# Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based Visual-Inertial Odometry
## 3 Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based VIO
### Summary of MCH\({}_{\infty}\)IF-Based VIO  
Combining the above methods, a mixed-order cubature \(H_{\infty}\) information filter (MOC\(H_{\infty}\)IF) was designed for the nonlinear non-Gaussian VIO system. The time-update phase was the same as Equations (30)-(34). The measurement-update phase was performed in the following steps.  
* Evaluate cubature points based on the 5-SSR rule with the time complexity of \(O\big{(}N_{p}\big{)}\) \[\hat{\mathbf{\xi}}_{k|k-1,j}^{\mathrm{m}}=\mathbf{S}_{1,k|k-1}\mathbf{\gamma}_{j}^{\mathrm{SSR }}+\hat{\mathbf{x}}_{1,k|k-1}\] (56)
* Evaluate the predicted measurements \(\hat{\mathbf{z}}_{k|k-1}\) and the cross-covariance matrix \(\mathbf{P}_{xz}\) with the time complexity of \(O\big{(}mN_{p}\big{)}\) \[\hat{\mathbf{z}}_{k|k-1}=\sum\limits_{j=1}^{n_{1}^{2}+3n_{1}+3}\mathrm{w }_{j}h\Big{(}\hat{\mathbf{\xi}}_{k|k-1,j}^{\mathrm{m}}\Big{)}\] (57) \[\mathbf{P}_{xz}=\left[\begin{array}{c}\sum\limits_{j=1}^{n_{1}^{2} +3n_{1}+3}\mathrm{w}_{j}\hat{\mathbf{\xi}}_{k|k-1,j}^{\mathrm{m}}h\Big{(}\hat\mathbf{{\xi}}_{k|k-1,j}^{\mathrm{m}}\Big{)}^{\mathrm{T}}-\hat{\mathbf{x}}_{1,k|k-1}\hat{ \mathbf{z}}_{k|k-1}^{\mathrm{T}}\\ \mathbf{P}_{21,k|k-1}\mathbf{P}_{11,k|k-1}^{-1}\Big{(}\sum\limits_{j=1}^{n_{1}^{2}+3n _{1}+3}\mathrm{w}_{j}\hat{\mathbf{\xi}}_{k|k-1,j}^{\mathrm{m}}h\Big{(}\hat{\mathbf{ \xi}}_{k|k-1,j}^{\mathrm{m}}\Big{)}^{\mathrm{T}}-\hat{\mathbf{x}}_{1,k|k-1}\hat{ \mathbf{z}}_{k|k-1}^{\mathrm{T}}\end{array}\right]\]
* Evaluate the measurement matrix \(\mathbf{H}_{k}\) with the third equation in Equation (17) and update the information matrix using the information filter with the time complexity of \(O(m)\) \[\mathbf{Y}_{k}{}^{\prime}=\mathbf{Y}_{k|k-1}+R^{-1}\mathbf{H}_{k}^{\mathrm{T}}\mathbf{H}_{k}\] (58)
* Evaluate the gain matrix \(\mathbf{K}_{\infty}\) and execute the measurement update of the \(H_{\infty}\) filter with the time complexity of \(O(m)\) \[\mathbf{K}_{\infty}=R^{-1}\mathbf{Y}_{k}{}^{\prime}{}^{-1}\mathbf{H}_{k}^{ \mathrm{T}}\] (59) \[\hat{\mathbf{i}}_{k}=R^{-1}\mathbf{H}_{k}^{\mathrm{T}}\Big{(}\mathbf{z}_{k}- \mathbf{h}\Big{(}\hat{\mathbf{x}}_{k|k-1}\Big{)}+\mathbf{H}_{k}\hat{\mathbf{x}}_{k|k-1}\Big{)}- \gamma^{-2}\Big{(}\hat{\mathbf{x}}_{k|k-1}+\mathbf{K}_{\infty}\Big{(}\mathbf{z}_{k}-\mathbf{h} \Big{(}\hat{\mathbf{x}}_{k|k-1}\Big{)}\Big{)}\Big{)}\] \[\hat{\mathbf{y}}_{k|k}=\hat{\mathbf{y}}_{k|k-1}+\mathbf{i}_{k}\] \[\mathbf{Y}_{k|k}=\mathbf{Y}_{k}{}^{\prime}-\gamma^{-2}\mathbf{1}_{n}\]In conclusion, the time complexity of the measurement-update phase of \(\text{MOCH}_{\infty}\text{IF}\) is \(O\big{(}mN_{p}\big{)}\). In the case of \(n_{1}=6\), we have \(N_{p}=57\). Generally, the number of measurements is much larger than that i.e., \(m\gg N_{p}\). Thus, the above measurement-update phase is a linear algorithm. In addition, the process of the measurement-update phase is highly parallelizable except for the Cholesky decomposition and the inverse of \(\Upsilon_{k}{{}^{\prime}}\) as it mainly consists of matrix additions and multiplications.