# Robotic Coverage Path Planning for Ultrasonic Inspection
## 6 Coverage Path Planning
```
1:procedureCCPP(mesh)
2:\(voxelgrid\leftarrow\) voxelize mesh\(\triangleright\) mark all voxels as not inspected
3:\(surface\leftarrow\) discretize outer surface
4:seeds\(\leftarrow\) calculate curvature vector field for surface
5:sort seeds using the curvature value in descending order
6:for\(s\) in seeds do
7:if\(s\) is too close to an existing streamline then
8: skip \(s\)
9:if curvature of \(s>\)thresholdthen
10:\(streamline\leftarrow\) expand streamline from s
11:else\(\triangleright\) Note: specimen is now subdivided into flat areas
12:\(A\leftarrow\) connected flat area containing \(s\)
13:\(c\leftarrow\) center of \(A\), re-projected onto the mesh
14:\(v\leftarrow\) vector pointing along the principal component of \(A\)
15:\(streamline\leftarrow\) expand streamline from \(c\) with initial direction \(v\)
16:for\(p\) in streamline do
17:\(inspected\) voxels \(\leftarrow\) run ray-tracing algorithm for pose \(p\)
18: update voxelgrid using inspected voxel
19:for\(s\) in seeds do\(\triangleright\) Only the seed points underneath the sensor are considered
20:\(n\leftarrow\) #voxels along the surface-normal marked as not inspected
21:if\(n=0\)then
22: remove \(s\)
```  
**Algorithm 1** Cartesian coverage path planning algorithm  
The complete algorithm proceeds as follows. Initially, the input mesh is voxelized, and the outer surface is discretized. For the discretized surface, the vector field based on curvature is calculated, and all surface points are initially marked as seed points. Afterwards, the algorithm iterates over all seed points in descending order of their curvature value. For each seed point, a streamline that follows the second principal curvature is expanded. Once the streamline expansion is terminated, the volumetric inspection is updated by running the ray tracing algorithm for every point on the streamline. Seed points are skipped if all voxels along the surface-normal (inward) are already inspected or if another streamline is in close proximity. If a seed point's curvature value is below a given threshold, the corresponding area is considered flat. Note that once a seed point meets this criterion, the not inspected area of the mesh is composed of unconnected flat areas. On flat areas, streamlines are expanded in a straight line from the areas center along its principal component. Once the Cartesian paths are created, the second phase of the algorithm is executed. We deploy a greedy search algorithm that generates the corresponding robot path. In a first step, the algorithm calculates multiple valid robot paths per streamline. In a second step, the separate streamline-paths are connected, yielding the final robot path that covers all points on the Cartesian paths exactly once. For a 6-axis robot, each Cartesian pose may have multiple inverse kinematics (IK) solutions, i.e., multiple joint configurations yield the same sensor pose. To generate valid robot paths for a given streamline, first, all possible joint configurations are calculated for each point on the streamline. Since the sensor is invariant under a 180\({}^{\circ}\) rotation around its z-axis, also the IK solutions for the rotated pose are calculated. In a next step, _connectable_ IK solutions for adjacent points on the streamline are connected, forming robot path segments that the robot can traverse without considerable re-configuration. More precisely, two points \(a\) and \(b\) (in joint space) are _connectable_ if the Cartesian midpoint \(m=(FK(b)-FK(a))/2\) and the point obtained from the forward kinematics (FK) of the midpoint in \(m^{\prime}=FK((b-a)/2)\) are sufficiently close in joint space. The path segments are constructed in such a way that multiple _streamline paths_ are generated. A streamline path consists of multiple path segments, which together, cover all points of the streamline exactly once. Note that a path segment may be part of multiple streamline paths. In the final stage of the algorithm, the generated path segments are connected.