# Jumping with Expandable Trunk of a Metamorphic Quadruped Robot--The Origaker II
## 3 Design of the Expandable Trunk with Jumping Motion  
The sketch of the robot is showed in Figure 2. The lengths of links on the trunk are defined as \(AF=BC=DC=EF=l_{1}+l_{2}=l\) and \(AB=DE=l_{3}\). Where \(l_{1}\) and \(l_{2}\) represent the distances between the hip and the link joints. The hip joints are labeled with \(P_{i}\) (\(i=1,2,3,4\)). The offset distance from the center of each hip to the link on which it is mounted is labeled with \(e\). Every leg has three joints. The axes of leg joints are illustrated in Figure 2 as \(\mathbf{s}_{1}\), \(\mathbf{s}_{2}\) and \(\mathbf{s}_{3}\). The \(\mathbf{s}_{1}\) is perpendicular to \(\mathbf{s}_{2}\) and \(\mathbf{s}_{2}\) is parallel to \(\mathbf{s}_{3}\). The lengths of leg links are represented by \(d_{1}\), \(d_{2}\), \(d_{3}\) respectively. The plane which the planar 6-bar linkage lies on is defined as the trunk plane. The geometry parameters of the robot are set as shown in Table 1.  
Figure 1: The observation of jumping motion of the frog. (**a**) Contracting trunk; the frog is in a crouched position and ready to jump; (**b**) Begin expanding trunk; the frog lifts its fore legs and begins to jump; (**c**) Expanding trunk; the frog gradually expands its trunk and hind legs; (**d**) Expanding trunk; the frog completely expands its trunk and jumps into the air. As the planar 6-bar linkage has 3 degrees of freedom (DOF), joints \(A\), \(E\) and \(C\) are chosen as active joints to fully control the trunk. In order to perform the expanding motion during the robot jumps, these three angles should be restricted by the following equations. \[\angle A+\frac{\angle C}{2}=\pi,\angle E+\frac{\angle C}{2}=\pi \tag{1}\]  
Similar to the method of [PERSON] and [PERSON] [14], the hip location can be derived by the matrix transformation. The body coordinate system \(O\)-_xyz_ of the robot is established at the center of the link \(AB\), point \(O\), shown in Figure 2, where the \(z\)-axis is perpendicular to the plane of trunk, the \(y\)-axis is perpendicular to the link \(AB\) on the trunk plane and the \(x\)-axis is determined by the right-hand rule. The hip locations will change as the trunk expands. For convenience, the expandable angle \(\theta\) is defined as the angle between the link \(ED\) and link \(DC\), shown in the Figure 2. The smaller the expandable angle \(\theta\) is, the more the trunk will contract. Thus, the coordinates of the hip locations can be derived by the method of matrix transformation. Therefore, the transformation matrices \(\mathrm{T}_{i}\) of the hip joints \(P_{i}\) (\(i=1\), \(2\), \(3\), \(4\)) will be  
\[\mathrm{T}_{1}=T\bigg{(}x,\frac{l_{3}}{2}\bigg{)}R(z,\theta)T(x,l_{2})T(z,-e) R\bigg{(}y,-\frac{\pi}{2}\bigg{)}, \tag{2}\]  
\[\mathrm{T}_{2}=T\bigg{(}x,-\frac{l_{3}}{2}\bigg{)}R(z,\pi-\theta)T(x,l_{2})T(z,-e)R\bigg{(}y,-\frac{\pi}{2}\bigg{)}, \tag{3}\]  
\[\mathrm{T}_{3}=T(y,2l\sin\theta)T\bigg{(}x,\frac{l_{3}}{2}\bigg{)}R(z,-\theta )T(x,l_{2})T(z,-e)R\bigg{(}y,-\frac{\pi}{2}\bigg{)}, \tag{4}\]  
\[\mathrm{T}_{4}=T(y,2l\sin\theta)T\bigg{(}x,-\frac{l_{3}}{2}\bigg{)}R(z,\theta) T(y,-l_{2})T(z,-e)R\bigg{(}x,-\frac{\pi}{2}\bigg{)}, \tag{5}\]  
where \(T\)(\(a\),\(b\)) represents translating along the axis \(a\) by distance \(b\) and \(R\)(\(\alpha\),\(\beta\)) represents rotating about the axis \(\alpha\) by angle \(\beta\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathrm{I}_{1}\)/**mm** & \(\mathrm{I}_{2}\)/**mm** & \(\mathrm{I}_{3}\)/**mm** & \(\mathrm{\varepsilon}\)/**mm** & \(\mathrm{\textit{d}_{1}/mm}\) & \(\mathrm{\textit{d}_{2}/mm}\) & \(\mathrm{\textit{d}_{3}/mm}\) \\ \hline
125 & 55 & 108 & 35 & 46 & 54 & 110 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: parameters of the metamorphic robot. Figure 2: Sketch of the Origaker II. Hence the hip locations \(\mathbf{P}_{i}\) (\(I=1,\,2,\,3,\,4\)) can be derived as below. \[\mathbf{P}_{1}=\left[ \begin{array}{cc}\frac{l_{1}}{2}+l_{2}\cos\theta&l_{2}\sin \theta&-e\end{array} \right]^{\text{T}}, \tag{6}\]  
\[\mathbf{P}_{2}=\left[-