# Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based Visual-Inertial Odometry
## 3 Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based VIO
### Robust \(H_{\infty}\) Filter
Applying the information-description of the Kalman filter on the above CH\({}_{\infty}\)F, a cubature \(H_{\infty}\) information filter (CH\({}_{\infty}\)IF) can be formulated as follows:  
\[ \begin{array}{l}\mathbf{P}_{\text{zz}}=\mathbf{H}_{k}\mathbf{P}_{k|k-1}\mathbf{H}_{k}^{\text {T}}+\mathbf{R}_{k}\\ \mathbf{i}_{k}=\mathbf{H}_{k}^{\text{T}}\mathbf{R}_{k}^{-1}\Big{(}\mathbf{z}_{k}-\textit{h} \Big{(}\hat{\mathbf{x}}_{k|k-1}\Big{)}+\mathbf{H}_{k}\hat{\mathbf{x}}_{k|k-1}\Big{)}- \gamma^{-2}\Big{(}\hat{\mathbf{x}}_{k|k-1}+\mathbf{P}_{\text{xz}}\mathbf{P}_{\text{zz}}^{- 1}\Big{(}\mathbf{z}_{k}-\textit{h}\Big{(}\hat{\mathbf{x}}_{k|k-1}\Big{)}\Big{)}\Big{)} \\ \hat{\mathbf{y}}_{k|k}=\mathbf{\hat{y}}_{k|k-1}+\mathbf{i}_{k}\\ \mathbf{Y}_{k|k}=\mathbf{Y}_{k|k-1}+\mathbf{H}_{k}^{\text{T}}\mathbf{R}_{k}^{-1}\mathbf{H}_{k}- \gamma^{-2}\mathbf{\mathcal{I}}_{n}\end{array}  \tag{49}\]  
where the covariance matrix of measurement error vector \(\mathbf{P}_{\text{zz}}\) is an \(m\)-dimension matrix. As previously stated, \(m\) is generally large in direct VO, which means that the inversion of \(\mathbf{P}_{\text{zz}}\) in the above equation is not practical. We used the expressions of the gain matrix in the Kalman filter to simplify the inversion. In this way, the measurement-update phase in CH\({}_{\infty}\)IF was divided into two steps as follows:  
(1) Evaluate the measurement matrix \(\mathbf{H}_{k}\) with a cubature rule and update the information matrix using the information filter. \[\mathbf{Y}_{k}{}^{\prime}=\mathbf{Y}_{k|k-1}+\mathbf{H}_{k}^{\text{T}}\mathbf{R}_{k}^{-1}\mathbf{ H}_{k} \tag{50}\]  
(2) Evaluate the gain matrix \(\mathbf{K}_{\infty}\) and execute the measurement update of the \(H_{\infty}\) filter. \[ \begin{array}{l}\mathbf{K}_{\infty}=\mathbf{Y}_{k}{}^{\prime}{}^{-1}\mathbf{H}_{k}^{ \text{T}}\mathbf{R}_{k}^{-1}\\ \mathbf{i}_{k}=\mathbf{H}_{k}^{\text{T}}\mathbf{R}_{k}^{-1}\Big{(}\mathbf{z}_{k}-\textit{h} \Big{(}\hat{\mathbf{x}}_{k|k-1}\Big{)}+\mathbf{H}_{k}\hat{\mathbf{x}}_{k|k-1}\Big{)}-\gamma^ {-2}\Big{(}\hat{\mathbf{x}}_{k|k-1}+\mathbf{K}_{\infty}\Big{(}\mathbf{z}_{k}-\textit{h} \Big{(}\hat{\mathbf{x}}_{k|k-1}\Big{)}\Big{)}\Big{)}\\ \hat{\mathbf{y}}_{k|k}=\hat{\mathbf{y}}_{k|k-1}+\mathbf{i}_{k}\\ \mathbf{Y}_{k|k}=\mathbf{Y}_{k}{}^{\prime}-\gamma^{-2}\mathbf{\mathcal{I}}_{n}\end{array}  \tag{51}\]  
It is easy to prove that the above two-step measurement-update process is equal to Equation (49). The inversion of the \(m\)-dimension matrix is replaced by the inversion of a 9-dimensional matrix \(\mathbf{Y}_{k}{}^{\prime}\). It is worth noting that the measurement-update phase of the proposed CH\({}_{\infty}\)IF was different to the one with the same abbreviation that [PERSON] derived in [31]. Compared with Equation (49), [PERSON]'s method omitted the second term of \(\mathbf{i}_{k}\) when using the technique to analyze the estimation error in [35; 36]. The error transition function of the measurement-update phase of our method is as follows:  
\[\widetilde{\mathbf{x}}_{k|k}=\widetilde{\mathbf{x}}_{k|k-1}-\mathbf{K}_{k}\widetilde{\mathbf{z}}_ {k} \tag{52}\]  
\[\mathbf{K}_{k}=\mathbf{P}_{k|k}\Big{(}\mathbf{H}_{k}^{\mathrm{T}}\mathbf{R}_{k}^{-1}-\gamma^{-2 }\mathbf{K}_{\infty}\Big{)} \tag{53}\]  
where \(\widetilde{\mathbf{z}}_{k}\) is the predicted error of the measurement \(\mathbf{z}_{k}\), and formulated as follows:  
\[\widetilde{\mathbf{z}}_{k}=\mathbf{z}_{k}-\mathbf{h}\Big{(}\hat{\mathbf{x}}_{k|k-1}\Big{)} \tag{54}\]  
While the error transition function of the measurement-update phase of [PERSON]'s method is related to the predicted mean value of the state vector \(\hat{\mathbf{x}}_{k|k-1}\) as follows:  
\[\widetilde{\mathbf{x}}_{k|k}=\widetilde{\mathbf{x}}_{k|k-1}-\mathbf{P}_{k|k}\mathbf{H}_{k}^{ \mathrm{T}}\mathbf{R}_{k}^{-1}\widetilde{\mathbf{z}}_{k}-\gamma^{-2}\mathbf{P}_{k|k}\hat {\mathbf{x}}_{k|k-1} \tag{55}\]  
This goes against the convergence of the filter.