# Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based Visual-Inertial Odometry
## 3 Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based VIO
### Bayes Filter for Nonlinear Visual-Inertial Navigation System
\begin{array}{rl}\boldsymbol{P}_{xz}&=\int_{\mathbb{R}^{n}}xh^{ \mathrm{T}}(x)\boldsymbol{\mathcal{N}}\Big{(}x;\hat{\mathbf{x}}_{k-1|k-1},P_{k -1|k-1}\Big{)}dx-\hat{\mathbf{x}}_{k|k-1}\hat{\mathbf{z}}_{k|k-1}^{\mathrm{T}}\\ &=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}\left[\begin{array}{c}x_ {1}\\ x_{2}\\ \end{array}\right]h^{\mathrm{T}}(x_{1})p(x_{1}x_{2})dx_{2}dx_{1}-\hat{\mathbf{x }}_{k|k-1}\hat{\mathbf{z}}_{k|k-1}^{\mathrm{T}}\end{array}  \tag{23}\]where \(p(x_{1},x_{2})\) is the joint probability density function of two random vectors \(x_{1}\) and \(x_{2}\) i.e., the probability density function of the Gaussian distribution. Based on the formula of conditional probability, one can obtain  
\[p(x_{1},x_{2})=p(x_{2}|x_{1})p(x_{1}) \tag{24}\]  
Substituting Equation (24) into Equation (23) yields  
\[\mathbf{P}_{xz}=\int_{\mathbb{R}^{n_{1}}}\int_{\mathbb{R}^{n_{2}}}\left[ \begin{array} []{c}x_{1}\\ x_{2}\end{array} \right]h^{\rm T}(x_{1})p(x_{2}|x_{1})dx_{2}p(x_{1})dx_{1}- \hat{\mathbf{x}}_{k|k-1}\hat{\mathbf{z}}_{k|k-1}^{\rm T} \tag{25}\]  
With the Gaussian assumption of the state vector, the distribution of \(\mathbf{x}_{2}\) conditional on \(x_{1}\) is a multivariate Gaussian with the mean value of \(\hat{\mathbf{x}}_{2}+\mathbf{P}_{21}\mathbf{P}_{11}^{-1}(x_{1}-\hat{\mathbf{x}}_{1})\) and the covariance matrix of \(\mathbf{P}_{22}-\mathbf{P}_{21}\mathbf{P}_{11}^{-1}\mathbf{P}_{12}\). Thus, the inner integral in Equation (25) is solved as  
\[\int_{\mathbb{R}^{n_{2}}}\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]h^{\rm T}(x_{1})p(x_{2}|x_{1})dx_{2}=\left[ \begin{array} []{c}x_{1}\\ \hat{\mathbf{x}}_{2}+\mathbf{P}_{21}\mathbf{P}_{11}^{-1}(x_{1}-\hat{\mathbf{x}}_{1})\end{array}  \right]h^{\rm T}(x_{1}). \tag{26}\]  
The cross-covariance matrix is evaluated as follows:  
\[\mathbf{P}_{xz} =\int_{\mathbb{R}^{n_{1}}}\left[ \begin{array}{c}x_{1}\\ \hat{\mathbf{z}}_{2k|k-1}+P_{21,k|k-1}\mathbf{P}_{11,k|k-1}^{-1}\Big{(}x_{1}-\hat{\mathbf{ x}}_{1,k|k-1}\Big{)}\end{array} \right]h^{\rm T}(x_{1})p(x_{1})dx_{1}-\hat{\mathbf{x}}_{k|k-1} \hat{\mathbf{z}}_{k|k-1}^{\rm T} \tag{27}\] \[=\left[ \begin{array}{c}\int_{\mathbb{R}^{n_{1}}}x_{1}h^{\rm T}( x_{1})p(x_{1})dx_{1}\\ \int_{\mathbb{R}^{n_{1}}}\Big{(}\hat{\mathbf{z}}_{2,k|k-1}+P_{21,k|k-1}^{-1}\Big{(}x _{1}-\hat{\mathbf{x}}_{1,k|k-1}\Big{)}\Big{)}h^{\rm T}(x_{1})p(x_{1})dx_{1}\\ \end{array} \right]-\hat{\mathbf{x}}_{k|k-1}\hat{\mathbf{z}}_{k|k-1}^{\rm T}\] \[=\left[ \begin{array}{c}\int_{\mathbb{R}^{n_{1}}}x_{1}h^{\rm T}( x_{1})p(x_{1})dx_{1}-\hat{\mathbf{x}}_{1,k|k-1}^{-1}\hat{\mathbf{z}}_{k|k-1}^{\rm T}\\ \mathbf{P}_{21,k|k-1}\mathbf{P}_{11,k|k-1}^{-1}\Big{(}\int_{\mathbb{R}^{n_{1}}}x_{1}h^ {\rm T}(x_{1})p(x_{1})dx_{1}-\hat{\mathbf{x}}_{1,k|k-1}\hat{\mathbf{z}}_{k|k-1}^{\rm T }\end{array} \right]\]  
In this way, the integrals in 9-dimensional space are simplified into integrals in 6-dimensional space. This is practically useful when applying high-degree cubature rules and will be discussed later.