# A Closed-Form Method for Simultaneous Target Localization and UAV Trajectory Optimization
## 1 Introduction
[PERSON] [10] studied the influence of the geometric configuration between the multiple stationary observation stations and the stationary target on target localization performance; [PERSON] [11] investigated the optimal sensor placement strategies for AOA localization in 3-dimensional space by using a resistor network analogy to determine the sensor-target geometry. The metric for optimizing the sensor-target geometry in the above work is mainly the Fisher information matrix (FIM). Furthermore, [PERSON] [12] proposed a two-UAV intersection target localization system for which the UAVs were equipped with airborne optoelectronic platforms. The optimal positions for the two UAVs were considered by changing the baseline between them. Compared with the sensor-target optimization problem for a stationary target as above, the counterpart problem for the moving target is a different situation, which is a problem on the UAV trajectory optimization. [PERSON]. [13] presented an early study that tried to optimize the observer's path to increase the localization accuracy for a moving target. Afterward, the problem on the localization of moving targets with measurements from different mobile sensors has attracted many researchers such as [14], who used using time differences of arrival (TDOA) and frequency differences of arrival (FDOA). In [15], the moving observer was illustrated to make a spiral motion around the moving target. Ref. [16] studied the vehicle planning problem for optimal target localization with range measurement by simultaneously considering the collision avoidance and vehicle maneuvering constraints. Ref. [17] proposed a non-causal trajectory optimization method for optimally localizing a moving target, which could achieve a better localization performance in some external constraint conditions than the causal trajectory optimization strategy, for example, the UAV travel length or the total travel time is restricted. Optimal vehicle trajectory that maximizes the observability for maneuvering target is also generated in [18]. The authors dealt with range observability for the target with a bearing-only measurement, which is also equivalent to the localization observability of the target as together with the bearing measurement, the target can be localized with the observable range. Ref. [19] utilized a geometry metric to measure the target localization performance, with which a 2-dimensional constrained robot trajectory optimization problem was addressed to increase the localization performance using bearing-only measurement. Furthermore, trajectory optimization also involves the stationary target localization problem. In [20], the problem of trajectory optimization for multiple UAVs with investigated with heterogeneous sensors, which can cooperatively localize a target to achieve a better performance. Ref. [21] investigated the UAV path planning problem to increase the localization performance to a signal emitter. The optimal observer trajectory that maximizes the observability of the target of interest was also studied in [22; 23; 24; 25]. However, the above work mainly deals with the stationary target or the moving target with an ideal motion model such as the constant velocity. In practical applications, the target motion model is not exactly known, and this can usually be modeled by the process noises of the dynamic model. If the error from the motion model is neglected, the resultant optimized trajectory will become unstable. Under the vehicle constraints, this will result in a different trajectory, which will degenerate the target localization performance. In this paper, we present a closed-form solution for simultaneous target localization and UAV trajectory optimization problem. With this method, the extended information filtering (EIF) [26; 27] is exploited to estimate the target state, which can compute the predicted information matrix required by the trajectory optimization explicitly and iteratively.