free_flying_vs_free_floating

Free-Flying vs Free-Floating Space Manipulators

Canonical taxonomy page

The single authority on the free-flying / free-floating distinction (the load-bearing
distinction for this thesis). Formerly a synthesis; collapsed here when the synthesis
type was retired.

Overview

A space manipulator is a robotic arm carried by a spacecraft base. The single design choice
that splits the entire literature — and that is load-bearing for this thesis — is what
actuates the base:

• Free-FLOATING (much of the classical literature). The base carries no actuation
  during arm motion: no thrusters, no attitude control. With no external wrench, the system’s
  linear and angular momentum are conserved. The base translates and rotates passively in
  reaction to the arm. Because momentum is conserved, the base rates are not independent: they
  are a dependent function of the joint rates, and they can be folded out of the kinematics.
  This is exactly what the generalized Jacobian matrix ${\mathbf{J}}_g$ does
  (generalized_jacobian, Umetani 1989).

• Free-FLYING (OUR system). The base is a fully-actuated 6-DOF spacecraft — it can
  command base force ${\mathbf{f}}_b$ and base torque ${\mathbf{\tau }}_b$ independently of the
  arm. Momentum is therefore not conserved (actuators inject external wrench), so there is no
  momentum constraint to fold out: the six base DOF are independent controls. The thesis uses
  the Giordano 2019 circumcentroidal coordinated formulation, whose central object is the
  circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$
  (circumcentroidal_motion).

The two are not notational variants of one system — they are different dynamics, with
different Jacobians, different singularity structures, and different control problems. The
distinction is so easy to blur that even foundational papers misuse the words: Umetani 1989’s
abstract calls its free-floating system “free-flying” — pre-standardization usage
(umetani1989resolved, abstract note). This page fixes the
vocabulary.

Intermediate regimes exist

The clean two-way split is a pedagogical anchor; real designs interpolate. Giordano 2019
contrasts three regimes explicitly: full base control (free-flying, base position and
attitude rigidly controlled), the proposed partial base control (attitude + CoM controlled,
base translation left free), and floating-base control (free-floating)
(giordano2019coordinated, §IV). An attitude-only
controlled base (reaction wheels, no thrusters) is a further intermediate that the
free-floating momentum formalism subsumes as a special case (Umetani §4.3; Nenchev’s broadened
${\mathcal{F}}_b$). Read “free-flying / free-floating” as the endpoints of a spectrum keyed on
how much base actuation authority is asserted.

The dynamics, side by side

The full coupled equations of motion are the same for both regimes — the difference is which
inputs are available and whether momentum is conserved. Starting from the full coupled dynamics in
base + joint coordinates (notation.md):

$$\mathbf{M}(\mathbf{q})\left[\begin{array}{c}{\dot{\mathbf{v}}}_b\\ {\dot{\mathbf{\omega }}}_b\\ \ddot{\mathbf{q}}\end{array}\right]+\mathbf{C}\left[\begin{array}{c}{\mathbf{v}}_b\\ {\mathbf{\omega }}_b\\ \dot{\mathbf{q}}\end{array}\right]=\left[\begin{array}{c}{\mathbf{f}}_b\\ {\mathbf{\tau }}_b\\ \mathbf{\tau }\end{array}\right]\text{\hspace{1em}(Giordano eq 4)}$$

Free-floating sets the base inputs to zero, ${\mathbf{f}}_b=0$,
${\mathbf{\tau }}_b=0$ (with zero initial momentum). The top six rows then become a
conservation constraint that ties the base rates to the joint rates. Eliminating
$[{\mathbf{v}}_b;{\mathbf{\omega }}_b]$ gives the end-effector velocity as a function of joint
rates alone:

$${\mathbf{\nu }}_e={\mathbf{J}}_g\dot{\mathbf{q}}+{\mathbf{\nu }}_{e,0},{\mathbf{J}}_g={\mathbf{J}}_{{\nu }_e}-{\hat{\mathbf{J}}}_S{\hat{\mathbf{I}}}_S^{-1}{\hat{\mathbf{I}}}_M\text{\hspace{1em}(Umetani eqs 18–19)}$$

where the second term is precisely the momentum-conservation fold (${\hat{\mathbf{I}}}_S,{\hat{\mathbf{I}}}_M$ are the base/manipulator inertia couplings). ${\mathbf{J}}_g$ is a
dynamic object: it depends on the system mass distribution, not kinematics alone.

Free-flying keeps ${\mathbf{f}}_b,{\mathbf{\tau }}_b$ as live inputs. There is no
conservation constraint to fold; the base is commanded. The thesis instead applies the
coordinate transform $\mathbf{\Gamma }$ to rewrite the EE twist about the system CoM
$\mathcal{C}$:

$${\mathbf{\nu }}_e={\mathbf{G}}_{v_c}{\mathbf{v}}_c+{\mathbf{\nu }}_e^{\oplus },{\mathbf{\nu }}_e^{\oplus }={\mathbf{G}}_{{\omega }_b}{\mathbf{\omega }}_b+{\mathbf{J}}_{{\nu }_e}^{\oplus }\dot{\mathbf{q}},{\mathbf{J}}_{{\nu }_e}^{\oplus }={\mathbf{J}}_{{\nu }_e}-\left[\begin{array}{c}{\mathbf{R}}_{eb}\\ 0\end{array}\right]{\bar{\mathbf{J}}}_v\text{\hspace{1em}(Giordano eqs 14–16)}$$

${\mathbf{J}}_g$ and ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ are DIFFERENT objects.

Both subtract a base-coupling term from the fixed-base Jacobian ${\mathbf{J}}_{{\nu }_e}$, and
the two are easy to conflate. They are not the same:

• ${\mathbf{J}}_g$ removes both base translation and rotation, because momentum
  conservation determines both passively. It only exists when the base is uncontrolled.
• ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ removes only the base translation (subtracts the
  mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v$); the base rotation
  ${\mathbf{\omega }}_b$ stays as an independent commanded coordinate, carried separately by
  ${\mathbf{G}}_{{\omega }_b}$ (Giordano §II). This is circumcentroidal, not generalized.

See notation.md (the ${\mathbf{J}}_g$ row explicitly flags the distinction).

Key Claims

Claim 1 — Free-floating folds momentum conservation into ${\mathbf{J}}_g$; free-flying does not

In the free-floating regime the base is uncontrolled and momentum is conserved, which supplies the
extra equations needed to eliminate the base DOF and collapse the kinematics to a history-aware but
velocity-linear map ${\mathbf{J}}_g$ (the generalized Jacobian). As base inertia
$\to \infty$, ${\mathbf{J}}_g\to {\mathbf{J}}_{{\nu }_e}$ (the Earth-fixed Jacobian) — “a
ground-fixed manipulator is a link system attached to the very large inertia of the Earth.”
Source support: umetani1989resolved (eqs 18–20, free-floating);
papadopoulos1993dynamic (eq 35, ${\mathbf{J}}^{\ast }$ via the
angular-momentum constraint eq 33c). In free-flying the base wrench is an independent input, the
conservation constraint disappears, and the thesis uses $\mathbf{\Gamma }$ /
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ instead. Source support:
giordano2019coordinated (§II, “spacecraft fully actuated”;
eqs 14, 19) and the project master (circumcentroidal_motion,
generalized_jacobian).

Claim 2 — Singularity structure differs: dynamic (path-dependent) vs algorithmic/coordinate

Free-floating suffers dynamic singularities: configurations where ${\mathbf{J}}_g$ (i.e.
${\mathbf{J}}^{\ast }$) loses rank because of the mass distribution, not the kinematic chain. Worse,
because spacecraft attitude is a non-integrable function of joint history, the inertial workspace
splits into a Path-Independent Workspace and a Path-Dependent Workspace; the same joint
configuration maps to different inertial points depending on the path taken, and all resolved-rate
schemes fail where $\det {\mathbf{J}}^{\ast }=0$. Source support:
papadopoulos1993dynamic (eqs 37–38; PIW/PDW partition);
dynamic_singularity. Free-flying has no dynamic
singularity in this sense — with the base actively controlled, ${\mathbf{\omega }}_b$ and the CoM
are commanded rather than passively determined, and the pathology that produced path-dependence is
“bought out.” What remains is the conditioning of the coordinate transform
$\mathbf{\Gamma }$, which goes singular exactly where its lower-right block
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ does (Spearman $\ge 0.9969$ between ${\sigma }_{\min }(\mathbf{\Gamma })$
and ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$). Source support:
giordano2019coordinated (Prop. IV.1, region $\Omega$);
singularity_threshold_cascade;
gamma_closed_form_inverse.

Claim 3 — The control problem differs: reaction management vs coordinated actuation

Free-floating control is dominated by managing reactions the arm imposes on the uncontrolled
base — e.g. reactionless motion via the reaction null space (the kernel of the coupling inertia
${\mathbf{M}}_{bm}$), which keeps the base undisturbed without spending base actuation. Source
support: nenchev2013reaction (eq 12,
${\mathbf{M}}_{bm}\dot{\mathbf{\theta }}=0$);
reaction_null_space. Free-flying control instead
coordinates base and arm actuators. Giordano’s triangular actuation map shows the base force
${\mathbf{f}}_b$ is needed only to relocate the system CoM; the EE and attitude tasks never
command ${\mathbf{f}}_b$, so a contact-free maneuver costs zero translational fuel (zero, if base
torque is supplied by momentum-exchange devices). Source support:
giordano2019coordinated (eqs 22, 32; §IV.C);
coordinated_control;
coordinated_control_lyapunov_stability.
Crucially, Nenchev flags the bridge: promote the base-force term ${\mathcal{F}}_b$ from a constraint
to an actuator wrench and the same equations model an attitude-controlled (free-flying) base — so
in the free-flying regime reactionless arm motion becomes a choice (fuel/torque economy) rather
than a necessity.

Claim 4 — Free-flying buys attitude pointing and singularity relief at a fuel/drift cost

The free-flying base is what lets the thesis point a camera (regulate base attitude and EE pose
simultaneously) and removes the path-dependent dynamic singularities — capabilities the
free-floating regime cannot offer. The price is actuator effort (thruster fuel) and, under
partial base control, a bounded base-position drift after each maneuver. Source support:
giordano2019coordinated (fuel ranking: full > partial >
floating; §IV.C, VI.A); papadopoulos1993dynamic
(mitigations — large base inertia, attitude control — map onto these trade-offs).

Comparison table

Aspect	Free-FLOATING (much of the literature)	Free-FLYING (OUR system)
Base actuation	None during arm motion	Fully actuated 6-DOF (${\mathbf{f}}_b,{\mathbf{\tau }}_b$ commanded)
Momentum	Conserved ($\mathbf{p},\mathbf{L}$ const.)	Not conserved (external wrench injected)
Base DOF	Dependent on joint rates (folded out)	Independent controls
Governing Jacobian	Generalized Jacobian ${\mathbf{J}}_g$ — removes base translation and rotation	Circumcentroidal ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ — removes base translation only
EE kinematics	${\mathbf{\nu }}_e={\mathbf{J}}_g\dot{\mathbf{q}}+{\mathbf{\nu }}_{e,0}$ (Umetani 18)	${\mathbf{\nu }}_e={\mathbf{G}}_{v_c}{\mathbf{v}}_c+{\mathbf{\nu }}_e^{\oplus }$ (Giordano 15)
Singularity type	Dynamic (mass-distribution-dependent), path-dependent (PIW/PDW)	$\mathbf{\Gamma }$ / ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ conditioning; no path-dependence
Singularity locus	$\det {\mathbf{J}}^{\ast }(\mathbf{q})=0$ (depends on $\mathbf{q}$ only)	${\sigma }_{\min }(\mathbf{\Gamma })\to 0\iff {\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })\to 0$
Central control problem	Manage base reactions (RNS, reactionless motion)	Coordinate base+arm actuators (triangular map)
Base attitude pointing	Limited (attitude is reaction-driven)	Native (regulated as a task)
Cost / penalty	Path-dependent workspace, no pointing	Fuel / actuator effort; base drift (partial control)
Canonical sources	Umetani 1989, Papadopoulos 1993, Nenchev (RNS)	Giordano 2019 (+ project master)

Source classification by regime

Free-FLOATING (uncontrolled / momentum-conserving base):

• umetani1989resolved — origin of the generalized Jacobian; folds momentum conservation (note: abstract mislabels it “free-flying”).
• papadopoulos1993dynamic — dynamic singularities, ${\mathbf{J}}^{\ast }$, PIW/PDW; explicitly free-floating.
• nenchev2013reaction — RNS / reactionless motion; free-floating base is the limiting case (but broadens ${\mathcal{F}}_b$ to cover an actuated base — see bridge note).
• nanos2011use · das2025understanding · dai2022review — free-floating-centric (verify each individually before citing the regime in the thesis).

Free-FLYING (fully / partially actuated base):

• giordano2019coordinated — the thesis template: fully-actuated base, circumcentroidal coordinated control, three-regime taxonomy.
• giordano2020coordination — companion coordinated-control work (confirm scope on the source page).

Mixed / hybrid / regime to confirm (cited on the duplicate stubs; do not assert a regime
without reading the source page):

• bruschi2025singularity · ellery2004engineering · rutkovskii2023control · seweryn2008optimization · sukhanov2015dynamic · virgili-llop2019convex · wilde2018equations · zhang2020adaptive

Tensions & Open Questions

• Terminology drift in the literature. “Free-flying” is used by several older sources
  (Umetani 1989 abstract) for what is, by the modern definition, free-floating. Any thesis
  citation must check the assumptions (is the base actuated?), not the word the source prints.
• Where does “partial base control” sit? Giordano’s proposed regime (attitude + CoM
  controlled, translation free) is neither textbook free-flying nor free-floating. The thesis must
  state precisely which DOF it actuates to satisfy the examiner.
• Does ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ inherit any dynamic-singularity flavor? It contains
  ${\bar{\mathbf{J}}}_v$ (mass-averaged), so it is not purely kinematic. Open question: how much
  of its singular structure is mass-distribution-driven vs kinematic, and whether actuating the
  base fully removes path-dependence or only suppresses it. (Giordano excludes the singular set
  from $\Omega$ rather than analyzing it.)
• Redundancy. Giordano assumes a nonredundant arm ($n=6$) so ${\mathbf{J}}_{{\nu }_e}^{\oplus }$
  is square. RNS / reactionless motion (free-floating) is defined on redundancy
  ($\mathrm{rank}{\mathbf{P}}_{bm}=n-6$). The free-flying redundant case — exploiting a null
  space while the base is actuated — is the project’s 7-DOF work and is not settled in the cited
  sources.

Connected Pages

Topics: generalized_jacobian ·
circumcentroidal_motion ·
free_floating_dynamics ·
ffsm_dynamics ·
dynamic_singularity ·
momentum_conservation ·
reaction_null_space ·
coordinated_control ·
coupling_inertia_matrix

Results: circumcentroidal_decoupling ·
gamma_closed_form_inverse ·
singularity_threshold_cascade ·
coordinated_control_lyapunov_stability

Sources: giordano2019coordinated ·
umetani1989resolved ·
papadopoulos1993dynamic ·
nenchev2013reaction