SAMPE Journal - November/December 2020 - 46

FEATURE / COMPLEX CFRP PARTS
tance to the surface. In addition, to obtain an ideal
sound coupling, the wave vector needs to be perpendicular to the specimen's surface. The rotation
about the surface normal has no impact on the
inspection process and is used as a degree of freedom for path planning. The following section addresses the problem of finding a path that meets
these requirements. We assume that each inspection position needs to be passed exactly once.
Hereinafter, path planning consists of three
steps: (i) the surface is segmented into connected
regions, which - without regarding collision avoidance - can be captured with meander-shaped
movements, (ii) valid paths in the robot's joint
space are calculated and (iii) the order of individual segments is optimized by finding a solution to
the associated traveling salesman problem.
Segmentation
As a starting point, we use a cellular decomposition2, which allows the decomposition of the specimen's surface into cells, which can be covered
individually by meander-shaped Cartesian paths.
These Cartesian paths are generated by sweeping
a slicing plane over the specimen. At points where
the connectivity changes (e.g. forking), the surface
is split into new cells (see Figure 3(a) and (b)).
Ideally, inspected points should be spatially
close to each other, whereas the sensor orientations should be similar in order to keep joint
movements small. We construct an undirected
graph consisting of vertices (p, n) with the position p and the sensor orientation n. An edge eij
connects two vertices Ki and Kj, if the distance
and the inner product of the sensor orientations
are below certain -to be defined -parameters
dmax and cmax (similar to7, see Figure 3(c)):
[1]
For the resulting segments formed by connected vertices, we plan paths by utilizing the method
described in the following subchapter.
Path Planning
Path planning is considered for non-redundant
six-axis industrial robots with the most common
architecture consisting of a trunk, shoulder, upper arm, forearm, and wrist. Regular end-effector
poses of these robots have eight unique inverse
kinematics (IK) solutions due to three joint singularities. The IK solutions can be grouped by
their appearance in the joint space (e.g. shoulder
right, wrist down, elbow up) referred to as manipulator poses (MP). Moving the robot from one MP

SAMPE JOURNAL

to another generally requires large joint motions
and therefore should be avoided. The presented
method generates inspection paths, which allow
for inspection of segments with minimal re-orientations.
Problem Definition
The vertex inspection order is predefined by the
meander-shaped Cartesian path. Moving from
one k-th inspection position to the next, ideally
causes small robot motions in the configuration
space. We denote the rotation about the sensor
axis at individual positions as jk.
The rotation about the sensor axis between
two consecutive vertices needs to be limited.
In order to find a solution, the following control
problem is defined:

[2]
with the control input uk and its box constraints
specifying the maximal accepted rotation angle
about the sensor axis between consecutive vertices, the path constraint c g {0,1} mapping to zero
if the configuration does not cause any collisions,
and the number of vertices N.
Path Construction
To find a solution, firstly a map m : k × j g {0,1}
with the discrete angles j E {0, umax, 2umax, ..., 2p}
is generated, which subsequently allows an easy
construction of a valid path backwards from k =
N - 1 to k = 0:
* For k = 0, the value is set equal to the value
of the collision detection m(0, j) = c(0, j).
* For k ≠ 0, the value is set to 0 if c(k, j) = 0 and
an angle jk-1 E {j - umax, 0, j + umax} exists such
that m(k - 1, jk-1) = 0. That means that pairs
(k,j) only map to zero if they represent collision free configurations and allow to reach
a valid point at time k - 1 in compliance with
the box constraints of the input variable.
For every valid terminal point (N - 1, jN-1), i.e.
m(N - 1, jN-1) = 0, finally a unique bang-bang solution can be found by constructing it backwards in
time:

[3]

N OV E M B E R / D E C E M B E R 2 0 2 0

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SAMPE Journal - November/December 2020

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