## 3 Robot Kinematics Model
### Jacobian Matrix Solution for Arm Frame  
The transformation between the Cartesian space motion velocity and the joint space motion velocity of a robotic arm is called the Jacobian matrix. The methods for constructing Jacobian matrices include vector product method and differential transformation method. In this paper, the differential transformation method was used to calculate the Jacobian matrix of the robot. The solution for the \(i\)th column of the Jacobian matrix is  
\[J_{i}=[(p\times n)_{z^{\prime}},(p\times o)_{z^{\prime}},(p\times a)_{z^{ \prime}},n_{z},o_{z},a_{z}]^{T}. \tag{14}\]  
In the above equation, the \(n\), \(o\), and \(a\) vectors form the rotation transformation matrix for the transformation from the \(i\)th link coordinate system to the robot end coordinate system, while \(p\) is the position vector for the transformation from the \(i\)th link coordinate system to the robot-end coordinate system.  
\[M_{34}=\begin{bmatrix}\cos(\theta_{4})&-\sin(\theta_{4})&0&a_{3}\\ \sin(\theta_{4})&\cos(\theta_{4})&0&0\\ 0&0&1&d_{4}\\ 0&0&0&1\end{bmatrix} \tag{15}\]  
\[M_{24}=M_{23}M_{34}=\begin{bmatrix}\cos(\theta_{3}+\theta_{4})&-\sin(\theta_{ 3}+\theta_{4})&0&a_{3}\times\cos(\theta_{3})+a_{2}\\ \sin(\theta_{3}+\theta_{4})&\cos(\theta_{3}+\theta_{4})&0&a_{3}\times\sin( \theta_{3})\\ 0&0&1&d_{4}\\ 0&0&0&1\end{bmatrix} \tag{16}\]  
\[M_{14}=\begin{bmatrix}\cos(\theta_{2}+\theta_{3}+\theta_{4})&-\sin(\theta_{2} +\theta_{3}+\theta_{4})&0&a_{3}\times\cos(\theta_{2}+\theta_{3})+a_{2}\times \cos(\theta_{2})+a_{1}\\ 0&0&-1&-d_{4}\\ \sin(\theta_{2}+\theta_{3}+\theta_{4})&\cos(\theta_{2}+\theta_{3}+\theta_{4}) &0&a_{3}\times\sin(\theta_{2}+\theta_{3})+a_{2}\times\sin(\theta_{2})\\ 0&0&0&1\end{bmatrix} \tag{17}\]  
For the fourth joint, Equation (15) is the transformation matrix from the connecting rod coordinate system to the end coordinate system of the joint. Values \(n\), \(o\), \(a\), and \(p\) in Equation (15) were substituted for Equation (14) to obtain the Jacobian column vector \(J_{4}\).  
\[J_{4}=[0,0,d_{4},0,0,1]^{T}. \tag{18}\]  
For the third joint, Equation (16) is the transformation matrix from the connecting rod coordinate system to the end coordinate system of the joint. Values \(n\), \(o\), \(a\), and \(p\) in Equation (16) were substituted for Equation (14) to obtain the Jacobian column vector \(J_{3}\).  
\[J_{3}=[0,0,d_{4},0,0,1]^{T}. \tag{19}\]For the second joint, Equation (17) is the transformation matrix from the connecting rod coordinate system to the end coordinate system of the joint. Values \(n\), \(o\), \(a\), and \(p\) in Equation (17) were substituted for Equation (14) to obtain the Jacobian column vector \(J_{2}\).  
\[J_{2}=[s_{234}(a_{3}s_{23}+a_{3}s_{2}),c_{234}(a_{3}s_{23}+a_{3}s_{2}),0,s_{234}, c_{234},0]^{\mathrm{T}}. \tag{20}\]  
For the first joint, Equation (11) is the transformation matrix from the connecting rod coordinate system to the end coordinate system of the joint. Values \(n\), \(o\), \(i\), and \(p\) in Equation (11) were substituted for Equation (14) to obtain the Jacobian column vector \(J_{1}\).  
\[J_{1}=[s_{234}(a_{3}s_{23}+a_{2}s_{2}+d_{1}),c_{234}(a_{3}s_{23}+a_{2}s_{2}+d_{ 1}),0,s_{234},c_{234},0]^{\mathrm{T}}. \tag{21}\]  
The corresponding Jacobian matrix velocity solution is  
\[J=[J_{1},J_{2},J_{3},J_{4}]=\begin{bmatrix}s_{234}(a_{3}s_{23}+a_{2}s_{2}+d_{ 1})&s_{234}(a_{3}s_{23}+a_{3}s_{2})&0&0\\ c_{234}(a_{3}s_{23}+a_{2}s_{2}+d_{1})&c_{234}(a_{3}s_{23}+a_{3}s_{2})&0&0\\ 0&0&d_{4}&d_{4}\\ s_{234}&s_{234}&0&0\\ c_{234}&c_{234}&0&0\\ 0&0&1&1\end{bmatrix}. \tag{22}\]