**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]. Guang [8] applied a line-feature extractor in the pre-processing of images and estimated the attitude error with line features. The IMU measurements were fused with the visual measurements loosely with the conventional Kalman filter. Mostafa [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]. Forster developed a hybrid approach with a direct method for initial alignment and an indirect method for joint optimization. Aladem [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. Bloesch [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, Bloesch 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]. Engel designed a direct sparse odometry (DSO) using a monocular camera [16]. The even sampling and keyframe management methods used in Engle'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, Arasaratnam 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, Jia designed a fifth-degree CKF for higher accuracy [18]. Wang developed another CKF method with the regular simplex and moment matching method [19]. 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. Zhang 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. Tseng [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. Pakki 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). Jia applied the fifth-degree spherical-radial cubature rule on the information filter for multi-sensor fusion [24,25]. Yin 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*∞ filter was developed for minimizing the estimation errors in the presence of unsatisfactory noises. Yang developed a solution of the *H*∞ filter in the form of the regular Kalman filter [29]. Chandra directly embedded Yang's *H*∞ 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*∞ information filter-based VIO (MC*H*∞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 MC*H*∞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.


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*∞ 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.
