Coordinated Incremental Trajectory Tracking of a Tailsitter Drone

Technical Summary: Coordinated Incremental Trajectory Tracking of a Tailsitter Drone

Problem Statement
Hybrid Unmanned Aerial Vehicles (UAVs), specifically tailsitters, offer the ability to transition between hover and forward flight. However, controlling these vehicles across their full flight envelope is challenging, particularly during coordinated flight regimes involving significant vertical velocity components. Existing state-of-the-art differential flatness-based controllers, such as the one proposed by Tal et al. [3], rely on assumptions valid primarily for level flight. When applied to trajectories with significant vertical velocity, these approaches encounter singularities or ambiguities, particularly when defining the yaw angle or when the vertical component of specific force approaches zero. Furthermore, alternative formulations using the $\pi$-theory aerodynamic model [6] often require costly wind-tunnel identification and introduce singularities near hover where angle of attack and sideslip are undefined.

Methodology
The authors propose a framework for the analytical derivation of a differential flatness transform for a quadrotor tailsitter under coordinated flight conditions. The methodology is built upon the following pillars:

• Aerodynamic Modeling: The work utilizes the $\phi$-theory parameterization [4], which offers a global model applicable across the entire flight envelope, including transition and post-stall regimes. Unlike the $\pi$-theory, this approach avoids singularities near hover and allows for parameter identification directly from flight data without explicit lift and drag coefficients. The net specific force is modeled as a sum of aerodynamic forces and specific thrust, expressed entirely in the body frame.
• Rotation Matrix Formulation: To avoid the singularities and ambiguities inherent in Euler angle representations, the attitude and rotational dynamics are formulated exclusively using rotation matrices. The vehicle's orientation is represented by the rotation matrix $R^i_b \in SO(3)$.
• Differential Flatness Derivation:
  • Flat Output: The vehicle position $p$ is selected as the flat output.
  • Attitude Reconstruction: For coordinated flight, the velocity relative to the airmass ($v_a$) and the specific force ($f$) must lie on the body XZ plane. The body Y-axis ($b_y$) is derived analytically as the normalized cross product of $v_a$ and $f$. An intermediate frame $E$ is constructed to facilitate the calculation of the rotation from the inertial frame to the body frame.
  • Analytical Transform: The paper derives closed-form analytical expressions for the attitude ($R^i_b$), angular rates ($\Omega$), and angular accelerations ($\dot{\Omega}$) based on the flat output and its derivatives up to the fourth order. This allows for the direct computation of feedforward signals.
  • Control Input Recovery: The control inputs (propeller speeds) are recovered analytically from the required angular accelerations and specific thrust using a control effectiveness matrix.
• Controller Architecture: The derived flatness transform is integrated into an Incremental Nonlinear Dynamic Inversion (INDI) controller. The controller uses the analytical transform to generate feedforward angular rate and acceleration signals, which are combined with feedback terms to track reference trajectories.
• Singularity Handling: The authors acknowledge that the formulation exhibits singularities when $\|v_a \times f\| = 0$ (e.g., near-hover, free-fall, or aligned velocity/force). While the primary strategy is to handle these at the trajectory generation level, a heuristic approach is adopted for the controller: near singularities, the commanded body Y-axis is held constant based on the previous time step. For near-hover conditions, the system switches to the uncoordinated flatness transform from [3].

Key Contributions

1. Analytical Differential Flatness Transform: The paper derives an analytical differential flatness transform specifically for coordinated flight of tailsitter UAVs using the $\phi$-theory parameterization.
2. Rotation Matrix Exclusivity: The formulation avoids Euler angles entirely, utilizing rotation matrices to ensure robustness against singularities and ambiguities in attitude representation.
3. Extension of Applicability: The framework extends the applicability of existing incremental control architectures to flight regimes with significant vertical velocity components, where previous methods (e.g., [3]) fail or become singular.
4. Experimental Validation: The approach is validated experimentally on a Parrot Swing quadrotor tailsitter. The experiments demonstrate successful tracking of complex maneuvers, including half-loops with vertical and cross-track motion, where uncoordinated methods would encounter singularities.

Results
Experimental trials were conducted in an indoor arena using a motion-capture system. Two primary scenarios were tested:

• Half-Loop Maneuver: A trajectory involving a 4m entry, a 1.5m radius half-loop, and a symmetric exit. The proposed controller successfully tracked the trajectory with an error norm below 0.4m. The experiment highlighted that the uncoordinated method would fail at the loop exit where the vertical specific force component approaches zero, causing a singularity in the yaw calculation. The proposed method remained well-defined.
• Half-Loop with Cross-Track Motion: A 3D maneuver involving a half-loop combined with lateral motion. This scenario further demonstrated the inability of uncoordinated methods to define a meaningful yaw angle for coordinated flight in 3D space. The proposed transform maintained stability, though minor sensitivity was observed near the condition where $v_a \parallel f$, attributed to noisy velocity estimates from the positioning system.

Significance and Claims
The authors position this work as a complementary advancement to the uncoordinated flight control framework presented in [3]. While [3] addresses uncoordinated flight, this paper addresses the coordinated flight regime, thereby enabling higher-speed maneuvers and more complex trajectories that were previously inaccessible to differential flatness-based controllers on tailsitters.

The paper modestly claims that the proposed framework:

• Removes the need for costly wind-tunnel identification by relying on flight-data-identified $\phi$-theory parameters.
• Avoids the singularities associated with Euler angles and near-hover conditions found in $\pi$-theory models.
• Provides a well-defined analytical solution for attitude and control inputs in coordinated flight, even when vertical velocity is significant.

The authors note that trajectory generation remains a challenge due to the vehicle's underpowered nature and the need to avoid actuator saturation, and they identify online trajectory generation and outdoor flight validation as necessary future work. They also propose the potential for a unified framework combining both coordinated and uncoordinated transforms to enable a broader range of aerobatic maneuvers.