## 2 Materials and Methods
### Estimation of Finite Finger Joint CoRs  
For the estimation of the finite CoRs of a finger joint, proximal bone surfaces of the joint at different postures were aligned to measure the relative motions of the distal bone of the joint from one posture to another, as illustrated in Figure 6. The alignment of the proximal bone surfaces of the joint was performed using the above-mentioned 3-point alignment method in RapidForm 2006. The proximal bone surfaces of the joint were exactly aligned with each other since the bone surfaces were the same.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Posture** & **RMSE** & **Posture** & **RMSE** & **Posture** & **RMSE** \\ \hline \multirow{2}{*}{2} & 0.001 & \multirow{2}{*}{5} & 0.002 & \multirow{2}{*}{8} & 0.002 \\  & (\(\pm\) 0.001) & & (\(\pm\) 0.001) & & (\(\pm\) 0.001) \\ \multirow{2}{*}{3} & 0.002 & \multirow{2}{*}{6} & 0.002 & \multirow{2}{*}{9} & 0.001 \\  & (\(\pm\) 0.002) & & (\(\pm\) 0.001) & & (\(\pm\) 0) \\ \multirow{2}{*}{4} & 0.001 & \multirow{2}{*}{7} & 0.002 & \multirow{2}{*}{10} & 0.002 \\  & (\(\pm\) 0) & & (\(\pm\) 0.003) & & (\(\pm\) 0.001) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mean (\(\pm\) S.D.) registration accuracy of the hand skeletons (\(n=15\)) in the template posture (posture 1) to those in the other postures (postures 2 to 10) measured by root mean square error (RMSE) of Hausdorff distance (unit: mm).  
Figure 4: Registration of each bone in the hand skeleton of posture 1 (template posture) to the corresponding bones in the hand skeletons of the other nine postures.  
Figure 5: Registration of a bone in the template posture to that in a target posture: (**a**) Selection of three points at a similar position, each from the template bone surface and the target bone surface; (**b**) Initial registration by aligning the selected points; (**c**) Fine registration.  
After the proximal bone surfaces were aligned, the centerlines of the distal bone surfaces of the joint at different postures were identified. First, a preliminary centerline of the distal bone surface at a posture was obtained using the principal component analysis (PCA) method. The PCA method consists of four steps: (1) calculation of the mean value of each of the three coordinates of all the vertices at the distal bone surface and subtraction of the corresponding mean value from each of the three coordinates of all the vertices to obtain new coordinates for the vertices, (2) calculation of a covariance matrix for the new coordinates, (3) calculation of the eigenvectors and eigenvalues of the covariance matrix, and (4) sorting the eigenvalues in a descending order to obtain the largest eigenvalue. The preliminary centerline of the distal bone surface was then identified as the line going through the axis determined by the corresponding eigenvector of the largest eigenvalue. Then, the fine centerline (Figure 7) of the distal bone surface was obtained by performing a linear fit of the centroids of the surface vertices at the perpendicular planes to the preliminary centerline along the distal bone shaft, the part of a bone excluding the two ends of the bone.  
The finite finger joint CoR from one posture to another was determined as the intersection point of the centerlines of the distal bone surfaces from the two postures, as shown in Figure 8. A program was coded by Matlab R2017b (MathWorks, Inc., Natick, MA, USA) to perform the procedure of finding the centerlines of the distal bone surfaces and finite finger joint CoRs.  
Figure 6: Alignment of the proximal bone surfaces of a joint at two postures to measure the relative motion of the distal bone of the joint between the two postures.  
Figure 7: Identification of the centerline for the distal bone of a joint by performing a linear fit of the centroids (red) of the surface vertices at the perpendicular planes to a preliminary centerline along the distal bone shaft.