### State of the Art in Path Generation with Skeletonization
For the theoretical background and alternative concepts in the topic of skeletonization, we refer the reader to the book [33], while in [34] the reader will find a more detailed practical explanation of skeletonization. The review articles [28; 35; 36] include explanations of more recent skeletonization methods, their main advantages and disadvantages, and the most important criteria for success. Particular for path environment reduction, there are several concepts based on skeletonization such as Voronoi diagrams [25; 26], grass-fire also referred to as wave propagation [37; 38], potential fields [39; 40], or thinning [12; 27]. The authors of [25; 26] use Voronoi diagrams in which both the environment and the obstacles are constructed as polygons. However, the use of Voronoi diagrams has some disadvantages. For example, if the polygon is noisy, there are many possible branches [28], resulting in a nonrobust path algorithm highly influenced by such noise. An alternative skeletonization method is called thinning, which can be used in pathfinding [27]. On thinning the boundary of the 2D or 3D image is eroded until only 1D lines for 2D images and 1D lines and 2D surfaces for 3D images results. Thinning is slightly more robust compared to a noisy boundary, but often leads not exactly to centered skeletons [28]. Compared to the cost function-based path algorithms, the skeleton-based algorithms offer the advantage of reducing the possible paths to only centered paths that are maximally far from the boundary. In contrast to the use of A* directly without simplified paths [10], these paths are suitable for robust fault management in the digital twin, since there is a quasi-maximal avoidance range to the boundary. Similarly, topologically alternative routes are chosen, e.g., to avoid an obstacle from either side. In addition to generating paths via thinning, the authors of [11] used thinning for preprocessing, which was then proceeded with A* for the final path selection [11]. While in [10] individual obstacles are touched, in [11] central paths in the traversable region result from the combination of A* with path simplification using skeletonization. Therefore, in this work, a thinning-based path-finding method analogous to [27] is used, whose shortestcentral path can subsequently be searched using A* or Dijkstra analogous to [27]. However, since 3D maps also play a role particularly for the flight zones of drones or uneven grounds and bridges for mobile robots, a 3D curve skeleton with multiple targets and starting positions based on a modification of the thinning method of [13] is chosen in contrast to [11; 27]. This modification is based on so-called anchor points. Anchor points are points that are accepted as skeletal points and thus not eroded during thinning [41; 42]. In our work, these anchor points are placed on task and robot positions to ensure connectivity to the path estimation. Likewise, alternative properties for controlling the robots are integrated into the path estimation, to separate recalibration and global path estimation. Another alternative for the calculation of these skeletons is the use of the extreme values of the so-called distance transformation [28]. In this case, the individual skeleton branches are recorded on the basis of extreme values. Since this information is used directly in the skeletonization, a maximum passage width along the path can be integrated automatically. Therefore, individual robots can be instructed with different-sized packages to only passable paths. In this work, in addition to path generation, the maximum passage width is also extracted based on skeletonization. Since both passage width and skeletonization are based on a similar theoretical model (Blum's maximally inscribed balls), this leads to a robust algorithm without the necessity to cover special cases. In addition to the actual pathfinding, it may be necessary to place certain safety zones around the robot to allow sufficient time for correction in case of intrusion, e.g., a person [43].