coordinated_circumcentroidal_control

Coordinated Circumcentroidal Control — End-to-End Spine

Last updated: 2026-06-22

Overview

This page is the canonical overview of how we control the free-flying space manipulator
(a fully-actuated 6-DOF base carrying a redundant arm with a camera end-effector — not the
free-floating regime of much of the literature). It threads the whole controller as one spine:

frames → coordinate transform $\mathbf{\Gamma }$ → reduced / decoupled EOM →
control law + working closed loop → guidance feedforward.

The organizing idea, due to Giordano 2019, is a
circumcentroidal change of velocity coordinates. The base velocity is eliminated in favour of the
whole-system centre-of-mass (CoM) velocity ${\mathbf{v}}_c$, the base angular velocity
${\mathbf{\omega }}_b$, and the end-effector velocity taken about the CoM, ${\mathbf{\nu }}_e^{\oplus }$.
In these coordinates the $3\times 3$ CoM translation loop decouples from a coupled $9\times 9$
attitude+EE block, the coupled block is passive, and a Cartesian-impedance law on it is
Lyapunov-stable. The project layers its own deterministic guidance (a spherical-helix CoM orbit
plus a look-at end-effector pose) on top, supplying an analytic acceleration feedforward into that
stable inner loop. All symbols are rendered from notation.md.

The spine, in five stages:

1. Frames (§0.1). Four frames — target/inertial $\mathcal{T}$, system-CoM $\mathcal{C}$
   (axes non-rotating w.r.t. $\mathcal{T}$, translating as the arm moves), end-effector/camera
   $\mathcal{E}$, base $\mathcal{B}$ — set up the CoM-relative (“circumcentroidal”) split.
2. Coordinate transform $\mathbf{\Gamma }$ (§2). A single $12\times (6+n)$ map
   $\mathbf{z}=\mathbf{\Gamma }\mathbf{x}$ takes $[{\mathbf{v}}_b;{\mathbf{\omega }}_b;\dot{\mathbf{q}}]$
   to $[{\mathbf{v}}_c;{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$; see
   circumcentroidal_motion.
3. Reduced / decoupled EOM (§3). The congruence under $\mathbf{\Gamma }$ (with a
   $\dot{\mathbf{\Gamma }}$ transport correction) block-diagonalizes the inertia and gives a passive
   coupled block; see circumcentroidal_decoupling.
4. Control law + working closed loop (§4). A Cartesian-impedance / coordinated law on the coupled
   block, proven asymptotically stable; see
   coordinated_control and
   coordinated_control_lyapunov_stability.
5. Guidance feedforward (§5). The project’s own CoM-orbit + look-at pose guidance, converted into
   the $\oplus$ frame to feed an analytic acceleration feedforward.

Key Claims

Claim 1 — The circumcentroidal transform $\mathbf{\Gamma }$ is the hinge of the whole design

Eliminating the base velocity ${\mathbf{v}}_b$ rewrites the EE twist as a CoM-translation part plus
motion about the CoM, ${\mathbf{\nu }}_e={\mathbf{G}}_{v_c}{\mathbf{v}}_c+{\mathbf{\nu }}_e^{\oplus }$,
with the circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }={\mathbf{J}}_{{\nu }_e}-[{\mathbf{R}}_{eb};0]{\bar{\mathbf{J}}}_v$
(Giordano eq 14). Stacking the CoM-velocity relation, the identity on ${\mathbf{\omega }}_b$, and the
$\oplus$ split gives the coordinate transform

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

(Giordano eq 19 / current_sota eq 2.4)

Because ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ sits in the lower-right block, $\mathbf{\Gamma }$ is
invertible iff ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is nonsingular — the two go singular together. This
single fact is what makes ${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$ the natural proximity
measure (Spearman $\ge 0.9969$ with ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$) and is the analytic
root of all the conditioning work. Source support:
circumcentroidal_motion;
giordano2019coordinated (eqs 12–19); closed-form inverse in
gamma_closed_form_inverse.

Regime — load-bearing

The $\oplus$ split is purely kinematic: it removes base translation only, never base rotation,
and does not rely on momentum conservation. This is the free-flying object
${\mathbf{J}}_{{\nu }_e}^{\oplus }$, distinct from the free-floating generalized Jacobian
${\mathbf{J}}_g$ (Umetani 1989), which folds momentum conservation for an uncontrolled base. See
generalized_jacobian and
free_flying_vs_free_floating. Our base is fully actuated.

Claim 2 — Under $\mathbf{\Gamma }$ the dynamics decouple into a CoM loop + a coupled $9\times 9$ block

The congruence $\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-T}\mathbf{M}{\mathbf{\Gamma }}^{-1}$,
$\hat{\mathbf{C}}={\mathbf{\Gamma }}^{-T}(\mathbf{C}-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}){\mathbf{\Gamma }}^{-1}$
block-diagonalizes the inertia, $\hat{\mathbf{M}}=\mathrm{blkdiag}(m\mathbf{E},\breve{\mathbf{M}})$,
and the centroid Coriolis vanishes. The system splits into

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

(Giordano eqs 22a, 22b / current_sota eqs 3.3, 3.4)

The $3\times 3$ CoM equation is an isotropic point mass; the coupled $9\times 9$ attitude+EE block carries
all the dynamic coupling. Source support:
circumcentroidal_decoupling;
giordano2019coordinated (eq 21, App. B).

Inconsistency — hat vs breve (canonical)

$\hat{\mathbf{M}},\hat{\mathbf{C}}$ are the full $12\times 12$ transformed matrices;
$\breve{\mathbf{M}},\breve{\mathbf{C}}$ are the reduced $9\times 9$ lower-right block (the
coupled attitude+EE subsystem surviving after the CoM decouples). They are different objects. A prior
7-DOF effort conflated them; the canonical distinction lives at
circumcentroidal_decoupling.

Claim 3 — The coupled block is passive, and this requires the $\dot{\mathbf{\Gamma }}$ transport term

The reduced block satisfies the skew-symmetry / passivity identity

$${\breve{\mathbf{v}}}^T(\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}})\breve{\mathbf{v}}=0,\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }].$$

(Giordano eq 23 / current_sota eq 3.5)

This is not a free congruence: it holds only because the transport term
$-\mathbf{M}{\mathbf{\Gamma }}^{-1}\dot{\mathbf{\Gamma }}$ is retained inside $\breve{\mathbf{C}}$
(it absorbs the $\dot{\hat{\mathbf{M}}}$ contribution from the time-dependence of $\mathbf{\Gamma }$).
Dropping that term — the most common shortcut — destroys the skew structure and with it the clean
Lyapunov bound. Source support:
coordinated_control_passivity;
giordano2019coordinated (eq 23, App. B).

Claim 4 — The coordinated (Cartesian-impedance) law is Lyapunov-asymptotically-stable on the singularity-free region

On the coupled block the control is a negative-feedback impedance law (gains SPD; uses the project’s
sign correction to Giordano eq 31):

$$\left[\begin{array}{c}{\mathbf{\tau }}_b^{\oplus }\\ {\mathbf{w}}_e^{\oplus }\end{array}\right]=-{\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}-\breve{\mathbf{D}}\breve{\mathbf{v}},$$

(Giordano eq 31, sign-corrected / current_sota eq 4.7)

where the error-rate Jacobian ${\mathbf{J}}_{\tilde{x}}=\mathrm{blkdiag}({\mathbf{J}}_{{\tilde{x}}_b},{\mathbf{J}}_{{\tilde{x}}_e})$
maps the stacked twist to the quaternion-based pose-error rate
($\dot{\tilde{\mathbf{x}}}={\mathbf{J}}_{\tilde{x}}\breve{\mathbf{v}}+{\mathbf{J}}_{\tilde{x}}{\breve{\mathbf{G}}}_{v_c}{\dot{\tilde{\mathbf{x}}}}_c$;
task_space_error_dynamics). With the energy certificate

$$V=\frac{1}{2}{\breve{\mathbf{v}}}^T\breve{\mathbf{M}}\breve{\mathbf{v}}+\frac{1}{2}{\tilde{\mathbf{x}}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}},\dot{V}=-{\breve{\mathbf{v}}}^T\breve{\mathbf{D}}\breve{\mathbf{v}}\le 0,$$

(Giordano eqs 37, 38 / current_sota eqs 4.15, 4.16)

the tracking-error origin is asymptotically stable on
$\Omega =\{{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })>0\}$ (Giordano eq 36). The passivity of Claim 3 is
the load-bearing step that collapses $\dot{V}$ to pure damping dissipation. Joint rates are recovered by
$\dot{\mathbf{q}}={\mathbf{J}}_{{\nu }_e}^{\oplus -1}{\breve{\mathbf{G}}}_{{\omega }_b}\breve{\mathbf{v}}$
with ${\breve{\mathbf{G}}}_{{\omega }_b}=[-{\mathbf{G}}_{{\omega }_b}\mathbf{E}]$ (eq 34d,
block-order-corrected). Source support:
coordinated_control_lyapunov_stability;
coordinated_control;
giordano2019coordinated (eqs 31, 34, 36–38).

Claim 5 — The project’s working closed loop adds damping-on-error and an acceleration feedforward

The paper certifies a regulator. The project’s working controller tracks a moving reference: the
damper acts on the velocity error and a matched acceleration feedforward is added to the RHS,

$$\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}=-\breve{\mathbf{C}}\breve{\mathbf{v}}-\breve{\mathbf{D}}(\breve{\mathbf{v}}-{\breve{\mathbf{v}}}_d)-{\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}-({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c+\breve{\mathbf{M}}{\dot{\breve{\mathbf{v}}}}_d.$$

(final.tex eq 34b / current_sota eq 4.12)

Under nonzero reference rates the equilibrium shifts off the raw origin to a steady-state cruise-lag
floor ${\tilde{\mathbf{x}}}_{ss}$ (a balance among the proportional term, the cruising arm’s Coriolis
force, and the CoM coupling; current_sota eq 4.14). The CoM error loop itself is an autonomous damped
second-order system $m{\ddot{\tilde{\mathbf{x}}}}_c+{\mathbf{D}}_c{\dot{\tilde{\mathbf{x}}}}_c+{\mathbf{K}}_c{\tilde{\mathbf{x}}}_c=0$.
Source support: coordinated_control;
steady_state_error_floor.

Claim 6 — Guidance supplies the feedforward in the $\oplus$ frame; the inner loop stays the same

Guidance is the project’s own deterministic POSE-mode plan, distinct from the borrowed control law. The
CoM tracks a spherical-helix orbit with a startup speed ramp; the desired EE pose is a look-at standoff
construction from the surface aim point (${\mathbf{z}}_{ed}=\hat{\mathbf{r}}$, the whole pose
governed by one direction $\hat{\mathbf{r}}$; current_sota eqs 5.7), with a roll-free desired twist
${\mathbf{\omega }}_{ed}=\hat{\mathbf{r}}\times \dot{\hat{\mathbf{r}}}$ (eq 5.12). The desired
twist is converted to local then circumcentroidal coordinates,

$${\mathbf{\nu }}_{ed}^{\oplus }={\mathbf{\nu }}_{ed}^{\mathrm{local}}-{\mathbf{G}}_{v_c}{\mathbf{v}}_{cd}+{\mathbf{G}}_{{\omega }_b}{\mathbf{\omega }}_{bd},$$

(final.tex eq nu_dot_des_2 / current_sota eq 5.17)

and differentiated (with ${\dot{\mathbf{\omega }}}_{bd}\approx 0$) to supply the analytic
acceleration feedforward $\breve{\mathbf{M}}[0_3;{\dot{\mathbf{\nu }}}_{ed}^{\oplus }]$ into
the RHS of Claim 5’s working loop — replacing the older finite-differenced demand. The maps
${\mathbf{G}}_{v_c},{\mathbf{G}}_{{\omega }_b}$ are already available from the dynamics (§2–3), so
guidance reuses the same circumcentroidal machinery rather than introducing new objects. Source support:
current_sota §5 (final.tex primary); coordinated_control.

Tensions & Open Questions

• Three project corrections to Giordano 2019 are load-bearing. The eq-31 negative-feedback sign,
  the eq-34d selector block-order $[-{\mathbf{G}}_{{\omega }_b}\mathbf{E}]$, and the
  $\dot{\mathbf{\Gamma }}$ transport term in $\breve{\mathbf{C}}$. With the paper’s printed forms
  the $\dot{V}\le 0$ bound does not follow. These are flagged on
  coordinated_control_lyapunov_stability and rolled
  up in _inconsistencies.md.
• Regulator vs tracker. The Lyapunov certificate proves the regulator. Under nonzero reference
  rates the equilibrium becomes the cruise-lag floor ${\tilde{\mathbf{x}}}_{ss}$, so $V$ certifies
  stability about a moving reference, not the raw origin. A rigorous tracking guarantee for the working
  loop (Claim 5) is open.
• Inner-loop idealization. The cascade is analyzed at the inner CoM loop’s equilibrium. In practice
  the measured CoM error settles near $10^{-3}\mathrm{m}$, leaving a small persistent coupling
  ${\mathbf{C}}_c{\mathbf{v}}_c$ that the bare regulator proof does not cover; a cascaded-systems
  argument is the rigorous route (cascaded_systems).
• Validity confined to $\Omega$. Everything above holds only where ${\mathbf{J}}_{{\nu }_e}^{\oplus }$
  (equivalently $\mathbf{\Gamma }$) is nonsingular. Near the boundary the joint-rate inverse is
  regularized (Tikhonov / damped-$J^{\oplus }$), which perturbs the very loop the certificate describes; see
  singularity_robust_inverse,
  dynamic_singularity,
  singularity_threshold_cascade.
• Nonredundant ($n=6$) assumption. The whole spine assumes a square ${\mathbf{J}}_{{\nu }_e}^{\oplus }$.
  The redundant ($n=7$) case adds a self-motion / null-space reconstruction (the redundant DOF swings
  violently near singularity and is frozen below $\sim 10^{-3}$); how the $\oplus$ split and the reduced
  block generalize is open.
• Regime hygiene. “Generalized Jacobian” is overloaded across the corpus: ${\mathbf{J}}_g$ (Umetani,
  free-floating, momentum-folded) vs ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (Giordano, free-flying, kinematic).
  This spine is free-flying throughout; do not import GJM dynamic-singularity intuition.

Connected Pages

Results:
circumcentroidal_decoupling ·
coordinated_control_passivity ·
coordinated_control_lyapunov_stability ·
gamma_closed_form_inverse ·
steady_state_error_floor ·
singularity_threshold_cascade

Topics:
coordinated_control ·
circumcentroidal_motion ·
generalized_jacobian ·
task_space_error_dynamics ·
cartesian_impedance_control ·
cascaded_systems ·
free_flying_vs_free_floating ·
dynamic_singularity ·
singularity_robust_inverse

Sources:
giordano2019coordinated ·
giordano2020coordination ·
umetani1989resolved

Notation: notation.md