# Kinematics Model Optimization Algorithm for Six Degrees of Freedom Parallel Platform
## 3 Results
### MPR-NR Algorithm Simulation

Table 3: Comparison between Newton iteration method and MPR-NR algorithm. and the Newton iteration stop condition is obtained. We designed algorithmic accuracy experiments. Assume that the Euclidean distance between the numerical solution of the algorithm and the target attitude is \(D(X_{k})=\sqrt{\left(X_{k}-X_{tar}\right)^{T}(X_{k}-X_{tar})}\). Let the stop condition be the Euclidean distance, \(D(L_{k})=\sqrt{\left[E(X_{k})\right]^{2}}=\sqrt{\left[G(X_{k})-L_{tar}\right] ^{T}[G(X_{k})-L_{tar}]}\), between the length of each group of manipulators after the kinematics forward solution \(X_{k}\) of the \(k\)-th iteration and the given length of each group of manipulators. Experiment on Test Set 2. The relationship between the iteration stop condition, \(\sqrt{\left[E(X_{k})\right]^{2}}\), of the MPR-NR and Newton iterative algorithm and the solution accuracy and number of iterations of the kinematics forward solution is shown in Table 4.  
It can be seen from Table 4 that, with the improvement of the accuracy of the stop condition, \(D(L_{k})\), the accuracy of the kinematics forward solution of the Newton iterative algorithm and the MPR-NR algorithm also increases accordingly. The precision controllability of the Newton iteration algorithm and MPR-NR is illustrated. When comparing the kinematics forward solution accuracy \(D(X_{k})\) of the MPR-NR algorithm and the Newton iterative algorithm, we can see that the solution accuracy of the two algorithms is at the same level. By comparing the number of iterations of the MPR-NR algorithm and the Newton iterative algorithm, we can obtain the following conclusions. Since the initial solution \(X_{-}0\) of MPR-NR is better than the Newton iteration algorithm, the average number of iterations is 2.41 times less than that of the Newton iteration algorithm. The relationship between \(D(L_{k})\) and \(D(X_{k})\) and the number of iterations is shown in Figure 5.  
Both \(D(L_{k})\) and \(D(X_{k})\) in Figure 5a are logarithmic images with base 10. It can be seen from Figure 5a that \(log_{10}D(L_{k})\) and \(log_{10}D(X_{k})\) of the two algorithms present a positive linear relationship. Therefore, every time the stop condition, \(D(L_{k})\), increases by an order of magnitude, the accuracy of the kinematics positive solution, \(D(X_{k})\), also increases by an order of magnitude. It can be seen from Figure 5b that \(log_{10}D(L_{k})\) has a negative linear relationship with the number of iterations of the two algorithms, and the slopes of the two algorithms are similar. Therefore, it can be considered that in the second test sample set, the MPR process in the MPR-NR algorithm is equivalent to completing 2.41 Newton iterative algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \(D(L_{k})\) & **Newton Algorithm \(D(X_{k})\)** & \ \begin{tabular}{c} **Average Number of** \\ **Iterations** \\ \end{tabular}  & **MPR-NR \(D(X_{k})\)** & \begin{tabular}{c} **Average Number of** \\ **Iterations** \\ \end{tabular}  \\ \hline \(\mathbf{1\times 10^{-3}}\) & 7.00 \(\times 10^{-4}\) & 4.16 & \(1.20\times 10^{-3}\) & 1.70 \\ \(\mathbf{1\times 10^{-4}}\) & 7.59 \(\times 10^{-5}\) & 5.53 & \(7.19\times 10^{-5}\) & 3.18 \\ \(\mathbf{1\times 10^{-5}}\) & 7.81 \(\times 10^{-6}\) & 6.91 &