## 2 Materials and Methods
### Theoretical Background  
It is current in biomechanics to represent the resulting ground reaction forces as a vector located under the foot at the COP. The COP is defined as the barycenter of the vertical pressure under the foot. It does not take into account friction forces in the horizontal plane (XZ). Thus, at the COP, the ground reaction moments are null alongside X and Z. Only the vertical free moment is not null. However, the COP is not the point where the moment is minimal. At this point Q of the ground, the moment is not vertical but collinear to the reaction forces. Let us denote \(\mathbf{M^{Q}}\) as the moment at this point and \(\mathbf{R}\) as the reaction force. \[\mathbf{M^{Q}}=\lambda\mathbf{R} \tag{1}\]  
where \(\lambda\) is a given real value. According to the [PERSON] theorem, we can compute the moment at another point P:  
\[\mathbf{M^{P}}=\mathbf{M^{Q}}+\mathbf{R}\times\mathbf{Q}\mathbf{P}=\lambda \mathbf{R}+\mathbf{R}\times\mathbf{Q}\mathbf{P} \tag{2}\]  
where \(\times\) denotes the cross product. The square of Equation (2) leads to:  
\[(\mathbf{M^{P}})^{2}=(\lambda\mathbf{R}+\mathbf{R}\times\mathbf{Q}\mathbf{P} )^{2}=(\lambda\mathbf{R})^{2}+2\lambda\mathbf{R}.(\mathbf{R}\times\mathbf{Q} \mathbf{P})+(\mathbf{R}\times\mathbf{Q}\mathbf{P})^{2} \tag{3}\]where. denotes the scalar product. It is clear that \(2\lambda\mathbf{R}. (\mathbf{R}\times\mathbf{Q}\mathbf{P})=0\) since \(\mathbf{R}\times\mathbf{Q}\mathbf{P}\) is orthogonal to \(\mathbf{R}\) Finally:  
\[(\mathbf{M}^{\mathrm{P}})^{2}=(\lambda\mathbf{R})^{2}+(\mathbf{R}\times\mathbf{ Q}\mathbf{P})^{2}=(\mathbf{M}^{\mathrm{Q}})^{2}+(\mathbf{R}\times\mathbf{Q} \mathbf{P})^{2} \tag{4}\]  
and  
\[(\mathbf{M}^{\mathrm{P}})^{2}\geq(\mathbf{M}^{\mathrm{Q}})^{2} \tag{5}\]  
Thus, the square-norm of the moment at point Q is minimal. One can note that this moment is constant alongside an axis colinear to \(\mathbf{R}\), named minimal moment axis (MMA). Indeed, if M is a point along this axis:  
\[\mathbf{Q}\mathbf{M}=\mathrm{a}\mathbf{R}\quad\text{ with }\mathrm{a}\in[- \infty,+\infty] \tag{6}\]  
\[\mathbf{M}^{\mathrm{M}}=\mathbf{M}^{\mathrm{Q}}+\mathbf{R}\times\mathbf{Q} \mathbf{M}=\mathbf{M}^{\mathrm{Q}}+\mathbf{R}\times(\mathrm{a}\mathbf{R})= \mathbf{M}^{\mathrm{Q}} \tag{7}\]  
Now, if A is any point in space, it is possible to localize this minimal moment axis according to A. We define the point B as:  
\[\mathbf{A}\mathbf{B}=(\mathbf{R}\times\mathbf{M}^{\mathrm{A}})/\mathbf{R}^{2} \tag{8}\]  
The moment at point B is:  
\[\mathbf{M}^{\mathrm{B}}=\mathbf{M}^{\mathrm{A}}+\mathbf{R}\times\mathbf{A} \mathbf{B}=\mathbf{M}^{\mathrm{A}}+\mathbf{R}\times(\mathbf{R}\times\mathbf{M} ^{\mathrm{A}})/\mathbf{R}^{2} \tag{9}\]  
Using the properties of the double cross product:  
\[\mathbf{R}\times(\mathbf{R}\times\mathbf{M}^{\mathrm{A}})/\mathbf{R}^{2}=[( \mathbf{R}.\mathbf{M}^{\mathrm{A}})\mathbf{R}-(\mathbf{R}.\mathbf{R})\mathbf{ M}^{\mathrm{A}}]/\mathbf{R}^{2}=(\mathbf{R}.\mathbf{M}^{\mathrm{A}})\mathbf{R}/ \mathbf{R}^{2}-\mathbf{M}^{\mathrm{A}}=\lambda\mathbf{R}-\mathbf{M}^{\mathrm{A}} \tag{10}\]  
Considering \(\lambda=(\mathbf{R}.\mathbf{M}^{\mathrm{A}})/\mathbf{R}^{2}\). Finally, the moment at point B is:  
\[\mathbf{M^{P}}=\mathbf{M^{Q}}+\mathbf{R}\times\mathbf{Q}\mathbf{P}=\lambda \mathbf{R}+\mathbf{R}\times\mathbf{Q}\mathbf{P} \tag{2}\]0  
Accordingly, it is possible to determine one point of the MMA from any other point given the reaction force and the moment at this point. Moreover, this MMA exists regardless of the conditions of the gait. Indeed, this axis is not defined only when walking on flat ground like the COP and can take into account all external mechanical actions: swing phase, double stance, contact with a cane, staircase ramp, etc. Moreover, the vector \(\mathbf{A}\mathbf{B}=(\mathbf{R}\times\mathbf{M}^{\mathrm{A}})/\mathbf{R}^{2}\) represents the shortest distance between A and the MMA (the orthogonal projection of A on the MMA).