# Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based Visual-Inertial Odometry
## 1 Introduction
The developed fifth-degree spherical simplex-radial cubature Kalman filter (SSRCKF) required fewer cubature points than the fifth-degree CKF in [18] in high-dimensional applications. [PERSON] developed an arbitrary degree interpolatory cubature Kalman filter (ICKF) with additional free parameters [20]. By adjusting the free parameters, CKF and UKF are special cases of ICKF. An estimator with an online measuring of non-linearity was developed in [21]. The method adaptively switches between cubature rules with different degrees. The CKF was used in [22] for a loosely coupled nonlinear attitude estimator. [PERSON] [23] embedded the Huber M-estimation method with CKF in [17] to deal with the outliers and non-Gaussian noises in the measurements. As an equivalent description of the Kalman filter, the information filter has also been extended with the cubature rules for nonlinear systems. [PERSON] directly applied the technique used in [17] on the information filter and obtained a third-degree spherical-radial cubature rule-based cubature information filter (CIF). [PERSON] applied the fifth-degree spherical-radial cubature rule on the information filter for multi-sensor fusion [24; 25]. [PERSON] developed the third-degree spherical simplex-radial cubature information filter. CIFs with different cubature rules have been used in the state estimation for the biased nonlinear system [26], bearing-only tracking with multi sensors [27], and the trajectory estimation for the ballistic missile [28]. The \(H_{\infty}\) filter was developed for minimizing the estimation errors in the presence of unsatisfactory noises. [PERSON] developed a solution of the \(H_{\infty}\) filter in the form of the regular Kalman filter [29]. [PERSON] directly embedded [PERSON]'s \(H_{\infty}\) filter in the CKF and CIF according to the computational procedures of the filters [30; 31] to improve the CKF and CIF in the presence of non-Gaussian noises. In this paper, motivated by the recently developed cubature information filters for nonlinear systems, a mixed-degree cubature \(H_{\infty}\) information filter-based VIO (M\(CH_{\infty}\)IF-VIO) was designed by fusing the measurements of IMU and intensity errors in a tightly coupled manner. Unlike previous studies that have focused on refining the feature descriptions, we developed the M\(CH_{\infty}\)IF-VIO system according to the system characteristics and used a simple frame-to-frame alignment model. The proposed VIO method mainly focuses on the nonlinearity and non-Gaussian noises in the VIO system,and is suitable for the camera-IMU system consisting of an IMU and an RGBD camera. The main contributions of this paper are as follows. 1. A mixed-degree nonlinear filter framework was proposed for fusing the IMU measurements and the intensity measurements of the camera. The proposed filter applied two cubature rules with different degrees to guarantee numerical stability with high accuracy. 2. The \(H_{\infty}\) filter was used in the measurement update phase for robustness toward the non-Gaussian noises in intensity errors. The remainder of this paper is organized as follows. The model of the visual-inertial odometry is presented in Section 2. The VIO system based on a mixed-order cubature \(H_{\infty}\) information filter is detailed in Section 3. The test results with a public dataset are presented in Section 4. The conclusions are presented in Section 5.