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Generalized n-Dimensional Rigid Registration: Theory and Applications.

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    This study introduces a novel closed-form linear least-square solution for rigid registration in high-dimensional spaces. The method efficiently provides accurate uncertainty information for rotation and translation, enhancing robotic navigation applications.

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    Area of Science:

    • Computational geometry and applied mathematics
    • Robotics and sensor fusion

    Background:

    • The generalized rigid registration problem in high-dimensional Euclidean spaces is a fundamental challenge in various scientific and engineering fields.
    • Existing methods, such as those using singular value decomposition (SVD) and linear matrix inequality (LMI), often face computational limitations or provide less detailed uncertainty information.

    Purpose of the Study:

    • To develop a novel, efficient, and accurate method for solving the generalized rigid registration problem.
    • To derive closed-form solutions that provide probabilistic descriptions of registration uncertainties (rotation and translation covariances).
    • To demonstrate the applicability and superiority of the proposed method in covariance-preserving interpolation and Lidar mapping for robotic navigation.

    Main Methods:

    • Minimization of the loss function using an equivalent error formulation based on the Cayley formula.
    • Derivation of a closed-form linear least-square solution to compute registration covariances.
    • Application of the method to interpolation on the special Euclidean group SE(n) and covariance-aided Lidar mapping.

    Main Results:

    • The proposed closed-form solution accurately determines registration covariances, offering precise probabilistic descriptions of rotation and translation uncertainty.
    • Simulation results confirm the method's correctness and demonstrate significant computational efficiency compared to SVD and LMI-based algorithms.
    • Experiments in Lidar mapping show practical superiority, highlighting the method's effectiveness in real-world robotic navigation scenarios.

    Conclusions:

    • The developed closed-form linear least-square method offers a computationally efficient and accurate approach to the generalized rigid registration problem.
    • The derived registration covariances provide valuable uncertainty information, crucial for robust probabilistic modeling in robotics.
    • The method's successful application in Lidar mapping demonstrates its practical utility and potential for advancing robotic navigation systems.