Conversion from Rotation Matrix to Quaternion

The conversion from rotation matrix to quaternion is a fundamental operation in attitude representation, particularly significant in robotics, 3D graphics, and aerospace applications. While rotation matrices describe 3D spatial rotations, quaternions offer advantages in computational efficiency and compact representation, making them more suitable for interpolation and continuous rotation descriptions.

In MATLAB, the conversion from rotation matrix to quaternion can be achieved by parsing the matrix elements. The fundamental algorithm involves: first calculating the four quaternion components (w, x, y, z), followed by normalization to ensure unit quaternion properties. The implementation typically follows these steps:

Trace Calculation: The trace of the rotation matrix (sum of diagonal elements) determines the primary calculation approach for quaternion components. If the trace is positive, the real component (w) is computed first; otherwise, the largest imaginary component (one of x, y, z) is prioritized to maintain numerical stability.

Quaternion Component Computation: Based on trace conditions, different combinations of calculations derive the quaternion components. For instance, when the trace is maximum, the real part is calculated first followed by derivation of imaginary parts; otherwise, the largest imaginary component is selected first to avoid numerical instability.

Normalization: The computed quaternion must undergo normalization to ensure it represents a valid rotation (unit quaternion), typically implemented using vector normalization: q_normalized = q / ||q||.

MATLAB's built-in `rotm2quat` function directly performs this conversion, but understanding the underlying mathematics enables optimization in custom implementations or adaptation to specific requirements. This transformation is particularly crucial in robotics kinematics, sensor fusion (such as IMU data processing), where it effectively avoids gimbal lock issues and enhances the robustness of attitude representation. The algorithm typically uses conditional branching based on trace values and employs square root operations for component calculations.