Geometry-Based Unconstrained Yaw and Pitch Angle Estimation From a Single Projectile Image

The knowledge of yaw and pitch angles can provide significant benefits while assessing the performance of a projectile as it approaches the terminal stage of flight. It is typical in contemporary applications to employ multiple camera-based systems to estimate yaw and pitch angles useful for finding the total yaw angle, a critical parameter in the investigation of various aeroballistic parameters during ammunition testing. In practice, however, precision alignment requiring multiple observations can be difficult to achieve, and it may produce incorrect results due to severe misalignment errors. This research presents a geometric modeling-based approach for accurately estimating the yaw and pitch angles of a projectile through a single image without the necessity of any prior information about intrinsic or extrinsic camera parameters or projectile dimensions. The captured image of the projectile tail is observed to be the 2D representation of an ellipse, which is generated due to the projection of the 3D circle that depicts the physical projectile's tail. The image of the projectile's tail is utilized to estimate the corresponding 3D circle, which forms the foundation for deriving the geometry-based models. The first method solves an overdetermined system of linear equations employing the least squares technique on estimated 3D circle coordinates to calculate the Total Rotation matrix. The formulations for the yaw and pitch angles are obtained subsequently through the relationship with the generalized rotation matrix. The second method utilizes the partial derivative-based minimization of perpendicular distances of ellipse coordinates from the major axis after orthogonally projecting the estimated 3D circle onto the X-Y and Y-Z planes. Using a precise geometric formulation to calculate the yaw and pitch angles, a series of image processing pipeline stages is designed. A projectile orientation dataset was created to evaluate the effectiveness of the proposed methodology in estimating yaw and pitch angles from a single projectile image, demonstrating its high accuracy.