# Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based Visual-Inertial Odometry
## 3 Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based VIO
### Mixed-Degree Cubature Information Filter  
In general, the four integrals in Equations (16) and (17) cannot be solved directly. A group of numerical integration-based approximation methods named by cubature rules are used for the filter process and the resulting filters are called cubature filters. Using a cubature rule, a Gaussian weighted integral of a nonlinear function \(\mathbf{g}(\mathbf{x})\) is approximated with a weighted sum as follows:  
\[\int_{\mathbb{R}^{n}}g(x)\mathbf{\mathcal{N}}\Big{(}x;\hat{\mathbf{z}}_{k-1|k-1},P_{k- 1|k-1}\Big{)}dx\approx\sum_{j=1}^{N_{p}}w_{j}g(X_{j}) \tag{28}\]  
where \(\mathbf{X}_{j}\) represents the cubature points; \(w_{j}\) is the corresponding weight that satisfies \(\sum\limits_{j=1}^{N_{p}}w_{j}=1\); and \(N_{p}\) is the number of the cubature points. The formulations of the cubature points and weights are different in different cubature rules. The spherical-radial (SR) cubature rules use the spherical-radial coordinates to transform the original integral into a double integral and select the cubature points on the spherical coordinate and the radial coordinate respectively. Based on the SR rules, the spherical simplex-radial (SSR) cubature rules use the n-simplex to evaluate the cubature points on the spherical coordinate. The degree of a cubature rule is defined based on the degree of used monomials in the Tayler polynomial of the nonlinear function. A \(p\)-degree cubature rule integrates all of the monomials in the Tayler polynomial of \(\mathbf{g}(\mathbf{x})\) up to degree \(p\) exactly but not exactly for some monomials of degree \(p+1\). In general, the cubature rule with a higher degree achieves higher accuracy, but suffers from more computation with more cubature points and worse numerical stability with potential negative weights. Practically, third-degree cubature rules and fifth-degree cubature rules are commonly used in the Bayes filter. The numbers of cubature points of the four cubature rules are listed in Table 2.  
A cubature rule with all weights being positive is more desirable than one with both positive and negative weights for the sake of numerical stability. A stability parameter was defined in [33] as the sum of absolute values of the weights:  
\[\Theta=\sum_{i=0}^{N_{p}}\lvert w_{i}\rvert \tag{29}\]  
When \(\Theta\) is larger than 1, the truncation error will decrease the accuracy of the numerical integration. The curves of \(\Theta\) with respect to \(n\) for different cubature rules are shown in Figure 2. The stability parameters of 3-SR and 3-SSR were maintained at 1, while the stability parameter of 5-SR increased since \(n>4\), and the stability parameter of 5-SSR increased since \(n>7\). According to the previous motion and measurement models, the time-update phase involves Gaussian-weighted integrals in 9-dimensional space while the measurement-update phase involves Gaussian-weighted integrals in 6-dimensional space. To obtain high precision and numerical stability with as few cubature points as possible, we applied the 3-SR rule in the time-update phase and the 5-SSR rule in the measurement-update phase. Substituting the chosen cubature rules, we formulated the VIO process based on the mixed-degree cubature information filter as follows. (1) Time-update phase Evaluate the cubature points based on the 3-SR rule:  
\[\mathbf{\zeta}_{k-1\lvert k-1,j}=\mathbf{S}_{k-1\lvert k-1}\mathbf{\gamma}_{j}^{\text{SR3} }+\mathbf{\hat{x}}_{k-1\lvert k-1},1\leq j\leq 2n \tag{30}\]  
where \(\mathbf{S}_{k-1\lvert k-1}\) is the square root of the covariance matrix \(\mathbf{P}_{k-1\lvert k-1}\) so that \(\mathbf{P}_{k-1\lvert k-1}=\mathbf{S}_{k-1\lvert k-1}\mathbf{S}_{k-1\lvert k-1}^{\text{T}}\) and could be evaluated using the Cholesky decomposition; and \(\mathbf{\gamma}_{j}^{\text{SR3}}\) is formulated as follows: