3

1

3

14

L

B014

J

J

J

B1

2

J

L15 L16

15

5

4

J

J

J

6

J8

16

7

J

J

J

9

J

L13

L12

10

J13

J

12

J

11

J

10

B

11

B

B9

B8

B6

B7

B4

B5

J1

J11

J4

J2

J16

J12

J5

J7

J14

J10

J13

J9

J6

J8

J3

J15 B2

B0

B6

B3

B1

B4

B11

B5

B8

B9

B7

B10

a)

b)

Fig. 5. a) Reference conﬁguration of Wunderlich’s linkage, and b) its topological graph.

K2,3

0={x∈R16|x1= 4s, x2=−2s, x3=−4s,

x4=−s, x5= 2s x6=−s,

x7=−4s, x8= 2s, x9=−4s,

x10 =s, x11 =s, x12 =s,

x13 =s, x14 =−2s, x15 =s

x16 =s}(24)

The K2,i

0are tangent spaces to the motion modes inter-

section at q=0. The K2,1

0is the tangent space to a 2-

dimensional motion mode. The K2,2

0and K2,3

0are tangent

spaces to 1-dimensional modes. Figure 6 shows represen-

tative conﬁgurations in either motion mode.

Once the tangent cone is known, also the tangent space

to Vat qis known as TqV= span CqV. For the Wun-

derlich linkage it is in fact TqV=K1

q= ker J(q). Since

T0V6=C0V, the conﬁguration q=0is a singular point of

V.

The parameterizations of the tangent spaces reveal the

feasible motions when the linkage branches from one to the

other mode. It follows from (22) and (23) that the mode

2 (with tangent space K2,2

0) can be entered smoothly from

mode 1 (with tangent space K2,1

q) by setting t= 6s. Hence

there is a regular kinematotropy between mode 1 and 2.

In other words, the linkage can be controlled from the 2-

dimensional to the 1-dimensional mode without the need to

stop its motion and without velocity or acceleration jumps.

This should be deemed an important ability. However, the

actual control through the singularity is still a difﬁcult task.

The Wunderlich linkage was analyzed in [16] using com-

putational algebraic geometry. There the different motion

modes were deduced from a global analysis of the c-space

variety.

a)

b)

c)

d)

Fig. 6. a) Wunderlich’s linkage in a) singularity, b) mode 1, c) mode 2,

and d) mode 3.

VII. Conclusion

The remarkable feature of kinematotropic linkages, that

they can change their DOF without disassembling them,

has already attracted some attention. If this is eventually

to be exploited in applications, it is important to study the

linkage’s kinematics during the transition between motion

modes with different DOF.

In this paper the notion of regular kinematotropy is intro-

duced as a smooth curve in c-space, i.e. a smooth motion,

between different-dimensional motion modes. There is yet

no criterion for the existence of regular kinematotropy, and

it remains open whether it exists for a given linkage. It

is shown in this paper that the existence of a regular kine-

matotropy can be established by means of a higher-order

analysis of the transition conﬁguration. This is based on

tangent cone to the c-space. The latter consists of the tan-

gents to feasible curves in V, and thus reveals the motion

characteristics. As an example for the existence of a regular

kinematotropy, the Wunderlich mechanism (a 12 bar multi-

loop linkage) is considered. It is shown explicitly that there

is a regular kinematotropy between a 1-dimensional and 2-

dimensional mode. Hence this linkage can transit between

different DOFs without exhibiting non-smooth motions.

In order to synthesize kinematotropic linkages with reg-

ular kinematotropy, further research is required aiming on

conditions on the linkage geometry.

Acknowledgement

The ﬁrst author acknowledges partial supported by the

Austrian COMET-K2 program of the Linz Center of

Mechatronics (LCM).

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