## 3 The Optimized Kinematic Modeling Method
### Optimized Kinematic Modeling of the Mobile Manipulator Robot
#### 3.3.1 Optimized Kinematic Modeling of the Manipulator
Therefore, the corresponding homogeneous transformation matrix is as follows:  
\[{}^{i-1}T_{i}=Tr_{x}(d_{x}^{i})Tr_{y}(d_{y}^{i})Tr_{z}(d_{z}^{i})R_{x}(q_{x}^{i })R_{y}(q_{y}^{i})R_{z}(q_{z}^{i}) \tag{3}\]  
where \(Tr_{i}()\) represents the translation transformation matrix along the corresponding coordinate axis, and \(R_{i}()\) represents the rotation transformation matrix around the corresponding coordinate axis. The specific representation of various transformation matrices is expressed as:  
\[ \begin{split} R_{x}=\left[\begin{array}{cccc}1&0&0&0\\ 0&C_{q_{x}}&-S_{q_{x}}&0\\ 0&S_{q_{x}}&C_{q_{x}}&0\\ 0&0&0&1\end{array}\right];R_{y}=\left[\begin{array}{cccc}C_{q_{y}}&0&S_{q_{y }}&0\\ 0&1&0&0\\ S_{q_{y}}&0&C_{q_{y}}&0\\ 0&0&0&1\end{array}\right];R_{z}=\left[\begin{array}{cccc}C_{q_{z}}&-S_{q_{z }}&0&0\\ S_{q_{z}}&C_{q_{z}}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right];\\ Tr_{x}=\left[\begin{array}{cccc}1&0&0&dx\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right];Tr_{y}=\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&dy\\ 0&0&1&0\\ 0&0&1\end{array}\right];Tr_{z}=\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&dz\\ 0&0&0&1\end{array}\right];\\ \end{split}  \tag{4}\]  
where \(C_{q_{i}}\) indicates \(cos(q_{i})\) and \(S_{q_{i}}\) indicates \(sin(q_{i})\). Step 6: Determine the position and posture of the coordinate frames in the base-fixed frames. The position and posture of each coordinate frame in the base-fixed coordinate system can be described by the homogeneous transformation matrix from the base-fixed coordinate frame to the corresponding coordinate frame, which can be expressed as:  
\[\left\{ \begin{array}{c}{}^{b}T_{i}{=}^{b}T_{1}{}^{1}T_{2}{}^{2}T_{3}\cdot.....^{i-1}T_{i},(i=1,2,...,n);\\ {}^{b}P_{i}=[{}^{b}T_{i}(1,4),{}^{b}T_{i}(2,4),{}^{b}T_{i}(3,4)]^{T};\\ \end{array} \right. \tag{5}\]  
where \({}^{b}T_{i}\) indicates the position and posture of each coordinate frame in the base-fixed coordinate frame, and \({}^{i-1}T_{i}\) indicates the homogeneous transformation matrix from the \((i-1)\)-th coordinate system to the \(i\)-th coordinate frame. By adding the point on the structure of the connecting link, the kinematic modeling result is similar to the actual physical model rather than the kinematic model based on the classical method, as shown in Figure 2. The position of the manipulator's joint met the actual position requirements, and the structure of the kinematic model has a small difference from the actual physical model. Meanwhile, the establishment of each local coordinate frame and the acquisition of key parameters are simpler than the classical methods. Many different coordinate frames are attached to each joint, some structures of the connecting links, and the end-effector. The relationship between each coordinate frame can be determined by the homogeneous transformation matrix. Therefore, the position and posture of the end-effector of the manipulator in the base-fixed frame can be directly solvedby the homogeneous transformation matrix multiplication from the base-fixed frame to the end-fixed frame, which can be described as  
\[{}^{b}T_{e}={}^{b}T_{1}\cdot{}^{1}T_{2}\cdot\ldots\cdot{}^{n-1}T_{n}\cdot{}^{n}T_ {e} \tag{6}\]  
where \({}^{b}T_{e}\) indicates the homogeneous transformation matrix from the end-fixed coordinate frame to the base-fixed frame, \({}^{n}T_{e}\) indicates the homogeneous transformation matrix from the end-fixed frame to the \(n\)-th joint-fixed frame, and \({}^{i-1}T_{i}\) indicates the homogeneous transformation matrix from the \(i\)-th joint-fixed frame to the \((i-1)\)-th joint-fixed frame.