Spectral Geometry of SPD Matrices: Geodesic Interpolation and Robotics Applications

This study examines the Riemannian geometry of symmetric positive-definite (SPD) matrices and their applications in robotics, focusing on geodesic interpolation to preserve positive-definiteness and numerical stability. SPD matrices naturally arise as robot inertia matrices, manipulability ellipsoids, and covariance matrices. Conventional Euclidean operations may compromise their physical validity, producing unstable or non-physical results.We leverage the affine-invariant Riemannian metric to compute geodesic paths and distances, ensuring all intermediate matrices remain SPD. Numerical simulations in Python employ eigen-decomposition to compute matrix square roots, inverses, logarithms, and exponentials, comparing geodesic and Euclidean interpolation. Results demonstrate that geodesic paths maintain strict positive-definiteness and physically meaningful eigenvalue trajectories.Applications include smooth inertia interpolation for motion planning, manipulability preservation, and extensions to Riemannian optimization, tensor interpolation, and machine learning with SPD data. All data are synthetic, and Python scripts are provided for full reproducibility.