Center Of Mass Jacobian

Definition

The center-of-mass (CoM) Jacobian ${\mathbf{J}}_c$ maps the system’s generalized velocities to the
velocity of the whole-system center of mass. It is the mass-weighted average of the per-link geometric
Jacobians: each link’s Jacobian is scaled by that link’s mass and the sum is divided by the total mass
(virgili-llop2019convex, eq 49). The construction is purely
kinematic/mass-geometric — it depends only on the configuration and the mass distribution, not on whether
the base is actuated — so it is regime-agnostic; what is regime-specific is its use. Here it is used in a
free-flying setting: the base thrusters are the only actuators able to impart linear momentum
(${\mathbf{f}}_0={\mathbf{f}}_c$), and ${\mathbf{J}}_c$ supplies the kineto-static map (Equation 48) that
re-expresses the equations of motion in a non-inertial frame $\mathcal{C}$ attached to the system CoM.

Key Equations

Symbols per notation.md.

Mass-averaged Jacobian (virgili-llop2019convex eq 49). With ${\mathbf{J}}_i$ the geometric Jacobian
and $m_i$ the mass of link $i$ ($i=0$ is the base), $m={\sum }_i{m}_i$ the total mass:

$${\mathbf{J}}_c=\frac{{\sum }_{i=0}^n{\mathbf{J}}_i{m}_i}{{\sum }_{i=0}^n{m}_i},{\dot{\mathbf{r}}}_c={\mathbf{J}}_c\mathbf{u}.$$

The translational, joint-only block of this object is the mass-averaged linear Jacobian
${\bar{\mathbf{J}}}_v\in {\mathbb{R}}^{3\times n}$ of notation.md. (Caveat: the source’s
${\mathbf{J}}_c$ spans the full generalized coordinates including the base — its base-translation
sub-block is the $3\times 3$ identity, eq 50 — whereas ${\bar{\mathbf{J}}}_v$ is the $3\times n$ joint
block. Same averaging, different column span.)

Kineto-static use (eq 48). The CoM force applied at the base is mapped to generalized inertial forces
by the transpose, exploiting kineto-static duality:

$${\mathbf{\tau }}_{\mathcal{I}}=-{\mathbf{J}}_c^{\mathsf{T}}{\mathbf{f}}_c^{\star },{\mathbf{f}}_c^{\star }={\mathbf{f}}_0^{\star }.$$

The joint block of this ${\mathbf{J}}_c$ — the mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v$ — is
also the ingredient that, in Giordano’s separate free-flying formulation, is subtracted to turn the
fixed-base manipulator Jacobian into the circumcentroidal Jacobian
${\mathbf{J}}_{{\nu }_e}^{\oplus }={\mathbf{J}}_{{\nu }_e}-[{\mathbf{R}}_{eb};0]{\bar{\mathbf{J}}}_v$
(see circumcentroidal_motion for that derivation). Note these are
different uses of the same mass-averaging — virgili-llop’s eq 48 is a transpose-mapped generalized
force correction, whereas Giordano’s $\oplus$ split is a velocity-Jacobian correction; neither
source equates them, only the shared ${\bar{\mathbf{J}}}_v$ links the two.

Source Support

• virgili-llop2019convex — defines ${\mathbf{J}}_c$ as the
  mass-weighted average of link Jacobians (eq 49) and uses it via kineto-static duality (eq 48) to write
  the equations of motion in a CoM-attached frame for the internal-reconfiguration sub-maneuver.

Related Topics

• generalized_jacobian — the free-floating GJM ${\mathbf{J}}_g$ is a
  distinct object (folds momentum conservation; do not conflate). It is linked here only because the
  joint block of this page’s ${\mathbf{J}}_c$ — the mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v$ —
  is the term that corrects the fixed-base Jacobian into the free-flying circumcentroidal Jacobian
  ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, the GJM’s free-flying analogue.
• center_of_mass_regulation — regulating ${\mathbf{r}}_c$ relies on this
  Jacobian to relate CoM motion to base/arm motion.
• coordinated_control — coordinated base+arm control uses the CoM split that
  ${\mathbf{J}}_c$ underlies to decouple whole-body translation from internal reconfiguration.
• circumcentroidal_motion — the $\oplus$ split referencing EE motion to the
  system CoM $\mathcal{C}$; ${\bar{\mathbf{J}}}_v$ is the term that removes base translation.
• dynamic_coupling — arm motion shifts the CoM and reacts on the base; ${\mathbf{J}}_c$
  quantifies how joint motion moves ${\mathbf{r}}_c$.

Open Questions

• The source’s ${\mathbf{J}}_c$ (full generalized-coordinate span, with a $3\times 3$ identity base block)
  and notation.md’s ${\bar{\mathbf{J}}}_v$ (joint-only $3\times n$) are the same averaging restricted
  differently — is the wiki’s canonical CoM Jacobian the joint block, or should the base column be retained
  for the free-flying actuated base? (Confirm with the user.)
• virgili-llop2019convex develops the CoM frame for an internal-reconfiguration sub-maneuver that borrows
  free-floating structure; does the mass-averaged Jacobian carry over unchanged when the fully-actuated base
  is driven simultaneously rather than in a two-step decoupling?