PHODCOS: Pythagorean Hodograph-based Differentiable Coordinate System

TL;DR

PHODCOS introduces an exact, differentiable coordinate system for curves using Pythagorean Hodograph functions, enabling precise geometric analysis and navigation for complex trajectories like lunar orbits.

Contribution

The paper presents PHODCOS, a novel algorithm that constructs an exact, differentiable coordinate system for nonlinear curves, suitable for autonomous navigation and geometric property computation.

Findings

Provides a finite coefficient-based representation of the coordinate system.

Guarantees the coordinate system matches the curve within a specified tolerance.

Demonstrates applicability with numerical examples in lunar orbit scenarios.

Abstract

This paper presents PHODCOS, an algorithm that assigns a moving coordinate system to a given curve. The parametric functions underlying the coordinate system, i.e., the path function, the moving frame and its angular velocity, are exact -- approximation free -- differentiable, and sufficiently continuous. This allows for computing a coordinate system for highly nonlinear curves, while remaining compliant with autonomous navigation algorithms that require first and second order gradient information. In addition, the coordinate system obtained by PHODCOS is fully defined by a finite number of coefficients, which may then be used to compute additional geometric properties of the curve, such as arc-length, curvature, torsion, etc. Therefore, PHODCOS presents an appealing paradigm to enhance the geometrical awareness of existing guidance and navigation on-orbit spacecraft maneuvers. The PHODCOS algorithm is presented alongside an analysis of its error and approximation order, and thus, it is guaranteed that the obtained coordinate system matches the given curve within a desired tolerance. To demonstrate the applicability of the coordinate system resulting from PHODCOS, we present numerical examples in the Near Rectilinear Halo Orbit (NRHO) for the Lunar Gateway.