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A Graphical
Method for Singularity Analysis of Planar
Parallel Manipulators
Introduction:
This project
provides a new method to investigate the singularities of planar
parallel mechanisms (PPMs).
Comprehensive
work was done in this area using differentiation of the inverse kinematic
equations [Tsai,
Merlet,
Gosselin et al].
Geometric approach,
i.e. screw theory, has recently been used as well to investigate the
singularities of parallel mechanisms [Wolf, Simaan,
Joshi],
and in
particular of PPMs [Bonev].
We will present a new graphical approach that combines this geometric approach
with
a theory that goes back
as far as the
nineteenth century - Maxwell’s theorem. Maxwell’s
theorem, that was later also introduced by Cremona [Maxwell,Cremona]
shows the duality between self stress frameworks and reciprocal
figures.
We will show the duality between these reciprocal figures
and type II
singularities of planar parallel manipulators.
Content:
1. What are planar
parallel manipulators
2.
Traditional method for finding singularities of parallel
manipulators
3. What are the
types of singularities in parallel manipulators
4. What is Maxwell's reciprocal graphs, and Maxwell's
reciprocal theorem.
5. How to build MLGs in order to apply
Maxwell's reciprocal theorem.
6. How to build the reciprocal graphs.
7. Examples of planar parallel
manipulators and their reciprocal
graphs and singularity loci.
1. What are Planar Parallel Manipulators
(PPMs):
Parallel mechanism: Closed-loop mechanism in which the end-effector (mobile
platform) is connected to the base by at least two independent kinematic
chains. (Merlet, J.-P )
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| 6-DOF Parallel Robot (CMU) |
3-DOF - 3-RRR Robot (Planar) (Nanyang technological
university) |
More information about parallel robots
can be found at: http://www.parallemic.org/
2. Traditional method for finding the singularities of parallel
manipulators - Analytic method:
Here is a quick overview of the analytic method for
finding the singularity configurations of
parallel manipulators.
In order to find the types of singularities we
first need to define the Jocobian of a parallel manipulator:
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I |
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II |
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III |
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IV |
Where q and x are the set of input and output
coordinates of the mechanism (respectively)
I - The relationship between the input and
output coordinates.
II - Differentiating I with respect to time.
III - The two Jacobians of parallel manipulator
IV - When the mechanism is fully parallel Jq can be
inverted. Fully-parallel mechanism:
Parallel mechanism with an n-DOF
end-effector connected to the
base by n independent kinematic
chains, each having a single
actuated joint.
Singular Configuration:
For simplicity we'll refer to singular
configuration as a point where the robot changes
the number of DOF.
A more accurate definition is:
A
Singular configuration
is a configuration in which the relation between
the input variables’ velocities (active joints’
speeds)
and the output variables’
velocities (linear/angular velocities of the end effector) is
not fully defined in a point
where the robot changes the number
of DOF instantaneously. [Simaan]
BACK TO TOP
3.
Types of singularities in parallel robots:
In contrast to serial robots, parallel robots
have more than one kind of singularity.
Gosselin
et al
defined three types of singularities:
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Singularity type I:
when Jq
is singular
(IKS)
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Singularity type
II: when Jx
is singular (FKS)
-
Singularity type
III: when Jq
and Jx
are singular
A simple 5 bar
mechanism will be used to show the
types of singularities.
Here is an applet showing this 5-R
mechanism,
the left applet is one branch of the inverse
kinematics and the right is the other branch.
Try moving around the
red joints. Part of the
mechanism will disappear
when it exits its
workspace.
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NOTE
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