## 2 Materials and Methods
### Analytic Model
#### 2.1.2 Trial Measurements
Algorithms used for calculating proximal forces in the GCS are shown in Equations (9a), (9b), and (9c):  
\[F_{XP}-\;F_{XD}=ma_{x} \tag{9a}\] \[F_{YP}-\;F_{YD}=ma_{y}\] (9b) \[F_{ZP}-\;F_{ZD}-mg=ma_{z} \tag{9c}\]  
where \(F_{XP}\), \(F_{YP}\), \(F_{ZP}\) and \(F_{XD}\), \(F_{YD}\), \(F_{ZD}\) are segments' proximal and distal forces in the \(x\), \(y\), and \(z\) directions on the \(i\)-th sample expressed in GCS, respectively, \(g\) is the gravitational acceleration, \(m\) is the segment mass, and \(a_{x}\), \(a_{y}\), and \(a_{z}\) are segment \(CoM\) linear accelerations on the \(i\)-th sample expressed in GCS, respectively. Forces were transformed to the ACS to describe loads in the joint as shown in Equation (10):  
\[\big{[}F_{xp}\;F_{yp}\;F_{zp}\;\big{]}=[GA_{i}][F_{XP}\;F_{YP}\;F_{ZP}\;] \tag{10}\]  
where \(F_{xp}\), \(F_{yp}\), and \(F_{zp}\) are segment proximal forces in the \(x\), \(y\), and \(z\) directions on the \(i\)-th sample expressed in the ACS, respectively. Torques were calculated in the ACS as shown in Equations (11a), (11b), and (11c):  
\[I_{x}\alpha_{x}+\big{(}I_{y}-I_{z}\big{)}\omega_{z}\omega_{y}=\;F_{yd}l_{d}+F _{yp}l_{p}+T_{xp}-T_{xd} \tag{11a}\] \[I_{y}\alpha_{y}+\big{(}I_{z}-I_{x}\big{)}\omega_{x}\omega_{z}=\;F_{xd}l_{d}+F _{xp}l_{p}+T_{yp}-T_{yd}\] (11b) \[I_{z}\alpha_{z}+\big{(}I_{x}-I_{y}\big{)}\omega_{x}\omega_{y}=\;T_{zp}-T_{zd} \tag{11c}\]  
where \(I_{x}\), \(I_{y}\), and \(I_{z}\) are the constant segment moments of inertia in the \(x\), \(y\), and \(z\) directions expressed in ACS, respectively, \(\alpha_{x}\), \(\alpha_{y}\), and \(\alpha_{z}\) are angular accelerations in the \(x\), \(y\), and \(z\) directions on the \(i\)-th sample expressed in ACS, respectively, \(F_{xd}\), \(F_{yd}\), and \(F_{zd}\) are segment distal forces in the \(x\), \(y\), and \(z\) directions on the \(i\)-th sample expressed in ACS, respectively, \(T_{xp}\), \(T_{yp}\), \(T_{zp}\) and \(T_{xd}\), \(T_{yd}\), \(T_{zd}\) are proximal and distal torques in the \(x\), \(y\), and \(z\) directions on the \(i\)-th sample expressed in ACS, respectively, and \(l_{p}\) and \(l_{d}\) are proximal and distal moment arms from the CoM to the joint center. It is worth noting that the torque around the \(z\)-axis does not involve forces because they have zero lever arms around the corresponding axis. Before calculating next joint loads, torques were transformed to GCS as shown in Equation (12):  
\[[T_{XP}\ T_{YP}\ T_{ZP}\ ]=\ [GA_{i}]^{\text{T}}\big{[}T_{xp}\ T_{yp}\ T_{zp}\ ] \tag{12}\]  
where \(T_{XP}\), \(T_{YP}\), and \(T_{ZP}\) are the proximal torques in the \(x\), \(y\), and \(z\) directions on \(i\)-th sample expressed in GCS. Then, torques were transformed back in ACS of the next segment to express distal torques (\(T_{xd}\), \(T_{yd}\), \(T_{zd}\)) in the ACS of the next segment. Joint ACS with forces' and torques' orientation expressed with respect to lower segment ACS are shown in Figure 2.0  
Figure 2.1 Solution of 3D inverse dynamics. \(l_{p}\) and \(l_{d}\) represent proximal and distal lengths of the segment, respectively. \(F_{xd}\), \(F_{yd}\), \(F_{zd}\), \(T_{xd}\), \(T_{yd}\), and \(T_{zd}\) represent distal forces and torques of the segment, respectively. Variables \(a_{x}\), \(a_{y}\), \(a_{z}\), \(\omega_{x}\), \(\omega_{y}\), \(\omega_{z}\), \(a_{x}\), \(a_{x}\), and \(a_{x}\) represent linear acceleration, angular velocity, and angular acceleration, respectively, while, \(x\), \(y\), and \(z\) are local segment coordinates. \(F_{xp}\), \(F_{yp}\), \(F_{zp}\), \(T_{xp}\), \(T_{yp}\), and \(T_{zp}\) are proximal forces and torques of segments, respectively.