IEEE Robotics & Automation Magazine - September 2014 - 110

(a)

(a)

(b)
Figure 8. (a) A 3-RRR manipulator from the University of
Hannover (photo courtesy of Jens Kotlarski http://www.imes.unihannover.de). (b) Typical singularity loci computed for such kind
of manipulators [6].
(b)

((c))
Figure 7. (a) A box approximation of the 2-D C-space of the
welding task when the tool can contact the lower part of the
beam with any orientation. The approximation contains ~15,000
boxes, which are here rendered with translucent material.
(b) The cyclooctane molecule, a ring of eight carbon atoms
pairwise connected through rotatable covalent bonds, and a box
approximation of its 2-D C-space with ~300,000 boxes projected
on three of the problem variables. (c) Different shapes are obtained
by projecting the boxes on alternative triplets of variables.

conjunction with the linear equation to derive a tiny box
around the solution in Figure 5(b) and a larger dark gray box
bounding the solution in Figure 5(c). This pruning operation
is implemented using linear programming, and it is repeated
for each box until no further significant reduction is possible.
In the end, if the largest side of the box is below a given
threshold, it is considered a solution box. Otherwise, it is
bisected and the process is recursively applied to the newly
created subboxes. The following command:
> cuik welding0D.cuik

automates this process. It reads the cuik file, simplifies the
equations when possible, and applies the pruning and
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september 2014

bisection operations until all solution points are isolated at
the desired accuracy. The problem in Figure 3 is simple, but
the method is also successful on complex problems involving
general 6R robots and Stewart platforms [21] or in mechanisms with more than 12 kinematic loops [20]. To have an
idea, elimination or resultant methods are finding their limit
in mechanisms of substantially less complexity [23].
The cuik command can also be used to isolate C-spaces
of positive dimension. For instance, if the robot in Figure 3
has to weld along a line in the lower part of the beam while
keeping the tool orientation fixed, then the problem exhibits
a one-dimensional manifold of solutions. Figure 6 shows several stages of the branch-and-prune method in this case
[Figure  6(a)] and the final box approximation returned as
output [Figure 6(b)]. The plots are obtained and visualized
by executing
> cuik welding1D.cuik
> cuikplot3d welding1D.sol 1 3 4
welding1D.gcl
> geomview welding1D.gcl

which projects the solution boxes to the subspace of the
u1_x, u2_x, and u2_y coordinates. These are the variables
appearing in the first, third, and fourth positions of the
weldinglD.cuik file describing the problem. The insets in
Figure 6(b) show the arm configurations corresponding to
some of the boxes.
If the welding task is further relaxed and the tool can contact the beam with any orientation, the problem exhibits a
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