#### Abstract

We study singularities for a parallel mechanism with a planar moving platform in , 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.

#### 1. Introduction

Kinematic singularities of parallel mechanisms have been studied extensively (cf. [1β4]), 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 [5β7]). 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 mechanism , we can think of the set of all its configurations as
a topological space , called its *configuration space*,
and try to endow it with the structure of a differentiable manifold. The points
of where this
cannot be done constitute the *topological* (or differentiable) *singularities* of (see Section 2).

These have been studied in detail mainly in the case of a single closed chain (see [8β11] and compare [12β14]). 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 of 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 (Theorem 2.4);(b)we show that topological singularities for these mechanisms always give rise to kinematic singularities (Proposition 3.1).

The occurence of such singularities is illustrated in a specific example in Section 4.

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
configuration is naturally
described by the vector whose
coordinates correspond to
the positions of the various joints and links of the mechanism. In practice, we
choose a subset of *input* coordinates (corresponding to the actuated joints), which can serve as local
coordinates for around . In addition, we often focus on a subset 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 coordinates ; in particular, we may assume thatidentically in a neighborhood of in . The Jacobian consists of two blocks and , where must be nonsingular if the are to serve as local coordinates near (cf. [15] or [16, Section 5]).

The kinematics of the mechanism are described by a time-dependent path in the configuration space. Differentiating (1.1) with respect to we getwhich can be written in the formwhere .

If is of maximal
rank and is not, 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, if is of maximal
rank and is not, 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 nor is of maximal
rank, 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 and , the mechanism may have additional βpassive coordinatesβ 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 the *impossible 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
consideredβfor example, *control*, *constraint*, and *architectural* (cf. [18, Section 6.2.1])βwhich 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 are *polygonal* mechanism in (), consisting
of a moving planar -polygonal
platform with chains attached
to its vertices. The th chain is a
sequence of concatenated
links of lengths (), connected
by spherical or universal joints, in the spatial case, and rotational joints, in
the planar case. One end of the th chain is
attached to the vertex of the moving
platform, and the other is fixed at .

The restriction to these specific types of joints is intended to simplify the study of , 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, denotes the
length of the th link of the th chain.

#### 2. Topological Singularities

From now on we consider a fixed polygonal mechanism as in Section 1.3, and construct its configuration space as follows.

*Definition 2.1. *A *chain configuration* for a single -link chain
consists of vectors in of specified
lengths: for .

A chain configuration is said to be *aligned* if all the vectors are scalar
multiples of , which is called the *direction vector* of . The *direction line* for is .

*Definition 2.2. *A
configuration for consists of a
set of chain
configurations for each of the chains, such
that the endpoints of the
corresponding chain configurations form a polygon congruent to the given moving
platform . Here,for , where the
points are the
vertices of the fixed platform (see Figure 1).

The set of all such
configurations, topologized in the obvious way, is the *configuration space* of .

*Definition 2.3. *A
configuration for is called *singular
of type (a)* if for two of its chainsβsay, numbers βthe corresponding chain configurations and are aligned,
with coinciding direction lines: (see Figure 1).

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).

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]).

We can now formulate our main result.

Theorem 2.4. *A necessary condition for a configuration of a polygonal
mechanism 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 of ; in particular, we know of no topological
interpretation of the three different types (a)β(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) If is singular of
type (a), as in Figure 1, consider the submechanism consisting of
the two aligned chains of , with the corresponding configuration . If we assume
that these two chains have one and two links, respectively, then has a
neighborhood in the configuration space which is a
one-point union of two 2-discs (see
[14, Proposition 4.1])βso is singular.

On the other hand, if is the
mechanism obtained from by omitting the
two aligned chains, the configuration corresponding
to is
nonsingular and has a Euclidean neighborhood in . Since the itself has a
neighborhood in equivalent
to , we see that 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 curve for the
corresponding 4-chain
mechanism (see [21,
Chapter 4]), while the workspace for the third chain is an annulus .

For suitable parameters, the boundary (where the
third chain is aligned) will be tangent to . The configuration corresponding
to the point of tangency will be singular of type (b), as in Figure 3.

Note that near each point in has two
corresponding configurations, associated to βelbow up/downβ positions of the
third chain, which coalesce at itself; thus has a singular
neighborhood in consisting of
two transverse intervals.

*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 multiplies by a circle, so
it does not affect the topological singularities.

#### 3. Kinematic Singularities

Theorem 2.4 gives a *necessary* condition for a
configuration to be singular (topologically), namely, that some subset of its chains
be aligned, with direction lines , so that the
PlΓΌcker vectors of these lines span certain types of *varieties*, of
positive codimension in (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 Section 1.2 that
kinematic singularities require an *actuation architecture*, that is, a choice of input coordinates (corresponding
to the actuated joints) and output coordinates (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., embedded in ), 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 having actuators for the th chain ().(b)The first chain having actuators; three more chains with actuators for the th chain ().(c)Two chains having actuators (); the third chain with actuators.

##### 3.2. Screw Theory

We use screw theory (see [24]) to describe the forces operating at each joint of our mechanism.

A *screw* is a PlΓΌcker vector in , describing a
lineβor equivalently, the position and direction of a vectorβin (cf. [18, Section 5]). Thus
Figure 4 depicts the equivalent kinematic chain of an arbitrary chain of a mechanism,
where the movement of each spherical joint is described by
three unit screws , , and attached to
its center. For a universal joint (or just before a passive joint), we can make
do with two screws.

Considering the th chain as an open chain, we can express the instantaneous twist of the end-effector aswhere is the input coordinate for the screw (cf. [16, Section 5.6]).

In order to eliminate the passive joints from (3.1), for the th chain, we must multiply both sides by appropriate reciprocal screws. More precisely, choose a basis for the space of common reciprocals of the screws of all passive joints for this chain.

(a)If the chain has actuators, at all joints but the last two, the reciprocal screw for these two joints corresponds to the line passing through them (i.e., is one-dimensional).(b)If the chain has actuators, at all but the last joint, is 3-dimensional, with the corresponding lines all passing through the last joint (see [16, Chapter 5]).

For the th chain we obtain a system of linear equations for :(), in which of course the unactuated inputs have zero coefficient.

Combining the equations (3.2) for all chains, we obtain (1.3) for the chosen architecture ( for the first two and for the third).

The matrix will take the formwhose rows are the reciprocal screws of all actuated chains.

The matrix is block-diagonal:with the th bloc (for the th actuated chain, with passive joints) and a matrix of the form:

Proposition 3.1. *For a polygonal mechanism (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 thus for these
chains. Since , is singular.

Now by Theorem 2.4, a topological singularity can
have the following:

(1)two coaligned
chains, each with a pair of unactuated joints; they have a common reciprocal
(and ), so 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
again 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
case 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 mechanism , namely, the 3-URU 3-DOF mechanism (in ), introduced in [25]. Zlatanov et al. studied the constraint singularities of 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 region of the configuration space , 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).

Since the topological singularity in question is located in this region , for simplicity we may regard as if it were a 3-RRR planar mechanism . The pose of the equilateral moving platform in is determined by the two coordinates of its barycenter , and the rotation of in . Denote the common distance from the vertices to the barycenter by ().

The work space for each vertex of is an annulus centered at the fixed endpoint of the th chain, with boundary radii . Thus, if we fix the orientation of , the resulting constrained work space for (the shaded area in Figure 6) is the intersection of three annuli (with centers at ), namely, the displacements () of by a vector of length .

The configuration space is described in a neighborhood of any configuration by(a)discrete data on the elbow up/down position of each chain at ;(b)the orientation (which takes value in an open interval );(c)the location of in .

The work spaces will generically all be homeomorphic to a fixed curvilinear polygonal region (for near ). Thus topologically will be a cube, that is, a product . The discrete data yield eight identical copies of identified along their boundaries (which represent chain alignments).

For some values of , may be empty, so in fact may split up into two disjoint cubes as above (see Figure 7), where the vertical gap between them represents values of for which . This corresponds to a situation where the platform cannot rotate continuously between two orientations, each of which is feasible in itself. No singularities arise in this case.

More care is required for the analysis of the full configuration space near , because there are actually eight regions , corresponding to the eight possible choices of βelbow up/downβ for the three chains. The boundaries of are arcs of the boundary circles of the annuli , where the links of the th chain are aligned. Therefore (as noted above), the boundaries of the βcubesβ () are glued together in 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 mechanism is planar (and thus equivalent to ); all we can conclude about in the region corresponding to Figure 7 is that it has two connected components, locally isomorphic to .

However, as the parameters for (and thus ) vary, we find that in certain cases the two connected components of approach each other, and, for an appropriate , they actually touch at one point (see Figure 8).

In this case, is homeomorphic to , so that is topologically singular in . In βand therefore, in too, at least locallyβwe 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.