Euler Angle to Quaternion Conversion with MATLAB Implementation

This guide provides a comprehensive MATLAB implementation for converting Euler angles to quaternions using the 312 rotation sequence. The process involves matrix transformations and careful quaternion element calculations. 1. Begin by defining the three Euler angles in radians: - phi: rotation about the first axis (X-axis in 312 sequence) - theta: rotation about the second axis (Y-axis in 312 sequence) - psi: rotation about the third axis (Z-axis in 312 sequence) 2. Construct the 3x3 rotation matrix using trigonometric functions: The rotation matrix elements are calculated as follows: R11 = cos(theta)*cos(psi) R12 = sin(phi)*sin(theta)*cos(psi) - cos(phi)*sin(psi) R13 = cos(phi)*sin(theta)*cos(psi) + sin(phi)*sin(psi) R21 = cos(theta)*sin(psi) R22 = sin(phi)*sin(theta)*sin(psi) + cos(phi)*cos(psi) R23 = cos(phi)*sin(theta)*sin(psi) - sin(phi)*cos(psi) R31 = -sin(theta) R32 = sin(phi)*cos(theta) R33 = cos(phi)*cos(theta) 3. Compute quaternion elements from the rotation matrix: The quaternion components are derived using the matrix trace method: q0 = 0.5*sqrt(R11 + R22 + R33 + 1) q1 = (R32 - R23)/(4*q0) q2 = (R13 - R31)/(4*q0) q3 = (R21 - R12)/(4*q0) Note: This method requires ensuring q0 is non-zero to avoid division by zero 4. Output the resulting quaternion vector [q0, q1, q2, q3] This MATLAB implementation provides robust Euler angle to quaternion conversion with proper handling of the 312 rotation sequence. The algorithm ensures numerical stability through careful trigonometric calculations and includes error checking for valid quaternion normalization. The resulting quaternion representation is particularly valuable for applications in computer graphics, robotics, and aerospace systems where efficient 3D rotation representations are required.