# 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  
The aim of the \(H_{\infty}\) filter is to obtain the state estimation minimizing the following cost function [34]. \[J_{\infty}=\frac{\sum_{k=0}^{N-1}\lVert\mathbf{x}_{k}-\hat{\mathbf{x}}_{k}\rVert_{M_{ k}}^{2}}{\lVert\mathbf{x}_{0}-\hat{\mathbf{x}}_{0}\rVert_{\mathbf{Y}_{0}}^{2}+\sum_{k=0}^{N-1 }\left(\lVert w_{k}\rVert_{Q_{k}}^{2}+\lVert n_{k}\rVert_{R_{k}}^{2}\right)} \tag{42}\]  
Based on game theory, the design of the \(H_{\infty}\) filter is a game between the filter designer and nature. Nature's goal is to maximize the estimation error by properly choosing \(\mathbf{x}_{0}\), \(\mathbf{w}_{k}\), and \(\mathbf{n}_{k}\). The cost function is defined in a fractional form to create a fair game by preventing nature using extremely large noises. The above optimization problem is difficult to solve directly. An alternative strategy is to choose an upper bound and estimate the state satisfying the following constraint. \[J_{\infty}<\gamma^{2} \tag{43}\]  
where \(\gamma\) is the performance bound to be designed. According to the above equation, a new cost function is defined as follows:  
\[J=-\frac{1}{\theta}\lVert\mathbf{x}_{0}-\hat{\mathbf{x}}_{0}\rVert_{\mathbf{Y}_{0}}^{2}+ \sum_{k=0}^{N-1}\bigg{(}\lVert\mathbf{x}_{k}-\hat{\mathbf{x}}_{k}\rVert_{M_{k}}^{2}- \frac{1}{\theta}\Big{(}\lVert\mathbf{w}_{k}\rVert_{Q_{k}}^{2}+\lVert\mathbf{n}_{k} \rVert_{R_{k}}^{2}\Big{)}\bigg{)}<1 \tag{44}\]  
Therefore, the original optimization problem is transformed into the following minimax problem. \[J^{*}=\min\limits_{\hat{\mathbf{x}}_{k}}\max\limits_{\mathbf{x}_{0},\mathbf{w}_{k},\mathbf{n}_{ k}}J \tag{45}\]  
[PERSON] [29] solved the minimax problem and proposed an extended \(H_{\infty}\) filter (E\(H_{\infty}\)F) for the nonlinear system, which was similar to EKF in form. \[ \begin{array}{l}\mathbf{x}_{k|k-1}=\textit{f}\Big{(}\mathbf{x}_{k-1|k-1}\Big{)}\\ \mathbf{P}_{k|k-1}=\
abla\textit{f}\mathbf{P}_{k-1|k-1}\
abla\textit{f}^{\text{T}}+\mathbf{ Q}\end{array}  \tag{46}\]  
\[ \begin{array}{l}\mathbf{K}_{k}=\mathbf{P}_{k|k-1}\
abla\textit{h}^{\text{T}}\Big{(} \
abla\textit{h}\mathbf{P}_{k|k-1}\
abla\textit{h}^{\text{T}}+\mathbf{R}_{k}\Big{)}^{- 1}\\ \mathbf{x}_{k|k}=\mathbf{x}_{k|k-1}+\mathbf{K}_{k}\Big{(}\mathbf{z}_{k}-\textit{h}\Big{(}\mathbf{x} _{k-1|k-1}\Big{)}\Big{)}\\ \mathbf{P}_{k|k}^{-1}=\mathbf{P}_{k|k-1}^{-1}+\
abla\textit{h}^{\text{T}}\mathbf{R}_{k}^{- 1}\
abla\textit{h}-\gamma^{-2}\mathbf{\mathcal{I}}_{n}\end{array}  \tag{47}\]  
where \(\mathbf{\mathcal{I}}_{n}\) is the \(n\)-dimension identity matrix. The EH\({}_{\infty}\)F is a Jacobian-based filter, and can be extended with cubature rules for higher precision in nonlinear systems as [PERSON] did in [30]. The time-update phase of the cubature \(H_{\infty}\) filter (CH\({}_{\infty}\)F) is the same as the CKF and expressed as the equation. Therefore, we only provided the measurement-update equations as follows:  
\[ \begin{array}{l}\mathbf{P}_{\text{zz}}=\mathbf{H}_{k}\mathbf{P}_{k|k-1}\mathbf{H}_{k}^{\text {T}}+\mathbf{R}_{k}\\ \mathbf{K}_{k}=\mathbf{P}_{\text{xz}}\mathbf{P}_{\text{zz}}^{-1}\\ \hat{\mathbf{x}}_{k|k}=\hat{\mathbf{x}}_{k|k-1}+\mathbf{K}_{k}\Big{(}\mathbf{z}_{k}-\textit{h} \Big{(}\hat{\mathbf{x}}_{k|k-1}\Big{)}\Big{)}\\ \mathbf{P}_{k|k}^{-1}=\mathbf{P}_{k|k-1}^{-1}+\mathbf{H}_{k}^{\text{T}}\mathbf{R}_{k}^{-1}\mathbf{ H}_{k}-\gamma^{-2}\mathbf{\mathcal{I}}_{n}\end{array}  \tag{48}\]  
where the cross-covariance matrix \(\mathbf{P}_{\text{xz}}\) and the measurement matrix \(H_{k}\) are evaluated using a cubature rule. It can be seen from Equations (47) and (48) that the main difference between the EH\({}_{\infty}\)F and the Kalman filter was the formula to update the covariance matrix \(\mathbf{P}_{k|k}\). In this way, the EH\({}_{\infty}\)F acts to reduce the confidence of the state estimation in case the unexpected noises negatively impact the performance.