IEEE Robotics & Automation Magazine - September 2014 - 107

method able to compute the whole solution
set. The method departs from an initial box
problem.param
problem.world
bounding this set and effectively prunes
cuikexplore
unfeasible portions of the box until all solucuikequations
tion points are isolated at the desired accuproblem.cuik
problem.joints
racy. Software toolboxes, such as ALIAS [16],
Bertini [1], and PHC [28], have been succescuikatlas
rmpicuik
cuik
cuikatlasrrt
fully applied to kinematics, but they implement general methods for solving systems of
problem.sol problem.atlas problem_path.sol problem.rrt
algebraic equations. In contrast, the CUIK
suite sacrifices generality to gain simplicity
cuikplot3d cuikplotatlas cuikplayer
cuikplotrrt
and efficiency in the implementation.
Opposite to [1] and [28], moreover, it
problem.gcl problem_atlas.gcl
problem_rrt.gcl
directly isolates the real roots instead of the
complex ones, which is beneficial in practice.
While many approaches can only identify
geomview
the solutions when they are isolated points,
the branch-and-prune methods in the CUIK
Figure 2. A representative set of the commands implemented in the CUIK
suite can also approximate positive-dimensuite (ellipses) and their input/output files (rectangles). The parameters file
sional solutions. This can be used to compute
problem.param is used by all the commands. The suite implements many
other tools not shown here for simplicity. Note: geomview is an external
the whole C-space or some of its subsets, provisualization program.
viding global information on the motion
capabilities of a manipulator. In particular, by
adequately defining the equations passed to the solver, the branch-and-prune methods proceed by discarding nonvalid
CUIK suite can isolate the workspace boundaries and any of configurations, continuation techniques march from a given
the singularity sets typically defined [7], becoming the first point in all directions identifying new feasible configurations.
general tool able to do so, to the best of our knowledge.
Local charts of the C-space are constructed and coordinated
Branch-and-prune methods are complete in the sense that along the way, defining a global atlas that is suitable to deterthey obtain all configurations, irrespective of whether they mine feasible motion ranges, optimal configurations, or paths
form one or several connected components. In many prob- between configurations. Packages to compute the latter exist,
lems, however, it may be sufficient to explore only those con- but they are oriented to open-chain robots [9], [15], [26], [27]
figurations that are path connected to a given point. To this or to specific classes of closed-chain devices [14]. The CUIK
end, the CUIK suite implements higher dimensional continu- suite complements these packages by providing the new
ation tools to trace arbitrary manifolds [10]. While methods to deal with the general closed-chain case. For

P = (x, y)

i
l3

u3

u2
Y
l2

u1
X
O
l1

Figure 3. A planar arm welding on a beam. The goal is to weld
on a particular point P with orientation i.

[constants]
11:= 1
l2:= 1
l3:= 1.3
x:= 0.25
y:= 2.05
theta:= pi/2
[system vars]
u1_x: [-1,1]
u1_y: [-1,1]
u2_x: [-1,1]
u2_y: [-1,1]
u3_x: [-1,1]
u3_y: [-1,1]
[system eqs]
u1_xˆ2+u1_yˆ2=1;
u2_xˆ2+u2_yˆ2=1;
u3_x=cos(theta);
u3_y=sin(theta);
l1*u1_x+l2*u2_x+l3*u3_x=x;
l1*u1_y+l2*u2_y+l3*u3_y=y;
Example 1. The welding.cuik file encoding the problem
in Figure 3.

september 2014

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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107
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