# Jumping with Expandable Trunk of a Metamorphic Quadruped Robot--The Origaker II
## 4 Effects of the Expandable Trunk on Jumping Motion
### Launching Velocity of the Origaker II  
At any moment, the left and right legs perform the same actions. Therefore, the forces, velocities and accelerations provided by the left and right legs are all bilaterally symmetry.  
The posture at the moment when the robot launches is shown in Figure 4a. The trunk plane is represented by the plane \(\Sigma_{1}\). The points \(O_{3}\) and \(O_{4}\) on the plane \(\Sigma_{1}\) are the projections of hip joints \(P_{3}\) and \(P_{4}\). The COG (center of gravity) of the robot is expressed by the point \(G\) on the plane \(\Sigma_{1}\). The moving reference frame is established at point \(O_{3}\). Therefore, the absolute velocity \(\mathbf{v}_{G}\) of the point \(G\) relative to the ground, that is velocity of the robot, can be obtained by adding vectorially the convected velocity \(\mathbf{v}_{e}\) of the convected point \(O_{3}\) and the relative velocity \(\mathbf{v}_{r}\) of the particle \(G\) with respect to the convected point \(O_{3}\). That is  
\[v_{\rm G}=v_{\rm e}+v_{\rm r} \tag{10}\]At the time when robot jumps, the left hind leg provides the velocity at \(O_{3}\) relative to the ground. That is \(\mathbf{v}_{c}\).  
Because the point \(G\) is on the plane \(\Sigma_{1}\), the velocity analysis is taken on the \(\Sigma_{1}\), demonstrated in Figure 4b. The velocities \(v_{\text{Gp}}\) and \(v_{\text{ep}}\) on the plane \(\Sigma_{1}\) are the projections of \(v_{\text{G}}\) and \(v_{\text{e}}\). A moving reference coordinate system \(O_{3}\)-\(x_{3}y_{3}z_{3}\) is established at \(O_{3}\), where \(x_{3}\)-axis is parallel to the link \(DE\) and \(y_{3}\)-axis is perpendicular to the \(x_{3}\)-axis and the \(z_{3}\)-axis is determined by the right hand rule. The coordinate of point \(G\) on this coordinate system is  
\[G\bigg{(}\frac{l_{3}}{2}+l_{2}\cos\theta,l_{1}\sin\theta\bigg{)} \tag{11}\]  
where \(\theta\) is the expandable angle between link \(CD\) and \(x_{3}\)-axis. After operating derivations of the point \(G\) coordinate about time \(t\), the relative velocity \(\mathbf{v}_{r}\) of \(G\) with respect to the convected point \(O_{3}\) can be derived as:  
\[v_{r}=\frac{\text{d}}{\text{d}t}G=(-l_{2}\sin\theta\theta^{\prime},l_{1}\cos \theta\theta^{\prime}) \tag{12}\]  
Therefore, velocity of the robot \(\mathbf{v}_{G}\) can be obtained by adding vectorially the convected velocity \(\mathbf{v}_{e}\) and the relative velocity \(\mathbf{v}_{r}\). As Equation (12) shows, it can be found that the trunk motion provides a velocity component in the forward direction (\(y_{3}\)-axis), \(l_{1}\text{cos}\theta\theta^{\prime}\). What's more, a bigger angular velocity \(\theta^{\prime}\) could increase this velocity component. The faster the trunk expands, the bigger forward velocity component is.  
The longer \(l_{1}\) will lead to the larger forward velocity. That means if we want a larger forward velocity, we can mount the hip joints nearer to the point \(A\), \(B\), \(D\) and \(E\), respectively. That is consistent with the conclusion in [14] about the effect of hip location on stride length.  
For the rigid trunk robot, the forward velocities are only provided by the legs. In contrast, for the Origaker II, both the legs and trunk offer forward velocities when it jumps. The forward velocities of the Origaker II and the rigid trunk robot provided by the legs are the same based on motion design. Thus, an extra velocity component is added by the trunk motion of the Origaker II in the forward direction. This extra velocity increases the launching velocity compared with the rigid trunk robot. That is a significant effect of the trunk motion.  
Figure 4: the velocity decomposition of the robot. (**a**) The velocity of the leg and its projection on the plane \(\Sigma_{1}\); (**b**) The moving reference frame on the plane \(\Sigma_{1}\) and the relative velocity.