coordinated_control_lyapunov_stability

Coordinated Control: Lyapunov Asymptotic Stability of the Tracking Error

Statement

For the free-flying space manipulator (fully-actuated 6-DOF base + redundant arm) under
the circumcentroidal coordinated control law, the closed-loop tracking-error equilibrium

$${\tilde{\mathbf{x}}}_c={\dot{\tilde{\mathbf{x}}}}_c=0,\tilde{\mathbf{x}}=\breve{\mathbf{v}}=0$$

is asymptotically stable on the singularity-free region

$$\Omega =\{\mathbf{z}:{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus }(\mathbf{q}))>0\}.$$

(Giordano eq 36)

The certificate is the total mechanical energy of the reduced (circumcentroidal) attitude+EE
subsystem in error coordinates,

V=\tfrac12\,\breve{\boldsymbol{v}}^{T}\breve{\boldsymbol{M}}\,\breve{\boldsymbol{v}} +\tfrac12\,\tilde{\boldsymbol{x}}^{T}\breve{\boldsymbol{K}}\,\tilde{\boldsymbol{x}}, \tag{current_sota eq 4.15 / Giordano eq 37}

whose derivative along the closed loop is negative semidefinite,

\dot{V}=-\,\breve{\boldsymbol{v}}^{T}\breve{\boldsymbol{D}}\,\breve{\boldsymbol{v}}\le 0. \tag{current_sota eq 4.16 / Giordano eq 38}

Here $\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]\in {\mathbb{R}}^9$ is the
stacked attitude+EE velocity, $\tilde{\mathbf{x}}=[{\tilde{\mathbf{x}}}_b;{\tilde{\mathbf{x}}}_e]\in {\mathbb{R}}^9$
the stacked outer-loop error, and $\breve{\mathbf{M}},\breve{\mathbf{K}},\breve{\mathbf{D}}$
the reduced inertia / stiffness / damping (notation.md).

hat vs breve

$V$ is built from the reduced $9\times 9$ matrices $\breve{\mathbf{M}},\breve{\mathbf{K}},\breve{\mathbf{D}}$
— the coupled attitude+EE block that survives once the $3\times 3$ CoM equation decouples — not
the full $12\times 12$ transformed $\hat{\mathbf{M}},\hat{\mathbf{C}}$. The accent distinction
is load-bearing here; see circumcentroidal_decoupling.

Assumptions

• Singularity-free region. ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ nonsingular on $\Omega$
  (${\sigma }_{\min }>0$); equivalently $\mathbf{\Gamma }$ invertible, since the two go singular
  together (lower-right block). Outside $\Omega$ the joint-rate recovery
  $\dot{\mathbf{q}}={\mathbf{J}}_{{\nu }_e}^{\oplus -1}{\breve{\mathbf{G}}}_{{\omega }_b}\breve{\mathbf{v}}$
  blows up and the claim does not hold.
• Nonredundant arm, $n=6$, so ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is square and invertible (the
  redundant $n=7$ self-motion / null-space case is reserved for later work).
• SPD gains. $\breve{\mathbf{K}}=\mathrm{blkdiag}({\mathbf{K}}_b,{\mathbf{K}}_e)\succ 0$ and
  $\breve{\mathbf{D}}=\mathrm{blkdiag}({\mathbf{D}}_b,{\mathbf{D}}_e)\succ 0$; likewise the
  CoM-loop ${\mathbf{K}}_c,{\mathbf{D}}_c\succ 0$.
• Decoupling holds. The congruence under $\mathbf{\Gamma }$ yields the block-diagonal
  $\hat{\mathbf{M}}=\mathrm{blkdiag}(m\mathbf{E},\breve{\mathbf{M}})$ and a skew off-diagonal
  $\hat{\mathbf{C}}$ — established in circumcentroidal_decoupling.
• Passivity / skew-symmetry. ${\breve{\mathbf{v}}}^T(\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}})\breve{\mathbf{v}}=0$
  (Giordano eq 23). This requires retaining the $-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$
  transport term inside $\breve{\mathbf{C}}$ (see Caveats).
• Inner CoM loop settled. The cascade is studied at the equilibrium of the inner loop,
  ${\tilde{\mathbf{x}}}_c={\dot{\tilde{\mathbf{x}}}}_c=0$, where the CoM forcing on the
  attitude+EE block vanishes; the CoM error obeys the autonomous damped second-order system
  $m{\ddot{\tilde{\mathbf{x}}}}_c+{\mathbf{D}}_c{\dot{\tilde{\mathbf{x}}}}_c+{\mathbf{K}}_c{\tilde{\mathbf{x}}}_c=0$
  (current_sota eq 4.9), itself exponentially stable.

Proof sketch

The closed-loop extension proofs — obligations P3, P4 (Coriolis cross-terms vanish; $\dot{V}\le 0$) —
are in lyapunov_extension_proofs.md, with the
P1–P4 framing in derivation_7dof.md and an adversarial
checklist in review_7dof.md. The clean $n=6$ cascade write-up
(eqs 36–38 with the three corrections) remains Phase D.

1. Reduce. Under $\mathbf{\Gamma }$ the dynamics split into a decoupled CoM equation
   $m{\dot{\mathbf{v}}}_c={\mathbf{f}}_c$ and the coupled attitude+EE block
   $\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}+\breve{\mathbf{C}}\breve{\mathbf{v}}+{\mathbf{C}}_c{\mathbf{v}}_c=[{\mathbf{\tau }}_b^{\oplus };{\mathbf{w}}_e^{\oplus }]$
   (current_sota eqs 3.3–3.4; circumcentroidal_decoupling).

2. Close the loop with the corrected-sign law. Insert the negative-feedback
   $[{\mathbf{\tau }}_b^{\oplus };{\mathbf{w}}_e^{\oplus }]=-{\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}-\breve{\mathbf{D}}\breve{\mathbf{v}}$
   (current_sota eq 4.7 — the corrected eq 31 sign, not the paper’s printed positive sign). At the
   inner-loop equilibrium (${\tilde{\mathbf{x}}}_c={\dot{\tilde{\mathbf{x}}}}_c=0$) the
   CoM coupling drops and the regulated block is
   $\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}+\breve{\mathbf{C}}\breve{\mathbf{v}}+\breve{\mathbf{D}}\breve{\mathbf{v}}+{\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}=0$
   (current_sota eq 4.10).

3. Differentiate $V$.
   $\dot{V}={\breve{\mathbf{v}}}^T\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}+\frac{1}{2}{\breve{\mathbf{v}}}^T\dot{\breve{\mathbf{M}}}\breve{\mathbf{v}}+{\tilde{\mathbf{x}}}^T\breve{\mathbf{K}}\dot{\tilde{\mathbf{x}}}$.
   Substitute $\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}$ from step 2 and use the error-rate
   map $\dot{\tilde{\mathbf{x}}}={\mathbf{J}}_{\tilde{x}}\breve{\mathbf{v}}$ (current_sota eq 4.3,
   inner-loop part; task_space_error_dynamics).

4. Cancel by passivity. The Coriolis quadratic cancels via skew-symmetry,
   ${\breve{\mathbf{v}}}^T(\frac{1}{2}\dot{\breve{\mathbf{M}}}-\breve{\mathbf{C}})\breve{\mathbf{v}}=0$
   (Giordano eq 23), and the stiffness cross-terms cancel:
   $-{\breve{\mathbf{v}}}^T{\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}+{\tilde{\mathbf{x}}}^T\breve{\mathbf{K}}{\mathbf{J}}_{\tilde{x}}\breve{\mathbf{v}}=0$.
   What remains is the damping dissipation
   $\dot{V}=-{\breve{\mathbf{v}}}^T\breve{\mathbf{D}}\breve{\mathbf{v}}\le 0$ (eq 4.16).

5. Asymptotic stability (LaSalle). $V>0$ (radially unbounded on $\Omega$ since
   $\breve{\mathbf{M}},\breve{\mathbf{K}}\succ 0$), $\dot{V}\le 0$. On the largest invariant set
   in $\{\dot{V}=0\}=\{\breve{\mathbf{v}}=0\}$, the closed loop forces
   ${\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}=0$; with
   $\breve{\mathbf{K}}\succ 0$ and ${\mathbf{J}}_{\tilde{x}}$ full rank on $\Omega$ this gives
   $\tilde{\mathbf{x}}=0$, so the only invariant set is the origin. The CoM cascade
   (eq 4.9) settles independently. Hence asymptotic stability on $\Omega$.

This is the tracking-error counterpart to the structural results of
circumcentroidal_decoupling (which licenses the cascade) — the
block-diagonal $\hat{\mathbf{M}}$ and skew $\hat{\mathbf{C}}$ are exactly what make $V$ a clean
energy certificate.

Source / provenance

• Literature: giordano2019coordinated — eqs 36, 37, 38
  (region $\Omega$, $V$, $\dot{V}$). Giordano, Ott, Albu-Schäffer assume a fully-actuated
  thrusters-and-arm spacecraft, i.e. the free-flying regime — the same regime as ours (their base
  is controlled, unlike the free-floating literature where momentum conservation is folded into a
  generalized Jacobian; see generalized_jacobian).
• Ours: the three corrections that make the certificate actually hold as printed cannot — the eq 31
  negative-feedback sign, the eq 34d selector block-order, and the $\dot{\mathbf{\Gamma }}$ transport
  term in $\breve{\mathbf{C}}$ (see Caveats). Master sheet: my_writing/equations/current_sota.md
  §4.6 (eqs 4.15–4.16), with the corrections recorded at §4.2 (eq 4.7) and §4.3 (eq 4.11).

Caveats

The proof depends on three project corrections to Giordano 2019; with the paper’s printed forms the
$\dot{V}\le 0$ bound does not follow. These are flagged in the frontmatter inconsistencies and rolled
up in _inconsistencies.md.

Inconsistency — Giordano eq 31 sign (load-bearing for stability)

The compact base+EE law must be negative feedback,
$[{\mathbf{\tau }}_b^{\oplus };{\mathbf{w}}_e^{\oplus }]=-{\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}-\breve{\mathbf{D}}\breve{\mathbf{v}}$
(current_sota eq 4.7). The published Giordano eq 31 prints positive signs; substituting those
would make the stiffness cross-terms add rather than cancel and the damping term inject energy
($\dot{V}=+{\breve{\mathbf{v}}}^T\breve{\mathbf{D}}\breve{\mathbf{v}}\ge 0$). The Lyapunov
argument here uses the corrected negative sign.

Inconsistency — Giordano eq 34d block-order

The joint-rate recovery is
$\dot{\mathbf{q}}={\mathbf{J}}_{{\nu }_e}^{\oplus -1}{\breve{\mathbf{G}}}_{{\omega }_b}\breve{\mathbf{v}}$
with the selector ${\breve{\mathbf{G}}}_{{\omega }_b}=[-{\mathbf{G}}_{{\omega }_b}\mathbf{E}]\in {\mathbb{R}}^{6\times 9}$
(current_sota eq 4.11) — so $-{\mathbf{G}}_{{\omega }_b}$ multiplies ${\mathbf{\omega }}_b$ (the first
block of $\breve{\mathbf{v}}$) and $\mathbf{E}$ multiplies ${\mathbf{\nu }}_e^{\oplus }$. The
published eq 34d transposes the block order; the corrected order is what keeps $\dot{\mathbf{q}}$
consistent with $\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$.

Inconsistency — Γ̇ transport term in C-breve

The reduced Coriolis is
$\breve{\mathbf{C}}=$ (lower-right block of) ${\mathbf{\Gamma }}^{-T}(\mathbf{C}-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}){\mathbf{\Gamma }}^{-1}$.
The $-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$ transport term is not a naive
congruence; dropping it breaks the skew-symmetry $\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}}$
(Giordano eq 23) on which the Coriolis cancellation in step 4 relies — so $\dot{V}$ would no longer
reduce to $-{\breve{\mathbf{v}}}^T\breve{\mathbf{D}}\breve{\mathbf{v}}$. Details:
circumcentroidal_decoupling.

Other boundaries of validity:

• Only stability, only on $\Omega$. Asymptotic, not exponential, as stated; and strictly inside the
  singularity-free region $\Omega =\{{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })>0\}$. Near the boundary
  the joint-rate inverse is regularized (Tikhonov / damped-$J^{\oplus }$), which perturbs the closed loop
  the certificate describes; see singularity_robust_inverse
  and dynamic_singularity.
• Regulation vs tracking. $V,\dot{V}$ certify the regulator (set-point) form. The project’s working
  controller (current_sota eq 4.12) adds damping-on-velocity-error $+\breve{\mathbf{D}}{\breve{\mathbf{v}}}_d$
  and acceleration feedforward $+\breve{\mathbf{M}}{\dot{\breve{\mathbf{v}}}}_d$; under nonzero
  reference rates the equilibrium shifts to the cruise-lag floor ${\tilde{\mathbf{x}}}_{ss}$
  (current_sota eq 4.14), so $V$ certifies stability about that moving reference, not about the raw
  origin.
• Inner-loop idealization. The cascade is analyzed at the inner CoM loop’s equilibrium. In practice
  the measured CoM error settles near $10^{-3}\mathrm{m}$, not exactly zero, leaving residual coupling
  ${\mathbf{C}}_c{\mathbf{v}}_c$ — a small persistent forcing the bare regulator proof does not cover
  (a cascaded-systems argument is the rigorous route; cascaded_systems).