## 3 The Optimized Kinematic Modeling Method
### Optimized Kinematic Modeling of the Mobile Manipulator Robot
#### 3.3.3 Optimized Kinematic Modeling of the Entire System  
Since the manipulator's base is fixed on the mobile platform subsystem, the position and posture of the manipulator's end-effector in the frame will inevitably be affected by the fixed position and posture. To determine the position of the manipulator's end-effector, we need to convert the reference coordinate system from the base-fixed frame to the world frame by considering the homogeneous transformation matrix expressing the different fixing positions. Therefore, the position and posture of the base-fixed frame can be denoted as:  
\[{}^{d}T_{b}={}^{d}T_{c}\cdot{}^{c}T_{b}={}^{d}T_{c}\cdot T_{r}(x_{ cb})T_{y}(y_{cb})Tr_{z}(z_{cb})R_{x}(q_{xcb})R_{y}(q_{ycb})R_{z}(q_{zcb}) \tag{8}\] \[=\left[\begin{array}{cccc}1&0&0&x_{dc}+x_{cb}\\ 0&1&0&y_{dc}+y_{cb}\\ 0&0&1&z_{dc}+z_{cb}\\ 0&0&0&1\end{array}\right]\]  
where \(Tr_{i}()\) indicates the translation transformation matrix; \(R_{x}()\), \(R_{y}()\), \(R_{z}()\) represent the rotation transformation matrix; \(x_{cb}\), \(y_{cb}\), and \(z_{cb}\) indicate the translational distance of the base-fixed frame along the three axes of the body-fixed frame; \(q_{xcb},q_{ycb}\), and \(q_{zcb}\) represent the rotation angle of the base-fixed frame around the three axes of the body-fixed frame; \(x_{dc},y_{dc}\), and \(z_{dc}\) indicate the translational distance from the midpoint of the driving axis to the body-fixed frame along the three axes.  
According to the above Equations (6)-(8), the position and posture of the end of the mobile manipulator can be specifically expressed with the homogeneous transformation matrix in Equation (2). For the designated task, the position and posture of the robot's end are mainly described as follows:  
\[{}^{o}T_{e}={}^{o}T_{d}\cdot{}^{d}T_{b}\cdot{}^{b}T_{e} \tag{9}\]  
where \({}^{o}T_{e}\) indicates the position and posture of the manipulator's end-effector in the world frame, \({}^{o}T_{d}\) indicates the position and posture of the midpoint of two driving wheels in the world frame, \({}^{d}T_{b}\) indicates the position and posture of the manipulator's base in the midpoint-fixed frame, and \({}^{b}T_{e}\) indicates the position and posture of the manipulator's end in the base-fixed frame.  
The position and posture of the manipulator's end are mainly described as six variables in the Cartesian coordinate system. To improve the efficiency of performing tasks, we generally only set the position requirements to complete the given task. Therefore, it is necessary to convert the homogeneous transformation matrix into the position and angle in the Cartesian space:  
\[\left[\begin{array}{c}P_{x}\\ P_{y}\\ P_{z}\\ O_{x}\\ O_{y}\\ O_{z}\end{array}\right]=\left[\begin{array}{c}{}^{o}T_{e}(1,4)\\ {}^{o}T_{e}(2,4)\\ {}^{o}T_{3}(3,4)\\ {}atan\left(\frac{T_{e}(3,2)}{\sqrt{\sigma_{T}(3,3)}}\right)\\ {}atan\left(-\frac{T_{e}(3,1)}{\sqrt{\sigma_{T}(3,2)^{2}+\sigma_{T}(3,3)^{2}} }\right)\\ {}atan\left(\frac{eT_{e}(2,1)}{\sigma_{T}(1,1)}\right)\end{array}\right] \tag{10}\]  
where \(P_{x},P_{y},P_{z}\) indicate the position of the end of the manipulator subsystem in the world frame, and \(O_{x},O_{y},O_{z}\) indicate the Euler angle of the end of the manipulator subsystem around the world frame.