Rotational Trajectory Synopsis Groups
Abstract
We construct finite-dimensional Lie-group summaries of based rotational paths that retain the exact terminal attitude and selected iterated angular-velocity integrals. These summaries, called rotational trajectory synopses, compose exactly under path concatenation. Building on the classical works of Magnus and Chen, we establish endpoint-retaining body, spatial, and hybrid synopsis groups at every finite order and give a uniform matrix realization for the evaluated Magnus hierarchy. We distinguish this hierarchy from the full Chen signature from order three onward and prove \chg{that the exact endpoint is independent of every finite truncation of the body, spatial, or combined angular-velocity signature: the synopsis maps are surjective at every finite order}. An underactuated satellite example connects the retained coordinates to geometric phase in spacecraft attitude dynamics.
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BibTeX
@misc{chirikjian2026rotational,
title = {{Rotational Trajectory Synopsis Groups}},
author = {Gregory S. Chirikjian and Taeyoung Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01691},
eprint = {2610.01691},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01691},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Rotational Trajectory Synopsis Groups
AU - Gregory S. Chirikjian
AU - Taeyoung Lee
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01691
UR - https://arxiv.org/abs/2610.01691
AB - We construct finite-dimensional Lie-group summaries of based rotational paths that retain the exact terminal attitude and selected iterated angular-velocity integrals. These summaries, called rotational trajectory synopses, compose exactly under path concatenation. Building on the classical works of Magnus and Chen, we establish endpoint-retaining body, spatial, and hybrid synopsis groups at every finite order and give a uniform matrix realization for the evaluated Magnus hierarchy. We distinguish this hierarchy from the full Chen signature from order three onward and prove \chg{that the exact endpoint is independent of every finite truncation of the body, spatial, or combined angular-velocity signature: the synopsis maps are surjective at every finite order}. An underactuated satellite example connects the retained coordinates to geometric phase in spacecraft attitude dynamics.
ER -