## 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
\begin{array}{ll}s_{11}=-b_{c}q2\frac{b_{1}^{3}}{\big{(}\frac{b_{1}}{(b_{0} \cos\phi)}\big{)}^{2}}\\ s_{12}=-b_{c}b_{1}\Big{(}q_{1}\frac{b_{1}}{\frac{b_{1}}{\cos\phi}\big{)}+2q_{2} \frac{b_{1}}{\frac{b_{1}}{\cos\phi}}\frac{b_{0}-f}{b_{0}\cos\phi}\Big{)}}\\ s_{13}=-b_{c}b_{1}\Bigg{[}q_{0}+q_{1}\frac{b_{0}-f}{b_{0}^{\text{max}}\cos\phi }+q_{2}\Big{(}\frac{b_{0}-f}{b_{0}\cos\phi}\Big{)}^{2}\Bigg{]}\\ s_{21}=q_{2}b(1+a)\frac{b_{1}^{2}}{\big{(}\frac{b_{1}}{\theta_{0}\cos\phi} \big{)}^{2}}\\ s_{22}=b(1+a)\Big{(}q_{1}\frac{b_{1}}{b_{1}^{\text{max}}\cos\phi}+2q_{2} \frac{b_{1}}{\frac{b_{1}}{\cos\phi}\big{)}}\frac{b_{0}-f}{b_{0}\cos\phi}\Big{)} \\ s_{23}=b(1+a)\bigg{[}q_{0}+q_{1}\frac{b_{0}-f}{b_{0}^{\text{max}}\cos\phi}+q_{ 2}\Big{(}\frac{b_{0}-f}{b_{0}^{\text{max}}\cos\phi}\Big{)}^{2}\Bigg{]}\\ s_{31}=\frac{2}{n}c_{1}\frac{b_{1}}{\big{(}\frac{b_{0}-f}{b_{0}\cos\phi}\big{)}} \\ s_{32}=\frac{2}{n}c_{1}\Big{(}\frac{b_{0}-f}{b_{0}\cos\phi}-l_{0}\Big{)}\end{array} \]calibration process using an inverse dynamic model [32], are considerable. Moreover, characteristics of the human-exoskeleton system always change with time, which makes the model that includes the time-varying interactive force more difficult to be utilized in practical applications [34; 37]. When fast prediction, real-time control and self-learning of parameters are often required in the development of biomechanical models, the use of machine learning method would be the best solution so far to meet the adaptivity requirements of the human-exoskeleton models [16]. To find out the relationship among the variables more clearly, we first change the time step in (24) from \(k\) to \(k+1\), which does not affect the correctness of the equation. Then, put aside the constant parameters and turn the concrete expressions in (10) and (24) into a more general form:  
\[\left\{ \begin{array}{l}x_{k+1}=f(x_{k},u_{k})\\ y_{k+1}=h(x_{k+1},d_{k+1})\end{array} \right. \tag{25}\]  
where \(u_{k}\) is the muscle activation, \(x_{k}=\left(\dot{\theta}_{k},\theta_{k}\right)\) is the vector containing the joint angle and its velocity, \(y_{k+1}\) is the interactive force between the exoskeleton and human and \(d_{k+1}\) is their angle difference. Given that the angle different \(d_{k+1}\) is a time-varying variable which cannot be acquired directly from muscle activation and joint angle as stated in Section 2.2.3, its variation can be merged into the model adaptivity and learning process through the machine learning approaches. The aims are to reduce the impact of this uninterpretable variable on the model development and find out the relationships among the variables under study more intuitively. Thus, a clear formulation of state space model can then be acquired:  
\[\left\{ \begin{array}{l}x_{k+1}=f(x_{k},u_{k})+\omega_{k}\\ y_{k+1}=h(x_{k+1})+v_{k+1}\end{array} \right. \tag{26}\]  
where \(\omega_{k}\) and \(v_{k}\) are the process noise and the measurement noise, respectively. Based on (26), GP and BPNN are adopted for the state model and measurement model, respectively, to combine the relative strengths of both the non-parametric model and parametric model. The reason to apply GP and BPNN here is that BPNN has already been widely applied in the fields of human motion intent learning [1], and GP offers superior flexibility compared with traditional modelling methods in learning [16]. There have been successful applications of GP in human motion intent learning and prediction [22; 23; 44]. Also, unscented Kalman filter (UKF) is used to develop the close loop of the state space model [45]. In the proposed SSM, sEMG signals, which are thought to be caused by the motor unit action potentials (MUAPs) spreading on muscle fiber, are selected as the vector \(u_{k}\) which is the input to the state model to indicate muscle activation. SEMG signals are selected due to their outstanding characters including noninvasiveness and safety [25; 26].