## 4 Joint Trajectory Equations Based on High Degree Polynomials  
The piecewise polynomial interpolation method is a common approach to trajectory planning. Its fundamental principle is the use of different numbers of polynomials to generate trajectories in different time periods, with the aim of ensuring the smoothness and continuity of the resulting trajectories. A single high-order polynomial is susceptible to fluctuations and oscillations during the fitting process, and the parameters are challenging to adjust when satisfying complex boundary conditions, which increases the computational complexity [28; 29; 30]. Additionally, low-order polynomial curves are unable to guarantee the higher-order continuity of the joint trajectories of the robotic arm [31; 32]. In contrast, the piecewise polynomial is better suited to adapt to local characteristics, enhance fitting accuracy, and reduce computational complexity. In this paper, to circumvent the discontinuities in acceleration that are inherent to 3-5-3 piecewise polynomial trajectory planning, a 5-7-5 piecewise polynomial is used for trajectory planning. Firstly, the time interval of the entire trajectory is divided into three sections, and the trajectory simulation is conducted using quintic, septic, and quintic polynomials in succession, as shown in Figure 3.  
A and D represent the initial and final points of the trajectory, respectively; B and C represent intermediate points along the trajectory. The fifth-degree polynomials are used for interpolation in the A-B and C-D segments, while in the B-C segment, a seventh-degree polynomial interpolation is used. Equation (8) denotes the interpolating function of a 5-7-5 segmented polynomial trajectory of the joint variables with respect to time \(t\), and Equations (9)-(11) denote (angle)velocity, (angle)acceleration, and (angle)jerk, respectively. \[\left\{ \begin{array}{l}\theta_{1}(t)=a_{10}+a_{11}t+a_{12}t^{2}+a_{13}t^{3} +a_{14}t^{4}+a_{15}t^{5}\\ \theta_{2}(t)=a_{20}+a_{21}t+a_{22}t^{2}+a_{23}t^{3}+a_{24}t^{4}+a_{25}t^{5}+a_ {26}t^{6}+a_{27}t^{7}\\ \theta_{3}(t)=a_{30}+a_{31}t+a_{32}t^{2}+a_{33}t^{3}+a_{34}t^{4}+a_{35}t^{5}\\ \end{array} \right. \tag{8}\]  
\[\omega_{i}(t)=d\theta_{i}(t)/dt \tag{9}\]  
\[\alpha_{i}(t)=d^{2}\theta_{i}(t)/dt^{2} \tag{10}\]  
\[j_{i}(t)=d^{3}\theta_{i}(t)/dt^{3} \tag{11}\]  
In order to guarantee the continuity and smoothness of the operational trajectories of the manipulator and worktable segments, it is essential to ensure that the following constraints and constraint boundaries are satisfied for the joint variables, namely the angle of velocity, the angle of acceleration, and the angle of jerk:  
\[\left\{ \begin{array}{l}\theta_{1}(0)=\theta_{A}\\ \theta_{2}(0)=\theta_{1}(1)=\theta_{B}\\ \theta_{3}(0)=\theta_{2}(1)=\theta_{C}\\ \theta_{3}(0)=\theta_{D}\\ \end{array} \right. \tag{12}\]  
\[\left\{ \begin{array}{l}\omega_{1}(0)=\omega_{A}=0\\ \omega_{2}(0)=\omega_{1}(1)=\omega_{B}\\ \omega_{3}(0)=\omega_{2}(1)=\omega_{C}\\ \omega_{3}(0)=\omega_{D}=0\\ \end{array} \right. \tag{13}\]  
Figure 3: 5-7-5 piecewise polynomial interpolation. \[G= \begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&2&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&6&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix}  \tag{22}\]  
\[H= \begin{bmatrix}0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0\\ 0&0&0&0&0&0&1\\ 0&0&0&0&0&1&0&0\end{bmatrix}  \tag{23}\]  
\[I= \begin{bmatrix}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 1&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix}  \tag{24}\]  
\[a=A^{-1}b^{T}= \begin{bmatrix}A_{1}&A_{2}&A_{3}\end{bmatrix} ^{T} \tag{25}\]  
\[A_{1}= \begin{bmatrix}a_{10}&a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\end{bmatrix}  \tag{26}\]  
\[A_{2}= \begin{bmatrix}a_{20}&a_{21}&a_{22}&a_{23}&a_{24}&a_{25}&a_{26}&a_{27} \end{bmatrix}  \tag{27}\]  
\[A_{3}= \begin{bmatrix}a_{30}&a_{31}&a_{32}&a_{33}&a_{34}&a_{35}\end{bmatrix}  \tag{28}\]  
\[b=