Fig. 1 Overall description of C3DP systems considering the key challenges of division of labor and collision avoidance. SafeZone* co-considers layer geometry and topology and co-solves division of labor and collision avoidance. More about this image found in Overall description of C3DP systems considering the key challenges of divis...

Fig. 2 Four-partitions of a circular layer geometry with different topologies. Each partition has progressively fewer shared boundaries. A boundary may consist of a single point. More about this image found in Four-partitions of a circular layer geometry with different topologies. Eac...

Fig. 3 Partition graphs and corresponding adjacency matrices of circular layer topologies. Nodes represent sub-regions and edges represent boundaries between sub-regions. More about this image found in Partition graphs and corresponding adjacency matrices of circular layer top...

Fig. 4 Optimal printing schedule for different partitioning topologies using graph coloring on their respective partition graphs More about this image found in Optimal printing schedule for different partitioning topologies using graph...

Fig. 5 General reachability issue in C3DP applications with robotic arms for degree-2 topologies. In this case, the partition graph is a path graph P 4 . More about this image found in General reachability issue in C3DP applications with robotic arms for degre...

Fig. 6 Degree-3 partitions produce triangular grid partition graphs, which are guaranteed to be three-colorable More about this image found in Degree-3 partitions produce triangular grid partition graphs, which are gua...

Fig. 7 Voronoi tessellation to partition the layer geometry into sub-regions More about this image found in Voronoi tessellation to partition the layer geometry into sub-regions

Fig. 8 Buffer region creation process via intersection of A i sub-regions with Minkowski sum of boundary line and printer nozzle planar projection More about this image found in Buffer region creation process via intersection of A i sub-regions...

Fig. 9 The topology of the sub-graph representing the buffer region is the same as the original topology of the partition. This allows us to optimally print the buffer regions since it is guaranteed to be three-colorable. More about this image found in The topology of the sub-graph representing the buffer region is the same as...

Fig. 10 Flowchart of the Safezone* methodology framework. Given the layer geometry and a set of constraints, the GA optimizer looks for Voronoi sites that minimize the makespan function calculated using the partition graph. The final schedule is sent to the control hub for printing. More about this image found in Flowchart of the Safezone* methodology framework. Given the layer geometry ...

Fig. 11 Final printer positions using the SZ* framework. Starting from top left and going clockwise: z-shape with N = 4, square with N = 5, triangle with N = 7, and Stanford bunny with N = 6 More about this image found in Final printer positions using the SZ* framework. Starting from top left and...

Fig. 12 Variation of the normalized A i sub-regions as a function of the generations. The symmetric clustering of the areas is highlighted, showing different ratios for different geometries. Plots are a one-to-one match with the ordering in Fig. 12 . More about this image found in Variation of the normalized A i sub-regions as a function of the g...

Fig. 13 Experimental setup for physical validation of SZ*. The printers can only be symmetrically arranged around the printer area as shown. More about this image found in Experimental setup for physical validation of SZ*. The printers can only be...

Fig. 14 Leveraging the idea of virtual robot positions into the SZ* framework. An optimal and feasible partition is found although the printer locations were originally fixed by the manufacturing system. More about this image found in Leveraging the idea of virtual robot positions into the SZ* framework. An o...

Fig. 15 Top: degree-4 symmetric partition and bottom: degree-3 partition with VRPs: ( a ) and ( c ) buffer regions; ( b ) and ( d ) completed print More about this image found in Top: degree-4 symmetric partition and bottom: degree-3 partition with VRPs:...

Fig. 16 Two partitions that yield the best possible printing time for three printers. Partition ( a ) may be created if we relax the configuration space constraint and partition ( b ) may be created if disjoint buffer regions are treated as multiple regions. More about this image found in Two partitions that yield the best possible printing time for three printer...

Fig. 17 Using virtual robot positions to find an optimal degree-3 partition while satisfying the reachability constraints of a grid-like manufacturing system More about this image found in Using virtual robot positions to find an optimal degree-3 partition while s...

Fig. 18 Sample convergence graph for GA optimization, showing the fitness (makespan) of the z-beam geometry with four printers. A similar pattern is observed in the other numerical studies. More about this image found in Sample convergence graph for GA optimization, showing the fitness (makespan...