deruiter2014general

General Identities for Parameterizations of SO(3) With Applications

Authors: Anton H. J. de Ruiter, James Richard Forbes · Year: 2014 · Venue: ASME Journal of Applied Mechanics 81:071007 (doi 10.1115/1.4027144); corpus PDF is the 53 pp. author manuscript Raw: md · pdf

Summary

The published, fully-proved version of the identity family the attitude notes teach from: six identities for parameterizations $C(p)\in SO(3)$ — Identities 1–3 for unconstrained $p\in {\mathbb{R}}^n$ (Euler angles, Rodrigues parameters) and Identities 1’–3’ for constrained parameterizations (unit quaternions) — plus a treatment of perturbations across parameterizations. Three worked applications: rigid-body equations of motion via Lagrange/Boltzmann–Hamel machinery (with a careful virtual-work/generalized-force treatment), potential-energy sensitivity under attitude perturbations, and a rigorous minimal-representation EKF for spacecraft attitude determination. A distinctive example parameterizes the rotation matrix by itself, carrying the orthonormality constraint explicitly. The corpus copy is a LightOnOCR bf16 conversion, 53/53 pages zero-degeneracy.

Key Claims

• Identity 1: $\frac{\partial }{\partial p^{\top }}[C(p)v]\Gamma (p)=[C(p)v{]}^{\times }$ — the workhorse for generalized forces; reader-highlighted as (17) in the course notes.
• Identity 3 (and 3’): $(\dot{S}-\partial \omega /\partial p^{\top })\Gamma =-{\omega }^{\times }$ — stated without proof as eq (20) of deruiter2021attitude; proved here, for both unconstrained and constrained $p$.
• Perturbations in one parameterization map linearly (to first order) into any other parameterization of the same rotation — the basis for sensitivity/error analysis and the minimal-representation EKF.
• The virtual-work term in Lagrange’s equation reduces via Identity 2’ to a clean matrix-vector generalized torque — the step the paper flags as under-treated in the literature.

Method

Direct matrix algebra on $C\in {\mathbb{R}}^{3\times 3}$, same style as the course notes: cross-product-matrix identities, $S(p)$/$\Gamma (p)$ machinery, no Lie-group formalism — unusually close to what a Lean/Mathlib formalization would state on Matrix (Fin 3) (Fin 3) ℝ.

Regime note. Generic rigid-body attitude; no manipulator, no free-flyer specifics — regime-agnostic substrate.

Relevance to thesis

Matched pair with deruiter2021attitude: the notes state the identity family (16)–(21), this paper proves it in general $n$-parameter form. Together they charter the theorem-recon arc (SO(3) parameterization identities as Lean rungs). The EKF application is the earliest bridge in the boss corpus from these identities to estimation under uncertainty — relevant once the risk-aware layer needs attitude-error statistics.

Connections

Topics: Unit Quaternion · Kinematic Singularity · Exponential Coordinates Sources: deruiter2021attitude (states the family; eq (20) proved here) · deruiter2022analytical (the Lagrangian machinery the applications consume) · giordano2019coordinated (quaternion error machinery downstream)

Open Questions

• OCR fidelity spot-audit: sample 3 display equations (one per identity group) against the manuscript before any theorem statement is quoted from this md.
• Map Identities 1–3/1’–3’ to Mathlib statement shapes in the theorem recon (card 86bavg7e7).