# A Framework for Coverage Path Planning Optimization Based on Point Cloud for Structural Inspection
## 2 Proposed Framework
### Optimization Process
The problem has multiple objectives, being described through the sum of two factors: (i) Time, and (ii) Photometric fitness, as shown in Equation (6). Time fitness, as shown in Equation (8), is a function that tends to decrease mission time, with a maximum value of fitness equal to 10. The parameter \(vel_{UAV}\) is the average speed of the UAV during the mission and \(N_{Waypoints}\) is the total number of waypoints in the mission. \(D_{T}\) is the distance traveled in each horizontal transfer, calculated through the distances between the points:  
\[tan(\frac{\theta_{hor}}{2})=\frac{Dist_{hor}}{2D_{S}} \tag{1}\]0  
Many optimization problems involve several objectives that require simultaneous optimization. The difficulty in solving multi-objective problems (MOPs) is that these objectives are often contradictory to each other, which means that an improvement in one of the objectives implies the degradation of one or more of the remaining objectives. In time fitness and photometric fitness, the image objectives tend to create missions with closer waypoints and a greater number of flight layers, increasing the total mission time. There is no single ideal solution for such a situation instead of a set of optimal compensation solutions known as Pareto optimal solutions, called Pareto-optimal Front [40]. Several methods are present in the literature to explore the solutions present in the Pareto set. A prominent example is the use of scalarization functions, a way of combining multiple objectives into a scalar function, optimizing which will produce one solution to the original MOP [41], this being the method used in Equation (6) to highlight the importance of each of the objectives. Gain \(G_{Time}\) and \(G_{Photometric}\) control the importance between the time objective and the photometric objective. In the case studied, both values are unitary so that both objectives are explored. Other areas of interest that can be highlighted are Evolutionary Multi-objective Optimization (EMO) [42; 43] and Multi-Criteria Decision Making (MCDM) [44; 45]:  
\[tan(\frac{\theta_{hor}}{2})=\frac{Dist_{hor}}{2D_{S}} \tag{1}\]1  
\[tan(\frac{\theta_{hor}}{2})=\frac{Dist_{hor}}{2D_{S}} \tag{1}\]2  
The photometric fitness goals are to increase coverage (i.e., horizontal and vertical) and improve the characteristics of the intersection and incidence angles. This objective is represented by a sum of the coverages' fitness multiplied by the intersection angle gain,  
\[tan(\frac{\theta_{hor}}{2})=\frac{Dist_{hor}}{2D_{S}} \tag{1}\]3 Intersection and incidence angles. as shown in Equation (9). Coverage fitness, as shown in Equation (10), tends to increase coverage, with a maximum value of 5 used for horizontal and vertical coverage. The choice for double maximum value in time fitness highlights this objective concerning photometric, being closely linked to mission security. The values 10 and 5 were chosen empirically:  
\[tan(\frac{\theta_{hor}}{2})=\frac{Dist_{hor}}{2D_{S}} \tag{1}\]4  
\[tan(\frac{\theta_{hor}}{2})=\frac{Dist_{hor}}{2D_{S}} \tag{1}\]5  
The incidence angle can be calculated from three ranges and points from the point cloud. Figure 7 shows the three tracks with different heights. The yellow region is the surface with the points highlighted in green, blue, and purple as the captured points. The incidence angle must be the same as the red angle. In this way, the incidence angle will always be close to 0 degrees, added to the camera's gimbal angle. The intersection angle is the difference, in angles, of two previous and two posterior positions of the UAV (\(Dif_{Ang}\)). A gain is added to the cover fitness to identify how far it is from the 90-degree angle. Equation (11) shows the Gaussian function of the intersection gain.