# Manipulator Smooth Control Method Based on LSTM-XGboost and Its Optimization Model Construction
## 3 Inverse Kinematics Problem Solving Method of Serial Manipulator  
The problem of inverse kinematics in a serial manipulator is a fundamental aspect of trajectory planning. It necessitates the calculation of joint rotation angles and displacements based on the desired final position of the manipulator. Two prevalent strategies for solving this problem are the analytical and the numerical methods. The analytical method utilizes the manipulator's geometric configuration to produce a solution. This method is efficient, yielding accurate results swiftly. However, it comes with a precondition: the geometric configuration of the multi-axis serial manipulator must conform to the Pieper criterion. According to this criterion, either three consecutive joint axes must intersect at one point, or they should be parallel to each other. As a result, the applicability of the analytical method is restricted to manipulators with specific configurations. On the other hand, the numerical method views the inverse kinematics problem as a system of nonlinear equations, utilizing iterative techniques for its resolution. Unlike the analytical method, the numerical method does not necessitate the geometric configuration of the manipulator to meet the Pieper criterion. However, it struggles with situations involving multiple solutions and relies on suitable initial conditions for convergence.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Criteria** & **Value** \\ \hline \(\lvert a\rvert_{ave}\) & 2.7139 \\ \(\delta_{a}\) & 5.2828 \\ \(a_{p}\) & 9.0056 \\ \(\delta_{w}\) & 2.2541 \\ jerk & 55.7436 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Score Criteria.  
It's crucial to recognize that the final pose of a multi-axis serial manipulator is a cumulative result of the rotation angles and displacements of each joint. By comprehend the features and constraints of both the analytical and numerical methods, researchers can select the most fitting solution for the inverse kinematics problem, considering the particular configuration and requirements of the manipulator. In conclusion, the inverse kinematics problem of a serial manipulator can be addressed through either the analytical or numerical method. The analytical method offers precise solutions but necessitates specific geometric configurations, while the numerical method provides flexibility but struggles with multi-solution problems. By judiciously selecting the appropriate method, researchers can efficiently tackle the inverse kinematics problem and facilitate trajectory planning for serial manipulators. By incorporating these improvements, the text establishes clearer connections between the paragraphs and enhances the overall logic and coherence of the content. The combination of different joint rotation angles and displacements can make the end of the manipulator reach the same pose, which is the multi-solution problem in the inverse kinematics of the multi-axis series manipulator [28].  
The trajectory planning method of the manipulator with the time-optimal trajectory of running stability constraint is usually divided into the construction method, dynamic programming method, numerical integration method and convex optimization method.