An Analytic Solution to the Optimal Spherical Dubins Path Problem with Geodesic Curvature Constraints

Computing shortest paths for curvature-constrained Dubins vehicles on the unit sphere is fundamental to many engineering applications, including long-range flight planning, persistent surveillance patterns, and global routing problems where great circles are natural routes. Numerical optimization methods on $\SO(3)$ suffer from sensitivity to initialization, may converge to local minima, and often miss feasible solution branches. This paper proposes a unified analytic computational approach for spherical Dubins CGC and CCC paths that overcomes these limitations. By exploiting the axis-fixing property of rotations and developing a closed-form back-substitution method using geometric projection, the three-dimensional boundary value problem is reduced to solving a quadratic polynomial equation. The proposed analytic solver achieves machine precision accuracy with errors on the order of $10^{-16}$, is approximately $717$ times faster than numerical methods under the same computational environment, and systematically enumerates all feasible solution branches without requiring exhaustive multi-start initialization. The method provides closed-form solutions for optimal path computation in the regime where turning radius $\Rturn \in (0, 1/2]$, corresponding to $U_{\max} \geq \sqrt{3}$.