In this paper, we present a control design methodology, stability criteria, and performance bounds for autonomous helicopter aerial refueling. Autonomous aerial refueling is particularly difficult due to the aerodynamic interaction between the wake of the tanker, the contact-sensitive nature of the maneuver, and the uncertainty in drogue motion. Since the probe tip is located significantly away from the helicopter's center-of-gravity, its position (and velocity) is strongly sensitive to the helicopter's attitude (and angular rates). In addition, the fact that the helicopter is operating at high speeds to match the velocity of the tanker forces it to maintain a particular orientation, making the docking maneuver especially challenging. In this paper, we propose a novel outer-loop position controller that incorporates the probe position and velocity into the feedback loop. The position and velocity of the probe tip depend both on the position (velocity) and on the attitude (angular rates) of the aircraft. We derive analytical guarantees for docking performance in terms of the uncertainty of the drogue motion and the angular acceleration of the helicopter, using the ultimate boundedness property of the closed-loop error dynamics. Simulations are performed on a high-fidelity UH60 helicopter model with a high-fidelity drogue motion under wind effects to validate the proposed approach for realistic refueling scenarios. These high-fidelity simulations reveal that the proposed control methodology yields an improvement of 36% in the 2-norm docking error compared to the existing standard controller.