^{1}

^{2}

^{2}

^{3}

^{1}

^{2}

^{3}

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

Kinematic
singularities of parallel mechanisms have been studied extensively (cf.
[

By choosing
appropriate local coordinates for a given mechanism

These have been studied in detail mainly in the case
of a single closed chain (see [

Evidently, topological singularities of

we give a
necessary geometric condition for a topological singularity to arise in such a
mechanism (Theorem

we show that
topological singularities for these mechanisms always give rise to kinematic
singularities (Proposition

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

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.

In general, a
configuration

The structure of the mechanism imposes relations which
must hold among the coordinates

The kinematics of the mechanism are described by a
time-dependent path

If

On the other hand, if

This classification is widely used in the robotics
literature (see, e.g., [

The kinematic singularities which arise in this paper
are all of type I or III, corresponding to the

Other types of singularities for mechanisms have been
considered—for example,

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

The restriction to these specific types of joints is
intended to simplify the study of

For each component, we use parenthesized
superscripts to indicate the chain number, and subscripts to indicate the link
number. For example,

From now on we consider a fixed polygonal mechanism

A

A chain configuration

A
configuration for

The set

A singular configuration of type (a).

A
configuration

A singular configuration of type (b).

We can now formulate our main result.

A necessary condition for a configuration

For the proof, see [

The three types of singular configurations defined
above need not be topological singularities of

Consider the following two examples.

(1) If

On the other hand, if

(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 suitable parameters, the boundary

Note that near

Coupler curve tangent to workspace boundary.

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

Theorem

Recall from Section

The planar case (with more general joints) was
analyzed by Bonev et al. in a series of papers, summarized in
[

For the spatial case (i.e.,

Six chains
having

The first chain
having

Two chains
having

We use screw theory (see [

A

Equivalent kinematic structure of a chain.

Considering the

In order to eliminate the passive joints from (

If the chain
has

If the chain
has

For the

Combining the

The

The matrix

For a polygonal mechanism

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

Now by Theorem

two coaligned
chains, each with a pair of unactuated joints; they have a common reciprocal
(and

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

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

The sort of conditions in the Grassmann algebra used
here to identify singularities is of course well known in literature (see,
e.g., [

To round off the discussion we now present an example
of a topological singularity of type (b) for a specific real-life mechanism

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

A singular configuration for the mechanism

Since the topological singularity in question is
located in this region

The work space for each
vertex

Constrained work space

The configuration space

discrete data
on the elbow up/down position of each chain at

the orientation

the location of

The work spaces

For some values of

Configuration space for 3-RRR mechanism.

More care is required for the analysis of the
full configuration space

Of course, this is only true in the region where the
original mechanism

However, as the parameters for

Configuration space is locally

In this case,

The aligned poses can be calculated analytically using
the algorithm in Gosselin and Merlet (cf. [