# Kinematics Model Optimization Algorithm for Six Degrees of Freedom Parallel Platform
## 3 Results
### Single Algorithm Simulation
The average single solution time of MPR3 is about 28 microseconds. Comparing the solution time of MPR2 and MPR3, it can be seen that, with the increase of the polynomial degree, the calculation time consumption of a single solution is significantly improved. Compared with the Newton iterative algorithm, it can be seen that MPR has a significant advantage in solution efficiency. The solution time of MPR2 in the small space Test Set 1 is about 14.7% of Newton iteration, and the solution time of MPR3 with higher solution accuracy is only 25.1% of Newton iteration. Comparing the time consumption of MPR to solve Test Set 1 and Test Set 2, it can be seen that another characteristic of the MPR algorithm is that it will not significantly increase the calculation time consumption as the solution range increases. This is because the forward solution algorithm of MPR kinematics is composed of a fixed number of multiplications and additions. Therefore, the calculation time of the same algorithm for Test Set 1 and Test Set 2 does not change much. When comparing the calculation time consumption of Newton iterative algorithm in Test Set 1 and Test Set 2, the following conclusions can be drawn. The calculation time of the Newton iterative algorithm to solve Test Set 2 is significantly higher than that of Test Set 1, which increases from 1.144 s to 1.815 s. This is because Test Set 2 has a larger spatial range, so the deviation between the initial value and the target attitude is also greater. The Newton iterative algorithm requires a corresponding increase in the number of iterations, which eventually leads to an increase in calculation time. In terms of solution accuracy, when comparing the Newton iterative algorithm and the MPR algorithm, it can be seen that the solution accuracy of the Newton iterative algorithm is better than that of the two MPR algorithms. Moreover, the Newton iterative algorithm can set more iterations and a smaller \(\sqrt{\left[E(X_{k})\right]^{2}}\) by changing the stop condition to meet the higher precision requirements. The calculation result of the MPR algorithm is only determined by the algorithm itself and the polynomial coefficients. The polynomial coefficients are calculated offline, so the solution accuracy cannot be improved by changing the parameters of the algorithm itself. When comparing the performance of the MPR algorithm in Test Set 1 and Test Set 2, it can be seen that the accuracy of the MPR algorithm decreases by two orders of magnitude as the sample space increases. MPR2 achieved an average angular error of 0.95\({}^{\circ}\) and an average displacement error of 2 mm in Test Set 2. In contrast, MPR3 in Test Set 2 has an average angular error of 0.1\({}^{\circ}\) and an average displacement error of 0.5 mm. By comparing the data of MPR2 and MPR3, it can be seen that, with the increase of the polynomial degree, the accuracy of the solution was significantly improved. In theory, increasing the polynomial degree of the MPR algorithm can obtain higher solution accuracy. However, with the increase of polynomial coefficients, the amount of calculation for kinematics solution shows a geometric progression. As a result, the advantage of MPR in real-time performance over Newton iterative algorithm is reduced. Therefore, the strategy of increasing the solution accuracy by increasing polynomial coefficients is not sustainable. Through the simulation experiment, it is found that the Newton iteration method has high solution accuracy, and the polynomial regression solution time is short. In order to achieve a balance between the solution time and solution accuracy, the advantages of controllable solution accuracy of the Newton iterative algorithm and controllable calculation time of the MPR algorithm are utilized at the same time. This paper proposes a kinematics positive solution optimization algorithm (multivariate polynomial regression/NewtonRaphson iteration, MPR-NR) that uses a combination of polynomial regression and Newton iteration.