# Probabilistic Sensitivity Amplification Control for Lower Extremity Exoskeleton
## 2 Distributed Hidden-State Identification
### Distributed Identification Process  
To identify the latent state \(\mathbf{x_{k}}\), we have to compute the posterior distribution \(p(\mathbf{x_{k}}|\mathbf{z_{\mathrm{i}:k}})\). Note that, even if the system states can be obtained from the observation, the measurement data are not accurate owing to the system noise. Thus, the system state \(\mathbf{x_{k}}\) is seen as a hidden variable that should be identified. Moreover, the \(k\) time step system state only depends on \(k-1\) time step state and the \(k\) time step observation, which hence is a typical hidden Markov model. According to the previous measurements and current time step prediction, [PERSON] law yields:  
\[p(\mathbf{x_{k}}|\mathbf{z_{\mathrm{i}:k}})=\frac{p(\mathbf{z_{k}}|\mathbf{x_{k}})p(\mathbf{x_{k}}| \mathbf{z_{\mathrm{i}:k-1}})}{p(\mathbf{z_{k}}|\mathbf{z_{\mathrm{i}:k-1}})}, \tag{6}\]  
Figure 3: The graphic interpretation of the Monte Carlo sampling. The real distribution in the dashed blue line is not the GP type, but approximated as GP in the solid blue line with the mean and covariance of the corresponding data. where \(p(\mathbf{x_{k}}|\mathbf{z+1};\mathbf{k})\) is defined as a prior of the posterior distribution and \(p(\mathbf{z_{k}}|\mathbf{x_{k}})\) is determined by the measurements. In addition, to improve the identification process, we reuse the observation data, i.e., current, previous and future:  
\[\gamma(\mathbf{x_{k}})=p(\mathbf{x_{k}}|\mathbf{z_{1:T}}). \tag{7}\]  
For the recursive process, the smooth hidden states \(\gamma(\mathbf{x_{k-1}})\) are given from \(k=T\) to 2:  
\[\gamma(\mathbf{x_{k-1}})=\int p(\mathbf{x_{k-1}}|\mathbf{x_{k}},\mathbf{z_{1:k-1}})\gamma(\mathbf{ x_{k}})dx_{\mathbf{k}}. \tag{8}\]  
In light of the work in [22], the updated posterior distribution of the latent state for the \(m\)-th (\(m=1,\cdots,M\)) GP can be obtained:  
\[(\mathbf{r_{k|k}^{x}})^{m} =(\mathbf{r_{k|k-1}^{x}})^{m}+(\mathbf{C_{k|k-1}^{xz}})^{m}((\mathbf{C_{k|k-1} ^{z}})^{m})^{-1}(z_{k}-(\mathbf{r_{k|k-1}^{z}})^{m}), \tag{9}\] \[(\mathbf{C_{k|k}^{x}})^{m} =(\mathbf{C_{k|k-1}^{x}})^{m}-(\mathbf{C_{k|k-1}^{xz}})^{m}((\mathbf{C_{k|k-1 }^{z}})^{m})^{-1}((\mathbf{C_{k|k-1}^{x}})^{m})^{T},\] (10) \[(\mathbf{r_{k-1|T}^{x}})^{m} =(\mathbf{r_{k-1|k-1}^{x}})^{m}+(\mathbf{I_{k-1}})^{m}((\mathbf{r_{k|T}})^{m} -(\mathbf{r_{k|k}^{x}})^{m}),\] (11) \[(\mathbf{C_{k-1|T}^{x}})^{m} =(\mathbf{C_{k-1|k-1}^{x}})^{m}+(\mathbf{I_{k-1}})^{m}((\mathbf{C_{k|T}^{x}} )^{m}-(\mathbf{C_{k|k}^{x}})^{m})((\mathbf{I_{k-1}})^{m})^{T}. \tag{12}\]  
Not akin to [22], to infer a hidden state with a corresponding observation input, we should combine all the M GPs to fusion of the M predictions. Therefore, in this paper, we compare four fusion algorithms, i.e., product of expert (PoE) [23], generalized product of expert (gPoE)[24], Bayesian committee machine (BCM) [25] and robust Bayesian committee machine (rBCM) [20]. The only difference between the gPoE and PoE is the different precisions, since the important weights are set to \(\beta_{k}=1/M\) to avoid an unreasonable error bar. Moreover, the improvement made in the BCM is that it incorporates the GP prior explicitly when predicting the joint distribution. Hence, the prior precision should be added to the posterior predictive distribution. The rBCM seeks to add the flexibility of gPoE to BCM to enhance predictive performance. Moreover, we conduct an experiment with a one dimension problem [26]. Figure 4 shows the performance comparison of four distributed identification process (PoE, gPoE, BCM, rBCM) and robust Rauch-Tung-Striebel smoother for GP dynamic systems (GP-RTSS) [18], which is implemented under the full GP scheme. The PoE and BCM distributed schemes overestimate the variances in the regime of the series hidden states, which leads to overconfident precision compared with GP-RTSS. Moreover, the prediction of the rBCM is too conservative, although the rBCM aims to provide a more robust solution.