## 2 Materials and Methods
### Tone Identification Problem Formulation  
The tone identification relies on published experimental trunk stiffness data, collected from multidirectional trunk tests [18]. The muscle tones are treated as input parameters in the numerical trunk model. Since the position perturbations ideally start from the same sitting posture the muscle tones to be identified are accordingly presumed to be independent of the direction of perturbation. Thus, the tones are sought such that the model-generated trunk stiffness matches the experimental stiffness \(K_{exp,j}\) for each (the \(j\)-th) of the five directions of displacement perturbation. Given the uncertainty associated with the loss of muscle activation during the displacement perturbation, the model stiffness  
Figure 2: Trunk musculoskeletal model. Muscles of each group are with the same color. Bones are with brown color. can only be approximated. Therefore, instead of attempting to match the experimental with the model stiffness, we will only require it to lie within the interval \([K_{min,j},K_{max,j}]\) of model trunk stiffness. As a result, the identification amounts to determining a domain \(D\) of \"feasible\" RMTT values. The boundary of \(D\) is constructed pointwise by finding it intersections with selected lines crossing it. Along a line in the space T, the RMTTs are represented by a scalar parameter \(\alpha\). A line crossing the domain \(D\) intersects it with a segment defined by an interval \([\alpha_{min},\alpha_{max}]\). The bounds \(\alpha_{min}\) and \(\alpha_{max}\) are solutions to the problems \(\max\{\alpha_{j}/K_{min,j}(\alpha)=K_{exp,j},j=1,\ldots,5\}\) and \(\min\{\alpha_{j}/K_{max,j}(\alpha_{j})=K_{exp,j},j=1,\ldots,5\}\), respectively. The search process involves a repetitive calculation of the trunk stiffness bounds \(K_{min,j}\) and \(K_{max,j}\). This requires solving a series of linear equilibrium problems (Equation (1)) for the trunk subjected to small horizontal perturbations in the specified directions. For a given set \(\beta\) of RMTTs and a selected set of muscles assumed to remain active during the perturbation, the equilibrium problem for the \(j\)-th direction is stated as:  
\[[K(\{k\})]\{\cup\}=\{\Delta F_{j}\} \tag{1}\]  
\[k_{m}=q\ \beta_{m}\ F_{m0,max}/L_{m}\ m=1,\ldots,M_{a}\]  
where  
\([K]\) denotes the stiffness matrix of the trunk system;  
\(k_{m}\): the stiffness of the \(m\)th retained muscle;  
\(\{k\}\): the M-vector of retained muscle stiffnesses;  
\(\{\cup\}\): the vector of nodal displacements (output parameter);  
\([\Delta F_{j}]\): the perturbation force vector representing a single horizontal force applied at  
\(\mathrm{T10}\) in the \(m\)-th direction;  
\(q\): a dimensionless coefficient taken equal to 10 [10; 24];  
\(\beta_{m}\): the RMTT of the \(m\)-th muscle;  
\(F_{m0,max}\): the maximal force developed by the \(m\)-th muscle;  
\(L_{m}\): the length of the \(m\)-th muscle. The product \(\beta_{m}\ F_{0m}\) expresses the active muscle force, that is, the muscle tone \(T_{m}\) in the present case of free sitting posture. The trunk flexural stiffness \(K_{j}\), in the \(j\)-th direction, is deduced from  
\[K_{j}=\Delta F_{j}/\delta_{j} \tag{2}\]  
For the selected set of retained active muscles, where \(\Delta F_{j}\) is the horizontal force applied at T10 in the direction \(j\), and \(\delta_{j}\) the associated displacement component in the solution \(\{U\}\). The calculated stiffness \(K_{j}\) corresponds to the upper bound \(K_{max,j}\) when all the muscles are assumed to remain active, and to the lower bound \(K_{min,j}\) when all the shortened muscles are suspended from the model. The shortened muscles are identified from the solution of the equilibrium problem, and run with all the muscles assumed to be active. The tone identification is conducted according to two distinct formulations. In the first, it is carried out with the crude assumption that the RMTT is identical for all muscles.