# A Data-Driven Intermittent Online Coverage Path Planning Method for AUV-Based Bathymetric Mapping
## 3 Proposed Method
### Problem Statement  
In this research, AUVs are assumed to navigate at a constant depth, in a rectangular workspace. We define three frames in our work, namely the earth frame, vehicle frame, and sensor frame, as illustrated in Figure 1. As the path is planned in the horizontal plane, only x-axis and y-axis are shown in the figure. The origin of the earth frame is set at a known latitude and longitude on the sea surface. The x-axis is pointed north, y-axis is pointed east and z-axis is pointed downward. The origin of the vehicle frame is set at the center of gravity, x-axis pointing forward and y-axis pointing at starboard side. As the sensor is fixed to the vehicle, we assume the sensor frame has the same orientation as the vehicle frame. We could further assume the origin of the sensor frame is coincident with the vehicle frame for small AUVs with the sonar mounted near (<1 m) the origin of the vehicle frame. The workspace is set in the earth frame and decomposed into square cells. The cell size determines the resolution of the resultant bathymetric map, which is 1 m in this paper. The bathymetry data obtained by the multibeam sonar is a set of points in the sensor (vehicle) frame. For a point \(\mathbf{p}_{k}^{\mathrm{p}}\) in the vehicle frame, we can transfer it into the earth frame by:  
\[\mathbf{p}_{k}^{e}=\mathbf{T}_{v}^{e}\mathbf{p}_{k}^{v} \tag{1}\]  
where the subscript \(k\) denotes the data index, superscripts, \(v\) and \(e\), denote the affiliated frames, and \(\mathbf{T}\), the transformation matrix, is obtained based on the location of the vehicle frame and its orientation in the earth frame, which are normally estimated by global navigation satellite system (GNSS) and attitude and heading reference system (AHRS).  
The sonar data point in the earth frame, \(\mathbf{p}_{k}^{e}\), can be projected into a cell in the workspace \(<x,y>\). We define the observation condition of a cell as a binary variable, \(O(M_{x,y})\). If there is an observation made in the cell, \(O(M_{x,y})\) become 1 marked as observed, otherwise it remains 0 noted as unexploited. The mission goal is to explore the workspace until the overall observation condition have exceeded the user-defined goal, i.e., the overall percentage of the number of observed cells.  
In our method, the successive transect is calculated during the mission instead of preprogrammed. The conception is illustrated in Figure 2. The vehicle will plan the successive pair of waypoints after one transect is finished. In this way, our path planning algorithm runs intermittently at the end of each transect based on the collected data. Initially, user could program the starting point of the mission. When the algorithm is analyzing the candidate transects, it will consider both zonal and meridional transects. The flexibility allows the user to further include other factors into the objective function,  
Figure 1: The workspace and the configuration of the frames.  
e.g., the adversarial effects posed by the ocean current. The distance between the candidate transects is also adjustable. A large distance could be selected for a large workspace for computation efficiency. Herein, we select 1 m distance that matches the gridded map resolution.