# Efficient CORDIC Iteration Design of LiDAR Sensors' Point-Cloud Map Reconstruction Technology +
## 2 Related Work
\begin{array}{c}X_{0}\\ Y_{0}\end{array} \right] \tag{1}\]  
\[Z_{\mathrm{n}}=\theta\sum_{\mathrm{i=0}}^{n-1}\sigma_{\mathrm{i}}\tan^{-1}2^{ -i} \tag{2}\]  
Finally, the matrix of Equation (1) is expanded, and from Equation (2), then Equations (3)-(5) can be obtained. \[Z_{\mathrm{i+1}}=Z_{\mathrm{i}}-\sigma_{\mathrm{i}}\tan^{-1}2^{-\mathrm{i}} \tag{3}\]  
\[X_{\mathrm{i+1}}=X_{\mathrm{i}}-\sigma_{\mathrm{i}}\;Y_{\mathrm{i}}\;2^{- \mathrm{i}} \tag{4}\]  
\[Y_{\mathrm{i+1}}=Y_{\mathrm{i}}-\sigma_{\mathrm{i}}\;X_{\mathrm{i}}\;2^{- \mathrm{i}} \tag{5}\]  
In the above formula, \(\sigma\) is the direction of the selected rotation angle, clockwise rotation is a positive sign, and counterclockwise rotation is a negative sign. \(Z\) is the sum of the angular rotation operations. The final calculation results need to make the angle of the sum after \(Z\) rotation approach zero. The number of iterations is determined by the number of bits. For example, if the input angle \(Z\) is 16 bits, it needs to be rotated 16 times. X represents the result of the COS operation and Y represents the result of the SIN operation. Assuming the input angle is 30 degrees, \(Z\) will approach zero after the rotation. X will approach 0.866 and Y will approach 0.5. After a series of rotations, the rotated vector will be different from the original vector. To make the rotated vector equal to the original vector, it is necessary to multiply the vector factor K of the corrected length. Let tan\(\alpha_{\mathrm{i}}\)= \(\sigma_{\mathrm{i}}2^{-\mathrm{i}}\) and K = \(\prod_{i=0}^{n-1}\cos\alpha_{\mathrm{i}}\). With the Pythagorean theorem, we can calculate cos = \(\frac{1}{\sqrt{1+2^{-3}}}\) and bring it into the K related formula. Finally, after expanding the infinite series of K, the approximate constant value is 1.64676, as shown in Equation (6). \[\mathrm{K}=\prod_{\mathrm{i=0}}^{n-1}\left(\frac{1}{\sqrt{1+2^{-2\mathrm{i}}}} \right)=1.64676 \tag{6}\]  
For the calculation of CORDIC, many iterations are required. In the literature, many papers propose ways to reduce the number of iterations. HA CORDIC [22; 23; 24] is mainly used to reduce the number of iterations of a traditional coordinate rotation digital computer. The principle is to use the traditional coordinate rotation system to fix the angle between the two angles. The angle \(\theta\) is a fixed angle, and c(0) is 45 degrees plus 26.565051177 degrees. After addition, the value is divided by 2 to 35.782525588 degrees, and the values of c(0) to c(15) are obtained in this way. Z(i) represents the input rotation angle, and N represents the value of inputting a few bits, indicating that the maximum number of rotations is only N. The two variables are set to i and j, respectively, and the initial value is set to zero. When the first input angle Z(0) is greater than c(15) and j is less than N, the condition is established and the loop is established. Finally, i will be equal to i plus 1. If the condition is not established, it will not jump to the loop. j will be equal to j plus 1. By changing the two variables i and j into Formulas (7)-(9), the pre-selected iteration angle can be achieved. \[Z_{\mathrm{i+1}}=Z_{\mathrm{i}}-\mathrm{d_{i}}\theta_{\mathrm{j}} \tag{7}\]  
\[X_{\mathrm{i+1}}=X_{\mathrm{i}}-\mathrm{d_{i}}\ Y_{\mathrm{i}}\ 2^{-\mathrm{j}} \tag{8}\]  
\[Y_{\mathrm{i+1}}=Y_{\mathrm{i}}-\mathrm{d_{i}}\ X_{\mathrm{i}}\ 2^{-\mathrm{j}} \tag{9}\]  
The HA CORDIC algorithm uses the angle between the two intermediate values of the fixed angles to compare the comparison values, and the input angle is compared to the intermediate values, thus achieving the function of pre-rotation angle. In [25], [PERSON] and [PERSON] proposed a CORDIC processor with three computation modes. The CORDIC core uses 16-bit fixed point numbers. [PERSON] and [PERSON] performed one iteration per bit of the input data. In this circuit, 16 vector rotations are needed and 16 stages are constructed in the CORDIC pipeline circuit.