Analysis of the Geometric Heat Flow Equation: Computing Geodesics in Real-Time with Convergence Guarantees
This provides a numerically stable and efficient solution for real-time geodesic computation, which is important for fields like control and motion planning, though it appears incremental as it builds on existing geometric heat flow methods.
The paper tackles the problem of computing geodesics on Riemannian manifolds in real time by analyzing the geometric heat flow equation, proving exponential stability and asymptotic convergence under curvature bounds, and demonstrating a pseudospectral method that computes geodesics in milliseconds for non-contrived manifolds.
We present an analysis on the convergence properties of the so-called geometric heat flow equation for computing geodesics (extremal curves) on Riemannian manifolds. Computing geodesics numerically in real time has become an important capability across several fields, including control and motion planning. The geometric heat flow equation involves solving a parabolic partial differential equation whose solution is a geodesic. In practice, solving this PDE numerically can be done efficiently, and tends to be more numerically stable and exhibit a better rate of convergence compared to numerical optimization. We prove that the geometric heat flow equation is exponentially stable in $L_2$ if the curvature of the Riemannian manifold does not exceed a positive bound and that asymptotic convergence in $L_2$ is always guaranteed. We also present a pseudospectral method that leverages Chebyshev polynomials to accurately compute geodesics in only a few milliseconds for non-contrived manifolds. Our analysis was verified with our custom pseudospectral method by computing geodesics on common non-Euclidean surfaces, and in feedback for a contraction-based controller with a non-flat metric for a nonlinear system.