# Dynamic Balance of the Head in a Flexible Legged Robot for Efficient Biped Locomotion
## 3 Design and Validation of the Rollo Model
### Numerical Modal Analysis of the Rollo Robot  
Modal analysis [41] is a well-established mathematical technique to find out the modal properties of a mechanical system. Numerical modal analysis (FEA) is performed on the undamped Rollo model [41, 42]. Global mass \(\mathbf{M}_{S}\) and stiffness \(\mathbf{K}_{S}\) matrices are obtained from the [PERSON] model, which uses classical FEM matrix assembly technique [43] to generate the global matrices. The total number of degrees-of-freedom (dofs) is \(N=3432\), thus \(\mathbf{M}_{g},\mathbf{K}_{g^{\prime}}\in\mathbb{R}^{N,N}\). Numerical modal analysis is performed on the reduced system considering only the \(n\) active dofs. The reduction from the global number of dofs \(N\) to the active dofs \(n\) considers the slave dofs depending on other masters (and active) dofs, through kinematic relationships, and dofs on which boundary conditions are applied. The reduced model posses a total of \(n=2382\) active/independent dofs. The mass \(\mathbf{M}\in\mathbb{R}^{n,n}\) and stiffness \(\mathbf{K}\in\mathbb{R}^{n,n}\) describing the dynamic of the independent dofs are obtained from the global matrices as:  
\[\mathbf{M}=\mathbf{T}^{T}\mathbf{M}_{g}\mathbf{T}\qquad\mathbf{K}=\mathbf{T}^ {T}\mathbf{K}_{g}\mathbf{T} \tag{1}\]  
where \(\mathbf{T}\in\mathbb{R}^{N,n}\) is a transformation matrix taking into account kinematic master/slave relationships and boundary conditions. The discrepancies between the total number of dofs \(N\) and the independent number of dofs \(n\) is mainly due to the rigid connections used to lock the deformation of two spires for each extremity of each spring, as discussed in Appendix A. The eigenproblem in Equation (2) is solved for the \(R=20\) lowest numerical natural frequencies \(\omega_{r}\) and mode shapes \(\mathbf{\phi}_{r}\in\mathbb{R}^{n,1}\):  
\[(\mathbf{K}-\omega_{r}^{2}\mathbf{M})\mathbf{\phi}_{r}=\mathbf{0}\qquad r=1,...,20 \tag{2}\]  
The problem is solved in almost free-free conditions, although soft lumped springs are linked to the middle of the top beam to replicate the experimental setup, see Section 3.2. Mode shapes \(\mathbf{\phi}_{r}\) computed on the \(n\) active dofs are expanded to the total number of dofs \(N\):  
\[\mathbf{\varphi}_{r}=\mathbf{T}\mathbf{\phi}_{r} \tag{3}\]  
where \(\mathbf{\varphi}_{r}\in\mathbb{R}^{N,1}\) is the \(r\)th mode shape expanded to the system global dimension \(N\). The numerical natural frequencies \(f_{n}=\frac{1}{2\pi}\omega_{n}\) of flexible modes are listed in Table 1 and the numerical mode shapes \(\mathbf{\varphi}_{r}\), obtained through Equations (2) and (3), are shown in Figure 3 (on the left side of each sub-figure).  
The mode shapes \(\mathbf{\varphi}_{r}\) represent the vibration shape in terms of displacement, velocity, and accelerations at each natural frequency of the system. They are defined unless of a multiplicative constant; therefore, the absolute value is not essential, but only the overall shape makes sense. In the figure, the colours represent the amount of deviation of the mode shape with respect to the undeformed configuration: cold colours, i.e., toward blue, represent a portion that is not moving, while hot colours represent portions of the system which are increasingly vibrating from blue to red.