# Tracking of Moving Targets Through Asynchronous Measures
## 1 Introduction  
Nowadays accurate 3D localization is crucial to enable many industrial applications, from robotics to agriculture [1, 2], surveillance systems, and Unmanned Aerial Vehicles (UAVs). However, the requirements on positioning accuracy are getting more and more restrictive amidst varying and more challenging operational conditions, such as the lack of guaranteed periodic and reliable measurements from all anchors. Our motivation for deriving and exploring a novel 3D tracking algorithm in fact comes from the necessity of addressing some of the impairments the algorithms currently available are not able to effectively deal with. Particularly, in this article, we will focus on UAVs and explore scenarios in which the distance measurements from the anchors could reach the target asynchronously and without guarantee, due for instance to impairments like Non-Line-of-Sight (NLoS) or unreliable links. In the context of UAVs, a study [3], published in 2024, explores how carrier frequency offset, phase noise, and down-tilted base station antennas affect UAV positioning and tracking. Another article [4] addresses the problem of estimating the orientation, position, and linear velocity of a rigid body moving in 3 Dspace with six degrees of freedom. This is achieved by combining inertial motion units and cameras using an Unscented Kalman Filter (UKF) in GPS-denied environments. Two additional studies have focused on the issue of mutual 3D localization between multiple UAVs, also deriving the Cramer-Rao Bound for relative localization accuracy [5; 6]. The most widely employed tracking algorithms usually estimate the 3D position through multilateration, eventually combined with angular and power measurements, and rely on Extended Kalman Filters (EKFs) and Particle Filters (PFs) to track the target over time [7; 8; 9; 10]. However, the multilateration process is commonly performed assuming that Time of Arrival (ToA) or other measurements are available at the processing unit in clusters associated with the same time instant or target position. Since a new 3D position update requires a set of 3 or 4 (if target and anchors are mutually asynchronous) ToA measurements, it is usually assumed that the target position does not change during the time required to collect these measurements. Here, we remove this assumption and consider a tracking system in which the measurements associated with different anchors can arrive asynchronously and without a specific order. Thus, each measurement is associated with a different target position, and as a result, the measurements cannot be organized into clusters, which would be necessary to generate a new 3D position update in the tracking process. To the best of the authors' knowledge, this scenario is not commonly considered in the literature, except for a few contributions. An article [11] published in 2001 provided an efficient strategy to localize and synchronize the nodes of a wireless sensor network, taking into account both clock offsets and clock drifts. However, this method is designed for static targets, which are not impacted by the issue of asynchronous ranging measures. A study published in 2014 [12] provided an algorithm to combine asynchronous state predictions and updates through EKF and PF. However, this article does not tackle the problem of the unavailability of direct state estimates, which occurs when the system has fewer than 3 reliable range measurements available to perform the update. Another study [13] published in 2021 proved that asynchronous GPS and IMU sensors can be effectively interpolated through a smoothing filter, before the application of a standard EKF. Nevertheless, this study assumes that the anchors (satellites in this case) and thus the corresponding range measurements are synchronous. In this article, we propose a novel 3D tracking algorithm, denoted as the Asynchronous Tracking Algorithm, that relies on the multilateration of asynchronous anchors.