# Kinematics Model Optimization Algorithm for Six Degrees of Freedom Parallel Platform
## 2 Materials and Methods
### Forward Solution Algorithm of Newton-Raphson Kinematics  
The Newton-Raphson method is an approximate solution method proposed by [PERSON] and [PERSON] in the 17 th century to find the root of a nonlinear function [24]. Its core idea is to solve iteratively based on the first-order [PERSON] expansion of the function \(f(x)\) to be solved, as shown in Equation (10):  
\[\mathbf{x_{n+1}}=\mathbf{x_{n}}-\frac{f(\mathbf{x_{n}})}{f(\mathbf{x_{n}})} \tag{10}\]  
In Equation (10), \(f\) is the function to be solved, and \(x_{n}\) is the solution at the nth step. After the [PERSON] iterative method gives the initial solution \(X\_0\), the next approximate solution, \(x_{1}\), can be obtained through \(f(x_{0})\) and its first derivative. By analogy, gradually make \(f(x_{n})\) converge to zero.  
With the development of computer technology, [PERSON]'s iterative method has become a general and effective method for solving nonlinear equations. At the same time, it is also a classic numerical solution for the forward kinematics solution of parallel platforms [25]. In order to apply the [PERSON] iterative method to the solution of the forward kinematics solution of the parallel platform, it is necessary to obtain the iterative formula of the parallel platform. Suppose we have a set of robotic-arm-length data, \(L_{tar}=[l_{1},l_{2},l_{3},l_{4},l_{5},l_{6}]^{T}\). The corresponding pose data on the parallel platform workspace are \(X_{tar}=[x,y,z,\alpha,\beta,\gamma]^{T}\). Denote the parallel platform kinematics inverse solution operator as \(G\). At \(X_{k}\), the error between the length of the mechanical arm obtained by the inverse kinematics solution and the given length, \(L_{tar}\), is \(E(X_{k})\). Therefore, the error \(E(X_{tar})\) is zero when the parallel platform is at \(X_{tar}\), as shown in Equation (11):  
\[E(X_{tar})=G(X_{tar})-L_{tar}=0 \tag{11}\]  
In Equation (11), the first-order Taylor expansion of \(E(X_{tar})\) at \(X_{k}\) can help us obtain Equation (12):  
\[E(X_{tar})=E(X_{k})+E^{\prime}(X_{k})(X_{tar}-X_{k})+o(X_{tar}-X_{k}) \tag{12}\]  
In Equation (12), \(E^{\prime}(X_{k})\) is the Jacobian matrix, \(J\), when the moving platform and the manipulator are at \(X_{k}\). Discard the high-order infinitesimal term for Equation (12). The combination Equation (11) can be arranged to obtain the relational expression (13) between the target pose, \(X_{tar}\), and the current pose, \(X_{k}\):  
\[X_{tar}\approx X_{k}+J^{-1}(L_{tar}-G(X_{k})) \tag{13}\]Using the idea of [PERSON]'s iterative method, the initial pose, \(X_{0}\), of the moving platform is given. Continuously and iteratively approximate \(X_{tar}\), and then the iterative formula of step \(k+1\) of the moving platform is Equation (14):  
\[X_{k+1}=X_{k}+J^{-1}(L_{tar}-G(X_{k})) \tag{14}\]  
It can be seen from Equation (14) that this algorithm takes advantage of the fact that the forward kinematics solution of the parallel platform is difficult, but the kinematics inverse solution is simple. Therefore, it is a classical numerical solution for the forward kinematics solution of the parallel platform.