## Natural Frequencies and Modes of Vibration  
### Modes of Vibration Due to Elastic Deformability  
According to the previously mentioned assumptions, the proposed model consists of lumped masses connected by elastic elements, as shown in Figure 2. The first of these lumped masses (\(m_{top}\)) is located in the center of the spine; it represents the motor and spine masses, and a share (50%) of leg masses. The moment of inertia \(I_{top}\) is the moment of inertia about the center of the spine. The other two masses (\(m_{f}\) and \(m_{h}\)) represent the masses of the front and hind feet and a share (50%) of leg masses. They are still. The length of the spine is denoted by \(L_{h}\), the lengths of the front and hind legs are \(L_{v}\), and the angles between the legs and the vertical direction are \(\theta_{f}\) and \(\theta_{h}\), respectively. The spine and the legs are connected by torsional springs with stiffness \(k_{\theta}\). The legs of the robot are modeled as linear springs with stiffness \(k_{L}\). When the robot is in contact with the ground and friction prevents any tangential motion between the feet and the ground itself, the legs with the spine and the ground make up a flexible four-bar linkage [40]. Figure 2 shows that the configuration of this linkage depends on five variables: the rotations of the two legs with respect to the \(y\)-axis \(\left(\theta_{h},\ \theta_{f}\right)\), the rotation of the spine with respect to the \(x\)-axis (\(\varphi\)), and the lengths of the two legs (\(L_{v}+s_{h}\), \(L_{v}+s_{f}\)), in which \(s_{h}\) and \(s_{f}\) are length variations due to axial deformability. Actually, these five parameters are linked by the closure equation of the linkage and only three variables are independent. The components of the closure equation in \(x\) and \(y\) directions are:  
\[(L_{v}+s_{h})\cos\Bigl{(}\theta_{h}+\frac{\pi}{2}\Bigr{)}+L_{h}cos(\varphi)- \Bigl{(}L_{v}+s_{f}\Bigr{)}\cos\Bigl{(}\theta_{f}+\frac{\pi}{2}\Bigr{)}-L_{h}=0 \tag{1}\]  
\[(L_{v}+s_{h})\sin\Bigl{(}\theta_{h}+\frac{\pi}{2}\Bigr{)}+L_{h}sin(\varphi)- \Bigl{(}L_{v}+s_{f}\Bigr{)}\sin\Bigl{(}\theta_{f}+\frac{\pi}{2}\Bigr{)}=0 \tag{2}\]  
Figure 2: Model for the study of the modes of vibration on the ground. and expanding the trigonometric functions:  
\[-(L_{v}+s_{h})\sin(\theta_{h})+L_{h}cos(\varphi)+\left(L_{v}+s_{f}\right)\sin( \theta_{f})-L_{h}=0 \tag{3}\]  
\[(L_{v}+s_{h})\cos(\theta_{h})+L_{h}sin(\varphi)-\left(L_{v}+s_{f}\right)\cos( \theta_{f})=0 \tag{4}\]  
Since the variations in the lengths and angles caused by robot elastic deformation are small, these equations can be linearized around the reference configuration:  
\[-L_{v}\theta_{h}+L_{h}+L_{v}\theta_{f}-L_{h}=0 \tag{5}\]  
\[L_{v}+s_{h}+L_{h}\varphi-L_{v}-s_{f}=0 \tag{6}\]  
Equation (5) leads to:  
\[\theta_{h}=\theta_{f}\equiv\theta \tag{7}\]  
Coordinate \(\theta\) is associated with the torsion angle of the linkage. Equation (6) links three variables (\(s_{h}\), \(s_{f}\), and \(\varphi\)). Therefore, two independent parameters can be chosen:  
\[\frac{s_{f}-s_{h}}{L_{h}}=\varphi \tag{8}\]  
\[(L_{v}+s_{h})\cos\Bigl{(}\theta_{h}+\frac{\pi}{2}\Bigr{)}+L_{h}cos(\varphi)- \Bigl{(}L_{v}+s_{f}\Bigr{)}\cos\Bigl{(}\theta_{f}+\frac{\pi}{2}\Bigr{)}-L_{h}=0 \tag{1}\]0  
Variable \(s\) represents the heave of the spine along the leg's direction, which may be tilted by angle \(\theta\). Variables \(\theta\), \(s\), and \(\varphi\) are the generalized coordinates describing the motion of the deformable linkage. The equations of the vibrations of the robot can be derived with the [PERSON]'s method [41]:  
\[(L_{v}+s_{h})\cos\Bigl{(}\theta_{h}+\frac{\pi}{2}\Bigr{)}+L_{h}cos(\varphi)- \Bigl{(}L_{v}+s_{f}\Bigr{)}\cos\Bigl{(}\theta_{f}+\frac{\pi}{2}\Bigr{)}-L_{h}=0 \tag{1}\]1  
in which \(q_{1}=\theta\), \(q_{2}=s\), and \(q_{3}=\varphi\). The Lagrangian components of forces (\(Q_{k}\)) include the forces generated by the eccentric mass and are zero when free vibrations are considered.