*3.4. Summary of MCH*∞*IF-Based VIO*

Combining the above methods, a mixed-order cubature *H*∞ information filter (MOC*H*∞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 Np* 

$$\mathfrak{S}\_{k|k-1,j}^{\rm m} = \mathfrak{S}\_{1,k|k-1} \gamma\_j^{\rm SSR5} + \mathfrak{X}\_{1,k|k-1} \tag{56}$$

• Evaluate the predicted measurements *z*ˆ *<sup>k</sup>*|*k*−<sup>1</sup> and the cross-covariance matrix *Pxz* with the time complexity of *O mNp* 

$$\begin{aligned} \hat{\mathbf{z}}\_{k|k-1} &= \sum\_{j=1}^{n\_1^2 + 3n\_1 + 3} \mathbf{w}\_j h\left(\mathbf{x}\_{k|k-1,j}^{\rm m}\right) \\ \mathbf{P}\_{\mathbf{X}\mathbf{z}} &= \begin{bmatrix} n\_1^2 + 3n\_1 + 3 \\ \sum\_{j=1}^{n\_1} \mathbf{w}\_j \xi\_{\mathbf{k}|k-1,j}^{\rm m} h\left(\xi\_{\mathbf{k}|k-1,j}^{\rm m}\right)^{\rm T} - \hat{\mathbf{x}}\_{1,k|k-1} \hat{\mathbf{z}}\_{k|k-1}^{\rm T} \\ \mathbf{P}\_{21,k|k-1} \mathbf{P}\_{11,k|k-1}^{-1} \left( \sum\_{j=1}^{n\_1^2 + 3n\_1 + 3} \mathbf{w}\_j \xi\_{\mathbf{k}|k-1,j}^{\rm m} h\left(\xi\_{\mathbf{k}|k-1,j}^{\rm m}\right)^{\rm T} - \hat{\mathbf{x}}\_{1,k|k-1} \hat{\mathbf{z}}\_{k|k-1}^{\rm T} \right) \end{bmatrix} \end{aligned} \tag{57}$$

• Evaluate the measurement matrix *H<sup>k</sup>* with the third equation in Equation (17) and update the information matrix using the information filter with the time complexity of *O*(*m*)

$$\mathbf{Y}\_{k}^{\prime} = \mathbf{Y}\_{k|k-1} + \boldsymbol{\mathcal{R}}^{-1} \boldsymbol{H}\_{k}^{\mathrm{T}} \mathbf{H}\_{k} \tag{58}$$

• Evaluate the gain matrix *K*<sup>∞</sup> and execute the measurement update of the *H*<sup>∞</sup> filter with the time complexity of *O*(*m*)

$$\begin{aligned} \mathbf{K}\_{\infty} &= R^{-1} \mathbf{Y}\_{k}^{\prime \prime - 1} \mathbf{H}\_{k}^{\mathrm{T}} \\ \dot{\mathbf{z}}\_{k} &= R^{-1} \mathbf{H}\_{k}^{\mathrm{T}} \left( \mathbf{z}\_{k} - \hbar \left( \mathbf{\hat{x}}\_{k|k-1} \right) + \mathbf{H}\_{k} \mathbf{\hat{x}}\_{k|k-1} \right) - \gamma^{-2} \left( \mathbf{\hat{x}}\_{k|k-1} + \mathbf{K}\_{\infty} \left( \mathbf{z}\_{k} - \hbar \left( \mathbf{\hat{x}}\_{k|k-1} \right) \right) \right) \\ \dot{\mathbf{y}}\_{k|k} &= \mathbf{\hat{y}}\_{k|k-1} + \mathbf{i}\_{k} \\ \mathbf{Y}\_{k|k} &= \mathbf{Y}\_{k}^{\prime} - \gamma^{-2} \mathbf{I}\_{\mathrm{ll}} \end{aligned} \tag{59}$$

In conclusion, the time complexity of the measurement-update phase of MOC*H*∞IF is *O mNp* . In the case of *n*<sup>1</sup> = 6, we have *Np* = 57. Generally, the number of measurements is much larger than that i.e., *m Np*. 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 *Y<sup>k</sup>*  as it mainly consists of matrix additions and multiplications.
