# Safe Three-Dimensional Assembly Line Design for Robots Based on Combined Multiobjective Approach
## 3 A Combined Multiobjective Approach for Safety Layout: SE-NSGA2
### Principle of SE-NSGA2
#### 3.3.3 Verification  
Verification is a critical step in our approach. Unsafe layout plans are discarded in this step. The \(\gamma\) safety indicator reflects the assembly statuses of the robots. Accurate collision detection is needed when two robots are involved in cooperative assembly. Collision detection is based on a database that stores factory configuration data and historical data for assembly processes. The effectiveness of the initial layout plan can be verified for actual production. Unsafe extreme solutions are discarded in order to ensure that the new best Pareto solution set is the solution that meets the constraints of the safety index. The best layout design method is determined according to the target weights or user preferences. The safety indicators that are to be considered in the design of human-machine collaborative work scenes are visual monitoring, robot force, impedance control, safety zone setting, and collision detection [35]. We focus on collision detection by the threshold condition.  
In safety layout modeling, the layout of equipment is based on a global coordinate system; additionally, rough collision detection uses the AABB method, coordinated assembly occurs for irregular equipment, and other intricate tasks are performed. AABB collision detection cannot meet the accuracy requirements of many layouts that require precise detection. When robots are involved in cooperative assembly or grasp complex curved objects, a grid collision detection method can be employed. When a robotic arm is grasping concave objects, an implicit function collision detection method for complex curved surfaces is used. A safety indicator can be used in order to effectively ensure the safety of the layout plan for the Pareto solution set.  
Rough collision detection: axis-aligned bounding box collision detection [36] is a method that has the advantages of applying a constant direction and providing a fast detection speed. The construction of bounding boxes for nonrotating objects parallel to the coordinate axis is fast, and the detection results are quickly updated. This approach is ideal for collision detection for regular devices on assembly lines.  
Detailed collision detection: mesh collision detection methods and implicit function collision detection methods are available for complex surfaces [37]. For complex concave surfaces, the voxel feature points of an object to be detected are optimized, which can prevent the voxel point leakage phenomenon. The implicit function surface at time t is given in (9). T\({}_{t}\)(t) represents the transformation matrix, for which the voxels that are obtained by surface detection are translated from the initial time to time t. T\({}_{x}\)(t) represents the transformation matrix, for which the voxels that are obtained by surface detection are rotated from the initial time to time t. M\({}_{s}\)(S) represents the collection of voxels for the tool surface. According to the Gaussian theorem, the polynomial equation of an order complex coefficient only has n roots. Therefore, the number of roots can reflect the number of collisions. If no root is obtained, then no collision occurs.  
\[\text{F}(\text{x},\text{y},\text{z},\text{t})=\text{U}_{t=0}^{t}T_{t}(\text{t })T_{x}(\text{t})M_{s}(\text{S})t^{2} \tag{9}\]