On Regular Kinematotropies

Abstract

inematotropic linkage are fascinating as
they can switch between motion mode with different DOF.
Beside the mere curiosity, this phenomenon has also application 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 between motion modes with different DOF. Given a particular 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 brackets 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.

Full text

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=Vp1(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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