Aircraft Kinematic Equations with Full-Angle Quaternion to Euler Conversion
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Resource Overview
Aircraft attitude kinematic equations represented by Euler angles exhibit singularity at large angles, whereas quaternion representation avoids this issue. Therefore, quaternions are universally adopted for aircraft kinematic equations. However, control laws predominantly use Euler angles due to their intuitive visualization of attitude, making them more comprehensible for human interpretation. Consequently, aircraft control system simulations require bidirectional conversion between quaternions and Euler angles. While converting Euler angles to quaternions is straightforward with a one-to-one correspondence, the reverse conversion is complex due to non-unique mapping where one quaternion may correspond to one or two Euler angle sets. This necessitates robust algorithmic implementation for full-angle range coverage.
Detailed Documentation
In aircraft control systems, both Euler angles and quaternions are crucial concepts. Euler angles provide intuitive visualization of attitude angles, making them widely adopted in aircraft control laws. However, aircraft attitude kinematic equations based on Euler angles exhibit singularity during large-angle maneuvers, necessitating quaternion representation to circumvent this limitation. Thus, quaternions become the standard for aircraft kinematic equations.
The conversion from quaternions to Euler angles presents computational challenges due to non-bijective mapping—a single quaternion typically corresponds to one or two Euler angle sets. Conventional conversion formulas in literature only accommodate Euler angles within -90° to +90° ranges, whereas practical applications require handling full -180° to +180° variations across roll, pitch, and yaw axes.
To address this, we propose a full-angle conversion algorithm validated through digital simulations, proving both mathematically rigorous and practically applicable. Key implementation aspects include:
1. Normalization of quaternion components before conversion
2. Conditional branching based on trigonometric quadrant analysis
3. Sequential calculation of pitch (θ), roll (φ), and yaw (ψ) angles using atan2 functions for sign ambiguity resolution
4. Edge-case handling for gimbal lock scenarios near ±90° pitch angles
This algorithm enables robust Euler angle extraction from quaternions throughout the entire operational envelope, facilitating accurate attitude representation in aircraft control system simulations. The MATLAB-compatible implementation typically involves:
- quat2eul function customization with quadrant checks
- wrapToPi operations for angle normalization
- Direction Cosine Matrix (DCM) intermediate representations for numerical stability
Such implementation ensures reliable bidirectional conversion between intuitive Euler-based control interfaces and singularity-free quaternion kinematic computations.
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