# Mixed-Degree Cubature \(H_{\infty}\) Information Filter-Based Visual-Inertial Odometry
## 1 Introduction  
Visual odometry (VO) is the process of estimating the egomotion of an agent with a single or multiple visual sensors in mobile robot navigation. The aim of VO is to estimate the pose incrementally. Compared with the simultaneous localization and mapping (SLAM) method seeking globally consistent estimation, VO is concerned with the local consistency of the pose estimation [1]. As an inexpensive method independent of external reference systems such as global positioning systems (GPS) and motion capture systems, VO is widely applied in domains including robotics, automotive, and wearable computing [2]. In robotic applications, VO is the basis of studies on visual control [3], obstacle avoidance in robot navigation [4; 5], and unmanned aerial vehicle navigation [6]. In practical applications, VO is commonly combined with the inertial measurement units (IMUs) with the aid of Kalman filters. As the visual-inertial integrated navigation system serves as an egomotion estimator and is similar to VO functionally, we used the term of visual-inertial odometry (VIO) in this paper. The IMU generally operates independently of the environmental conditions at a higher rate than cameras. Therefore, the robustness and accuracy of the visual odometry can be enhanced by fusing the IMU measurements. The VO methods are grouped into indirect methods and direct methods, depending on the usage of the images. The indirect methods preprocess the images of the camera to extract features that are used for motion estimation with a matching process. Direct methods use the raw measurements of the visual sensors without feature extraction and matching processes. Indirect methods have dominated the research field for a long time, and many effective descriptors of features have been developed as shown in [7]. [PERSON] [8] applied a line-feature extractor in the pre-processing of images and estimated the attitude error with line features. The IMU measurementswere fused with the visual measurements loosely with the conventional Kalman filter. [PERSON] [9] constructed a multi-sensor fusion system based on the extended Kalman filter (EKF). An indirect VO algorithm with speeded up robust features (SURF) was applied to process the visual measurements. A tightly coupled nonlinear optimization-based indirect VO was applied on the micro aerial vehicle (MAV) in [10; 11]. [PERSON] developed a hybrid approach with a direct method for initial alignment and an indirect method for joint optimization. [PERSON] [12] built a light-weight VO using the adaptive and generic accelerated segment test (AGAST) corner detector and the binary robust independent elementary features (BRIEF) descriptor. Recently, direct methods have gained more attention due to their independence of extra feature extraction and matching processes. [PERSON] [13] used the intensity errors of image patches as measurements and fused the measurements of IMU with the direct VO with EKF. The resulting VIO method was applied in the navigation of a MAV in [14]. Furthermore, [PERSON] used the iterated EKF method that operates in a recursive form like the Gauss-Newton optimization to reduce the linearization errors in the regular EKF. A binocular vision system was used to construct a direct VIO in [15]. [PERSON] designed a direct sparse odometry (DSO) using a monocular camera [16]. The even sampling and keyframe management methods used in [PERSON]'s DSO contributed to improving the accuracy and robustness of the photometric-error optimization. The Jacobian-based EKF method suffers from linearization errors in nonlinear systems. To improve its accuracy, [PERSON] designed the cubature Kalman filter (CKF) by applying the third-degree spherical-radial cubature rule on the nonlinear Bayes filter [17]. The resulting CKF outperformed the EKF and unscented Kalman filter (UKF) on nonlinear systems. Based on the third-degree CKF, [PERSON] designed a fifth-degree CKF for higher accuracy [18]. [PERSON] developed another CKF method with the regular simplex and moment matching method [19].