## 3 Formation Cooperative Search and Transformation Strategy
### Formation Comprehensive Cost (FCC) Model  
Various evaluation models have been proposed by researchers to address the energy consumption problem in dynamic formation transformation. The optimal formation convergence time (FCT) efficiency model [11] aims to find the robot that consumes the most time \(t_{i}\) in the formation transformation and minimizes its moving distance, and the shortest convergence time model that is obtained is:  
\[T=\text{minmax}\{t_{1},t_{2},t_{3}\dots t_{n}\} \tag{6}\]  
The objective of the optimal formation energy consumption (FEC) efficiency model [11] is to calculate and minimize the sum of all robots moving in the formation transformation, and the resulting minimum energy consumption model is (with \(s_{k}\) representing the sum of paths for each AUV in the \(k\)-th permutation):  
\[D=\text{min}\{s_{1},s_{2},s_{3}\dots s_{k}\} \tag{7}\]  
Most of the existing formation transformation schemes are based on the above two models; however, the minimum time and the minimum energy consumption obtain different optimal results in some environments, and the power consumption of the sensors inside the platform during the operation (e.g., forward-looking sonar sensing the surrounding environment) is ignored. Therefore, this paper further considers the power consumption brought by hardware systems such as sensor networks inside the robot, combines the energy consumption caused by motion transformation with the inherent hardware energy consumption of the transformation process, and proposes a formation comprehensive cost (FCC) model that integrates the convergence time, transformation distance and load energy consumption, as shown in Equation (8):  
\[\text{min}E=\text{min}\sum_{i=1}^{n}\left((\sum_{j=1}^{k_{i}-1}\lambda v_{i} ^{3}\frac{\left|\overrightarrow{d_{i,j}d_{i,j+1}}\right|}{v_{g}}\right)+\frac {D_{\text{max}}}{v_{g}}p_{0}\right) \tag{8}\]  
The first half of Equation (8) represents the energy consumption caused by the distance the AUV moves when changing formation, and the second half represents the energy consumption of the system load, which is determined by the formation convergence time. \(\lambda\) represents the water resistance coefficient, which is related to the AUV shape and underwater environment, \(v_{i}\) represents the propulsion speed of the AUV propeller, \(v_{g}\) represents the motion speed of the AUV relative to the geodesic coordinate system, \(p_{0}\) represents the inherent power consumption of the AUV, which is determined by the consumption of the internal components of the robot, \(k_{i}\) represents the number of path points passed by the \(i\)-th AUV, and \(d_{i}\) is the position of each path point during the transformation. \(D_{\text{max}}\) denotes the maximum distance the robot moves during the formation change and is given by Equation (9):  
\[D_{\text{max}}=\text{max}\{\sum_{j=1}^{k_{1}-1}\left|\overrightarrow{d_{1,j}d _{1,j+1}}\right|,\sum_{j=1}^{k_{2}-1}\left|\overrightarrow{d_{2,j}d_{2,j+1} }\right  
\[\text{min}E=\begin{cases}\min\sum\limits_{i=1}^{n}\left((\sum\limits_{t=1}^{t_{i}-1 }\lambda v_{1}^{3}\frac{\left\|BT_{l_{i}-x}(t+1)-BT_{l_{i}-x}(t)(1)\right\|}{v _{g}}\right)+\frac{D_{\text{max}}}{v_{g}}p_{0})\\ \min\sum\limits_{i=1}^{n}\left((\sum\limits_{t=1}^{t_{i}-1}\lambda v_{1}^{3} \frac{\left\|BT_{l_{i}-y}(t+1)(2)-BT_{l_{i}-y}(t)(2)\right\|}{v_{g}}\right)+ \frac{D_{\text{max}}}{v_{g}}p_{0})\\ \min\sum\limits_{i=1}^{n}\left((\sum\limits_{t=1}^{t_{i}-1}\lambda v_{1}^{3} \frac{\left\|BT_{l_{i}-x}(t+1)(3)-BT_{l_{i}-x}(t)(3)\right\|}{v_{g}})+\frac{D_ {\text{max}}}{v_{g}}p_{0})\\ \end{cases} \tag{10}\]  
For energy consumption analysis of the formation transformation, this paper uses the position of each robot in the formation within the grid network as the basis for solving the position of the robot at the minimum fitness value (the formation transformation with the lowest integrated cost) using the FCC model.