circumcentroidal_decoupling

Circumcentroidal Decoupling (Derivation of Equation 21)

Statement

For a nonsingular circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (here $n=6$,
nonredundant), the coordinate transform $\mathbf{\Gamma }$ (notation.md) maps the
full coupled dynamics $\mathbf{M}\dot{\mathbf{x}}+\mathbf{C}\mathbf{x}=\mathbf{F}$
into

$$\hat{\mathbf{M}}\dot{\mathbf{z}}+\hat{\mathbf{C}}\mathbf{z}=\mathbf{G},\hat{\mathbf{M}}=\left[\begin{array}{cc}m\mathbf{E}& 0\\ 0& \breve{\mathbf{M}}\end{array}\right],\hat{\mathbf{C}}=\left[\begin{array}{cc}0& -{\mathbf{C}}_c^{\top }\\ {\mathbf{C}}_c& \breve{\mathbf{C}}\end{array}\right],$$

with $\mathbf{z}=[{\mathbf{v}}_c;{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$. The block
structure means the $3\times 3$ CoM equation decouples,

$$m{\dot{\mathbf{v}}}_c={\mathbf{f}}_c,$$

leaving the coupled $9\times 9$ attitude+EE subsystem

$$\breve{\mathbf{M}}\left[\begin{array}{c}{\dot{\mathbf{\omega }}}_b\\ {\dot{\mathbf{\nu }}}_e^{\oplus }\end{array}\right]+\breve{\mathbf{C}}\left[\begin{array}{c}{\mathbf{\omega }}_b\\ {\mathbf{\nu }}_e^{\oplus }\end{array}\right]+{\mathbf{C}}_c{\mathbf{v}}_c=\left[\begin{array}{c}{\mathbf{\tau }}_b^{\oplus }\\ {\mathbf{w}}_e^{\oplus }\end{array}\right].$$

Inconsistency — hat vs breve (canonical)

$\hat{\mathbf{M}},\hat{\mathbf{C}}$ are the full $12\times 12$ transformed matrices
$\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-T}\mathbf{M}{\mathbf{\Gamma }}^{-1}$,
$\hat{\mathbf{C}}={\mathbf{\Gamma }}^{-T}(\mathbf{C}-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}){\mathbf{\Gamma }}^{-1}$.
$\breve{\mathbf{M}},\breve{\mathbf{C}}$ are the reduced $9\times 9$ lower-right block of those
hat matrices, i.e. the coupled attitude+EE subsystem that survives once the $3\times 3$ CoM ($m\mathbf{E}$)
decouples. They are not the same object. A prior 7-DOF effort conflated them; this page is the
canonical home for the distinction.

Assumptions

• ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ nonsingular ⇒ $\mathbf{\Gamma }$ invertible (the two go singular
  together; see singularity_threshold_cascade).
• $n=6$ (nonredundant). The redundant ($n=7$) case adds a self-motion / null-space reconstruction —
  reserved for later work.
• Rigid bodies; the Γ̇ transport term $-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$
  is retained in $\hat{\mathbf{C}}$ (dropping it breaks passivity).

Proof sketch

The augmented-form Eq-21 block computation ($\hat{\mathbf{M}},\hat{\mathbf{C}}$ blocks) is in
dynamics_modifications_7dof.md; pedagogical
build-up in walkthrough_v3.md. The canonical $n=6$ hat/breve
derivation is deferred to Phase D — to be lifted from my_writing/crucial/cc_detailed/com_att_writeup.tex §2
into generated/math/com_att_writeup.md.

1. Form $\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-T}\mathbf{M}{\mathbf{\Gamma }}^{-1}$ and
   $\hat{\mathbf{C}}={\mathbf{\Gamma }}^{-T}(\mathbf{C}-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}){\mathbf{\Gamma }}^{-1}$.
2. Compute the blocks of $\hat{\mathbf{M}}$: ${\hat{M}}_{11}={\mathbf{R}}^{\top }m\mathbf{R}=m\mathbf{E}$,
   ${\hat{M}}_{12}={\hat{M}}_{13}=0$ (CoM inertia isotropic and decoupled); the lower-right
   $9\times 9$ collects ${\hat{M}}_{22},{\hat{M}}_{23},{\hat{M}}_{33}$ — this is $\breve{\mathbf{M}}$.
3. Show the centroid Coriolis vanishes,
   $-{\mathbf{C}}_{bc}^{\top }{\mathbf{\omega }}_b-{\mathbf{C}}_{ec}^{\top }{\mathbf{\nu }}_e^{\oplus }=0$,
   giving ${\hat{C}}_{11}=0$ and the skew off-diagonal coupling $\pm {\mathbf{C}}_c$.
4. Read off the decoupled CoM equation and the coupled $9\times 9$ block; $\breve{\mathbf{C}}$ is the
   lower-right block of $\hat{\mathbf{C}}$.

This block-diagonal $\hat{\mathbf{M}}$ is exactly what licenses the cascade (inner CoM loop +
outer attitude/EE loop) whose stability is proved in
coordinated_control_lyapunov_stability; the skew structure
of $\hat{\mathbf{C}}$ gives the passivity in
coordinated_control_passivity.

Source / provenance

• Literature: giordano2019coordinated (eq 21, App. B).
• Ours: the explicit block derivation and the $\dot{\mathbf{\Gamma }}$ transport correction.
  Private derivation master: my_writing/crucial/cc_detailed/com_att_writeup.tex §2 “Derivation of
  Equation 21” (to be lifted into generated/math/ in Phase D).

Caveats

• The $-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$ term makes $\breve{\mathbf{C}}$
  more than a naive congruence; it is what lets the passivity property hold.
• Holds only on the singularity-free region; see
  singularity_threshold_cascade.
• Related Giordano typo corrections (eq 31 sign, eq 34d block-order) live with the control law, not
  here — cross-referenced from
  coordinated_control_lyapunov_stability.

Implementation (sims wiki)

External — into the code wiki via the sims_wiki/ symlink (resolves in Obsidian, not GitHub).

• com_controller — the decoupled $3\times 3$ CoM loop this derivation isolates.