# Design and Analysis of the Rolling and Jumping Compound Motion Robot
## 3 The Kinetic Analysis of the Rolling and Jumping Robot
### Climbing Dynamics Equation  
Climbing motion refers to the upward rolling motion of the robot along the slope with a certain inclination angle to the horizontal plane. The climbing angle is a basic index to describe the climbing performance of robot. The maximum climbing angle is used to measure the ability of robot to climb over obstacles. Based on the theory of geometry and dynamics, the dynamic equation of robot climbing was established through the second kind of Lagrange equation. The robot climbed the slope without relative sliding along the inclined plane with angle \(\lambda\), shown in Figure 13. The pendulum mass was \(m\); the center of mass was fixedly connected with the geometric center of the shell through a light rod with a length of \(L\); the approximate equivalent mass of the remaining part without the pendulum was \(M_{1}\), and the geometric radius of the robot was \(R\); At that current moment, the deflection angle of the pendulum was \(\varphi\). When the robot rolls upward along the slope, the rolling constraints without sliding and bouncing can be expressed as:  
\[v=\omega R \tag{10}\]  
Figure 13: Schematic diagram of climbing. Since the linear velocity \(v\) is the first derivative of the displacement x with respect to time, and the angular velocity \(\omega\) is the first derivative of angle \(\Psi\) to time, integrating Equation (10) provides:  
\[x=\Psi R \tag{11}\]  
According to Equation (11), in the process of the robot rolling upward along the slope, the position and attitude of t at any time can be uniquely determined by the rolling angle \(\Psi\) of shell and the swing angle \(\varphi\). According to the Lagrange equation (Equation (3)), the robot system was divided into two parts: a mean thin-walled shell and a pendulum. The kinetic energy and potential energy of each part were calculated respectively, and then the kinetic energy and potential energy of the system were solved by superposition. The position coordinates of the robot were expressed as:  
\[\overset{\rightharpoonup}{Q}_{C}=x\overset{\rightharpoonup}{e}_{x}+R \overset{\rightharpoonup}{e}_{y} \tag{12}\]  
where: \(\overset{\rightharpoonup}{Q}_{C}\) is the vector from the coordinate origin A to the center of the sphere and \(\overset{\rightharpoonup}{e}_{x}\) and \(\overset{\rightharpoonup}{e}_{y}\) are unit vectors of ground coordinates. From Equation (12), the coordinate representation of the center of mass of the pendulum in the ground coordinate system was obtained:  
\[\overset{\rightharpoonup}{Q}_{p}=(x+L\sin\phi)\overset{\rightharpoonup}{e}_{ x}+(R-L\cos\phi)\overset{\rightharpoonup}{e}_{y} \tag{13}\]  
where: \(\overset{\rightharpoonup}{Q}_{p}\) is the vector form from the pendulum center of mass to center of the sphere. According to Formulas (11) and (12):  
\[\overset{\rightharpoonup}{Q}_{C}=\Psi R\overset{\rightharpoonup}{e}_{x}+R \overset{\rightharpoonup}{e}_{y} \tag{14}\]  
According to Formulas (11) and (13):  
\[\overset{\rightharpoonup}{Q}_{p}=(\Psi R+L\sin\varphi)\overset{\rightharpoonup }{e}_{x}+(R-L\cos\varphi)\overset{\rightharpoonup}{e}_{y} \tag{15}\]  
We calculated the derivative of time t on both sides of Equations (14) and (15), so the velocity vector of the ball center and the velocity vector of the pendulum were obtained as follows:  
\[\frac{d\overset{\rightharpoonup}{Q}_{C}}{dt}=\dot{\Psi}R\overset{\rightharpoonup }{e}_{x} \tag{16}\]  
\[\frac{d\overset{\rightharpoonup}{Q}_{p}}{dt}=\left(R\ddot{\Psi}+\varphi L\cos \varphi\right)\overset{\rightharpoonup}{e}_{x}+\varphi L\sin\varphi\overset{ \rightharpoonup}{e}_{y}, \tag{17}\]  
The kinetic energy \(E_{K1}\) of the shell included the translational kinetic energy \(E_{k1t}\) of the translational motion of the shell and the rotational kinetic energy \(E_{k1r}\) of the shell.