# Visual-Inertial Method for Localizing Aerial Vehicles in GNSS-Denied Environments
## 1 Introduction  
Navigation methods for unmanned vehicles can be broadly categorized into satellite-based, inertial, and vision-based navigation systems. Global Navigation Satellite Systems (GNSSs), such as the Global Positioning System (GPS), estimate positions using time measurements of satellite signals and triangulation techniques. Inertial navigation systems (INSs) rely on data from Inertial Measurement Unit (IMU) sensors such as accelerometers, gyroscopes, altimeters, and magnetometers [1]. However, both GNSS and INS have notable limitations. GNSS signals can be jammed [2], and obstacles between the UAV and satellites, such as urban structures and dense forests [3], can disrupt reception [4]. Indoors, GNSS may be completely unavailable, though alternative solutions like Ultra-Wideband (UWB) can be an effective localization method [5]. Solutions for cooperative aerial assets within swarms in GNSS-denied environments were also developed [6]. In contrast, INS suffers from drift error due to the cumulative nature of numerical integration required to convert acceleration data into displacement estimations [7]. Vision-based navigation utilizes data from visual sensors, often enhanced by computer vision algorithms [8]. The integration of visual data with IMU data is known as visual-inertial navigation [9]. Vision-based navigation can be divided into relative visual localization, which operateswithin a local coordinate system (visual odometry) [10] and absolute visual localization, which estimates positions in world coordinates using landmarks of known positions visible in 2D images [11]. For three or four noncollinear points, exact solutions can be retrieved: a fourth or fifth-degree polynomial system can be defined using geometrical invariants of the observed points, and the problem can be solved by finding roots of the polynomial system [12]. However, these methods are sensitive to noise and outliers, and no closed-form solutions exist for more than four points [13]. In computer vision, this problem is sometimes referenced as the Perspective-n-Point (PnP) problem, where n stands for the number of available reference points [14]. It has been shown that at least three reference points are needed to find a solution to this problem, although using more points can further enhance the accuracy of the results. Typically, a direct approach to the P3P problem may not yield a unique solution, depending on the configuration of the reference points. Direct solutions for the P4P problem with arbitrary reference point configurations were proposed in [15]. Algorithms for solving the PnP problem with \(n\) equal to or greater than four are more robust to noise but require iterative methods, which increase computational complexity. Research has focused on reducing the number of necessary points, as illustrated by approaches that incorporate gravity vectors or augment points with tangent information [16; 17]. These approaches demonstrate that even with a reduced number of points, it is possible to achieve sub-meter positional accuracy with aerial assets. A recent method proposed using only two known world coordinate points by incorporating IMU data such as Pitch and Roll (Euler) angles [18]. This approach, developed for unmanned aerial vehicles (UAVs) equipped with optical cameras with known intrinsic parameters and IMUs capable of estimating camera orientation during image acquisition, addresses the challenge of limited reference points in environments like vegetated areas or over the sea, where identifiable landmarks are sparse. The method proposed in this contribution requires camera parameters such as focal length, detector dimensions, and Pitch and Roll angles. While Pitch and Roll angles are accurately estimated using gyroscopes and accelerometers, Yaw angles, estimated using magnetometers, are susceptible to local magnetic disturbances [6]. Euler angles are estimated on-board, thus self-contained and resistant to radio frequency interference, like jamming and spoofing [19].