Generalized Jacobian

Definition

A generalized Jacobian maps the manipulator’s joint rates $\dot{\mathbf{q}}$ to a task-space
end-effector velocity in a way that accounts for base motion. Two distinct objects share this name and
must not be conflated. (1) The generalized Jacobian matrix (GJM) ${\mathbf{J}}_g$ of
Umetani & Yoshida (1989) is a free-floating construct: it
folds the conservation of system linear/angular momentum (the uncontrolled base reacts to arm motion)
directly into the kinematics, so that ${\mathbf{\nu }}_e={\mathbf{J}}_g\dot{\mathbf{q}}$ holds for
the momentum-constrained base. (2) Our free-flying system has a fully-actuated 6-DOF base, so
momentum is not conserved and the GJM does not apply; the analogous object is the circumcentroidal
Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (Giordano 2019), which
removes only the base translation and describes EE motion about the system centre of mass $\mathcal{C}$.
The two coincide only in form, never in regime: ${\mathbf{J}}_g$ encodes a dynamic (momentum) constraint;
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is a purely kinematic re-parametrisation of a fully-actuated system.

Inconsistency

“Generalized Jacobian” is overloaded across the corpus. The free-floating GJM ${\mathbf{J}}_g$
(Umetani 1989) folds momentum conservation and is singular at dynamic singularities that depend on
inertial parameters. The free-flying circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (Giordano
2019) carries no momentum constraint. This page treats them as separate objects; equations are tagged by
regime.

Key Equations

All symbols are rendered from notation.md.

Body-$j$ and end-effector velocities (free-flying, fully-actuated base). With the base twist
$[{\mathbf{v}}_b;{\mathbf{\omega }}_b]$ an independent input (for $j=0$:
${\mathbf{A}}_{jb}=\mathbf{E}$, ${\mathbf{J}}_{{\nu }_j}=0$):

$${\mathbf{\nu }}_j={\mathbf{A}}_{jb}\left[\begin{array}{c}{\mathbf{v}}_b\\ {\mathbf{\omega }}_b\end{array}\right]+{\mathbf{J}}_{{\nu }_j}\dot{\mathbf{q}},{\mathbf{\nu }}_e={\mathbf{A}}_{eb}\left[\begin{array}{c}{\mathbf{v}}_b\\ {\mathbf{\omega }}_b\end{array}\right]+{\mathbf{J}}_{{\nu }_e}\dot{\mathbf{q}}.$$

(Giordano eqs 2, 3 / current_sota eqs 1.1, 1.2)

Here ${\mathbf{J}}_{{\nu }_j}=[{\mathbf{J}}_{v_j};{\mathbf{J}}_{{\omega }_j}]\in {\mathbb{R}}^{6\times n}$ is the
body-$j$ Jacobian and ${\mathbf{J}}_{{\nu }_e}\in {\mathbb{R}}^{6\times n}$ is the fixed-base manipulator
Jacobian (alias ${\mathbf{J}}_m$). The base contribution rides through the adjoint twist transform

$${\mathbf{A}}_{xy}=\left[\begin{array}{cc}{\mathbf{R}}_{xy}& [{\mathbf{p}}_{xy}{]}^{\wedge }{\mathbf{R}}_{xy}\\ 0& {\mathbf{R}}_{xy}\end{array}\right]\in {\mathbb{R}}^{6\times 6}.$$

(Giordano eq 1 / current_sota eq 1.3)

Circumcentroidal Jacobian (free-flying analogue of the GJM). Eliminating the base translation by
referencing EE motion to the system CoM $\mathcal{C}$ gives the
circumcentroidal Jacobian

$${\mathbf{J}}_{{\nu }_e}^{\oplus }={\mathbf{J}}_{{\nu }_e}-\left[\begin{array}{c}{\mathbf{R}}_{eb}\\ 0\end{array}\right]{\bar{\mathbf{J}}}_v,$$

(Giordano eq 14 / current_sota eq 2.2)

where ${\bar{\mathbf{J}}}_v\in {\mathbb{R}}^{3\times n}$ is the
mass-averaged linear Jacobian. Only the translational block of
${\mathbf{J}}_{{\nu }_e}$ is corrected (via $[{\mathbf{R}}_{eb};0]$): unlike the GJM, the base
rotation remains an independent actuated DOF, carried by ${\mathbf{G}}_{{\omega }_b}$ in
${\mathbf{\nu }}_e^{\oplus }={\mathbf{G}}_{{\omega }_b}{\mathbf{\omega }}_b+{\mathbf{J}}_{{\nu }_e}^{\oplus }\dot{\mathbf{q}}$.
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is the lower-right block of the coordinate transform $\mathbf{\Gamma }$, and
the two go singular together — see Γ closed-form inverse and
circumcentroidal decoupling.

Free-floating GJM (contrast). When the base is uncontrolled, conservation of system momentum
$[\mathbf{p};\mathbf{L}]=\text{const}$ removes the base twist as an independent input, collapsing
eqs (1.1–1.2) into a single momentum-constrained map ${\mathbf{\nu }}_e={\mathbf{J}}_g\dot{\mathbf{q}}$
with ${\mathbf{J}}_g$ the GJM (Umetani 1989). ${\mathbf{J}}_g$ is
not instantiated in our free-flying formulation; the symbol ${\mathbf{J}}_g$ is reserved for it in
notation.md precisely to keep it distinct from ${\mathbf{J}}_{{\nu }_e}^{\oplus }$.

Free-flying vs free-floating

Umetani 1989 and the GJM assume a free-floating base (momentum conserved, base reacts passively).
Giordano 2019 and ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ assume a free-flying base (fully actuated, momentum
not conserved). Our system is free-flying.

Source Support

• alali2024reinforcement
• bruschi2025singularity
• calzolari2020singularity
• dai2022review
• das2025understanding
• ellery2004engineering
• giordano2018workspace
• giordano2019coordinated
• giordano2020coordination
• holt1994inertial
• jin2017reaction
• khatib1987unified
• misra2017optimal
• nanos2011use
• nanos2015avoiding
• papadopoulos1993dynamic
• rybus2017control
• sentis2005control
• seweryn2008optimization
• sze2024obstacle
• umetani1989resolved
• virgili-llop2016spacecraft
• wilde2018equations

Related Topics

• circumcentroidal_motion — the $\oplus$ split that defines ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, the free-flying analogue of the GJM.
• free_floating_dynamics — the regime in which the GJM ${\mathbf{J}}_g$ is the correct map (momentum-constrained base).
• momentum_conservation — the conserved quantity that the GJM folds into the kinematics and that our free-flying base breaks.
• center_of_mass_jacobian — the mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v$ that corrects ${\mathbf{J}}_{{\nu }_e}$ into ${\mathbf{J}}_{{\nu }_e}^{\oplus }$.
• dynamic_singularity — singularities of ${\mathbf{J}}_g$ are inertial-parameter-dependent; ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ carries no such momentum constraint.
• resolved_motion_rate_control — the resolved-rate scheme Umetani 1989 built on the GJM.

Open Questions

• Does any source give a clean limit map showing the GJM ${\mathbf{J}}_g$ degenerating to ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (or vice versa) as base actuation is added/removed, or are they only related by analogy?
• Confirm with the user that ${\mathbf{J}}_{{\nu }_e}$ in eq (1.2) is the fixed-base manipulator Jacobian (notation.md alias ${\mathbf{J}}_m$) and not a base-corrected object, to avoid downstream confusion with ${\mathbf{J}}_{{\nu }_e}^{\oplus }$.

Implementation (sims wiki)

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

• robot — assembles the Jacobian and the conditioning ${\sigma }_G$ that gets measured and logged.