Solving Quaternion Transformation Matrices for Common Measurement Points Across Different Coordinate Systems

In engineering and computer vision applications, coordinate transformations between different reference systems are frequently required. Suppose we have a set of identical physical points measured in both Coordinate System A and Coordinate System B. How can we determine the transformation relationship from Coordinate System A to Coordinate System B using these points? Quaternions provide an efficient method for representing rotations, offering advantages over Euler angles by avoiding gimbal lock issues and ensuring greater computational stability.

### Problem Formulation Given corresponding point sets in Coordinate System A and Coordinate System B, we aim to compute the rotation matrix (or quaternion) and translation vector from A to B. This process is commonly known as Point Set Registration, with typical implementation steps including:

Centroid Calculation: First compute the centroids (mean positions) of both point sets separately, then subtract these centroids from all points to eliminate translational effects. In code: centroid_A = mean(points_A, axis=0); points_A_centered = points_A - centroid_A

Covariance Matrix Construction: Using the centered point sets, compute the covariance matrix H = Σ(points_A_centered[i] · points_B_centered[i]^T). This matrix helps identify optimal rotational relationships through mathematical optimization.

Singular Value Decomposition (SVD): Perform SVD on the covariance matrix H = UΣV^T. The optimal rotation matrix is obtained as R = VU^T, with special handling required if det(R) = -1 to ensure proper rotation.

Quaternion Conversion: If quaternion representation is preferred, convert the rotation matrix to quaternion form using standard conversion algorithms. This representation facilitates subsequent computations and interpolation operations.

### Advantages of Quaternions Quaternions consist of four components (real part + three imaginary parts) that compactly represent rotations. Compared to 3×3 rotation matrices, quaternions require less storage and avoid orthogonality constraint issues. Furthermore, quaternion interpolation techniques like Spherical Linear Interpolation (Slerp) prove highly useful in animation and path planning applications.

### Practical Applications Robotic Navigation: Sensors such as LiDAR and IMUs often use different coordinate systems that must be unified into a world coordinate framework.

3D Reconstruction: Multi-view camera captured data requires alignment to a global coordinate system.

Pose Estimation: UAVs and AR/VR devices typically rely on quaternions for efficient attitude updates and orientation tracking.

Through this methodology, we can accurately establish transformation relationships between two coordinate systems, enabling seamless data alignment across different sensors or systems. The implementation typically involves numpy arrays for point manipulation and scipy for SVD computations in Python environments.