Coordinated Control

Definition

Coordinated control of a free-flying space manipulator regulates the base attitude and
end-effector pose together through the coupled base–arm dynamics, rather than controlling arm and
base independently. In the circumcentroidal formulation (Giordano 2019), a coordinate transform
$\mathbf{\Gamma }$ splits the dynamics into a decoupled centre-of-mass (CoM) translation loop and a
coupled attitude+EE loop — see circumcentroidal_decoupling —
yielding a provably stable cascade with near-zero translational fuel in contact-free maneuvers.

Key Equations

Symbols per notation.md. Equation tags reference the project equation sheet.

Coordinate transform $\mathbf{\Gamma }$ — eliminate the base velocity in favour of CoM + circumcentroidal coordinates:

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

Reduced equations of motion — the CoM decouples, leaving the coupled $9\times 9$ block (full derivation: circumcentroidal_decoupling):

$$m{\dot{\mathbf{v}}}_c={\mathbf{f}}_c,\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}+\breve{\mathbf{C}}\breve{\mathbf{v}}+{\mathbf{C}}_c{\mathbf{v}}_c=\left[\begin{array}{c}{\mathbf{\tau }}_b^{\oplus }\\ {\mathbf{w}}_e^{\oplus }\end{array}\right]\text{(Giordano eqs 22a, 22b)}$$

Compact task control law (gains SPD; negative-feedback form):

$$\left[\begin{array}{c}{\mathbf{\tau }}_b^{\oplus }\\ {\mathbf{w}}_e^{\oplus }\end{array}\right]=-{\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}\tilde{\mathbf{x}}-\breve{\mathbf{D}}\breve{\mathbf{v}}\text{(Giordano eq 31, sign-corrected)}$$

Working closed-loop equation (the project’s form, with damping-on-error and acceleration feedforward):

$$\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}=-\breve{\mathbf{C}}\breve{\mathbf{v}}-\breve{\mathbf{D}}(\breve{\mathbf{v}}-{\breve{\mathbf{v}}}_d)-{\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}\tilde{\mathbf{x}}-({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c+\breve{\mathbf{M}}{\dot{\breve{\mathbf{v}}}}_d\text{(final.tex eq 34b)}$$

Stability and passivity of this loop:
coordinated_control_lyapunov_stability,
coordinated_control_passivity.

Source Support

• giordano2019coordinated
• giordano2020coordination
• holt1994inertial
• zhang2020adaptive
• dambrosio2024redundant — merged from coordinated_base_manipulator_control.

Related Topics

• circumcentroidal_motion — the $\oplus$ velocity split this control uses.
• generalized_jacobian — the base-reaction Jacobian background.
• task_space_error_dynamics — the error-rate Jacobian ${\mathbf{J}}_{\tilde{x}}$.
• operational_space_formulation — the terrestrial antecedent.

Open Questions

• How does the cascade stability margin degrade as ${\sigma }_G\to {\sigma }_{\mathrm{crit}}$? (Phase D.)

Implementation (sims wiki)

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

• breve_controller — the live circumcentroidal coordinated base+arm control law.