Circumcentroidal Motion

Definition

Circumcentroidal motion is the part of the end-effector (camera) twist that is taken relative to
the whole-system centre of mass $\mathcal{C}$ rather than relative to the inertial target frame:
the EE twist is split into a rigid CoM-translation contribution plus the residual motion about the
CoM, the latter carrying the superscript $\oplus$ (notation.md). It is the change
of velocity coordinates underlying the Giordano (2019) coordinated formulation: eliminating the base
velocity ${\mathbf{v}}_b$ in favour of $[{\mathbf{v}}_c;{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$
removes base translation from the EE Jacobian and lets the $3\times 3$ CoM translation loop decouple
from the coupled attitude+EE block (see circumcentroidal_decoupling).
Because the CoM axes are held non-rotating w.r.t. the inertial target frame $\mathcal{T}$ (they
only translate as the arm moves), the $\oplus$ split removes only base translation, never base
rotation — distinguishing ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ from the free-floating generalized Jacobian
${\mathbf{J}}_g$.

Regime

Giordano (2019) assumes a base actuated by thrusters and reaction wheels — a fully-actuated
base, the same free-flying regime as this thesis. This is not the momentum-conserving
free-floating setting of Umetani’s generalized Jacobian; the $\oplus$ decomposition is purely
kinematic (a CoM-relative velocity split), so it does not rely on momentum conservation.

Key Equations

Symbols per notation.md. Equation tags reference the project equation sheet
(current_sota §2, itself anchored to Giordano 2019).

EE velocity in CoM coordinates — eliminate ${\mathbf{v}}_b$ from the EE twist ${\mathbf{\nu }}_e$:

$${\mathbf{\nu }}_e={\mathbf{G}}_{v_c}{\mathbf{v}}_c+{\mathbf{G}}_{{\omega }_b}{\mathbf{\omega }}_b+{\mathbf{J}}_{{\nu }_e}^{\oplus }\dot{\mathbf{q}}\text{(Giordano eqs 12–14)}$$

The maps and the circumcentroidal Jacobian — ${\mathbf{G}}_{v_c}$ injects pure CoM translation,
${\mathbf{G}}_{{\omega }_b}$ carries base rotation, and ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is the fixed-base
EE Jacobian with the mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v$ subtracted off (base
translation removed):

$${\mathbf{G}}_{v_c}=\left[\begin{array}{c}{\mathbf{R}}_{ec}\\ 0\end{array}\right],{\mathbf{G}}_{{\omega }_b}=\left[\begin{array}{c}[{\mathbf{p}}_{ec}{]}^{\wedge }{\mathbf{R}}_{eb}\\ {\mathbf{R}}_{eb}\end{array}\right],{\mathbf{J}}_{{\nu }_e}^{\oplus }={\mathbf{J}}_{{\nu }_e}-\left[\begin{array}{c}{\mathbf{R}}_{eb}\\ 0\end{array}\right]{\bar{\mathbf{J}}}_v\text{(Giordano eqs 12, 13, 14)}$$

The twist split — the EE twist is CoM motion plus motion about the CoM, the latter being the
circumcentroidal velocity ${\mathbf{\nu }}_e^{\oplus }$:

$${\mathbf{\nu }}_e={\mathbf{G}}_{v_c}{\mathbf{v}}_c+{\mathbf{\nu }}_e^{\oplus },{\mathbf{\nu }}_e^{\oplus }\triangleq {\mathbf{G}}_{{\omega }_b}{\mathbf{\omega }}_b+{\mathbf{J}}_{{\nu }_e}^{\oplus }\dot{\mathbf{q}}={\mathbf{\nu }}_e-\left[\begin{array}{c}{\mathbf{R}}_{ec}\\ 0\end{array}\right]{\mathbf{v}}_c\text{(Giordano eqs 15–18)}$$

The coordinate transform $\mathbf{\Gamma }$ — stacking the CoM-velocity relation (eq 1.7),
the identity on ${\mathbf{\omega }}_b$, and the $\oplus$ split gives the full velocity map whose
conditioning the singularity work targets:

$$\underset{\mathbf{z}}{\underbrace{\left[\begin{array}{c}{\mathbf{v}}_c\\ {\mathbf{\omega }}_b\\ {\mathbf{\nu }}_e^{\oplus }\end{array}\right]}}=\underset{\mathbf{\Gamma }}{\underbrace{\left[\begin{array}{ccc}{\mathbf{R}}_{cb}& -{\mathbf{R}}_{cb}[{\mathbf{p}}_{bc}{]}^{\wedge }& {\mathbf{R}}_{cb}{\bar{\mathbf{J}}}_v\\ 0& \mathbf{E}& 0\\ 0& {\mathbf{G}}_{{\omega }_b}& {\mathbf{J}}_{{\nu }_e}^{\oplus }\end{array}\right]}}\underset{\mathbf{x}}{\underbrace{\left[\begin{array}{c}{\mathbf{v}}_b\\ {\mathbf{\omega }}_b\\ \dot{\mathbf{q}}\end{array}\right]}},\mathbf{z}=\mathbf{\Gamma }\mathbf{x}\text{(Giordano eq 19)}$$

Because ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ sits in the lower-right block, $\mathbf{\Gamma }$ is
invertible iff ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is nonsingular — they go singular together, which
is why ${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$ tracks ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$
(Spearman $\ge 0.9969$). Closed-form inverse:
gamma_closed_form_inverse.

Source Support

• giordano2019coordinated — defines the $\oplus$ split, the maps ${\mathbf{G}}_{v_c},{\mathbf{G}}_{{\omega }_b}$, the circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, and the transform $\mathbf{\Gamma }$ (eqs 12–19).
• giordano2020coordination — companion development of the coordinated circumcentroidal control.

Related Topics

• coordinated_control — the control law built on the $\oplus$ velocity coordinates.
• circumcentroidal_decoupling — uses the $\mathbf{\Gamma }$ transform to decouple the $3\times 3$ CoM loop from the $9\times 9$ attitude+EE block.
• center_of_mass_jacobian — the mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v$ subtracted off to form ${\mathbf{J}}_{{\nu }_e}^{\oplus }$.
• generalized_jacobian — the free-floating ${\mathbf{J}}_g$, a distinct object: ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is kinematic (free-flying), ${\mathbf{J}}_g$ folds in momentum conservation (free-floating).
• task_space_error_dynamics — the error-rate Jacobian ${\mathbf{J}}_{\tilde{x}}$ that maps $\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$ to pose-error rates.

Open Questions

• How does the $\oplus$ split generalize for the redundant ($n=7$) case, where ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is non-square and the self-motion (null-space) DOF must be reconstructed?
• Quantify how ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ governs the conditioning of $\mathbf{\Gamma }$ as the arm approaches a stretched configuration. (Phase D.)