## 2 Materials and Methods
### Analytic Model
#### 2.1.2 Trial Measurements  
In the second phase, data from one trial were analyzed. The segment marker coordinate system on the \(i\)-th sample was established using the same procedure used for calculating the LCS in the calibration phase, with a difference that markers coordinates were in the GCS. The unit vector of the coordinate system was written in matrix (\(GM_{i}\)) using the same notation as in Equation (3). The transformation matrix (\(GA_{i}\)) between the GCS and ACS were defined from the angles \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) calculated based on Winter [14]. Matrix \(MA\) and \(GM_{i}\) were multiplied to define the \(GA_{i}\) matrix as shown in Equation (4):  
\[GA_{i}=\ [MA][GM_{i}] \tag{4}\]  
where \(GA_{i}\) is the transformation matrix between the GCS and ACS on the \(i\)-th sample, \(MA\) is the segment constant transformation matrix between the LCS and ACS expressed in the ACS, and \(GM_{i}\) is transformation matrix between the GCS and LCS on the \(i\)-th sample expressed in the GCS. CoM coordinates were defined as shown in Equation (5):  
\[V_{c}=p+[GA_{i}]^{T}[-c_{o}] \tag{5}\]  
where \(V_{c}\) are coordinates of the CoM on the \(i\)-th sample, \(p\) are coordinates of the segment's LCS origin marker expressed in the GCS on \(i\)-th sample, \(GA_{i}\) is defined in Equation (4), and \(c_{o}\) is a constant vector between the origin marker of the segment's LCS and CoM expressed in the ACS and calculated using Equation (2). At this point linear acceleration and angular velocity and acceleration of segment could be calculated using the finite-difference technique. The linear acceleration in GCS was calculated as shown in Equation (6):  
\[a_{i}=\ \frac{x_{i+1}-2x_{i}+x_{i-1}}{\Delta t^{2}} \tag{6}\]  
where \(a_{i}\) is the linear acceleration on \(i\)-th sample, \(x_{i+1}\), \(x_{i}\), and \(x_{i-1}\) are coordinates in \(x\), \(y\), or \(z\) directions on the \(i+1\), \(i\) and \(i-1\) samples, and \(\Delta t\) is a change in time. The linear acceleration in the GCS was needed because all the forces were calculated in GCS and then were expressed in segment ACS. Angular velocity in the ACS was calculated as shown in Equation (7):  
\[ \begin{array}{rcl}\omega=&[\omega_{x}\;\omega_{y}\;\omega_{z}\;]&=&[cos\theta _{2}cos\theta_{3}\;sin\theta_{3}\;0\\ &&-cos\theta_{2}sin\theta_{3}\;cos\theta_{3}\;0\;sin\theta_{2}\;0\;1\;][\dot{ \theta}_{1}\;\dot{\theta}_{2}\;\dot{\theta}_{3}\;]\end{array}  \tag{7}\]  
where \(\omega_{x}\), \(\omega_{y}\), and \(\omega_{z}\) are angular velocity in the \(x\), \(y\), and \(z\) directions, \(\theta_{1}\), \(\theta_{2}\), and \(\theta_{3}\) are Cardan rotational angles, and \(\theta_{1}^{\prime}\), \(\dot{\theta}_{2}\), and \(\dot{\theta}_{3}\) are first derivates of Cardan angles. The angular accelerations \(\alpha_{x}\), \(\alpha_{y}\), and \(\alpha_{z}\) were calculated using the finite-difference technique as shown in Equation (8):  
\[\alpha_{i}=\;\frac{\omega_{i+1}-\omega_{i-1}}{2t\Delta t} \tag{8}\]  
where \(\alpha_{i}\) is linear acceleration on the \(i\)-th sample, \(\omega_{i-1}\) and \(\omega_{i+1}\) are angular velocities in the \(x\), \(y\), or \(z\) directions on the \(i+1\) and \(i-1\) sample, and \(\Delta t\) is a change in time. The last phase was the calculation of joint loads. Newton-Euler 3D equations of motion for segments were used to calculate these loads. The solution for one segment is presented in Figure 2.  
All forces were calculated in the GCS, while torques were calculated in ACS.