PHODCOS: Pythagorean Hodograph-based Differentiable Coordinate System
Jon Arrizabalaga, Fausto Vega, Zbyn\v{e}k \v{S}\'IR, Zachary, Manchester, Markus Ryll

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…
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Taxonomy
TopicsIntelligent Tutoring Systems and Adaptive Learning
