## 2 Forward Dynamics and Human Exoskeleton Interaction Mechanism Interpretation of The Knee Joint
### SP Model-Based Human-Exoskeleton Interaction Mechanism Interpretation
#### 2.2.1 Active Force from AE  
The force kinetics affected by contraction velocity of AE formulates the variation dynamics of \(F_{a}\) under the fluctuation of contraction velocity [37]. In quasi-static cases, such a description is consistent with the Hill-type force-velocity relationship of muscle [31]. Specifically, the contractile force of the AE can be defined as  
\[F_{a}=\beta(t)G(l)(1+\delta) \tag{12}\]  
where \(\beta(t)\) is the activation degree ranging from 0 to 1; \(G(l)\), ranging from 0 to 1 at different \(l\)s, represents the tetanic force-length (\(F-l\)) relationship, \(\delta\) denotes the fluctuation magnitude of \(F_{a}\) induced by sarcomere shortening or lengthening at the tetanus state and slack length. While the \(G(l)\) used in the SP model and the \(f_{A}(l)\) defined in the former section are all based on the force-length relationship described by [PERSON] et al. [38], here the specific form of \(G(l)\) is treated to be the same as \(f_{A}(l)\). Based on (4), \(f_{A}(l)\) can be further represented as:  
\[ \begin{array}{c}f_{A}(l)=q_{2}\bigg{(}\frac{b_{1}}{l_{0}^{m}\cos\phi}\bigg{)} ^{2}\theta^{2}+\bigg{(}q_{1}\frac{b_{1}}{l_{0}^{m}\cos\phi}+2q_{0}\frac{b_{1}} {l_{0}^{m}\cos\phi}\frac{b_{0}-t}{l_{0}^{m}\cos\phi}\bigg{)}\theta\\ \hskip 14.226378 pt+\bigg{(}q_{0}+q_{1}\frac{b_{1}}{l_{0}^{m}\cos\phi}+q_{2} \bigg{(}\frac{b_{0}-t}{l_{0}^{m}\cos\phi}\bigg{)}^{2}\bigg{)}\end{array}  \tag{13}\]  
When \(\beta(t)\) in (12) and \(a(k)\) in (1) represent the muscle activation in different models, a proportional relationship is assumed for simplification when \(\beta(t)=k_{act}a(k)\) and \(k_{act}\) is a scale factor. Although the muscle activation is not strictly linearly related to muscular force [39; 40], the muscle activation-force relationship can still be assumed to be linear for some muscles, especially for the last 60-70% of their relationship curve [41]. The linear relation has also been used in some practical applications to estimate muscle force [42]. Thus, the relation here is assumed as:  
\[a(t)=Cof\cdot F^{mt}(t) \tag{14}\]  
where the \(Cof\) is the scale coefficient. \(F^{mt}(t)\) can be estimated based on (1). Thus, we have  
\[\Big{[}\Big{(}q_{0}+q_{1}l+q_{2}l^{2}\Big{)}a(t)+e^{10l-15}\Big{]}F_{0}^{mt}cos \phi\cdot Cof=a(t) \tag{15}\]  
\[\beta=\frac{k_{act}e^{10l-15}}{1/\Big{(}F_{0}^{m}cos\phi\cdot Cof\Big{)}-(q_{0 }+q_{1}l+q_{2}l^{2})} \tag{16}\]  
Substituting \(l\) with (4), the relation between muscle activation \(\beta\) and joint angle \(\theta\) could be developed as  
\[\beta=\frac{k_{act}e^{N_{21}\theta+N_{22}}}{N_{11}\theta^{2}+N_{12}\theta+N_{ 13}} \tag{17}\]  
where  
\[\left\{ \begin{array}{c}N_{11}=-q_{2}\bigg{(}\frac{b_{1}}{l_{0}^{m}\cos\phi} \bigg{)}^{2}\\ N_{12}=-\bigg{[}\frac{q_{1}b_{1}}{l_{0}^{m}\cos\phi}+2q_{2}\frac{b_{1}\left(b_ {0}-t\right)}{\left(l_{0}^{m}\cos\phi\right)^{2}}\bigg{]}\\ N_{13}=\frac{1}{F_{0}^{m}\cos\phi\cdot Cof}-q_{1}\frac{b_{0}-t^{m}}{l_{0}^{m} \cos\phi}-q_{2}\frac{\left(b_{0}-t\right)^{2}}{\left(l_{0}^{m}\cos\phi\right) ^{2}}\\ N_{21}=\frac{10b_{1}}{l_{0}^{m}\cos\phi}\\ N_{22}=\frac{10\left(b_{0}-t\right)}{l_{0}^{m}\cos\phi}-15\end{array} \right.\]  
For the force kinetics affected by the muscle contraction velocity, the measurable velocity of the half-sarcomere is defined as the external velocity (\(V\)) and the sliding velocity \(v\) as the internal velocity [31]. Without losing generality, the viscous drag force can be regarded as linearly correlated with \(V\) during active shortening, and this drag force is assumed to be negligibly small when the muscle is stretched [43]. Thus, the drag force \(F_{v}\) can be formulated as  
\[F_{v}=\left\{ \begin{array}{cc}b_{c}V&V\leq 0\\ 0&\text{ }^{\prime}&V>0\end{array} \right. \tag{18}\]  
where \(b_{c}\) is the drag coefficient and also a fitting parameter. According to the transformation between \(V\) and \(v\) and the steady state condition of \(\delta=\
u\tau_{t}\) where \(\tau_{t}\) is the time constant, we have  
\[\delta=\left\{