Internal Velocity

Definition

The internal velocity ${\mathbf{\nu }}_x^{int}\in {\mathbb{R}}^6$ of a frame $\mathcal{X}$ is the part of its
twist ${\mathbf{\nu }}_x$ that remains when the system’s kinetic momentum is zero — i.e. the motion produced
by the internal actuators (arm joints, and here reaction wheels) rather than by the conserved momentum
(giordano2020coordination, §3.1). It splits any frame’s velocity
into an external part (driven by the system momentum $\mathbf{h}$) and an internal part: when
$\mathbf{h}=0$, the two coincide. Although the concept originates in the free-floating
literature (Giordano’s ref. [24], joint-only), Giordano restates it “for general frames and for the case
in which reaction wheels are present” (§3.1) — i.e. for a system whose base is fully actuated by thrusters
that exert force and torque (§2; the free-flying regime studied here, though Giordano does not use
that label), generalizing the joint-only formulation to include wheel rates.
Its purpose is to inertially decouple the momentum dynamics from the end-effector and base-attitude
dynamics, which the raw velocities ${\mathbf{\nu }}_e,{\mathbf{\omega }}_b$ do not.

Key Equations

Symbols per notation.md.

Defining property — internal velocity equals the full velocity exactly when momentum vanishes (Eq. 6;
$\mathbf{h}$ is the stacked kinetic momentum, source symbol — registry currently lists $\mathbf{p},\mathbf{L}$ separately):

$\mathbf{h}=0⟺{\mathbf{\nu }}_x={\mathbf{\nu }}_x^{int}.$

External/internal split of the end-effector twist, and the internal velocity itself (Eqs. 8b, 8a):

\qquad \boldsymbol v_e^{int}=-\boldsymbol A_{eb}\boldsymbol M_b^{-1}\boldsymbol M_{bw}\dot{\boldsymbol q}_w+\boldsymbol J_{em}^{*}\dot{\boldsymbol q}_m,$$ where $\boldsymbol J_{em}^{*}=\boldsymbol J_{em}-\boldsymbol A_{eb}\boldsymbol M_b^{-1}\boldsymbol M_{bm}$ is the **generalized Jacobian** of the end-effector (source symbol; cf. $\boldsymbol J_g$ in [notation.md](../notation.md)). The wheel term shows that under reaction wheels the internal velocity depends on wheel rates $\dot{\boldsymbol q}_w$, not arm rates alone. Stacked compactly, $\boldsymbol v_e^{int}=\boldsymbol T_e\dot{\boldsymbol q}$, $\boldsymbol\omega_b^{int}=\boldsymbol T_b\dot{\boldsymbol q}$ (Eq. 11); the transform $\boldsymbol\Gamma$ that maps $\dot{\boldsymbol q}$ to $[\boldsymbol h;\boldsymbol v_e^{int};\boldsymbol\omega_b^{int}]$ is singular exactly when $\boldsymbol T=[\boldsymbol T_e;\boldsymbol T_b]$ loses rank (Eq. 12). > **Notation flag.** This page keeps the source's $\boldsymbol h$ (combined momentum), $\boldsymbol J_{em}^{*}$, > and $\boldsymbol M_{bw},\boldsymbol M_{bm}$ (base–wheel / base–arm coupling inertias). The registry's > $\boldsymbol\nu_e^{\oplus}$ (circumcentroidal EE velocity) is a *related but distinct* construct from a > different coordinate split — do not conflate the two; see [circumcentroidal_motion](circumcentroidal_motion.md). > **The $\boldsymbol\Gamma$ here is *not* the registry's $\boldsymbol\Gamma$.** In [notation.md](../notation.md) > (and in [circumcentroidal_motion](circumcentroidal_motion.md)) $\boldsymbol\Gamma$ is the Giordano-2019 > *circumcentroidal* transform mapping $\dot{\boldsymbol q}\!\to\![\boldsymbol v_c;\boldsymbol\omega_b;\boldsymbol\nu_e^\oplus]$, > singular iff $\boldsymbol J_{\nu_e}^\oplus$ loses rank. Eq. 12 here is Giordano-2020's distinct > *internal-velocity* transform $\dot{\boldsymbol q}\!\to\![\boldsymbol h;\boldsymbol v_e^{int};\boldsymbol\omega_b^{int}]$, > singular iff $\boldsymbol T=[\boldsymbol T_e;\boldsymbol T_b]$ loses rank. Same source letter, two different maps. ## Source Support - [giordano2020coordination](../sources/giordano2020coordination.md) — primary; §3.1 defines internal velocity for general frames in the actuated (thruster + reaction-wheel + arm) regime, derives the end-effector and base-attitude internal velocities (Eqs. 6–11), and uses them to inertially decouple momentum from the internal dynamics (Eq. 14). Notes the original joint-only formulation came from its ref. [24]. ## Related Topics - [dynamic_coupling](dynamic_coupling.md) — internal velocity is precisely the bookkeeping that exposes base–arm (and here base–wheel) coupling; the wheel-rate dependence in Eq. 8a *is* an indirect coupling. - [coordinated_control](coordinated_control.md) — internal velocities are the state on which the coordinated thruster/wheel/arm controller is built; they let attitude and EE inputs avoid the thrusters. - [coordinated_base_manipulator_control](coordinated_base_manipulator_control.md) — the broader control architecture that this decomposition feeds; cascaded external (momentum/CoM) then internal loops. - [ffsm_dynamics](ffsm_dynamics.md) — the external/internal split is a reparametrization of the full $(6+n)$-DOF coupled FFSM dynamics. - [null_space_control](null_space_control.md) — §3.3 projects the base internal angular velocity into the null space of $\boldsymbol T_e$ to fully decouple the internal dynamics; relates to redundant self-motion. Also relevant on first mention: [generalized_jacobian](generalized_jacobian.md) ($\boldsymbol J_{em}^{*}$), [momentum_conservation](momentum_conservation.md) (the $\boldsymbol h=\boldsymbol 0$ premise), [circumcentroidal_motion](circumcentroidal_motion.md) (the distinct $\oplus$ split used elsewhere in this wiki). ## Open Questions - Eqs. 12–14 assume a **nonredundant** arm and wheels (Assumptions 2–3, $n_m=6,\,n_w=3$); for our redundant arm, how does the internal-velocity split extend, and which null-space resolution applies? (Source defers redundancy to future work.) - The internal velocity here folds in **reaction-wheel** rates; does the same external/internal decoupling hold cleanly when the base is actuated by **continuous thrusters** for attitude (not just CoM), or is the triangular allocation structure specific to the thruster-CoM-only assignment? - This split decouples momentum from internal dynamics but Eq. 14 is still *fully coupled* internally; how much does the further null-space decoupling (§3.3) cost in singularity margin of $\boldsymbol T_e$?