# Probabilistic Sensitivity Amplification Control for Lower Extremity Exoskeleton
## 3 Evolving Learning of the Sensitivity Factor
### Constraint Optimization  
Since the sensitivity factor \(\mathbf{\alpha^{-1}}\in(0,1)\) is constrained regarding the optimization domain, to deal with this problem, the variance of the offspring distribution is reduced in the normal vector direction in the vicinity of the corresponding parental candidate. For the constraints \(\mathbf{v}_{j},j=1,\cdots,m\), an exponential fading vector is defined following [29] and updated when the optimization is lessened by the boundary:  
\[\mathbf{v}_{j}\leftarrow(1-c_{c})\mathbf{v}_{j}+c_{c}\mathbf{A}\mathbf{z}, \tag{20}\]  
where \(c_{c}\in(0,1)\) is a parameter that determines the convergence speed. Equation (20) can be seen as a low-pass filter, which guarantees that the tangential components of \(\mathbf{A}\mathbf{z}\) have been canceled out. In these situations, the Cholesky factor \(\mathbf{A}\) should also be rewritten to meet the demand of the restrained boundary:  
\[\mathbf{A}\leftarrow\mathbf{A}-\frac{\beta}{\sum_{j=1}^{m}\Gamma_{g_{j}(\mathbf{y})>0}} \sum_{j=1}^{m}\Gamma_{g_{j}(\mathbf{y})>0}\frac{\mathbf{v}_{i}\mathbf{w}_{j}^{\mathbf{T}}}{ \mathbf{w}_{j}^{\mathbf{T}}\mathbf{w}_{j}}, \tag{21}\]  
With \(\mathbf{w}_{j}=\mathbf{A^{-1}}\mathbf{v}_{j}\) and \(\Gamma_{g_{j}(\mathbf{y})}=1\) if \(g_{j}(\mathbf{y})>0\), otherwise zero. The update size is controlled by the parameter \(\beta\). Moreover, compared with Equations (17) and (19), Cholesky decomposition \(\mathbf{A}\) without a scalar greater than one aims to decrease the update step size to approach the boundary of constraints. The solution to generating offspring candidates is explained in Figure 7. In the direction of the normal vector, the variance of the offspring is reduced as shown in Figure 7a. After the operation, the evolved variance is tangential to the constraint boundary in Figure 7b. Thus, the offspring candidates can be sampled and derived from the evolved variance \(\mathbf{A}\). Note that, in Figure 7c, after constraint optimization, the constraint angle has been increased due to the change of the offspring distribution. Figure 6: Visualization of CMA-ES exploration and updating rules. In (**a**), the new distribution is shown in the red ellipse, while the last-step distribution is in blue. The quantity of the contributions of each sample is defined with the area of the green circles. Consequently, these contributions are also ranked in (**b**). (**a**) Sampling, estimation and new distribution; (**b**) contributions of the samples. In terms of the sequence hidden-states \((\vec{q}_{i}^{des},\hat{q}_{i}^{des},\mathbf{q}_{i}^{des})\), as well as actual data, the penalty function for updating of the sensitivity factor is given:  
\[J(\tau)=\sum_{i=1}^{i=k}|\vec{q}_{i}^{des}|+(\vec{q}_{i}^{des}-\dot{q}_{i})^{2}+ (\mathbf{q}_{i}^{des}-\mathbf{q}_{i})^{2}, \tag{22}\]  
where \(\dot{\mathbf{q}}_{i}\) and \(\mathbf{q}_{i}\) are the joint velocity and angle, separately. Note that not akin to the control algorithm in [27], where the penalty function consists of the previous loop trajectory data, the desired control input \((\vec{q}_{i}^{des},\mathbf{q}_{i}^{des},\mathbf{q}_{i}^{des})\) is computed and predicted from the identification process. Therefore, our scheme is more suitable for improving real-time control performance. The first term in Equation (22) is to avoid high jerk. Furthermore, the last two terms are penalized for tracking error. From our intuition, with less interaction force between the human and the exoskeleton, the pilot will feel more comfortable. According to this point view, punishing the last two terms may enhance the tracking performance. The main routine of the evolving learning of sensitivity factors scheme is presented in Algorithm 3.