Analysis of the Geometric Heat Flow Equation: Computing Geodesics in Real-Time with Convergence Guarantees

We present an analysis on the convergence properties of the so-called geometric heat flow equation for computing geodesics (shortest-path~curves) on Riemannian manifolds. Computing geodesics numerically in real-time has become an important capability in 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 globally exponentially stable in $L_2$ if the curvature of the Riemannian manifold is not too positive, 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.