## 3 Genetic Algorithm-Based Optimization of the Station Placement Scheme  
The objective of the station placement scheme optimization is to optimize the deployment of stations according to the target trajectory to be measured so that the total positioning error is minimized [28], and the CRLB is used as the objective function of the optimized placement due to the consistency of the CRLB and LS positioning errors. The positioning error function can be expressed as  
\[\text{positioningerror}=\frac{1}{V}\int_{x_{a}}^{x_{b}}\int_{y_{b}(x)}^{y_{b}( x)}\int_{z_{a}(x,y)}^{z_{b}(x,y)}\sqrt{\mathit{Tr}(CRLB)}dxdydz \tag{30}\]  
where \(x_{a}\), \(x_{b}\), \(y_{a}(x)\), \(y_{b}(x)\), \(z_{a}(x,y)\), \(z_{b}(x,y)\) are corresponding to the upper and lower integral limits of variable \(x\), \(y\), \(z\). \(V\) is the volume of the entire integral space.  
Since the function in Equation (30) is nonlinear and nonconvex, we use a Genetic Algorithm (GA) to solve for the optimal station deployment scheme. A GA algorithm is a stochastic search algorithm that draws on natural selection and natural genetic mechanisms in biology and is very suitable for dealing with complex, nonconvex, and nonlinear optimization problems that are difficult to solve by traditional search algorithms. Unlike traditional search algorithms, GA algorithms start from a randomly generated initial solution and iterate step by step through a series of selection, crossover, and mutation operations to generate a new solution. A certain number of sound individuals are selected from the previous generation according to their fitness, and the next generation population is formed by crossover and mutation. After several generations of evolution, the algorithm converges on the best chromosome, which is the optimal or suboptimal solution to the problem. Genetic algorithm has strong global search ability, but weak local search ability, and usually can only get the suboptimal solution of the problem, but not the optimal solution. Classical nonlinear programming algorithms mostly use the gradient descent method to solve the problem, which has a stronger local search ability but a weaker global search ability. Therefore, this paper combines the advantages of both algorithms, using a Genetic Algorithm for global search on the one hand and a nonlinear programming algorithm for local search on the other hand, so that the global optimal solution of the station placement scheme can be obtained.  
In practical applications, TOA and TDOA measurement stations are generally deployed on the ground, and the range of station placement is limited to a certain range due to the influence of radio link receiving distance. Therefore, it is necessary to put additional constraints on the station range in the GA algorithm, i.e., the station location is constrained to be within the square formed by four points in the XOY plane (\(-8\times 10^{3}\), \(-5\times 10^{3}\)), (\(-5\times 10^{3}\), \(5\times 10^{3}\)), (\(5\times 10^{3}\), \(-5\times 10^{3}\)), and (\(5\times 10^{3}\), \(5\times 10^{3}\)). The target trajectory is to be located in a line from point (\(5\times 10^{2}\), \(5\times 10^{2}\)) to point (\(5\times 10^{3}\), \(5\times 10^{3}\)) in the YOZ plane and the positioning error is calculated according to Equation (30) to find the optimal station location estimation vector \(\hat{\theta}\) to minimize the positioning error.  
\[\hat{\theta}=\underset{\theta}{\text{argmin}}\bigg{(}\frac{1}{V}\int_{x_{a}}^ {x_{b}}\int_{y_{a}(x)}^{y_{b}(x)}\int_{z_{a}(x,y)}^{z_{b}(x,y)}\sqrt{Tr(CRLB)} dxdydz\bigg{)} \tag{31}\]  
where \(\theta=[x_{1}\ y_{1}\ x_{2}\ y_{2}\cdot\ \cdot\ \cdot x_{N}\ y_{N}]\), \(N\) is the number of ground TOA and TDOA stations.