# Design and Analysis of the Rolling and Jumping Compound Motion Robot
## 3 The Kinetic Analysis of the Rolling and Jumping Robot  
A theoretical model based on the space coordinate system was established to analyze the influence of structural design parameters on the dynamic characteristics of the robot, which was of guiding significance to improve its motion performance. In addition, the physical meanings of the letters involved in the establishment of the dynamic model are shown in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline \(M\) & Mass of whole robot \\ \(M_{1}\) & Mass of the remaining part without counterweight \\ \(M_{2}\) & Mass of remaining part without energy storage \\ \(m\) & Mass of counterweight \\ \(m_{0}\) & Mass of control module \\ \(m_{1}\) & Mass of energy storage \\ \(R\) & Geometric radius \\ \(d\) & Distance between the center of mass and the geometric center of the shell \\ \(l\) & Distance between the limit position on both sides of the counterweight and the shape center \\ \(L\) & Length of swing arm \\ \(D\) & Energy storage compression distance \\ \(\beta\) & Angle between the major shaft and the horizontal plane \\ \(f\) & Friction force \\ \(k\) & Jumping spring coefficient \\ A & Origin point of the ground coordinate system \\ C & Origin point of the moving coordinate system \\ P & Center of mass of counterweight \\ \(\alpha\) & Deflection angle of the pendulum in rolling motion \\ \(r\) & Turning radius \\ \(\varphi\) & Deflection angle of the pendulum in climbing motion \\ \(\lambda\) & Climbing angle \\ \(E\) & Kinetic energy of the robot in rolling motion \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the robot. Figure 9: The jump process of the rolling and jumping robot: (**a**) The process of posture adjustment; (**b**) The process of energy release and jump; (**c**) The process of landing; (**d**) The process of energy storing for the next jump. The space coordinate system of the robot was generally composed of a carrier coordinate system and ground coordinate system. The law of inertia applied to the ground coordinate system, which could be used to describe the position change of the robot, while the carrier coordinate system was a non-inertial coordinate system moving with the robot. In this coordinate system, the moment of inertia and the product of inertia were constant, which could easily describe the attitude change of the robot. In the process of the space motion of the robot, a spatial coordinate system was established, shown in Figure 10. The ground coordinate system was recorded as: \(\sum_{\text{A}}(\text{X}_{\text{A}}\text{Y}_{\text{A}}\text{Z}_{\text{A}})\) and the carrier coordinate system was recorded as: \(\sum_{\text{C}}(\text{X}_{\text{C}}\text{Y}_{\text{C}}\text{Z}_{\text{C}})\). Taking the motion starting point as the origin A of the ground coordinate system, it was specified that the vertical upward direction was +Z\({}_{\text{A}}\), and the rolling forward direction of the robot at the starting time was +Y\({}_{\text{A}}\). The geometric center of the robot system was taken as the origin C of the moving coordinate system. The rolling forward direction of the robot at the current time was +Y\({}_{\text{C}}\). The vertical upward direction was +Z\({}_{\text{C}}\). The components of the attitude angle in X\({}_{\text{C}}\), Y\({}_{\text{C}}\) and Z\({}_{\text{C}}\) axes were defined as \(\theta\), \(\varphi\) and \(\Psi\). The components of the external torque vector in X\({}_{\text{C}}\), Y\({}_{\text{C}}\) and Z\({}_{\text{C}}\) axes were defined as \(\tau_{1}\), \(\tau_{2}\), \(\tau_{3}\). In order to simplify the analysis of the problem, the robot was equivalent to a multibody system, and the following assumptions were made:  
1. The spatial motion of the robot follows the constraint condition of rigid body sliding free rolling;
2. The shell is equivalent to a homogeneous thin-walled shell with uniform mass distribution;
3. In the initial state, the centroid of the robot coincides with the geometric center;
4.