Optimal UAV Trajectory Design for Moving Users in Integrated Sensing and Communications Networks

In this paper, we consider a unmanned aerial vehicle (UAV) aided integrated sensing and communications (ISAC) network with moving ground users in constant-velocity trajectory. A global optimal trajectory design scheme is proposed including a continuous analytic solution as well as optimization condition, which is theoretically different from numerical schemes obtaining a discrete piece-wise solution with approximate optimality. However, it is challenging to maximize the performance over entire infinite time slots in moving-user scenarios. By projecting the trajectory onto a user-relative coordinate frame, we reduce the performance to a location-determined function, which is in physical equivalence to an artificial potential field (APF). Accordingly, the optimization problem is reformulated to the shape determination problem of a density-varying catenary in the APF. By performing force analysis, we describe the topology of the catenary via a second-order differential equation determined by a boundary, i.e., any three combination of the location, orientation and turning curvature at arbitrary waypoints. Equivalently, the representation of the continuous solution is minimized in ultra-low-dimension parameter space and offers a flexible and lightning-speed design practice. Through complexity analysis, we find that the complexity of the proposed analytic scheme is significantly lower than the traditional discrete schemes. Further, we also prove that the global optimality, existence and uniqueness of the solution holds under a condition of a strong applicability to general sensing and communications (S&C) services. In simulation, the equivalence is confirmed and the results show global optimality, low-complexity and the high flexibility under avoidance, crossing and G-force limit.