## 2 Forward Dynamics and Human Exoskeleton Interaction Mechanism Interpretation of The Knee Joint
### SP Model-Based Human-Exoskeleton Interaction Mechanism Interpretation
#### 2.2.3 Joint Motion Model under Human-Exoskeleton Interaction  
Considering that the angular acceleration can be expressed as the function of angle in a discrete form as \(\ddot{\theta}_{k}=\frac{1}{\tau_{s}^{2}}(\theta_{k+1}-2\theta_{k}+\theta_{k-1})\), the joint motion model can be expressed as follows  
\[\frac{1}{T_{s}}\ddot{\theta}_{k}=\frac{1}{T_{s}}^{2}(\theta_{k+1}-2\theta_{k} +\theta_{k-1})=\frac{1}{I_{e}}(\tau-\tau_{ex})=\frac{1}{I_{e}}(F_{ext}\tau- \tau_{ex})=f_{sp}\Big{(}\theta_{k},\dot{\theta}_{k}\Big{)} \tag{22}\]  
where \(f_{sp}\) is a nonlinear function whose specific expression can be derived from (11)-(21) based on the SP muscle model. Its concrete expression will be included in the final expression. \(\theta_{k+1}\) and \(\dot{\theta}_{k+1}\) indicate the predicted joint angle and its velocity. Note that the predicted angle is also used as the controlling input to the exoskeleton system. Thus, the predicted angle and angular velocity are also regarded as the actual values of the exoskeleton. Supposing that the attachment between human and exoskeleton is rigid, which has been proved to be feasible in human-exoskeleton systems [21], the current and past joint angles of both human and exoskeleton can be thought of as the same. However, there must be errors in the predicted joint angle of humans, which would result in the interactive torque between the human and exoskeleton, as illustrated in Figure 2. The joint motion model is simplified as driven by two muscle groups, rectus femoris (RF) and biceps femoris (BF). Consequently, the actual human dynamics considering the interactive torque can be acquired based on (22) and are described as:  
\[\frac{1}{T_{s}}^{2}\ddot{\overline{\theta}}_{k}=\frac{1}{T_{s}}^{2}\Big{(} \overline{\theta}_{k+1}-2\theta_{k}+\theta_{k-1}\Big{)}=f_{sp}\Bigg{(} \overline{\theta}_{k},\widetilde{\overline{\theta}}_{k}\Bigg{)}+T_{int(k)} \tag{23}\]  
where \(\widetilde{\theta}_{k}\) and \(\widetilde{\overline{\theta}}_{k}\) denote the joint angle and angular velocity of a human, \(T\_int(k)\) is the interactive torque. Note that according to the assumption that the attachment between the human exoskeleton is rigid, the current angle of exoskeleton \(\theta_{k}\) in (22) and the current angle of human \(\widetilde{\theta}_{k}\) in (23) are the same. However, the angular velocity of human \(\widetilde{\theta}_{k}\) is different from that of the exoskeleton \(\dot{\theta}_{k}\) when the discrete form of angular velocity is given as \(\dot{\theta}_{k}=\frac{1}{T_{s}}(\theta_{k+1}-\theta_{k})\) which is defined at the beginning and the values of \(\theta_{k+1}\) and \(\widetilde{\theta}_{k+1}\) are different. Denote the joint angle difference in the time step \(k\) between the human and exoskeleton as \(J_{d}(k)=\widetilde{\theta}_{k}-\theta_{k}\) which is a time-varying variable and cannot be acquired directly from muscle activation and joint angle. Subtracting (22) from (23), we have  
\[ \begin{array}{ll}T_{int}(k)=\frac{l_{k}l_{d}(k)}{{T_{s}}^{2}}&-\frac{b_{1}l_ {d}(k)}{{T_{s}}}\frac{k_{\text{d}\omega}l^{N_{21}\theta_{k}+N_{22}}}{N_{11} \theta_{k}^{2}+N_{12}\theta_{k}+N_{13}}(s_{11}\theta_{k}^{2}+s_{12}\theta_{k} +s_{13})\\ &-b_{1}\frac{k_{\text{d}\omega}l^{N_{21}\theta_{k}+N_{22}}}{N_{11}\theta_{k}^ {2}+N_{12}\theta_{k}+N_{13}}\big{(}s_{21}\theta_{k}^{2}+s_{22}\theta_{k}+s_{2 3}\big{)}\Bigg{[}\frac{1}{b_{1}\big{(}\theta_{k}+\frac{l_{d}(k)}{T_{s}}\big{)} +b}\\ &-\frac{1}{b_{1}\theta_{k}+b}\Bigg{]}\\ &-b_{1}(s_{31}\theta_{k}+s_{32})\Big{\{}\text{arctan}\Big{[}c_{3}b_{1}\Big{(} \dot{\theta}_{k}+\frac{l_{d}(k)}{T_{s}}\Big{)}\Big{]}-\text{arctan}\Big{(}c_{ 3}b_{1}\dot{\theta}_{k}\Big{)}\Big{\}}\end{array}  \tag{24}\]  
where \(N_{11},N_{12},N_{13},N_{21}\) and \(N_{22}\) are the parameters defined in (17), and other parameters are listed as follows:  
\[