# Safe Three-Dimensional Assembly Line Design for Robots Based on Combined Multiobjective Approach
## 2 Modeling of Assembly Line Layouts Based on the Safety Indicator
### Modeling of Safe Layout
#### 2.2.1 Modeling Rules for Irregularly Shaped Equipment  
The irregularly shaped equipment types on an assembly line include industrial robots and specially shaped machines and tools. The reasonable modeling of irregular 3D target objects is the basis of the quantitative safety indicator. Irregular devices, such as industrial robots, are widely employed in assembly lines. In the initial layout of an assembly line, in actual engineering, the maximum bounding box of a robot is utilized for multiobjective optimization, and the layout plan is output. If the distance between robots is too large, then it will be impossible to coordinate assembly, and adjustments will undoubtedly increase the design cost and time. These issues can result in a waste of layout space and affect the progress of assembly tasks. Assume that a robot base is employed for multiobjective optimization and a layout plan is output. If the center point of the base is too close to another base, then the possibility of an unsafe collision will be high, and the small collaborative assembly area may be problematic. From the algorithm implementation perspective, the inconsistency among entity input shapes can be difficult to consider in encoding and decoding tasks. Therefore, all of the optimized objects in this paper are quadrilaterals. In order to illustrate the modeling rules of robots, we use a KUKA KR210 R2700 robot as an example in this case study. We build an industrial robot simulation model while using the Monte Carlo method to simulate the actual physical process in a workspace, as shown in Figure 2. Based on the Denavit-Hartenberg (DH) parameter table, as shown in Table 2, the cloud point model of the robot workspace is constructed. Figure 1: Schematic of the assembly line layout. Figure 3a shows the 3D point cloud diagram of the assembly range of two robots. The point cloud overlap area is the collaborative assembly area of two robots. The initial position of the robot layout affects the size of the collaborative assembly area. Figure 3b-d show the ground projection, which has an extensive range of motion. When two horizontally adjacent robots are cooperatively assembled, the maximum range of the first robot's assembly area passes through the center of that of the second robot, as shown in Figure 3b. At this time, the cooperative assembly ranges of the two robots reach maximums, and the skill of each robot is maximized. Figure 3c shows a case in which two robots are too close in the layout, and the active assembly area is small. Figure 3d shows a case in which the layout of two robots is too large, and the effective assembly area is small. The larger the overlap of the assembly areas, the higher the level of dexterity, and vice versa. Therefore, the length of the robot is \(\mathrm{L_{r}}=\mathrm{A}\mathrm{\SIUnitSymbolDegree}=\mathrm{C}\mathrm{ \SIUnitSymbolDegree}^{\prime}=\mathrm{E}\mathrm{\SIUnitSymbolDegree}^{\prime}= \mathrm{G}\mathrm{\SIUnitSymbolDegree}^{\prime}\mathrm{H}^{\prime}=\mathrm{R_{i}}\). In modeling, the vertical y-direction line spacing must meet the safe layout requirements, so the quadrilateral width of an optimized entity is \(\mathrm{W_{r}}=\mathrm{E}\mathrm{\SIUnitSymbolDegree}^{\prime}=\mathrm{F} \mathrm{\SIUnitSymbolDegree}^{\prime}\mathrm{G}^{\prime}=2\mathrm{R_{0}}\). \(\mathrm{R_{0}}\) is the base radius of a robot. \(\mathrm{R_{i}}\) is the maximum movement radius in the actual assembly of the robot. The optimized entity of the robot is the quadrilateral E'F'G'H', as shown in Figure 1.  
The irregular parts of the special-shaped machine tools \(\mathrm{M_{p}}\) and \(\mathrm{M_{q}}\) that are shown in Figure 1 represent the special-shaped contact areas between the irregularly shaped equipment and the ground. The irregular shadow areas are treated as the effective areas, and a circle is drawn with the device center at the circle center.