coordinated_control_passivity

Passivity / Skew-Symmetry of the Coupled 9×9 Attitude+EE Block

Statement

For the reduced circumcentroidal attitude+end-effector subsystem with inertia
$\breve{\mathbf{M}}\in {\mathbb{R}}^{9\times 9}$ and Coriolis
$\breve{\mathbf{C}}\in {\mathbb{R}}^{9\times 9}$ (notation.md), the matrix
$\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}}$ is skew-symmetric on the stacked velocity
$\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]\in {\mathbb{R}}^9$:

$$\left[\begin{array}{cc}{\mathbf{\omega }}_b^{\top }& {\mathbf{\nu }}_e^{\oplus \top }\end{array}\right](\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}})\left[\begin{array}{c}{\mathbf{\omega }}_b\\ {\mathbf{\nu }}_e^{\oplus }\end{array}\right]=0,\forall {\mathbf{\omega }}_b,{\mathbf{\nu }}_e^{\oplus }\in {\mathbb{R}}^3,{\mathbb{R}}^6.$$

(Giordano eq 23; current_sota eq 3.5)

Equivalently, ${\breve{\mathbf{v}}}^{\top }(\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}})\breve{\mathbf{v}}=0$.
This is the standard Lagrangian passivity (energy-conservation) identity, but established here for the
reduced $9\times 9$ block of the circumcentroidal dynamics — not for the original
$(6+n)\times (6+n)$ system. It is the structural property that drives the Lyapunov derivative of the
coordinated controller down to $\dot{V}=-{\breve{\mathbf{v}}}^{\top }\breve{\mathbf{D}}\breve{\mathbf{v}}\le 0$.

hat vs breve — load-bearing

The identity is stated on $\breve{(\cdot )}$, the reduced $9\times 9$ attitude+EE block, not on
$\widehat{(\cdot )}$, the full $12\times 12$ transformed matrices. The two are different objects: the
$\stackrel{˘}{}$ block is the lower-right corner of the $\hat{}$ matrices that survives once the isotropic
$3\times 3$ CoM equation ($m\mathbf{E}$) decouples. See
circumcentroidal_decoupling for the canonical hat/breve distinction
(a prior 7-DOF effort conflated them).

Assumptions

• Free-flying regime. The base is fully actuated (6-DOF). Giordano 2019 explicitly takes a
  fully-actuated spacecraft (external forces/torques from the spacecraft’s own actuators); the
  controller then chooses to leave base translation free (“partial base control”). This is not
  the free-floating generalized-Jacobian setting, where momentum conservation — not actuation —
  closes the dynamics. See free_flying_vs_free_floating.
• ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ nonsingular so the coordinate transform $\mathbf{\Gamma }$
  (notation.md) is invertible — $\mathbf{\Gamma }$ and ${\mathbf{J}}_{{\nu }_e}^{\oplus }$
  go singular together (lower-right block). Holds only on the singularity-free region
  $\Omega =\{{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })>0\}$.
• $n=6$ (nonredundant arm), so ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is square. The redundant ($n=7$) case
  carries a self-motion / null-space term reserved for later work.
• Rigid bodies; the $\dot{\mathbf{\Gamma }}$ transport term
  $-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$ is retained in the definition of
  $\hat{\mathbf{C}}$ (and hence $\breve{\mathbf{C}}$). Dropping it breaks this identity.

Proof sketch

The full Lemma-1 derivation (skew survives the augmented $\dot{\mathbf{\Gamma }}$-corrected congruence)
is in passivity_proofs.md; conceptual entry point
walkthroughv2.md ($M$-orthogonality and projectors from
scratch). The underlying $12\times 12\to 9\times 9$ Equation-21 derivation it builds on is deferred to
Phase D (target generated/math/com_att_writeup.md).

The property is not automatic from a congruence: a naive transform of the original passive
$(\dot{\mathbf{M}}-2\mathbf{C})$ would not give a skew $\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}}$,
because $\mathbf{\Gamma }$ is time-varying. The $\dot{\mathbf{\Gamma }}$-correction is what restores it.

1. Start from the original passive structure. In base+joint coordinates the full
   $\mathbf{M},\mathbf{C}$ (notation.md) satisfy the classical Lagrangian
   identity ${\mathbf{x}}^{\top }(\dot{\mathbf{M}}-2\mathbf{C})\mathbf{x}=0$ for a suitable
   Christoffel factorization of $\mathbf{C}$ (Giordano eq 4 system).

2. Apply the $\dot{\mathbf{\Gamma }}$-corrected congruence. The transformed Coriolis is
   $\hat{\mathbf{C}}={\mathbf{\Gamma }}^{-\top }(\mathbf{C}-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}){\mathbf{\Gamma }}^{-1}$
   with $\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-\top }\mathbf{M}{\mathbf{\Gamma }}^{-1}$
   (Giordano App. B; current_sota eqs 3.1, 3.2). The transport term
   $-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$ is exactly the piece that absorbs
   the $\dot{\hat{\mathbf{M}}}$ contribution coming from the time-dependence of $\mathbf{\Gamma }$.

3. Skewness survives the corrected congruence. Differentiating
   $\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-\top }\mathbf{M}{\mathbf{\Gamma }}^{-1}$ produces
   $\dot{\hat{\mathbf{M}}}$ terms in $\dot{\mathbf{\Gamma }}$; the matched
   $-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$ inside $\hat{\mathbf{C}}$
   cancels their symmetric part, leaving $\dot{\hat{\mathbf{M}}}-2\hat{\mathbf{C}}$ skew-symmetric
   in the full $12\times 12$ coordinates.

4. Restrict to the $9\times 9$ block. With the isotropic CoM block $m\mathbf{E}$ decoupling
   ($\dot{(m\mathbf{E})}=0$, off-diagonal Coriolis $\pm {\mathbf{C}}_c$ skew by
   construction — see circumcentroidal_decoupling), the skew property
   inherits onto the surviving lower-right block: $\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}}$
   is skew, hence the displayed quadratic form vanishes.

Why it matters (the payoff). With $V=\frac{1}{2}{\breve{\mathbf{v}}}^{\top }\breve{\mathbf{M}}\breve{\mathbf{v}}+\frac{1}{2}{\tilde{\mathbf{x}}}^{\top }\breve{\mathbf{K}}\tilde{\mathbf{x}}$,
the term $\frac{1}{2}{\breve{\mathbf{v}}}^{\top }\dot{\breve{\mathbf{M}}}\breve{\mathbf{v}}$ from
$\dot{V}$ is replaced (via this identity) by ${\breve{\mathbf{v}}}^{\top }\breve{\mathbf{C}}\breve{\mathbf{v}}$,
which then cancels the Coriolis term from the closed-loop dynamics, collapsing $\dot{V}$ to
$-{\breve{\mathbf{v}}}^{\top }\breve{\mathbf{D}}\breve{\mathbf{v}}\le 0$. This is the load-bearing
step in the cascade stability argument (sibling result, planned:
coordinated_control_lyapunov_stability; current_sota eqs
4.15–4.16). The closed-loop subsystem this feeds is the Cartesian-impedance law
(cartesian_impedance_control) acting on the coupled block.

Source / provenance

• Literature: giordano2019coordinated (eq 23, App. B).
  Regime: free-flying (fully-actuated base; “partial base control” leaves translation free).
• Master sheet: my_writing/equations/current_sota.md §3.5 (eq 3.5), derived from Giordano eq 23 and
  the §3.1–3.2 $\dot{\mathbf{\Gamma }}$-corrected congruence.
• Ours: the explicit attribution of the identity to the $\dot{\mathbf{\Gamma }}$ transport term
  (it is not a bare congruence) and the block-inheritance argument. The full derivation rides with
  circumcentroidal_decoupling (Phase D
  generated/math/com_att_writeup.md).

Caveats

• Not a free congruence. Dropping the $-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$
  term — the most common shortcut — destroys the skew structure and with it the clean
  $\dot{V}\le 0$. This is the same correction flagged on
  circumcentroidal_decoupling.
• Singularity-free only. The identity presumes $\mathbf{\Gamma }$ (equivalently
  ${\mathbf{J}}_{{\nu }_e}^{\oplus }$) invertible; it says nothing inside the singular set. Near a
  circumcentroidal-Jacobian singularity the controller loses actuation in the singular direction
  (dynamic_coupling / circumcentroidal_motion);
  the project’s singularity-handling layers live with the control law, not here.
• Skewness is factorization-dependent. As always with the $\dot{\mathbf{M}}-2\mathbf{C}$
  property, it holds for the Christoffel-symbol $\breve{\mathbf{C}}$; an arbitrary $\breve{\mathbf{C}}$
  satisfying the equations of motion need not be skew. The wiki uses the Christoffel factorization
  throughout (coordinated_control,
  lyapunov_stability).
• Nonredundant ($n=6$). The $9\times 9$ block size assumes a square ${\mathbf{J}}_{{\nu }_e}^{\oplus }$.
  Extending to $n=7$ (self-motion / null-space) is open and may alter the reduced-block structure.