gamma_closed_form_inverse

Closed-Form Inverse of the Coordinate Transform $\mathbf{\Gamma }$

Statement

The coordinate transform $\mathbf{\Gamma }\in {\mathbb{R}}^{12\times (6+n)}$ (here $n=6$,
nonredundant) maps the physical generalized velocities to the circumcentroidal stack,
$\mathbf{z}=\mathbf{\Gamma }\mathbf{x}$ with
$\mathbf{x}=[{\mathbf{v}}_b;{\mathbf{\omega }}_b;\dot{\mathbf{q}}]$,
$\mathbf{z}=[{\mathbf{v}}_c;{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$
(notation.md; see circumcentroidal_motion):

$$\left[\begin{array}{c}{\mathbf{v}}_c\\ {\mathbf{\omega }}_b\\ {\mathbf{\nu }}_e^{\oplus }\end{array}\right]=\underset{\mathbf{\Gamma }}{\underbrace{\left[\begin{array}{ccc}{\mathbf{R}}_{cb}& -{\mathbf{R}}_{cb}[{\mathbf{p}}_{bc}{]}^{\wedge }& {\mathbf{R}}_{cb}{\bar{\mathbf{J}}}_v\\ 0& \mathbf{E}& 0\\ 0& {\mathbf{G}}_{{\omega }_b}& {\mathbf{J}}_{{\nu }_e}^{\oplus }\end{array}\right]}}\left[\begin{array}{c}{\mathbf{v}}_b\\ {\mathbf{\omega }}_b\\ \dot{\mathbf{q}}\end{array}\right].$$

(Giordano eq 19)

Result. $\mathbf{\Gamma }$ admits the closed-form inverse

$${\mathbf{\Gamma }}^{-1}=\left[\begin{array}{ccc}{\mathbf{R}}_{bc}& [{\mathbf{p}}_{bc}{]}^{\wedge }+{\bar{\mathbf{J}}}_v({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}{\mathbf{G}}_{{\omega }_b}& -{\bar{\mathbf{J}}}_v({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}\\ 0& \mathbf{E}& 0\\ 0& -({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}{\mathbf{G}}_{{\omega }_b}& ({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}\end{array}\right],$$

(Giordano App. B / current_sota eq 2.6)

and this inverse exists if and only if the circumcentroidal Jacobian
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is nonsingular. Every entry of ${\mathbf{\Gamma }}^{-1}$ that
can blow up does so through the single factor $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}$: it sits in
the lower-right block of $\mathbf{\Gamma }$, so $\mathbf{\Gamma }$ and
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ go singular together. Empirically the two minimum singular
values are near-monotone images of one another:
${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$ and
${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ have Spearman rank correlation $\ge 0.9969$
(current_sota §6). This is what licenses using the cheap-to-evaluate ${\sigma }_G$ as the single
proximity-to-singularity currency throughout the conditioning stack
(see dynamic_singularity,
singularity_robust_inverse).

Assumptions

• Free-flying regime. The base is fully actuated (6-DOF) and $\mathbf{\Gamma }$ is the
  Giordano circumcentroidal coordinate transform — the $\oplus$ split removes base
  translation only, mapping the EE twist to motion about the system CoM $\mathcal{C}$. This is
  not the free-floating generalized Jacobian ${\mathbf{J}}_g$ (Umetani 1989), which folds
  momentum conservation for an uncontrolled base; that is a different object in a different
  regime (notation.md). Giordano 2019 explicitly treats the thrusters-and-arm
  (actuated-base) system, matching ours.
• Nonredundant arm, $n=6$. Then ${\mathbf{J}}_{{\nu }_e}^{\oplus }\in {\mathbb{R}}^{6\times 6}$ is
  square and $\mathbf{\Gamma }\in {\mathbb{R}}^{12\times 12}$ is square; “inverse” is literal.
  The redundant ($n=7$) case replaces $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}$ with a generalized
  inverse plus a self-motion (null-space) reconstruction — reserved for later work.
• ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ nonsingular (${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })>0$),
  i.e. the configuration lies in the singularity-free region $\Omega$.
• ${\mathbf{R}}_{cb}\in \mathrm{SO}(3)$ (so ${\mathbf{R}}_{cb}^{-1}={\mathbf{R}}_{cb}^{\top }={\mathbf{R}}_{bc}$),
  and $\mathbf{E}$ in the middle row is a genuine identity (the ${\mathbf{\omega }}_b$
  channel passes through untouched).

Proof sketch

The block back-substitution is given in full above. The $\mathbf{\Gamma }$-construction build-up is
in walkthrough_v3.md (arm pseudoinverse → circumcentroidal
Jacobian → self-motion) and the augmented-case derivation in
derivation_7dof.md; a dedicated $n=6$ write-up is a
low-priority Phase-D item.

1. Block-triangular structure. $\mathbf{\Gamma }$ is block upper-triangular in the
   partition $({\mathbf{v}}_c\mid {\mathbf{\omega }}_b\mid {\mathbf{\nu }}_e^{\oplus })$ vs.
   $({\mathbf{v}}_b\mid {\mathbf{\omega }}_b\mid \dot{\mathbf{q}})$ once the trivial
   middle row ${\mathbf{\omega }}_b\mapsto {\mathbf{\omega }}_b$ is recognized. The first column
   contains only ${\mathbf{R}}_{cb}$ (and zeros below it), so $\det \mathbf{\Gamma }=\det {\mathbf{R}}_{cb}\cdot \det \mathbf{E}\cdot \det {\mathbf{J}}_{{\nu }_e}^{\oplus }=\det {\mathbf{J}}_{{\nu }_e}^{\oplus }$ (since $\det {\mathbf{R}}_{cb}=1$). Hence
   $\mathbf{\Gamma }$ is invertible iff ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is — the
   “go singular together” claim is exact, not merely empirical; the Spearman figure only quantifies
   how tightly the magnitudes track.
2. Recover $\dot{\mathbf{q}}$. From the third row
   ${\mathbf{\nu }}_e^{\oplus }={\mathbf{G}}_{{\omega }_b}{\mathbf{\omega }}_b+{\mathbf{J}}_{{\nu }_e}^{\oplus }\dot{\mathbf{q}}$,
   solve $\dot{\mathbf{q}}=({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}({\mathbf{\nu }}_e^{\oplus }-{\mathbf{G}}_{{\omega }_b}{\mathbf{\omega }}_b)$
   — the source of the $-({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}{\mathbf{G}}_{{\omega }_b}$ and
   $+({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}$ entries in the bottom row of ${\mathbf{\Gamma }}^{-1}$.
   (This is exactly the joint-rate-recovery map
   $\dot{\mathbf{q}}=({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}{\breve{\mathbf{G}}}_{{\omega }_b}\breve{\mathbf{v}}$,
   current_sota eq 4.11.)
3. Pass ${\mathbf{\omega }}_b$ through. The middle row is the identity, so ${\mathbf{\omega }}_b$
   maps to itself ($\mathbf{E}$ in the centre of ${\mathbf{\Gamma }}^{-1}$).
4. Recover ${\mathbf{v}}_b$. From the first row
   ${\mathbf{v}}_c={\mathbf{R}}_{cb}{\mathbf{v}}_b-{\mathbf{R}}_{cb}[{\mathbf{p}}_{bc}{]}^{\wedge }{\mathbf{\omega }}_b+{\mathbf{R}}_{cb}{\bar{\mathbf{J}}}_v\dot{\mathbf{q}}$,
   left-multiply by ${\mathbf{R}}_{bc}={\mathbf{R}}_{cb}^{\top }$ and substitute the
   $\dot{\mathbf{q}}$ from step 2: $${\mathbf{v}}_b={\mathbf{R}}_{bc}{\mathbf{v}}_c+([{\mathbf{p}}_{bc}{]}^{\wedge }+{\bar{\mathbf{J}}}_v({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}{\mathbf{G}}_{{\omega }_b}){\mathbf{\omega }}_b-{\bar{\mathbf{J}}}_v({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}{\mathbf{\nu }}_e^{\oplus },$$ which reads off the entire top row of ${\mathbf{\Gamma }}^{-1}$. The
   ${\bar{\mathbf{J}}}_v({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}{\mathbf{G}}_{{\omega }_b}$ cross-term
   is precisely where the joint motion couples back into the base-velocity recovery.
5. Verification. $\mathbf{\Gamma }{\mathbf{\Gamma }}^{-1}=\mathbf{E}$ follows by
   direct multiplication; the only nontrivial cancellations are the
   $[{\mathbf{p}}_{bc}{]}^{\wedge }$-vs-${\bar{\mathbf{J}}}_v({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}{\mathbf{G}}_{{\omega }_b}$
   pairing in the (1,2) block and the ${\mathbf{G}}_{{\omega }_b}$ cancellation in the (3,2) block.

The dual statement — that generalized forces transform contragrediently,
$[{\mathbf{f}}_b;{\mathbf{\tau }}_b;\mathbf{\tau }]={\mathbf{\Gamma }}^{\top }[{\mathbf{f}}_c;{\mathbf{\tau }}_b^{\oplus };{\mathbf{w}}_e^{\oplus }]$
(Giordano eq 20) — means ${\mathbf{\Gamma }}^{-T}$ is what congruence-transforms the inertia into
$\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-T}\mathbf{M}{\mathbf{\Gamma }}^{-1}$. This is the
entry point to circumcentroidal_decoupling, where the resulting
block-diagonal $\hat{\mathbf{M}}$ splits the dynamics.

hat vs breve — do not conflate

This page concerns ${\mathbf{\Gamma }}^{-1}$ itself, the full $12\times 12$ velocity inverse.
The matrices built from it — $\hat{\mathbf{M}},\hat{\mathbf{C}}$ (full $12\times 12$) and
their reduced lower-right $\breve{\mathbf{M}},\breve{\mathbf{C}}$ ($9\times 9$,
attitude+EE) — are distinct objects. The canonical accent rule
($\widehat{(\cdot )}=$ full $12\times 12$; $\breve{(\cdot )}=$ reduced $9\times 9$) and the warning
against conflating them live in
circumcentroidal_decoupling. A prior 7-DOF effort confused the
two; render ${\mathbf{\Gamma }}^{-1}$ as the $12\times 12$ object and reach $\breve{(\cdot )}$
only through the decoupling result.

Source / provenance

• Literature: giordano2019coordinated — eq 19
  ($\mathbf{\Gamma }$), eq 20 (force dual), and App. B (the closed-form inverse).
• Equation sheet: current_sota §2.6 (eq 2.6) for ${\mathbf{\Gamma }}^{-1}$; §6 for the
  ${\sigma }_G$ / ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ Spearman $\ge 0.9969$ coupling.
• Ours: the explicit “iff ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ nonsingular” determinant argument
  and the use of ${\sigma }_G$ as the singularity currency it underwrites.

Caveats

• Holds only on $\Omega$, the singularity-free region
  $\{\mathbf{q}:{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus }(\mathbf{q}))>0\}$. As
  ${\mathbf{J}}_{{\nu }_e}^{\oplus }\to$ singular, every $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}$ entry of
  ${\mathbf{\Gamma }}^{-1}$ diverges; the velocity reconstruction amplifies by
  $1/{\sigma }_{\min }$. The conditioning stack (Tikhonov $\mathbf{\Gamma }$ regularization,
  impedance derate, damped ${\mathbf{J}}^{\oplus }$ inverse) exists precisely to manage this
  shared divergence — see singularity_robust_inverse.
• The Spearman $\ge 0.9969$ figure is an empirical, monotone statement about how the two
  minimum singular values co-vary; the exact coupling is the determinant identity
  $\det \mathbf{\Gamma }=\det {\mathbf{J}}_{{\nu }_e}^{\oplus }$ (step 1). Do not over-read $0.9969$
  as an algebraic equality of $\sigma$-values — only the singular/nonsingular event coincides
  exactly; the magnitudes track tightly but are not identical.
• Nonredundant only. For $n>6$ the literal inverse does not exist; substitute a
  damped/generalized inverse and a null-space term.
• A $\dot{\mathbf{\Gamma }}$ (time-derivative) correction is needed when this inverse feeds the
  dynamics congruence; that correction lives with
  circumcentroidal_decoupling, not here — this page is the
  static (instantaneous) velocity inverse only.