## 1 Introduction  
The mobile manipulator robot fixes the traditional manipulator on the mobile platform, thereby extending the substantial potential applications of the fixed manipulator [1, 2]. Due to the combination of the high mobility of the mobile platform and the dexterous maneuverability of the manipulator, it can be widely used in current life and work, such as in planetary exploration, nuclear reactor maintenance, landmine detection, agriculture missions, and fire rescue operations [3, 4, 5]. The mobile manipulator will be able to smoothly complete several complex tasks if the degree of freedom of the manipulator system is increased and a more reasonable structure for the entire system is designed. [6, 7, 8]. However, improving the accuracy of the kinematic model and avoiding self-collision remain challenging problems in robot operations [9; 10]. The resolution of these challenges is extremely promising for the development of mobile manipulator systems. Kinematic modeling is a fundamental problem and a commonly discussed issue in regard to the high-precision control of the mobile manipulator robot [11; 12; 13]. The current methods of kinematic modeling include the standard Denavit-Hartenberg (SDH) method, the modified Denavit-Hartenberg (MDH) method, the screw theory method, and the dual-quaternions method. As the most typical methods, SDH and MDH are based on the homogeneous transformation matrix, but the description of the two matrices are very different [14; 15; 16; 17]. The difference between the two methods is mainly caused by the differences in selecting the origin of the joint-fixed frames and the direction of three axes. Meanwhile, due to the difference in the number of input and output variables, the mobile platform includes holonomic and non-holonomic constraint equations. The non-holonomic constraints of the mobile platform increase the complexity of system control [18; 19; 20]. Researchers combined screw theory and Lie algebra to simplify the kinematic modeling of the manipulator robot in order to solve the kinematic modeling of the manipulator robot. However, the computational complexity of screw theories is not significantly reduced since the analysis process is still carried out via matrix operations [21; 22; 23; 24]. With the rapid development of mathematical theory, researchers have successfully established the kinematic modeling of the manipulator robot based on dual-quaternions to reduce computational complexity. However, the dual-quaternions method is still in the research stage due to incompleteness of the theory [25; 26; 27]. Therefore, the SDH and MDH methods are the main methods in the kinematic modeling process. The kinematic modeling of the mobile manipulator robot consists of two parts: mobile platform modeling and manipulator modeling. For the mobile platform subsystem, researchers regarded the mobile platform as two translation joints and one rotation joint. Then, they added the nonholonomic constraints in the kinematic modeling to establish the kinematic model of the mobile platform. For the manipulator subsystem, researchers have successfully developed and designed a variety of mature arms, including UR series robots, KUKA LBR series robots, and the Jaco arm robot. According to the difficulty in completing tasks and the requirements of work performance, UR series manipulators have become more commonly used manipulator systems. For the UR-serial robot, researchers have established the corresponding coordinate system directly at the end position of the link according to the SDH method. Then, they solved the position and posture of the tool end of the robotic arm via the homogeneous transformation matrix. Nevertheless, the kinematic modeling result is notably different from the actual model of the mobile manipulator.