# A Novel Method for Field Measurement of Ankle Joint Stiffness in Hopping
## 2 Materials and Methods
### Calculating AJS Using New Method  
In this method, lower extremity was modeled as a three linked rigid segments (including thigh, shank, and foot) robot as a two-dimensional hopping model in the sagittal plane. The segments were interconnected with three torsional springs as three torsional frictionless hinge joints, one between thigh and shank (knee), the other between shank and foot (ankle), and the third one between the fifth metatarsophalangeal and the floor (only the contact phase of hopping was analyzed). These springs represented the action of the muscles, tendons, and ligaments across the joint. The hip was considered as the end effector of the robot and assumed that the total body mass was concentrated on it. The hip was assumed to follow a periodic vertical displacement (sinusoidal wave) during hopping based on which joints kinematics were calculated using inverse kinematics method. However, this assumption excluded some movement in horizontal plane that happens which may be considered as a limitation for this model. To determine the position and orientation of each limb related to other limbs, the Denavit-Hartenberg (D-H) method was used [27]. This model and the coordinate systems are illustrated in Figure 2.  
Figure 1: The placement four reflective markers at the legs are shown. In this method (Figure 3), to find the angles at knee and ankle, an inverse kinematic problem was solved. In first step, the reference frame transition matrixes were calculated by direct kinematic. The matrix that converts reference frame 3 (attached to hip) to reference frame 0 (attached to metatarsal-floor connection) is  
\[{}^{0}T_{3}=\left[ \begin{array}{cccc}\cos\varphi&-\sin\varphi&0&x\\ \sin\varphi&\cos\varphi&0&z\\ 0&0&1&0\\ 0&0&0&1\\ \end{array} \right] \tag{1}\]  
In which  
\[\varphi=\sum_{i=1}^{i=3}\theta_{i} \tag{2}\]  
\[x=\sum_{i=1}^{i=3}\left(l_{i}\cos\left(\sum_{j=1}^{i=i}\theta_{j}\right)\right) \tag{3}\]  
\[z=\sum_{i=1}^{i=3}\left(l_{i}\sin\left(\sum_{j=1}^{j=i}\theta_{j}\right)\right) \tag{4}\]  
In Equations (2)-(4), \(\theta\) and \(l\) represents joint angle and segment length respectively. To solve the inverse kinematic, the resolved motion rate control (RMRC) method was used [28]. The Jacobian matrix, that related the angles to the position of end effector, was calculated as:  
\[J[\theta]=\left[ \begin{array}{cccc}-l_{3}s_{123}-l_{2}s_{12}-l_{1}s_{1}&-l _{3}s_{123}-l_{2}s_{12}&-l_{3}s_{123}\\ l_{3}c_{123}+l_{2}c_{12}+l_{1}c_{1}&l_{3}c_{123}+l_{2}c_{12}&l_{3}c_{123}\\ \end{array} \right] \tag{5}\]  
In which  
\[S_{123}=\sin\left(\theta_{1}+\theta_{2}+\theta_{3}\right) \tag{6}\]  
\[C_{123}=\cos\left(\theta_{1}+\theta_{2}+\theta_{3}\right) \tag{7}\]  
Figure 2: Model of lower extremity and the coordinate systems. Because this is a redundant problem, the Pseudo inverse of Jacobian matrix should be calculated  
\[J^{\#}=J^{T}\big{(}JJ^{T}\big{)}^{-1} \tag{8}\]  
The horizontal linear velocity of end effector (hip) supposed to be zero and its vertical linear velocity was calculated from  
\[v(t)=\frac{F_{max}}{m}\ \frac{T_{c}}{\pi}\times\cos\bigg{(}\frac{\pi}{T_{c}} \times\ t\bigg{)}-gt+g\frac{T_{c}}{2} \tag{9}\]  
which is obtained of modeling ground reaction force in hopping task as a sine wave (Equations (10) and (11)), as fully detailed elsewhere [25]  
\[F(t)=\ F_{max}\times\sin\ (\frac{\pi}{T_{c}}\ \times\ t) \tag{10}\]  
which is in  
\[F_{max}=mg\ \times\frac{\pi}{2}\times\bigg{[}\frac{T_{f}}{T_{c}}+1\bigg{]} \tag{11}\]  
In these equations, \(m\) is the mass of subject, \(T_{c}\) and \(T_{f}\) represent the contact time and flight time in hoping, \(g\) used for gravitational acceleration, and \(t\) shows the time. By integrating the above expression of the velocity, Equation (9), vertical displacement of hip was calculated  
\[z(t)=-\frac{F_{max}}{m}\ \frac{T_{c}{}^{2}}{\pi^{2}}\times\sin\bigg{(}\frac{ \pi}{T_{c}}\times\ t\bigg{)}-\frac{1}{2}\ g\times t^{2}+g\frac{T_{c}}{2}\ t \tag{12}\]  
where \(z\) (0) = 0 is the vertical position of center of mass at the time of first touch with ground.