## 3 The Optimized Kinematic Modeling Method
### Optimized Kinematic Modeling of the Mobile Manipulator Robot
#### 3.3.1 Optimized Kinematic Modeling of the Manipulator  
When establishing kinematic modeling using the classical method, the physical structure of the actual links and the actual position of the joint are ignored to emphasize the relationship between the location of the end tool and the driving joints. At the same time, the establishment of the joint-fixed coordinate frame is very complicated. To meet the structural requirements of the mathematical model and reduce the complexity of kinematic modeling, we optimized the kinematic modeling method by re-establishing the coordinate frames on the joint and adding some coordinate frames on the physical structure of the connecting links. Therefore, we selected the key points both on each joint and on the conversion positions in the actual connecting link, which is shown in Figure 2b. Some key points are mainly used to express the location of joints, but two adjacent points can be used to describe the structure of each link. The specific steps of the optimized kinematic modeling method are as follows:  
Step 1: Number links and joints. For the manipulator robot with n-degree-of-freedom, the links are marked from the base \((0)\) to the final end-effector \((n)\). Furthermore, the joints are identified from \(1\) to \(n\), the joint \(i\) is established between the link \((i-1)\) and the link \(i\). Step 2: Judge whether to add the key point on the structure. After connecting the adjacent joints through a straight line, we need to observe whether the straight line between the corresponding joints is similar to the structure of the connecting link. If they are the same, we do not need to add other key points. Otherwise, the key points must be added to the location where the structure changes. After establishing the key points, we connect the joint points and the key point again to determine whether the structure of the connecting link can be expressed by these lines. If it cannot be expressed, we add the other key points again until the structure of connecting link can be expressed by these straight lines. Step 3: Establish all coordinate frames. (1) Establish the joint-fixed coordinate frame. The origin of the joint-fixed coordinate frame was established at the location of the corresponding joint. The direction of one axis needs to be the same as the rotation direction of the joint, and the other axis directions need to meet the left-hand rule. (2) Establish the key points-fixed coordinate frames. We establish the coordinate frame at the positions of key points, and the three-axis directions of those coordinate frames need to meet the left-hand rule. (3) Establish the other coordinate frames. We establish the base-fixed coordinate frame at the base position according to the actual requirement. The origin of the base-fixed  
Figure 3: The kinematic modeling of the mobile platform. (**a**) Kinematic modeling based on the DH method; (**b**) Kinematic modeling based on the optimized method. coordinate frame is fixed in the convenient position of the robot's base, and three-axis directions need to meet the left-hand rule. Step 4: Identify transformation parameters. (1) Obtain three distances \((d_{x}^{i},d_{y}^{i},d_{z}^{i})\). The origin of the coordinate frame \((O_{i})\) moves to the origin of the next coordinate frame \((O_{i+1})\) along the three coordinate axes with the order of \(x_{i}\)-axis, \(y_{i}\)-axis, and \(z_{i}\)-axis. (2) Obtain three angles \((q_{x}^{i},q_{y}^{i},q_{z}^{i})\). The three-axis directions of the \(i\)-th coordinate frame are rotated to the three-axis directions of the \((i+1)\)-th coordinate frame around the three coordinate axes with the order of \(x_{i}\)-axis, \(y_{i}\)-axis, and \(z_{i}\)-axis. (3) Determine the angle variable which is controlled by the driving joint. Step 5: Determine the homogeneous transformation matrix. The current coordinate frame can reach the adjacent coordinate frame through rotation and translation relative to the current coordinate frame.