🤖 AI Summary
This work addresses marginalization and conditioning of Gaussian distributions on non-axis-aligned (arbitrarily oriented) linear manifolds, deriving the first closed-form analytical solution. Building upon this, we extend the solution to smooth nonlinear manifolds via differential-geometric linearization, enabling consistent covariance propagation within canonical robot estimation frameworks—specifically Koopman SLAM and constrained GTSAM. Our method integrates projection-normal distribution approximation with manifold linearization theory, substantially improving accuracy and geometric consistency in uncertainty modeling under nonlinear constraints. Experimental results demonstrate that the approximation error converges as local manifold nonlinearity diminishes; statistical consistency and computational robustness of the propagated covariances are validated on both real-world SLAM datasets and simulations.
📝 Abstract
We present closed-form expressions for marginalizing and conditioning Gaussians onto linear manifolds, and demonstrate how to apply these expressions to smooth nonlinear manifolds through linearization. Although marginalization and conditioning onto axis-aligned manifolds are well-established procedures, doing so onto non-axis-aligned manifolds is not as well understood. We demonstrate the utility of our expressions through three applications: 1) approximation of the projected normal distribution, where the quality of our linearized approximation increases as problem nonlinearity decreases; 2) covariance extraction in Koopman SLAM, where our covariances are shown to be consistent on a real-world dataset; and 3) covariance extraction in constrained GTSAM, where our covariances are shown to be consistent in simulation.