Notation

Canonical symbol registry for the wiki. Every equation-bearing page renders symbols only from
this table; if a page needs a symbol not listed here, it must be added centrally, not invented
locally. This is the defense against the examiner’s notation objections. Symbols are written in
LaTeX ($...$), which both Obsidian (KaTeX) and GitHub render.

review: draft — promoted from the project’s canonical equation sheet
(my_writing/equations/current_sota.md §0, itself a synthesis of final.tex and Giordano 2019).
The user should verify these against the thesis conventions and flip to verified.

Conventions (locked ✓ — confirmed 2026-06-22)

• Inertia symbol is $\mathbf{M}$ (not $H$), repo-wide.
• Attitude error uses the unit quaternion: vector part $\mathbf{ϵ}$, scalar part $\eta$.
• Skew / cross-product: $[\cdot {]}^{\wedge }$, identical to $[\cdot {]}^{\times }$. Giordano writes
  $[\cdot {]}^{\wedge }$; final.tex writes $[\cdot {]}^{\times }$ and ${\mathbf{\omega }}^{\times }$ — treat as the same operator.
• Frame subscripts: ${\mathbf{R}}_{xy},{\mathbf{p}}_{xy}$ map from frame $\mathcal{X}$ to
  $\mathcal{Y}$, expressed in $\mathcal{X}$. Twist ${\mathbf{\nu }}_{xy}$ is of $\mathcal{Y}$ relative to
  $\mathcal{X}$; a single subscript ${\mathbf{\nu }}_y$ means relative to inertial.
• Circumcentroidal quantities carry the superscript $\oplus$ (motion about the system CoM).
• Accents: $\widehat{(\cdot )}$ = full $12\times 12$ transformed; $\breve{(\cdot )}$ = reduced $9\times 9$
  attitude+EE block. See circumcentroidal_decoupling.

Frames

Frame	Label	Location / orientation
Target satellite	$\mathcal{T}$	global inertial frame
System CoM	$\mathcal{C}$	at the whole-system CoM; axes non-rotating w.r.t. $\mathcal{T}$ (shifts as the arm moves)
Camera / end-effector	$\mathcal{E}$	at the end-effector; $+z$ along the camera aim direction
Base spacecraft	$\mathcal{B}$	at the base CoM, aligned with its principal axes
Body-$j$ CoM	$\mathcal{J}$	CoM of body $j$ ($j=0$ is the base)

Core matrices & maps

Symbol	Dim	Meaning
${\mathbf{R}}_{xy},{\mathbf{p}}_{xy}$	$\mathrm{SO}(3),{\mathbb{R}}^3$	rotation / position from $\mathcal{X}$ to $\mathcal{Y}$, expressed in $\mathcal{X}$
$[\cdot {]}^{\wedge }$ ($\equiv [\cdot {]}^{\times }$)	—	skew/cross-product matrix of a 3-vector
${\mathbf{A}}_{xy}$	${\mathbb{R}}^{6\times 6}$	adjoint (twist) transform
${\mathbf{J}}_{{\nu }_j}=[{\mathbf{J}}_{v_j};{\mathbf{J}}_{{\omega }_j}]$	${\mathbb{R}}^{6\times n}$	body-$j$ Jacobian (linear / angular)
${\mathbf{J}}_{{\nu }_e}$	${\mathbb{R}}^{6\times n}$	manipulator (end-effector) Jacobian; fixed-base part (alias: ${\mathbf{J}}_m$)
${\bar{\mathbf{J}}}_v$	${\mathbb{R}}^{3\times n}$	mass-averaged linear Jacobian
${\mathbf{J}}_{{\nu }_e}^{\oplus }$	${\mathbb{R}}^{6\times n}$	circumcentroidal Jacobian: motion of $\mathcal{E}$ about $\mathcal{C}$ (removes base translation)
${\mathbf{J}}_g$	—	generalized Jacobian matrix (GJM, Umetani 1989) — folds momentum conservation for the free-floating regime. Distinct from ${\mathbf{J}}_{{\nu }_e}^{\oplus }$: different object, different regime (our free-flying system uses ${\mathbf{J}}_{{\nu }_e}^{\oplus }$).
${\mathbf{G}}_{v_c},{\mathbf{G}}_{{\omega }_b}$	${\mathbb{R}}^{6\times 3}$	CoM-velocity / base-angular-velocity maps onto the EE twist
${\breve{\mathbf{G}}}_{v_c}=[0;{\mathbf{G}}_{v_c}]$	${\mathbb{R}}^{9\times 3}$	CoM$\to \breve{\mathbf{v}}$ velocity map (stacked)
${\breve{\mathbf{G}}}_{{\omega }_b}=[-{\mathbf{G}}_{{\omega }_b}\mathbf{E}]$	${\mathbb{R}}^{6\times 9}$	joint-rate recovery selector
$\mathbf{\Gamma }$	${\mathbb{R}}^{12\times (6+n)}$	coordinate transform $[{\mathbf{v}}_b;{\mathbf{\omega }}_b;\dot{\mathbf{q}}]\to [{\mathbf{v}}_c;{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$
$\mathbf{M},\mathbf{C}$	${\mathbb{R}}^{(6+n)\times (6+n)}$	full coupled inertia / Coriolis
$\hat{\mathbf{M}},\hat{\mathbf{C}}$	${\mathbb{R}}^{12\times 12}$	full transformed (stacked) inertia / Coriolis: $\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-T}\mathbf{M}{\mathbf{\Gamma }}^{-1}$
$\breve{\mathbf{M}},\breve{\mathbf{C}}$	${\mathbb{R}}^{9\times 9}$	reduced inertia / Coriolis of the coupled attitude+EE (circumcentroidal) subsystem — the lower-right block of $\hat{\mathbf{M}},\hat{\mathbf{C}}$
${\mathbf{C}}_c=[{\mathbf{C}}_{bc};{\mathbf{C}}_{ec}]$	${\mathbb{R}}^{6\times 3}$	CoM-coupling Coriolis
${\mathbf{J}}_{\tilde{x}}=\mathrm{blkdiag}({\mathbf{J}}_{{\tilde{x}}_b},{\mathbf{J}}_{{\tilde{x}}_e})$	${\mathbb{R}}^{9\times 9}$	error-rate Jacobian
$\breve{\mathbf{K}}=\mathrm{blkdiag}({\mathbf{K}}_b,{\mathbf{K}}_e)$	${\mathbb{R}}^{9\times 9}$	block-diagonal stiffness (SPD)
$\breve{\mathbf{D}}=\mathrm{blkdiag}({\mathbf{D}}_b,{\mathbf{D}}_e)$	${\mathbb{R}}^{9\times 9}$	block-diagonal damping (SPD)
${\mathbf{K}}_c,{\mathbf{D}}_c$	${\mathbb{R}}^{3\times 3}$	CoM-loop stiffness / damping (SPD)
${\mathbf{I}}_e$	—	EE integral gain (folded into the stiffness term)
$\mathbf{E},\mathbb{I}$	—	identity matrix

Velocities, wrenches, errors

Symbol	Dim	Meaning
${\mathbf{\nu }}_{xy}=[{\mathbf{v}}_{xy};{\mathbf{\omega }}_{xy}]$	${\mathbb{R}}^6$	body twist of $\mathcal{Y}$ rel. $\mathcal{X}$
${\mathbf{v}}_c$	${\mathbb{R}}^3$	whole-system CoM velocity
${\mathbf{\nu }}_e^{\oplus }$	${\mathbb{R}}^6$	circumcentroidal EE velocity ($={\mathbf{\nu }}_{ce}$)
$\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$	${\mathbb{R}}^9$	stacked attitude+EE velocity (coupled-subsystem state)
${\mathbf{f}}_c,{\mathbf{\tau }}_b^{\oplus },{\mathbf{w}}_e^{\oplus }$	${\mathbb{R}}^3,{\mathbb{R}}^3,{\mathbb{R}}^6$	transformed inputs: CoM force, base torque, EE wrench
${\mathbf{f}}_b,{\mathbf{\tau }}_b,\mathbf{\tau }$	${\mathbb{R}}^3,{\mathbb{R}}^3,{\mathbb{R}}^n$	physical base force, base torque, joint torques
${\tilde{\mathbf{x}}}_c,{\tilde{\mathbf{x}}}_b,{\tilde{\mathbf{x}}}_e$	${\mathbb{R}}^3,{\mathbb{R}}^3,{\mathbb{R}}^6$	CoM / base-attitude / EE-pose tracking errors
$\tilde{\mathbf{x}}=[{\tilde{\mathbf{x}}}_b;{\tilde{\mathbf{x}}}_e]$	${\mathbb{R}}^9$	stacked outer-loop error
${\tilde{\mathbf{x}}}_{ss}$	${\mathbb{R}}^9$	steady-state (cruise-lag) pose-error floor
$\mathbf{ϵ},\eta$	—	unit-quaternion vector / scalar parts (attitude error)
$\mathbf{p},\mathbf{L}$	${\mathbb{R}}^3$	linear / angular momentum (conserved in the free-floating regime)
$m,m^{(j)}$	—	total mass; mass of body $j$

Guidance & singularity scalars

Symbol	Units	Meaning
$s$	—	runtime progress variable along the discrete helix $\Lambda =\{{\mathbf{\lambda }}_i{\}}_{i=1}^N$
$u\in [0,1]$	—	analytic helix parameter ($\theta =w_{\theta }u$, $\varphi ={\varphi }_0+w_{\varphi }u$)
$R,w_{\theta },w_{\varphi },{\varphi }_0$	m, rad, rad, rad	helix radius, longitude rate, latitude rate, initial latitude
$v_{\mathrm{ref}}$	m/s	configured cruise speed
${\tau }_{\mathrm{su}},{\tau }_{\mathrm{ff}}$	s	startup-ramp / feedforward-blend time scales
$r_{\mathrm{cam}}$	m	camera standoff radius
$\mathbf{\kappa }$	1/m	path curvature vector
${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$	—	min singular value of $\mathbf{\Gamma }$ — proximity to singularity
${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$	—	min singular value of the circumcentroidal Jacobian (Spearman $\ge 0.9969$ with ${\sigma }_G$)
${\sigma }_{\mathrm{crit},1},{\sigma }_{\mathrm{crit},2},{\sigma }_{\mathrm{crit},3}$	—	singularity threshold cascade
${\lambda }_{\Gamma },{\lambda }_J,\beta$	—	Tikhonov damping ($\mathbf{\Gamma }$), damped-$J^{\oplus }$ damping, isotropic floor
$\gamma ({\sigma }_G)\in [{\sigma }_{\mathrm{crit},1},1]$	—	impedance-derate ramp (scales gains; emergent slowdown)
$\alpha$	—	CVaR confidence level (risk layer)

Disambiguation. ${\sigma }_G$ always means ${\sigma }_{\min }(\mathbf{\Gamma })$. A bare subscripted
${\sigma }_i$ means an SVD singular value in the singularity sections, but a normalized score term in
the guidance scorer — disambiguate locally. “Self-motion (null-space)” denotes the redundant-DOF
reconstruction (7-DOF work).

Additions (equations fan-out — confirmed; fold into the tables above at next pass)

Symbols harvested centrally from the fan-out (agents flagged these as missing rather than coining them
per-page). Conventions confirmed 2026-06-22; these rows are canonical and may be merged into the
tables above whenever the registry is next reorganized.

Symbol	Meaning
$\Omega$	singularity-free region $\{\mathbf{q}:{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })>0\}$ (stability domain)
$x_{\max }$	highest tolerable generalized velocity (e.g. UR3 wrist rate $\approx 2\pi$ rad/s) — sets ${\sigma }_{\mathrm{crit},1}=v/x_{\max }$
$w$	Yoshikawa manipulability $w=\sqrt{\det (\mathbf{J}{\mathbf{J}}^{\top })}$ (volume measure; contrast ${\sigma }_{\min }$)
$s_{\min ,J}$	observed minimum singular value of ${\mathbf{J}}^{\oplus }$ (empirical, e.g. $0.0268$)
$\mathbf{\Lambda }$	operational-space (task) inertia $\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1}$ (Khatib)
$\bar{\mathbf{J}}$	dynamically-consistent generalized inverse (Khatib); distinct from ${\bar{\mathbf{J}}}_v$ (mass-averaged linear Jacobian)
$\mathbf{N}$	null-space projector, $\mathbf{N}=\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top }$
${\mathbf{M}}_{bm}$ (≡ coupling block; Giordano ${\mathbf{M}}_{tr},{\mathbf{M}}_{tm}$)	base–arm coupling inertia; its kernel is the reaction null space
${\mathbf{P}}_{\mathrm{RNS}}$	reaction null-space projector ${\mathbf{P}}_{\mathrm{RNS}}=\mathbf{E}-{\mathbf{M}}_{bm}^{+}{\mathbf{M}}_{bm}$
${\mathbf{J}}^{\ast }$	free-floating end-effector Jacobian (Papadopoulos); dynamic singularity at $\det {\mathbf{J}}^{\ast }=0$. Distinct from ${\mathbf{J}}_g$ and ${\mathbf{J}}_{{\nu }_e}^{\oplus }$
${\mathbf{H}}_0,{\mathbf{H}}_m,{\mathbf{H}}_{0m}$	base ($6\times 6$), arm ($N\times N$), and base–arm coupling ($6\times N$) inertia blocks of the full GIM (wilde2018equations writes $H$; rendered $\mathbf{M}$ here per M not H). ${\mathbf{H}}_{0m}\equiv$ coupling_inertia_matrix, kernel = reaction null space
${\mathbf{M}}^{\star }={\mathbf{H}}_m-{\mathbf{H}}_{0m}^{\top }{\mathbf{H}}_0^{-1}{\mathbf{H}}_{0m}$	free-floating reduced GIM ($N\times N$ Schur complement of ${\mathbf{H}}_0$; wilde2018equations eq 45, conserved-momentum mode). Distinct from the full free-flying $\mathbf{M}$
${\mathbf{C}}^{\star }$	free-floating reduced Coriolis/centrifugal block paired with ${\mathbf{M}}^{\star }$ (dambrosio2024redundant’s full-coupled analogue is the “convective inertia” $\mathbf{C}$)
${\mathbf{M}}_0$	initial momentum 6-vector (free-floating); ${\mathbf{M}}_0=0\Rightarrow$ base uncontrolled (the regime our free-FLYING base leaves)

Convention — desired/reference quantities carry the subscript ${}_d$ (e.g. ${\breve{\mathbf{v}}}_d$,
${\mathbf{\nu }}_{ed}^{\oplus }$, ${\mathbf{v}}_{cd}$, ${\mathbf{a}}_{cd}$, ${\mathbf{\omega }}_{bd}$); the
per-frame quaternion error instances (${\mathbf{ϵ}}_{e{e}_d}$, ${\eta }_{b{b}_d}$, …) instantiate the
generic $\mathbf{ϵ}/\eta$ + ${}_d$ convention.

Generic systems-theory symbols (class-$\mathcal{K}$/${\mathcal{K}}_{\infty }$ functions, cascade
subsystem states $x_1,x_2$, Lie derivative $L_{g}V$) used on the stability pages are reproduced
source-faithfully from Panteley/Loria and are not canonicalized here.

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Source: promoted from my_writing/equations/current_sota.md §0 (private master). Add rows centrally
as new symbols appear during ingestion; never coin notation on a per-page basis.