# Topological and Kinematic Singularities for  a Class of Parallel Mechanisms

Authors: Nir Shvalb; Moshe Shoham; Hagay Bamberger; David Blanc
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We study singularities for a parallel mechanism with a planar moving
platform inℝd(d=2,3), with joints which are universal, spherical (spatial case),
or rotational (planar case). For such mechanisms, we give a necessary condition
for a topological singularity to occur, and describe the corresponding kinematic
singularity. An example is provided.


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## Body

## 1. Introduction

Kinematic
singularities of parallel mechanisms have been studied extensively (cf.
[1–4]), but less attention has
been paid to topological singularities in this context. The topological theory
of singularities is an extensive and rapidly growing field of mathematics, with
connections to complex and real analysis, algebraic geometry, and differential
topology (see [5–7]). In this note, we
investigate the connection between the two types of singularities (for a
certain class of mechanisms), thus illustrating their relevance to robotics.


### 1.1. Topological Singularities

By choosing
appropriate local coordinates for a given mechanismΓ, we can think of the set of all its configurations as
a topological space 𝒞=𝒞(Γ),  called its configuration space,
and try to endow it with the structure of a differentiable manifold. The points
of 𝒞 where this
cannot be done constitute the topological (or differentiable) singularities of Γ (see Section 2).These have been studied in detail mainly in the case
of a single closed chain (see [8–11] and compare [12–14]). Here we consider parallel
mechanisms with a planar moving platform, in any dimension. In the spatial
case, we require the joints to be universal or spherical; in the planar case,
all joints are rotational.Evidently, topological singularities of𝒞 are an inherent
feature of the mechanism, independent of the actuation scheme. However, at
first glance they appear to have no mechanical significance. The main results
of this paper areas follows:(a)
we give a
necessary geometric condition for a topological singularity to arise in such a
mechanism (Theorem2.4);(b)
we show that
topological singularities for these mechanisms always give rise to kinematic
singularities (Proposition3.1).The occurence of such singularities is illustrated in
a specific example in Section4.These results are intended to exemplify the power of
(higher-dimensional) topological methods for the study of singularities in
robotics. In the future we hope to show how they apply to more general types of
mechanisms.


### 1.2. Kinematic Singularities

In general, a
configuration𝒱 is naturally
described by the vector x=(x1,…,xN) whose
coordinates xi correspond to
the positions of the various joints and links of the mechanism. In practice, we
choose a subset xin of input coordinates (corresponding to the actuated joints), which can serve as local
coordinates for 𝒞 around 𝒱. In addition, we often focus on a subset xout of output coordinates of interest (which may describe the position of the end effector).
The remaining joints (if any) are passive.The structure of the mechanism imposes relations which
must hold among the coordinatesxi;  in
particular, we may assume that(1.1)F(xin,xout)=0,identically in a neighborhood of 𝒱 in 𝒞. The Jacobian J:=∂F/∂x consists of
two blocks ∂F/∂xin and Jout:=∂F/∂xout,  where Jout must be
nonsingular if the xin are to serve
as local coordinates near 𝒱 (cf. [15] or [16, Section 5]).The kinematics of the mechanism are described by a
time-dependent pathx(t) in the configuration space.
Differentiating (1.1)  with respect to t we
get(1.2)∂F∂xx˙=0,which can be written in the
form(1.3)Jinx˙in=Joutx˙out,where Jin:=−∂F/∂xin.IfJout is of maximal
rank and Jin is not, 𝒱 is called a (instantaneous)
kinematic singularity of type I; this means that not every infinitesimal
change in output can be obtained by changing the actuated joints.On the other hand, ifJin is of maximal
rank and Jout is not, 𝒱 is called
singular of of type II; in this case the actuated joints do not determine
uniquely the behavior of the outputs. Finally, if neither Jout nor Jin is of maximal
rank, 𝒱 is called
singular of type III (cf. [17]). Gosselin and Angeles give examples of all three
types of singularities for a parallel 3-RRR  planar mechanism.This classification is widely used in the robotics
literature (see, e.g., [18, Section 6.2]). Note
that in addition to the (somewhat arbitrary) choice of xin and xout,  the mechanism
may have additional “passive coordinates” of interest. Thus, Zlatanov et al. provide a more detailed classification, listing six types of
singular configurations (see [2]).The kinematic singularities which arise in this paper
are all of type I or III, corresponding to theimpossible output (IO) of
[2, Section 5, Definition 7],
where there exists an infinitesimal output vector for which (1.2)
cannot be satisfied with any combination of active and passive input vectors.Other types of singularities for mechanisms have been
considered—for example,control, constraint, and architectural (cf. [18, Section 6.2.1])—which we do not attempt to discuss here. Note, however, that there is no
analogous classification of singularities for topological spaces, which can be
extremely complicated in general (see [5]). See [19] for a general discussion of singularities in
robotics.


### 1.3. Parallel Mechanisms with a Planar Platform

In this note
we begin a study of the relationship between topological and kinematic
singularities for a common class of mechanisms (which occur in applications).These arepolygonal mechanism Γ in ℝd (d=2,3),  consisting
of a moving planar k-polygonal
platform 𝒫 with k chains attached
to its vertices. The ith chain is a
sequence of n(i) concatenated
links of lengths ℓj(i) (j=1,…,n(i)),  connected
by spherical or universal joints, in the spatial case, and rotational joints, in
the planar case. One end of the ith chain is
attached to the vertex p(i) of the moving
platform, and the other is fixed at x(i)∈ℝd.The restriction to these specific types of joints is
intended to simplify the study of𝒞(Γ),  which can
then be interpreted as a space of immersions of the corresponding metric graph.
In future work, we hope to extend these results to more general parallel
mechanisms.Convention
For each component, we use parenthesized
superscripts to indicate the chain number, and subscripts to indicate the link
number. For example,ℓj(i) denotes the
length of the jth link of the ith chain.


## 1.1. Topological Singularities

By choosing
appropriate local coordinates for a given mechanismΓ, we can think of the set of all its configurations as
a topological space 𝒞=𝒞(Γ),  called its configuration space,
and try to endow it with the structure of a differentiable manifold. The points
of 𝒞 where this
cannot be done constitute the topological (or differentiable) singularities of Γ (see Section 2).These have been studied in detail mainly in the case
of a single closed chain (see [8–11] and compare [12–14]). Here we consider parallel
mechanisms with a planar moving platform, in any dimension. In the spatial
case, we require the joints to be universal or spherical; in the planar case,
all joints are rotational.Evidently, topological singularities of𝒞 are an inherent
feature of the mechanism, independent of the actuation scheme. However, at
first glance they appear to have no mechanical significance. The main results
of this paper areas follows:(a)
we give a
necessary geometric condition for a topological singularity to arise in such a
mechanism (Theorem2.4);(b)
we show that
topological singularities for these mechanisms always give rise to kinematic
singularities (Proposition3.1).The occurence of such singularities is illustrated in
a specific example in Section4.These results are intended to exemplify the power of
(higher-dimensional) topological methods for the study of singularities in
robotics. In the future we hope to show how they apply to more general types of
mechanisms.


## 1.2. Kinematic Singularities

In general, a
configuration𝒱 is naturally
described by the vector x=(x1,…,xN) whose
coordinates xi correspond to
the positions of the various joints and links of the mechanism. In practice, we
choose a subset xin of input coordinates (corresponding to the actuated joints), which can serve as local
coordinates for 𝒞 around 𝒱. In addition, we often focus on a subset xout of output coordinates of interest (which may describe the position of the end effector).
The remaining joints (if any) are passive.The structure of the mechanism imposes relations which
must hold among the coordinatesxi;  in
particular, we may assume that(1.1)F(xin,xout)=0,identically in a neighborhood of 𝒱 in 𝒞. The Jacobian J:=∂F/∂x consists of
two blocks ∂F/∂xin and Jout:=∂F/∂xout,  where Jout must be
nonsingular if the xin are to serve
as local coordinates near 𝒱 (cf. [15] or [16, Section 5]).The kinematics of the mechanism are described by a
time-dependent pathx(t) in the configuration space.
Differentiating (1.1)  with respect to t we
get(1.2)∂F∂xx˙=0,which can be written in the
form(1.3)Jinx˙in=Joutx˙out,where Jin:=−∂F/∂xin.IfJout is of maximal
rank and Jin is not, 𝒱 is called a (instantaneous)
kinematic singularity of type I; this means that not every infinitesimal
change in output can be obtained by changing the actuated joints.On the other hand, ifJin is of maximal
rank and Jout is not, 𝒱 is called
singular of of type II; in this case the actuated joints do not determine
uniquely the behavior of the outputs. Finally, if neither Jout nor Jin is of maximal
rank, 𝒱 is called
singular of type III (cf. [17]). Gosselin and Angeles give examples of all three
types of singularities for a parallel 3-RRR  planar mechanism.This classification is widely used in the robotics
literature (see, e.g., [18, Section 6.2]). Note
that in addition to the (somewhat arbitrary) choice of xin and xout,  the mechanism
may have additional “passive coordinates” of interest. Thus, Zlatanov et al. provide a more detailed classification, listing six types of
singular configurations (see [2]).The kinematic singularities which arise in this paper
are all of type I or III, corresponding to theimpossible output (IO) of
[2, Section 5, Definition 7],
where there exists an infinitesimal output vector for which (1.2)
cannot be satisfied with any combination of active and passive input vectors.Other types of singularities for mechanisms have been
considered—for example,control, constraint, and architectural (cf. [18, Section 6.2.1])—which we do not attempt to discuss here. Note, however, that there is no
analogous classification of singularities for topological spaces, which can be
extremely complicated in general (see [5]). See [19] for a general discussion of singularities in
robotics.


## 1.3. Parallel Mechanisms with a Planar Platform

In this note
we begin a study of the relationship between topological and kinematic
singularities for a common class of mechanisms (which occur in applications).These arepolygonal mechanism Γ in ℝd (d=2,3),  consisting
of a moving planar k-polygonal
platform 𝒫 with k chains attached
to its vertices. The ith chain is a
sequence of n(i) concatenated
links of lengths ℓj(i) (j=1,…,n(i)),  connected
by spherical or universal joints, in the spatial case, and rotational joints, in
the planar case. One end of the ith chain is
attached to the vertex p(i) of the moving
platform, and the other is fixed at x(i)∈ℝd.The restriction to these specific types of joints is
intended to simplify the study of𝒞(Γ),  which can
then be interpreted as a space of immersions of the corresponding metric graph.
In future work, we hope to extend these results to more general parallel
mechanisms.Convention
For each component, we use parenthesized
superscripts to indicate the chain number, and subscripts to indicate the link
number. For example,ℓj(i) denotes the
length of the jth link of the ith chain.


## 2. Topological Singularities

From now on we consider a fixed polygonal mechanismΓ as in Section 1.3, and
construct its configuration space 𝒞 as
follows.Definition 2.1.
Achain configuration for a single n-link chain
consists of n vectors V=(v1,…,vn) in ℝd of specified
lengths: ∥vj∥=ℓj for j=1,…,n.
A chain configurationV is said to be aligned if all the vectors vj are scalar
multiples of v, which is called the direction vector of V. The direction line for V is Line:={x+τv∣τ∈ℝ}.Definition 2.2.
A
configuration forΓ consists of a
set 𝒱=(V(1),…,V(k)) of chain
configurations for each of the k chains, such
that the k endpoints p(i) of the
corresponding chain configurations form a polygon congruent to the given moving
platform 𝒫. Here,(2.1)p(i):=x(i)+∑j=1nvj(i),for i=1,…,k,  where the
points x(i) are the
vertices of the fixed platform (see Figure 1).
The set𝒞=𝒞(Γ) of all such
configurations, topologized in the obvious way, is the configuration space of Γ.Figure 1
A singular configuration of type (a).Definition 2.3.
A
configuration𝒱 for Γ is called singular
of type (a) if for two of its chains—say, numbers i1,i2∈{1,…,k}—the corresponding chain configurations V(i1) and V(i2) are aligned,
with coinciding direction lines: Line(i1)=Line(i2) (see Figure 1).
𝒱 is singular
of type (b) if three of its chain configurations are aligned, with
direction lines in the same plane meeting in a single point (this is referred
to in literature as a planar pencil) (see Figure 2).
𝒱 is singular
of type (c) if (at least) four of its chain configurations are aligned,
with direction lines in the same plane (compare [18, Section 6.4.1, condition 3d]).Figure 2
A singular configuration of type (b).We can now formulate our main result.Theorem 2.4.
A necessary condition for a configuration𝒱=(V(1),…,V(k)) of a polygonal
mechanism Γ to be a
topological singularity is that it be singular of type (a), (b), or (c).For the proof, see [20].Remark 2.5.
The three types of singular configurations defined
above need not be topological singularities of𝒞; in particular, we know of no topological
interpretation of the three different types (a)–(c). As noted above, in general
the classification of topological or differentiable singularities is very
difficult (see [6]).Examples 2.6.
Consider the following two examples.
(1) If𝒱 is singular of
type (a), as in Figure 1, consider the submechanism Γ′ consisting of
the two aligned chains of Γ, with the corresponding configuration 𝒱′.  If we assume
that these two chains have one and two links, respectively, then 𝒱′ has a
neighborhood U′ in the configuration space 𝒞(Γ′) which is a
one-point union of two 2-discs (see
[14, Proposition 4.1])—so 𝒱′ is singular.
On the other hand, ifΓ′′ is the
mechanism obtained from Γ by omitting the
two aligned chains, the configuration 𝒱′′ corresponding
to 𝒱 is
nonsingular and has a Euclidean neighborhood U′′ in 𝒞(Γ′′).  Since the 𝒱 itself has a
neighborhood in 𝒞(Γ) equivalent
to U′×U′′,  we see that 𝒱 is a
topological singularity.
(2)  Consider a mechanism with a triangular
platform, and two chains with one link each. In this case, the workspace for
the third vertex of the platform is the coupler curveγ for the
corresponding 4-chain
mechanism (see [21,
Chapter 4]), while the workspace for the third chain is an annulus A.
For suitable parameters, the boundary∂A (where the
third chain is aligned) will be tangent to γ. The configuration 𝒱 corresponding
to the point of tangency will be singular of type (b), as in Figure 3.
Note that near𝒱 each point in γ has two
corresponding configurations, associated to “elbow up/down” positions of the
third chain, which coalesce at 𝒱 itself; thus 𝒱 has a singular
neighborhood in 𝒞 consisting of
two transverse intervals.Figure 3
Coupler curve tangent to workspace boundary.Remark 2.7.
For simplicity, in the spatial case we restricted
attention to mechanisms where all joints are universal. Replacing any such
joint by a spherical one simply multiplies𝒞 by a circle, so
it does not affect the topological singularities.


## 3. Kinematic Singularities

Theorem2.4 gives a necessary condition for a
configuration to be singular (topologically), namely, that some subset {i1,…,im} of its chains
be aligned, with direction lines (Line(ij))j=1m,  so that the
Plücker vectors of these lines span certain types of varieties, of
positive codimension in ℝ6 (see [18, Section 5]). We now
show how topological singularities give rise to kinematic singularities, for
our class of polygonal mechanisms.


### 3.1. Architectures

Recall from Section1.2 that
kinematic singularities require an actuation architecture, that is, a choice of input coordinates xin (corresponding
to the actuated joints) and output coordinates xout (corresponding
to the end effector of the mechanism: in our case, the position and orientation
of the moving platform).The planar case (with more general joints) was
analyzed by Bonev et al. in a series of papers, summarized in
[22]. For a
classification of the kinematic singularities of such mechanisms, using
instantaneous centers of rotation, see [23].For the spatial case (i.e.,Γ embedded
in ℝ3),  we assume for simplicity that all actuators
are universal or spherical, while the passive joints (located at the end of the
chain, say) are spherical. There are three architectures to consider.(a)
Six chains
havingn(i)−1 actuators for
the ith chain  (i=1,…,6).(b)
The first chain
havingn(1) actuators; three more chains with n(i)−1 actuators for
the ith chain  (i=2,3,4).(c)
Two chains
havingn(i) actuators  (i=1,2);  the third
chain with n(3)−1 actuators.


### 3.2. Screw Theory

We use screw theory (see [24]) to describe the forces
operating at each joint of our mechanism.Ascrew$ is a Plücker vector in ℝP5,  describing a
line—or equivalently, the position and direction of a vector—in ℝ3 (cf. [18, Section 5]). Thus
Figure 4 depicts the equivalent kinematic chain of an arbitrary chain i of a mechanism,
where the movement of each spherical joint j is described by
three unit screws $j,1(i), $j,2(i),  and $j,3(i) attached to
its center. For a universal joint (or just before a passive joint), we can make
do with two screws.Figure 4
Equivalent kinematic structure of a chain.Considering theith chain as an
open chain, we can express the instantaneous twist of the end-effector
as(3.1)$p=∑j=0n(i)∑k=13$j,k(i)⋅θ˙j,k(i),where θj,k(i) is the input
coordinate for the $j,k(i) screw (cf.
[16, Section 5.6]).In order to eliminate the passive joints from  (3.1),
for the ith chain, we
must multiply both sides by appropriate reciprocal screws. More precisely,
choose a basis {$t(i)⊥}t=1λ(i) for the
space V(i) of common
reciprocals of the screws of all passive joints for this chain.(a)
If the chain
hasn(i)−1 actuators, at
all joints but the last two, the reciprocal screw for these two joints
corresponds to the line passing through them (i.e., V(i) is
one-dimensional).(b)
If the chain
hasn(i) actuators, at
all but the last joint, V(i) is 3-dimensional,
with the corresponding lines all passing through the last joint (see [16, Chapter 5]).For theith chain we
obtain a system of λ(i) linear
equations for $p:(3.2)$t(i)⊥⋅$p=∑j=0n(i)∑k=13$t(i)⊥⋅$j,k(i)⋅θ˙j,k(i),(t=1,…,ℓ(i)),  in which of
course the unactuated inputs θ˙j,k(i) have zero
coefficient.Combining theλ=∑i=1kλ(i) equations (3.2)
for all k chains, we
obtain (1.3)  for the chosen architecture  (λ=6 for the first
two and λ=7 for the
third).Theλ×6 matrix Jout will take the
form(3.3)Jout=($1(1)⊥⋮$λ(1)(1)⊥⋮$λ(k)(k)⊥),whose rows are the reciprocal
screws of all actuated chains.The matrixJin is
block-diagonal:(3.4)Jin=(A(i1)⋯0⋮⋱⋮0⋯A(ik)),with the ith bloc (for
the ith actuated
chain, with d passive joints) and
a λ(i)×(n(i)−d) matrix of the
form:(3.5)A(i):=($1(i)⊥⋅$0,1(i)⋯$1(i)⊥⋅$n(i)−d,3(i)⋮⋮$λ(i)(i)⊥⋅$0,1(i)⋯$λ(i)(i)⊥⋅$n(i)−d,3(i)).Proposition 3.1.
For a polygonal mechanismΓ (with no
unactuated chains), there is an instantaneous kinematic singularity of type I
or III at any topological singularity.Proof.
At a topological singularity at
least two chains are aligned, so the reciprocals to the passive joint(s) are
reciprocal to all screws of these chains, and thusA(i)=0 for these
chains. Since λ≤7, Jin is singular.
Now by Theorem2.4, a topological singularity can
have the following:
(1)
two coaligned
chains, each with a pair of unactuated joints; they have a common reciprocal
(andλ=6),  so Jout has rank ≤5.  The same
holds in the second architecture whenever two chains are coaligned;(2)
three aligned
chains whose lines lie in a planar pencil, each with a pair of unactuated
joints; in this case the corresponding screws are linearly dependent, so
againJout has rank ≤5;(3)
four aligned
chains whose lines are in one plane; the lines of those with a pair of
unactuated joints; each lying in a planar pencil (rank 3). In the first
architecture, each of the last two lines adds at most 1 to the rank; in
the second, adding the last line forms a degenerate congruence (total rank 4). Thus in any
caseJout has rank ≤5;Remark 3.2.
The sort of conditions in the Grassmann algebra used
here to identify singularities is of course well known in literature (see,
e.g., [18, Section 6.4]). Our
point is that these are necessary conditions for topological
singularities, and sufficient for kinematic singularities, providing an
implication between two concepts of independent interest.


## 3.1. Architectures

Recall from Section1.2 that
kinematic singularities require an actuation architecture, that is, a choice of input coordinates xin (corresponding
to the actuated joints) and output coordinates xout (corresponding
to the end effector of the mechanism: in our case, the position and orientation
of the moving platform).The planar case (with more general joints) was
analyzed by Bonev et al. in a series of papers, summarized in
[22]. For a
classification of the kinematic singularities of such mechanisms, using
instantaneous centers of rotation, see [23].For the spatial case (i.e.,Γ embedded
in ℝ3),  we assume for simplicity that all actuators
are universal or spherical, while the passive joints (located at the end of the
chain, say) are spherical. There are three architectures to consider.(a)
Six chains
havingn(i)−1 actuators for
the ith chain  (i=1,…,6).(b)
The first chain
havingn(1) actuators; three more chains with n(i)−1 actuators for
the ith chain  (i=2,3,4).(c)
Two chains
havingn(i) actuators  (i=1,2);  the third
chain with n(3)−1 actuators.


## 3.2. Screw Theory

We use screw theory (see [24]) to describe the forces
operating at each joint of our mechanism.Ascrew$ is a Plücker vector in ℝP5,  describing a
line—or equivalently, the position and direction of a vector—in ℝ3 (cf. [18, Section 5]). Thus
Figure 4 depicts the equivalent kinematic chain of an arbitrary chain i of a mechanism,
where the movement of each spherical joint j is described by
three unit screws $j,1(i), $j,2(i),  and $j,3(i) attached to
its center. For a universal joint (or just before a passive joint), we can make
do with two screws.Figure 4
Equivalent kinematic structure of a chain.Considering theith chain as an
open chain, we can express the instantaneous twist of the end-effector
as(3.1)$p=∑j=0n(i)∑k=13$j,k(i)⋅θ˙j,k(i),where θj,k(i) is the input
coordinate for the $j,k(i) screw (cf.
[16, Section 5.6]).In order to eliminate the passive joints from  (3.1),
for the ith chain, we
must multiply both sides by appropriate reciprocal screws. More precisely,
choose a basis {$t(i)⊥}t=1λ(i) for the
space V(i) of common
reciprocals of the screws of all passive joints for this chain.(a)
If the chain
hasn(i)−1 actuators, at
all joints but the last two, the reciprocal screw for these two joints
corresponds to the line passing through them (i.e., V(i) is
one-dimensional).(b)
If the chain
hasn(i) actuators, at
all but the last joint, V(i) is 3-dimensional,
with the corresponding lines all passing through the last joint (see [16, Chapter 5]).For theith chain we
obtain a system of λ(i) linear
equations for $p:(3.2)$t(i)⊥⋅$p=∑j=0n(i)∑k=13$t(i)⊥⋅$j,k(i)⋅θ˙j,k(i),(t=1,…,ℓ(i)),  in which of
course the unactuated inputs θ˙j,k(i) have zero
coefficient.Combining theλ=∑i=1kλ(i) equations (3.2)
for all k chains, we
obtain (1.3)  for the chosen architecture  (λ=6 for the first
two and λ=7 for the
third).Theλ×6 matrix Jout will take the
form(3.3)Jout=($1(1)⊥⋮$λ(1)(1)⊥⋮$λ(k)(k)⊥),whose rows are the reciprocal
screws of all actuated chains.The matrixJin is
block-diagonal:(3.4)Jin=(A(i1)⋯0⋮⋱⋮0⋯A(ik)),with the ith bloc (for
the ith actuated
chain, with d passive joints) and
a λ(i)×(n(i)−d) matrix of the
form:(3.5)A(i):=($1(i)⊥⋅$0,1(i)⋯$1(i)⊥⋅$n(i)−d,3(i)⋮⋮$λ(i)(i)⊥⋅$0,1(i)⋯$λ(i)(i)⊥⋅$n(i)−d,3(i)).Proposition 3.1.
For a polygonal mechanismΓ (with no
unactuated chains), there is an instantaneous kinematic singularity of type I
or III at any topological singularity.Proof.
At a topological singularity at
least two chains are aligned, so the reciprocals to the passive joint(s) are
reciprocal to all screws of these chains, and thusA(i)=0 for these
chains. Since λ≤7, Jin is singular.
Now by Theorem2.4, a topological singularity can
have the following:
(1)
two coaligned
chains, each with a pair of unactuated joints; they have a common reciprocal
(andλ=6),  so Jout has rank ≤5.  The same
holds in the second architecture whenever two chains are coaligned;(2)
three aligned
chains whose lines lie in a planar pencil, each with a pair of unactuated
joints; in this case the corresponding screws are linearly dependent, so
againJout has rank ≤5;(3)
four aligned
chains whose lines are in one plane; the lines of those with a pair of
unactuated joints; each lying in a planar pencil (rank 3). In the first
architecture, each of the last two lines adds at most 1 to the rank; in
the second, adding the last line forms a degenerate congruence (total rank 4). Thus in any
caseJout has rank ≤5;Remark 3.2.
The sort of conditions in the Grassmann algebra used
here to identify singularities is of course well known in literature (see,
e.g., [18, Section 6.4]). Our
point is that these are necessary conditions for topological
singularities, and sufficient for kinematic singularities, providing an
implication between two concepts of independent interest.


## 4. An Example

To round off the discussion we now present an example
of a topological singularity of type (b) for a specific real-life mechanismΓ, namely, the 3-URU 3-DOF mechanism
(in ℝ3),  introduced in [25]. Zlatanov et al. studied the
constraint singularities of Γ extensively.
Here we do not attempt a comprehensive study, since our goal is merely to
illustrate topological methods.The mechanism consists of three two-link chains, and
both the base and moving platforms are equilateral triangles. It turns out that
in a certain region𝒰 of the configuration space 𝒞, Γ acts as a
planar mechanism (see [25]). In particular, the three R-joint axes of the base
platform meet in the base triangle but not in the center (otherwise
mobility would be increased by the additional spin dexterity of the extended
chain). Furthermore, in  this region the three intermediate R-joints in each chain
are parallel (see Figure 5).Figure 5
A singular configuration for the mechanismΓ.Since the topological singularity in question is
located in this region𝒰, for simplicity we may regard Γ as if it were a 3-RRR planar
mechanism Γ^. The pose of the equilateral moving platform 𝒫 in 𝒰 is determined
by the two coordinates (x,y) of its
barycenter p, and the rotation θ of 𝒫 in ℝ2.  Denote the
common distance  from the vertices to the
barycenter by r=d(p,p(i)) (i=1,2,3).The work space for each
vertexp(i) of 𝒫 is an annulus 𝒜(i) centered at
the fixed endpoint x(i) of the ith chain, with
boundary radii ℓ1(i)±ℓ2(i).  Thus, if we
fix the orientation θ of 𝒫, the resulting
constrained work space 𝒲θ for p (the shaded
area in Figure 6) is the intersection of three annuli (with centers at t(i)), namely, the displacements 𝒜^(i) (i=1,2,3)  of 𝒜(i) by a
vector p(i)p=x(i)t(i) of length r.Figure 6
Constrained work space𝒲θ for p with fixed θ.The configuration space𝒞 is described in
a neighborhood 𝒰 of any
configuration 𝒱 by(a)
discrete data
on the elbow up/down position of each chain at𝒱;(b)
the orientationθ (which takes
value in an open interval I⊆ℝ);(c)
the location ofp in 𝒲θ.The work spaces𝒲θ will
generically all be homeomorphic to a fixed curvilinear polygonal region 𝒲′=𝒲θ0 (for θ near θ0).  Thus topologically 𝒰 will be a
cube, that is, a product I×𝒲′.  The discrete
data yield eight identical copies of 𝒰 identified
along their boundaries (which represent chain alignments).For some values ofθ, 𝒲θ may be empty,
so in fact 𝒰 may split up
into two disjoint cubes as above (see Figure 7), where the vertical gap between
them represents values of θ for which 𝒲θ=∅.  This
corresponds to a situation where the platform 𝒫 cannot rotate continuously
between two orientations, each of which is feasible in itself. No singularities
arise in this case.Figure 7
Configuration space for 3-RRR
mechanism.More care is required for the analysis of the
full configuration space𝒞(Γ^) near 𝒱0,  because there
are actually eight regions 𝒰1,…,𝒰8,  corresponding
to the eight possible choices of “elbow up/down” for the three chains. The
boundaries of 𝒲θ are arcs of
the boundary circles of the annuli 𝒜^(i),  where the
links of the ith chain are
aligned. Therefore (as noted above), the boundaries of the “cubes” 𝒰j (j=1,…,8)  are glued
together in 𝒞(Γ^) according to a
combinatorial pattern represented by the colors in the two figures. For
example, gluing faces for the situation depicted in Figure 7 yields two
disjoint 3-dimensional
tori.Of course, this is only true in the region where the
original mechanismΓ is planar (and
thus equivalent to Γ^); all we can
conclude about 𝒞(Γ) in the region
corresponding to Figure 7 is that it has two connected components, locally
isomorphic to ℝ3.However, as the parameters forΓ (and thus Γ^) vary, we find
that in certain cases the two connected components of 𝒰 approach each
other, and, for an appropriate Γ, they actually touch at one point 𝒱0∈𝒞 (see Figure 8).Figure 8
Configuration space is locallyℝ3⋁cℝ3.In this case,𝒰 is homeomorphic
to ℝ3⋁𝒱0ℝ3,  so that 𝒱0 is
topologically singular in 𝒰. In 𝒞(Γ^)—and therefore, in 𝒞(Γ) too, at least
locally—we obtain a one-point union of two 3-tori.The aligned poses can be calculated analytically using
the algorithm in Gosselin and Merlet (cf. [26]), since each of the extreme situations can be treated
as an equivalent 3-RPR robot,
whose link lengths are fixed and known.



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*Source: 249349-2009-04-23.xml*