## 2 Basic Theory and Principle
### Feedforward Neural Networks  
In satellites, the azimuth and elevation of the laser terminal are influenced by mechanical vibrations, thermal deformation, coaxial errors, installation errors, servo errors, and other factors. The azimuth and elevation must be adjusted to maintain laser link stability. In this study, the adjustment was realized using an FNN Model, the architecture of which is shown in Figure 5. The green, blue and yellow represents input layer, hidden layer and output layer, separately. The adjustment of the azimuth and elevation was defined as the pointing error in the azimuth and elevation. The inputs of the FNN model include the local and target J2000 coordinate and velocities in the x, y, z direction, as well as the azimuth,  
Figure 3: Spiral scanning function (**a**) before the CFCC system and (**b**) after the CFCC system.  
Figure 2: Coarse–fine composite control system.  
pitch, yaw angles of the satellite platform, and CPA theoretical angle. The outputs are the pointing errors in azimuth and elevation. There isn activation function and a standard neuron structure between the input and output.  
In the FNN modeling process, more than \(1\times 10^{5}\) sample sets were used. Of these, 70% of the data were used as the training set, 15% as the test set, and the remaining 15% as the validation set. The choice of optimization method significantly affects the speed of the FNN modeling process. Several optimization methods are compared in Table 1. During modeling, the learning rate was set to 0.01, and performance evaluation was based on the mean square error (MSE). The minibatch size is settled as 100. Among the methods tested, the Adaptive Moment Estimation (AdamW) optimizer demonstrated the best performance, achieving the lowest MSE.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Optimization Method** & **Iterative Steps** & **MSE** \\ \hline Adam & 200 & 4551.5825 \\ Adamax & 200 & 7025.6172 \\ AdamW & 200 & 1374.5110 \\ Adagrad & 200 & 20,831.5346 \\ RMSprop & 200 & 9259.3991 \\ Adadelta & 200 & 17,830.1423 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of MSE between multiple optimization methods.  
Figure 4: Flowchart of the coordinate transformation step.  
To improve the performance and avoid overfitting, the MSEs of multiple FNN Models were evaluated at different numbers of hidden layer nodes and layers. The results of this analysis are presented in Figure 6. The FNN Model showed higher performance when the hidden layer was set to 70 nodes and 20 layers.  
Activation functions can introduce nonlinear transformations to enhance the expression of the FNN and improve its performance of the FNN Model. To obtain the appropriate activation functions, activation functions including PReLU, sigmoid, and tanh were compared, as shown in Table 2. We can observe that the PReLU function exhibits higher performance with a smaller MSE.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Activation Function** & **Iterative Steps** & **MSE (\(\boldsymbol{\mathrm{\mu rad^{2}}}\))** \\ \hline ReLu & 200 & 1602.0095 \\ Leaky ReLU & 200 & 1905.9974 \\ PReLU & 200 & 1374.5110 \\ RReLu & 200 & 2234.6063 \\ Sigmoid & 200 & 166,471.11 \\ Tanh & 200 & 166,175.66 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of MSE between multiple activation functions.  
Figure 5: Forward neural network architecture.  
Figure 6: MSE of multiple FNN Models at different numbers of hidden layer nodes and layers.