PHODCOS: Pythagorean Hodograph-based Differentiable Coordinate System

TL;DR

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.

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.

Figures (5)

• Figure 1: An illustrative example of the PHODCOS coordinate system applied to a highly nonlinear curve, specifically the Near Rectilinear Halo Orbit (NRHO) selected for the Lunar Gateway. The frame components are colored red, blue, and green, respectively. The Earth is represented by the green sphere in the middle, whereas the Moon's orbit is given by the dashed gray line. This depiction is presented in the Earth-Centered Inertial (ECI) frame.
• Figure 2: A) Comparison of the parametric paths resulting from the conversion of an exemplary analytical curve \ref{['eq:exemplary_curve']} by means of PHODCOS -- the algorithm \ref{['algo:papor']} presented in this work -- for 1, 2 and 8 segments. The original analytical curve is given by the dashed black line, while the converted parametric path is depicted by its adapted frame, with the first, second and third components shown in red, green and blue, respectively. The computed parametric paths are divided into columns, each of which displays the results associated with a specific number of segments. In each column, the top row provides an isometric view, while the middle and bottom rows offer side and top views, respectively. The conversion error and improvement ratios associated to every number of segments are given in Table \ref{['tab:benchmark']}. As stated in theorem \ref{['theorem:approximation_error']}, the conversion error decreases as the number of segments augments, resulting in a greater similarity between the parametric path and the original curve. B). Continuity analysis in the preimage ${\mathcal{A}}(\xi)$ (first column), adapted frame $\text{R}(\xi)$ (second, third and fourth columns) and angular-velocity $\bm{\omega}(\xi)$ (fifth column) for the 2 segments case. The transition between the first and second segments is given by the gray vertical line at $\xi=0.5$. The first row depicts the functions themselves, while the second and third relate to the first and second derivatives. The dashed lines intend to expose the loss on continuity in the parametric functions if the angle offset cancellation \ref{['eq:frame_cont']} is not conducted.
• Figure 3: Numerical validation for the planarity, invariance and reversion theorems \ref{['theorem:planarity']}, \ref{['theorem:invariance']} and \ref{['theorem:reversed']}. The original curve is a planar variant of \ref{['eq:exemplary_curve']}, whose $y$ component is set to $0$. In both figures the interpolation data is represented by black arrows. A) Within the $xz$ plane colored in yellow, the original curve is given in blue, the transformed curve in red and the curve resulting from rotating and translating back the transformed curve in dashed-red. This curve aligns with the original one, and thereby, numerically validates theorem \ref{['theorem:invariance']}. Additionally, in accordance with the planarity theorem \ref{['theorem:planarity']}, all curves remain in the same plane as the interpolation data. B) Planar view of the original and reversed curves, alongside their adapted frames, where the first and second components are colored in red and green, respectively. As stated by the reversion theorem \ref{['theorem:reversed']}, the curves obtained for the optimal interpolants $\bm{\phi}^*=0$ are equivalent and can only be distinguished by the direction of their adapted-frames.
• Figure 4: The PHODCOS coordinate system is applied to the Near Rectilinear Halo Orbit (NRHO) within the Earth-Moon Barycentric Rotating (EMBR) reference frame. This system is illustrated from three perspectives: A) isometric view, B) front view, and C) side view. The vector components corresponding to the coordinate system are represented in red, blue, and green, respectively. The parametric speed $\sigma$ and the angular velocity $\bm{\omega}$ associated with the coordinate system are depicted in panels D) and E). For an alternative view of the coordinate system from the Earth-Centered Inertial (ECI) frame, please refer to Fig. \ref{['fig:nrho_cover']}.
• Figure 5: Ascent maneuvers that take the lunar lander from five different positions at the Moon's surface back to the NRHO. The respective motions have been computed by a prediction-based controller whose dynamics evolve according to the PHODCOS coordinate system. The Moon and NRHO are given in gray, while the five lunar lander trajectories are colored blue, green, orange, red and purple. The left column provides a broad overview of the converging maneuvers towards the orbit, while panels A), B), C), and D) present more detailed isometric, front, side, and top views respectively. Panel E) illustrates the position and orientation errors between the lunar lander and the NRHO (rows 1 and 2), its velocity magnitude (row 3) and gimbal angles (rows 4 and 5), color coded according to the different initial conditions.

Theorems & Definitions (16)

• Definition 1: Commutative operation
• Remark 1
• Lemma 1
• Lemma 2
• Definition 2: PH curves
• Definition 3: Standard form
• Remark 2
• Theorem 1: Approximation error
• proof
• Corollary 1: Optimal interpolant $\bm{\phi^*}=\bm{0}$