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arXiv · 2610.01691

Rotational Trajectory Synopsis Groups

Gregory S. Chirikjian·Taeyoung Lee
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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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Gregory S. Chirikjian, Taeyoung Lee. 2026-10-01. Rotational Trajectory Synopsis Groups. https://arxiv.org/abs/2610.01691

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