Trajectory Planning for ISAC Systems: Variational Method to Minimization of Completion Time

Integrated sensing and communication (ISAC) enables a unified framework for high-capacity communications and accurate sensing, where trajectory optimization plays a crucial role in jointly enhancing communication quality and sensing performance. This paper explores trajectory planning for ISAC systems, aiming to minimize the completion time for a radio target (RT) to traverse between designated starting and ending points, while fulfilling a communication task with a minimum data transmission requirement. We introduce the concept of path exposure to characterize path sensitivity by capturing the cumulative sensing intensity along a trajectory. Furthermore, path capacity is proposed to measure the end-to-end throughput of a given path. The trajectory planning problem is formulated as the minimization of completion time while maintaining path sensitivity and path capacity requirements. By leveraging the Lagrangian approach, the problem is converted into a framework with calculus of variations and solved using the Euler–Lagrange equation. The boundary value problem (BVP) methodology is then employed to compute the optimal trajectory. Simulation results demonstrate the effectiveness of the proposed continuous trajectory planning approach compared to discrete methods and highlight the inherent trade-off between trajectory length and the quality-of-service requirements in both sensing and communication domains.