# A Closed-Form Method for Simultaneous Target Localization and UAV Trajectory Optimization
## 1 Introduction  
Target localization based on unmanned aerial vehicles (UAVs) has played an important role in many modern applications such as intelligence, reconnaissance, and surveillance missions (ISR) [1, 2]. Equipped with onboard infrared and optical cameras as well as communication and navigation hardware, modern UAVs are capable of tracking an anonymous target, surveying the environment and providing the operator with real-time visual information. A common ISR mission for UAV is to track and localize a target detected by airborne sensors. Exploiting the image data of the target from different observation stations of the UAV and the knowledge of the UAV position and orientation, the target position can be estimated by establishing the collinear equation of the optical center, imaged point, and the true position of the target. However, due to the noisy measurements, the obtained estimates always deviate from the true positions. In order to position the target with a high accuracy given the fixed noise levels, a feasible and effective solution is to choose the observation stations so that the optimal sensor-target geometry can be formed, which can minimize the estimate uncertainty. As the task is to localize a moving target, the optimized sequential UAV observation stations form a desired UAV trajectory through which the target can be observed with the minimized uncertainty. An off-line planned trajectory cannot well predict the behavior of the target, and it is also difficult for a human operator to fly the vehicle so that the target position can be estimated with the minimized error. Hence, an autonomous and on-line UAV path generation is important to target localization. The purpose of this paper was to provide a method for UAV trajectory generation that enables the optimized localization performance. Relevant works on target localization are primarily about the localization of a stationary target [3; 4; 5], for instance, [PERSON] [6] developed a total least square algorithm for two-dimensional localization of a stationary target using a bearing only measurement; [PERSON] [7] exploited the geometric constraint method to implement target position estimation, which can obtain a geometrically consistent estimation; [PERSON] [3] used the recursive least square (RLS) filtering and wind velocity estimation method to improve the target localization performance, showing that the target localization bias caused by measurement biases can be eliminated using an on-line learning algorithm; [PERSON] [4] studied how to improve localization performance using wind velocity estimation and pitch estimation method; [PERSON] [5] presented a method for localizing a stationary target using the pixel location of the target in the aerial image, with the measurement of UAV position and attitude, and camera pose angles. The obtained estimate of target position in the world coordinate has been demonstrated with an error less than 11 m. [PERSON] [8] studied the use of electro-optical stabilized imaging systems to localize multiple targets in real time, and the RLS method was used to estimate the positions of targets. However, the above works mainly focused on methods on the estimation or bias elimination. Apart from establishing a proper estimation method that can improve the target localization accuracy, another important factor that plays an important role in the localization performance should also be taken into account, namely, the UAV-target geometry. There have been many works on optimizing the target localization geometry. [PERSON] [9] established the angular separation requirement for the angle of arrival (AOA) sensors to achieve the best mean-squared-error (MSE) localization performance for arbitrary but fixed sensor ranges.