## 2 Engineering Problem for Laser Radar Station Planning of BIW Inspection  
The dimensional accuracy elements of the BIW primarily consist of three key features: points, holes, and slots. These measurement features are predefined in body dimension engineering design, with their fundamental attributes including positional coordinates and directional vector information (Figure 1). Due to the complex surface geometry of the measured BIW, the numerous measurement features, and the varying distances and irregular distribution between these features, a single measurement station cannot achieve full-coverage measurement. Multiple measurement stations are therefore required to comprehensively measure the BIW. Since measurement field errors may arise during the station transfer process of the Laser Radar, the station layout should be optimized to minimize the number of stations within a fixed measurement space. Furthermore, under constrained measurement resources, the measurement efficiency could be improved with fewer stations through the induction of the instrument repositioning time. Therefore, the goal of the Laser Radar station planning is to determine the minimum measurement stations required to achieve the full-coverage dimensional deviation measurement of the BIW.  
The ranging measurement between the Laser Radar and target features is accomplished through time-of-flight detection, where the receiver module precisely captures the phase-shifted optical signals retroreflected from measured surfaces. By combining the calibrated beam emission vector (azimuth (\(\theta\)) and elevation (\(\varphi\))) with the radar's geodetic coordinates [X, Y, Z] in the global frame, the spatial position of each feature is analytically derived through trigonometric transformation, achieving micron-level coordinate determination accuracy. Prior to the station planning of Laser Radar inspection, key definitions are established, as illustrated in Figure 2.  
Figure 1: Measurement elements in BIW inspection.  
The inspection coordinate system refers to the global coordinate system in the workspace. The features to be measured and Laser Radar stations are defined in this coordinate system. It can be denoted as (O\({}_{M}\), X\({}_{M}\),Y\({}_{M}\), Z\({}_{M}\)).  
A feature refers to a specific point or area to be inspected on the BIW. The center point of the i-th feature is denoted as F\({}_{i}\), and its normal vector is denoted as N\({}_{i}\).  
The inspection distance is the distance from the radar to the feature \(F_{i}\). It is denoted as \(D_{i}\) for \(F_{i}\).  
The inspection angle is the angle between the laser beam and normal vector of the feature. It is denoted as \(\theta_{i}\) for F\({}_{i}\).  
The inspection area defines the reachable region of the radar's laser beam. The pitch angle of the laser beam is \(\pm\)45\({}^{\circ}\), and the rotation angle is 0\({}^{\circ}\)\(\sim\)360\({}^{\circ}\).  
A station is the position and posture where the Laser Radar is placed in the workspace. It is denoted as \(S_{k}=(p_{k},v_{k})\). P\({}_{k}\) is the position of the radar station, and v\({}_{k}\) is the vector of the station posture.  
To guarantee the inspection accuracy, the inspection distance must be within a specific range. Similarly, the inspection angle should be maintained within a defined range. The optimal placement of the Laser Radar is within a region that meets the accuracy requirements for the inspection of multiple features.  
The constraints in this study are divided into three categories: position constraints, interference constraints, and spatial constraints. The position constraints include the inspection angle and distance. The purpose of the interference constraints checking is to prevent laser beam interference. The spatial constraints encompass the space interference and robot reachability requirements.