# Jumping with Expandable Trunk of a Metamorphic Quadruped Robot--The Origaker II
## 4 Effects of the Expandable Trunk on Jumping Motion
### Kinetic Energy of the Origaker II  
The kinetic energy is calculated by the dynamic method. Due to that the motions of legs of the two robots are the same, the dynamic analysis is focused on the trunk motion. Figure 6 presents the mass of each link. Where the mass of the link \(AB\) and \(DE\) is symbolized as \(m_{1}\), the mass of the link \(BC\) and \(DF\) is symbolized as \(m_{2}\), the mass of the link \(CD\) and \(FA\) is symbolized as \(m_{3}\) and the \(\theta\) expresses the generalized coordinates.  
For every link, the kinetic energy could be calculated by the following formula.  
\[E_{ki}=\frac{1}{2}m_{i}v_{si}^{2}+\frac{1}{2}J_{si}\omega_{si}^{2} \tag{20}\]  
where \(m_{i}\) is the mass of link \(i\), \(v_{si}\) is the velocity of the center of mass of link \(i\), \(J_{si}\) represents the moment of inertia of the link \(i\) relative to the center of mass and \(\omega_{si}\) means the angular velocity of link \(i\).  
Figure 6: The parameters of the robot trunk.  
The sum of the kinetic energy of all the links could be calculated.  
\[\begin{array}{ll}E_{k}&=\frac{1}{2}m_{1}(2l)^{2}\omega^{2}\cos^{2}\theta+\frac{ (m_{2}+m_{3})l^{2}}{12}\omega^{2}+(m_{2}+m_{3})\frac{l_{\omega}l_{\omega}}{4} \Big{(}1+4\cos^{2}\theta\Big{)}\\ &=(2m_{1}+m_{2}+m_{3})l^{2}\omega^{2}\cos^{2}\theta+\frac{(m_{2}+m_{3})l^{2}}{ 3}\omega^{2}\end{array} \tag{21}\]  
where \(\omega\) represents the angular velocity of \(\theta\).  
It is obvious that the kinetic energy increases with the length and the angular velocity of link \(CD\). The jumping stage starts at \(\theta=\theta_{1}\), and ends at \(\theta=90^{\circ}\). Assume that the angular velocity \(\omega\) during the jumping stage is constant. According to the theorem of kinetic energy, the work which has done by the motors on the trunk is  
\[\begin{array}{ll}W&=\Delta E_{k}=(2m_{1}+m_{2}+m_{3})l^{2}\omega^{2}\cos^{ 2}\theta_{1}+\frac{(m_{2}+m_{3})l^{2}}{3}\omega^{2}\\ &-\Big{(}(2m_{1}+m_{2}+m_{3})l^{2}\omega^{2}\cos^{2}90^{\circ}+\frac{(m_{2}+m_ {3})l^{2}}{3}\omega^{2}\Big{)}\\ &=(2m_{1}+m_{2}+m_{3})l^{2}\omega^{2}\cos^{2}\theta_{1}\end{array} \tag{22}\]  
The trunk work decreases with the increase of \(\theta_{1}\), shown in Figure 7. Under the condition of the constant angular velocity, at the beginning of jumping stage, the smaller \(\theta_{1}\) leads to the bigger work. It means that the expandable trunk outputs more work because of the motion design of the smaller \(\theta_{1}\). However, if the \(\theta_{1}\) is too small, the front legs will run into the hind legs and the motion of robot might be ruined. After plenty of tests, the initial angle \(\theta_{1}\) is chosen as \(60^{\circ}\), which ensures the correct motion of the Origaker II.  
After the above analyses, it can be found that the expandable trunk can provide a greater speed, a better elevation angle and extra work so that the robot has a good jumping performance. On the contrary, these analyses also guide our structure design and motion design, such as the smaller initial angle \(\theta_{1}\) and the hip joints farther from the COG.