# Dynamic Balance of the Head in a Flexible Legged Robot for Efficient Biped Locomotion
## 3 Design and Validation of the Rollo Model
### Model Validation  
The modal properties computed using the numerical model described in Section 2.2 and the EMA identified modal properties (Figure 3 and Table 1) are compared to assess the model goodness. Modal properties are compared using the modal assurance criterion for complex modes shapes (MACX) [46]. MACX index is used to perform pairing between two sets of modal vectors. The MACX is a real value between 0 and 1; it is MACX = 1 when two mode shapes are exactly the same and 0 when two mode shapes are totally different. MACX is defined as follows:  
\begin{table}
\begin{tabular}{c l l l l} \hline \hline
**Mode** & **Description** & **Num. \(f_{n}\) (Hz)** & **Exp. \(f_{n}\) (Hz)** & **Exp. \(\zeta\) (\%)** \\ \hline
1 & 1 st legs bending out of phase lateral direction & 0.527 & 0.492 & 54.92 \\
2 & 1 st legs torsion in phase & 4.063 & 2.796 & 6.78 \\
3 & 1 st legs torsion out of phase & 4.147 & 4.715 & 1.51 \\
4 & 1 st legs axial out of phase & 6.471 & 5.316 & 5.94 \\
5 & 2 nd legs bending in phase & 7.356 & 8.744 & 3.91 \\
6 & 2 nd legs bending out of phase lateral direction & 8.008 & 7.964 & 2.28 \\
7 & 1 st legs axial in phase & 8.175 & 7.431 & 3.62 \\
8 & 2 nd legs bending out of phase motion direction & 10.05 & 11.91 & 5.01 \\
9 & 2 nd legs bending in phase motion direction & 12.64 & 17.95 & 6.38 \\
10 & 3 rd legs bending in phase lateral direction & 23.99 & 28.67 & 5.60 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rollo numerical natural frequencies and experimental natural frequencies and damping ratios.  
Figure 4: (**A**) Experimental setup and accelerometer locations, (**B**) excited nodes.  
\[\text{MACX}\Big{(}\mathbf{\phi}_{\text{j}},\mathbf{\phi}_{\text{k}}\Big{)}=\frac{\Big{(} \big{|}\mathbf{\phi}_{\text{j}}^{\text{H}}\,\mathbf{\phi}_{\text{k}}|\,+\,\big{|}\mathbf{\phi }_{\text{j}}^{\text{T}}\,\mathbf{\phi}_{\text{k}}\big{|}\Big{)}^{2}}{\Big{(}\mathbf{ \phi}_{\text{j}}^{\text{H}}\,\mathbf{\phi}_{\text{j}}+\big{|}\mathbf{\phi}_{\text{j}}^ {\text{T}}\,\mathbf{\phi}_{\text{j}}\big{|}\Big{)}\,\big{(}\mathbf{\phi}_{\text{k}}^{ \text{H}}\,\mathbf{\phi}_{\text{k}}+\big{|}\mathbf{\phi}_{\text{k}}^{\text{T}}\mathbf{\phi} _{\text{k}}\big{|}\Big{)}} \tag{4}\]  
where \(\mathbf{\phi}_{j}\) and \(\mathbf{\phi}_{k}\) are the two complex modal vectors to be compared, \((\bullet)^{T}\) is the transpose operator, and \((\bullet)^{\text{H}}\) is the Hermitian operator (complex conjugate transpose). A matrix of MACX values is obtained comparing the experimental and numerical sets of mode shapes. In the ideal case that the two sets are the same, a matrix with 100% in the matrix diagonal is expected. The MACX matrix obtained comparing the numerical and experimental mode-shapes of Figure 3 is shown in Figure 5, where the spacing between the modes is proportional to the modes natural frequencies. If the mode paring is close to the iso-frequency line (black dashed-dot line), the mode shapes are similar, and the natural frequencies are close.  
The MACX matrix in Figure 5 shows a good correlation between numerical and experimental modes: the MACX value is generally high and close to the iso-frequency line. A good correlation between experimental and numerical modes is achieved for both mode shapes and natural frequencies, highlighted from both the MACX matrix in Figure 5 and the mode shapes graphical comparison in Figure 3. An exception is the first mode, which is at a low natural frequency of about 0.5 Hz and the experimentally identified mode shape is not reliable. Furthermore, numerical and experimental mode shapes 6 and 8 are switched; however, their natural frequencies are pretty close; hence the result is reasonable. The good correlation between numerical and experimental modal properties validates the structural model developed in Lupos.