## 3 Results and Analysis
### Identified Domain of Tone Ratios
#### 3.2.2 Case of Three-Tone Ratios  
Figure 4 visualizes the trunk stiffness, for the five directions of position perturbation, as functions of the tone ratios for classes C1 and C2. For each direction \(j\), the shown  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Classes** & **Muscles** \\ \hline C1 & RA—(2)/EO—(6)/LT—(24)/Ilpt—(16)/Ltpl—(10)/Ilp—(8) \\ \hline C2 & TM—(10)/M—(40) \\ \hline C3 & PM—(22)/QL—(36) \\ \hline \hline \end{tabular} Muscles abbreviations: RA: rectus abdominus, PM: pososa major, QL: quadratus lumborum, EO: external oblique, LT: longissimus thoracic, Ilpt: illocostalis lumborum pars thoracis, Ltpl: longissimus thoracis pars lumborum, Ilpl: illocostalis lumborum pars lumborum, TM: thoracic multifidus and M: multifidus. Numbers of shortened muscles are put between parentheses when muscles for left (L) and right (R) sides are identical.  
\end{table}
Table 2: Muscles of the three classes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{3}{c}{**RMTTs (\%)**} \\ \cline{2-3}
**Direction \#** & \(\beta_{min}\) & \(\beta_{max}\) \\ \hline
1 & 6.7 & 11.9 \\ \hline
2 & 5.4 & 10.8 \\ \hline
3 & 4.8 & 10.2 \\ \hline
4 & 8.6 & 31 \\ \hline
5 & 9.6 & 48 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Tone lower and upper limits for the five directions.  
three surfaces describe the variation of the model stiffnesses \(K_{min,j}\) and \(K_{max,j}\) and the experimental stiffness \(K_{exp,j}\) for \(\beta_{1}\) and \(\beta_{2}\) in the data range [0.05, 0.3] and a fixed value \(\beta_{3}\) = 0.15 (tone ratio for class C3).  
Figure 4: Experimental, minimal, and maximal trunk stiffness as functions of the tone ratios for classes C1 and C2 and directions 1 (**a**) and 3 (**b**), respectively. The class C3 tone ratio (\(\beta^{3}\)) is set to 15%.  
The plane representing the constant experimental trunk stiffness intersects the surfaces associated with the maximum and minimum model stiffnesses at two lines. These intersections are determined numerically for the five directions (Figure 4). The projections of the intersection lines onto the (\(\beta^{1}\), \(\beta^{2}\)) plane, as depicted in Figure 5, delimit a plane section of the 3D solution domain \(D\), feasible for the five directions. For instance, the section marked with yellow fill corresponds to a cut by the planes \(\beta^{3}=0.15\).  
Figure 6 visualizes the plane sections (Figure 6a) of the solution domain \(D\) for eight values of class C3 tone ratio, ranging between 2% and 33.5%, and the constructed domain \(D\) (Figure 6b). Each of the outer sections is determined through extrapolation of the vertex nodes from the second and third nearest end sections.  
The tone ratios of the external (C1) and the small internal (C2) muscles are found to lie, globally, in the intervals [5%, 12%] and [6%, 33%], respectively, as those of the large internal (C3) muscles are varied between 2% and 33.5%. More accurate, narrower ranges can be obtained over a section of \(D\), relative to a selected value of class C3 muscle tone ratio.  
There is a clear dependency between \(\beta_{mi}\). The more the \(\beta_{m}\) of the small internal muscles increases, the more the area of the solution domain is reduced.  
Figure 5: Lines where surfaces of experimental trunk stiffness intersect with those of maximum and minimum stiffnesses, for the five directions of displacement perturbation, with \(\beta^{3}\) set to 0.15. The yellow area represents the feasible solutions \(\beta^{1}\) and \(\beta^{2}\) for the given value of \(\beta^{3}\).