Center Of Mass Regulation

Definition

Center-of-mass (CoM) regulation drives the whole-system centroid $\mathcal{C}$ to a desired
inertial position ${\mathbf{p}}_{c_d}$ (and holds it there) by commanding a net centroidal force
${\mathbf{f}}_c$ on the system. Because the maximum reachable workspace of a space manipulator is
centered on the system CoM, fixing $\mathcal{C}$ in inertial space pins the workspace over a
stationary target so the end-effector can keep operating without a separate repositioning maneuver
(giordano2018workspace). In Giordano’s triangular (cascade)
formulation the centroidal equation decouples from the angular-momentum and end-effector equations,
so ${\mathbf{f}}_c$ regulates $\mathcal{C}$ independently of the EE controller.

Regime note. The source assumes a fully-actuated base (${\mathbf{w}}_b\in {\mathbb{R}}^6$) but
deliberately uses it in a near-free-floating fashion: base actuators stay off during contact-free
motion (zero fuel) and fire only to absorb contact-induced drift and restore $\mathcal{C}$. For our
free-flying system the base is actuated throughout, so CoM regulation is a continuously-active
coordinated-control task (one branch of coordinated_control), not a
fuel-saving contact-recovery measure.

Key Equations

Symbols per notation.md.

The centroidal velocity is the linear momentum scaled by total mass, and after the
energy-preserving coordinate transform the CoM channel is an inertially-decoupled double integrator
(Giordano Eq. 9, 13a):

${\mathbf{v}}_c=\frac{1}{m}\mathbf{p},m{\dot{\mathbf{v}}}_c={\mathbf{f}}_c.$

The regulation law is a PD (stiffness/damping) controller on the centroidal error
${\tilde{\mathbf{x}}}_c={\mathbf{p}}_c-{\mathbf{p}}_{c_d}$ (Giordano Eq. 19; the source
writes this error ${\tilde{\mathbf{p}}}_c$):

\qquad \boldsymbol{K}_c,\boldsymbol{D}_c \succ 0 .$$ Through the dual of the coordinate transform $\boldsymbol{\Gamma}$ (Giordano Eq. 11), this centroidal force is realized **only** by the base force $\boldsymbol{f}_b$ (plus a base-torque term correcting the $\mathcal{B}$-to-$\mathcal{C}$ lever arm): the base actuators are devoted to the outer CoM/attitude loop and never counterbalance the internal end-effector wrench — the source's fuel-saving "triangular" actuation split. ## Source Support - [giordano2018workspace](../sources/giordano2018workspace.md) — primary. Derives the triangular CoM / angular-momentum / EE dynamics, the PD CoM regulation law (Eq. 19), and the triangular actuation map; validated on the DLR OOS-Sim (7-DOF arm on a 6-DOF base). Shows that without CoM control, repeated contacts drift the CoM, stretch the arm, and collapse manipulability into a practical singularity around $t\approx140$ s; with CoM regulation the workspace and manipulability are restored after every contact. ## Related Topics - [center_of_mass_jacobian](center_of_mass_jacobian.md) — the mass-averaged map ($\bar{\boldsymbol J}_v$ / centroidal Jacobian) relating joint and base rates to $\boldsymbol{v}_c$; it is the kinematic object that CoM regulation acts through. - [coordinated_control](coordinated_control.md) — CoM regulation is the centroidal/outer branch of the base–arm coordinated controller; the cascade lets it run independently of the EE task. - [circumcentroidal_motion](circumcentroidal_motion.md) — the complementary EE-about-$\mathcal{C}$ channel. In giordano2018workspace the $\boldsymbol{\Gamma}$ transform (Eq. 10) decouples the CoM from the angular-momentum and *internal*-EE ($\boldsymbol{\nu}_{e,int}=\boldsymbol{J}_m^*\dot{\boldsymbol q}$) dynamics; the circumcentroidal $\oplus$ reformulation (Giordano 2019, not cited here) is the closely related change of velocity coordinates. - [dynamic_coupling](dynamic_coupling.md) — arm motion shifts the CoM and disturbs the base; regulating $\mathcal{C}$ explicitly manages that coupling instead of leaving it to react freely. - [base_disturbance_minimization](base_disturbance_minimization.md) — the free-floating counterpart: there the goal is *zero reaction* on the base (reaction null-space); here the actuated base is used to *hold the CoM*, a different objective for a different regime. Also relevant: [momentum_dumping](momentum_dumping.md) (the angular-momentum branch of the same cascade) and [manipulability_measure](manipulability_measure.md) (the workspace-quality scalar that CoM regulation protects). ## Open Questions - The source keeps the base near-floating (actuators off except at contact) for fuel economy. For our free-flying system the base is continuously actuated — does the triangular decoupling and its stability argument still hold when $\boldsymbol{f}_c$ is active at all times, or does the always-on base force interact with the EE loop in ways the contact-only analysis does not cover? - The controller "shares the singularity properties of the transposed-Jacobian free-floating controller": near a dynamic singularity of $\boldsymbol{J}_m^*$ it loses actuation in the singular direction rather than failing numerically. How does CoM regulation compose with our [singularity_robust_inverse](singularity_robust_inverse.md) / damped-inverse handling of the circumcentroidal Jacobian? - CoM and momentum feedback are reconstructed from the inertia model plus arm kinematics. How sensitive is regulation accuracy to inertial-parameter and CoM-location uncertainty, which the nominal analysis assumes away? </content> </invoke>