Minimum-Effort Differential Geometric Guidance with Angle Constraints and Obstacle Avoidance

This paper develops a minimum-effort waypoint-following guidance law for unmanned aerial vehicles (UAVs) in constant-altitude flight. By extending a differential geometry-based formulation, the guidance problem is posed in the arc-length domain with curvature as the optimization variable, resulting in a closed-form command that avoids iterative computation and remains robust to speed variations. To ensure computational scalability for long-range tasks, a receding horizon strategy is adopted. By optimizing over a fixed window of subsequent waypoints, the approach transforms the cubic complexity of global planning into a deterministic computational load independent of the task's total scale. Analysis indicates that a window size of N_w = 3 serves as the minimum topological requirement to resolve a complete S-turn primitive, providing a robust balance between geometric foresight and path-to-go estimation accuracy. Beyond this, the application scope is extended to incorporate flight-path angle constraints and fixed obstacle avoidance directly into the guidance formulation. Nonlinear simulations and real quadrotor flight experiments demonstrate reliable waypoint following, satisfaction of angular constraints, and smooth, efficient trajectories while explicitly avoiding predefined obstacles.