Generalized Inertia Matrix

Definition

The generalized inertia matrix (GIM) is the configuration-dependent, symmetric
positive-definite matrix that multiplies the generalized accelerations in the equations of motion of
a space manipulator. Caution — the name denotes two different objects depending on regime. In the
free-FLYING treatment (dambrosio2024redundant), the GIM is the full coupled inertia
$\mathbf{M}(\mathbf{q})\in {\mathbb{R}}^{(6+N)\times (6+N)}$ over the stacked base-and-joint
coordinate $\mathbf{q}=[{\mathbf{q}}_0;{\mathbf{q}}_m]$, with the base fully actuated. In the
free-FLOATING treatment (wilde2018equations), the GIM is instead the reduced
$N\times N$ matrix ${\mathbf{M}}^{\star }$ obtained after eliminating the base velocity under
conserved momentum (a Schur complement of the base-inertia block) — a fundamentally different matrix.
Either way it inherits the role of the fixed-base mass matrix: SPD, hence always invertible, and the
metric for inertia-weighted (dynamically-consistent) operations.

Key Equations

Symbols per notation.md. (Both sources write the inertia as $H$; notation.md mandates
$\mathbf{M}$ — the same object — so the full coupled matrix is rendered $\mathbf{M}$ here.
The reduced free-floating matrix keeps the source’s $\star$ superscript as ${\mathbf{M}}^{\star }$.)

Free-flying GIM = full coupled inertia. Over the generalized coordinate
$\mathbf{q}=[{\mathbf{q}}_0;{\mathbf{q}}_m]$ (base pose stacked on $N$ joint angles), the
coupled equations of motion are

$$\mathbf{M}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\dot{\mathbf{q}}=\mathbf{\tau },\mathbf{M}={\mathbf{M}}^T\succ 0,\mathbf{M}\in {\mathbb{R}}^{(6+N)\times (6+N)}.$$

(dambrosio2024redundant eq 1, where the SPD GIM is $H$ and $\mathbf{C}$ is their “convective inertia matrix”,
the Coriolis/centrifugal block.)

Block structure. Partitioning along $[{}^{\mathcal{J}}{\dot{\mathbf{x}}}_0;\dot{\mathbf{q}}]$
(base twist, joint rates), the GIM carries the base block, the arm block, and the base-arm coupling
block on its off-diagonal:

$$\mathbf{M}=\left[\begin{array}{cc}{\mathbf{H}}_0& {\mathbf{H}}_{0m}\\ {\mathbf{H}}_{0m}^T& {\mathbf{H}}_m\end{array}\right],{\mathbf{H}}_0\in {\mathbb{R}}^{6\times 6},{\mathbf{H}}_m\in {\mathbb{R}}^{N\times N},{\mathbf{H}}_{0m}\in {\mathbb{R}}^{6\times N}.$$

(wilde2018equations eqs 27, 28, 31, 36 — ${\mathbf{H}}_{0m}$ is the
coupling_inertia_matrix.)

Free-floating reduced GIM = Schur complement. When the base is uncontrolled and momentum is
conserved, the base acceleration is eliminated, condensing the dynamics to $N$ equations whose inertia
is the Schur complement of ${\mathbf{H}}_0$ in $\mathbf{M}$:

$${\mathbf{M}}^{\star }={\mathbf{H}}_m-{\mathbf{H}}_{0m}^T{\mathbf{H}}_0^{-1}{\mathbf{H}}_{0m}\in {\mathbb{R}}^{N\times N},{\mathbf{M}}^{\star }\ddot{\mathbf{q}}+{\mathbf{C}}^{\star }(\mathbf{q},\dot{\mathbf{q}})=\mathbf{\tau }.$$

(wilde2018equations eqs 45, 50 — derived for the floating mode with zero initial momentum
${\mathbf{M}}_0=0$; this ${\mathbf{M}}^{\star }$ is what survives eliminating the
coupling block, not the full $\mathbf{M}$.)

Notation map (source ↔ canonical)

Both sources write the inertia as $H$; this page and notation.md render it canonically
as $\mathbf{M}$ (M not H). The block symbols are registered centrally in notation.md.

Canonical (here)	wilde2018equations	dambrosio2024redundant	meaning
$\mathbf{M}$	(built from blocks)	$H$, full $(6+N)\times (6+N)$	the generalized inertia matrix
${\mathbf{H}}_0$	$H_0$	—	base inertia block ($6\times 6$)
${\mathbf{H}}_m$	$H_m$	—	arm inertia block ($N\times N$)
${\mathbf{H}}_{0m}$	$H_{0m}$	—	base–arm coupling block ($6\times N$)
${\mathbf{M}}^{\star }$	$H^{\star }$ (their “GIM”)	—	reduced free-floating GIM (Schur complement)
${\mathbf{C}}^{\star }$	$C^{\star }$	$C$ (“convective inertia”)	Coriolis/centrifugal block

Source Support

• dambrosio2024redundant — explicitly free-FLYING (base
  actively controlled in translation and rotation); defines the GIM as the full SPD
  $(6+N)\times (6+N)$ matrix $H$ in $H(\mathbf{q})\ddot{\mathbf{q}}+C\dot{\mathbf{q}}=\tau$
  (eq 1), built via the direct-path/SPART formulation. The primary source for our regime.
• wilde2018equations — free-FLOATING (floating mode,
  ${\mathbf{M}}_0=0$); full symbolic Lagrangian derivation of every inertia block
  (${\mathbf{H}}_0,{\mathbf{H}}_m,{\mathbf{H}}_{0m}$, eqs 28–36) and the reduced GIM
  ${\mathbf{M}}^{\star }={\mathbf{H}}_m-{\mathbf{H}}_{0m}^T{\mathbf{H}}_0^{-1}{\mathbf{H}}_{0m}$
  (eq 45) plus its time/joint-angle derivatives. The bookkeeping reference for the block structure.

Related Topics

• ffsm_dynamics — the coupled equations of motion in which the GIM is the inertia
  term; the canonical free-flying form uses the full $\mathbf{M}(\mathbf{q})$.
• coupling_inertia_matrix — the off-diagonal block ${\mathbf{H}}_{0m}$
  of the GIM; the reduced free-floating GIM is exactly what remains after Schur-eliminating it.
• operational_space_formulation — uses the GIM to build the task
  (operational-space) inertia $\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1}$
  and the dynamically-consistent inverse; relies on $\mathbf{M}\succ 0$.
• generalized_jacobian — the free-floating velocity map (Umetani–Yoshida)
  paired with ${\mathbf{M}}^{\star }$; both arise from the same momentum-elimination step, so the
  generalized Jacobian and the reduced GIM are companion constructs of the floating regime.
• free_floating_dynamics — the regime in which the reduced $N\times N$
  GIM is the valid form; momentum conservation is what makes the Schur reduction legitimate.

Open Questions

• Naming hazard for the examiner: dambrosio2024redundant (free-flying) and wilde2018equations
  (free-floating) both say “generalized inertia matrix” but mean different matrices (full $(6+N)$ vs
  reduced $N\times N$). Which definition should this thesis adopt as canonical, and how is the
  cross-walk recorded so one symbol authority governs?
• wilde2018equations forms ${\mathbf{M}}^{\star }$ only under conserved momentum
  (${\mathbf{M}}_0=0$). For our actuated-base (free-flying) system the base wrench is
  retained on the RHS and momentum is not conserved — does a reduced GIM exist at all, or is the
  full $\mathbf{M}(\mathbf{q})$ the only well-posed inertia (cf. Giordano’s circumcentroidal
  $\hat{\mathbf{M}}$ split in ffsm_dynamics)?
• Both sources assert SPD-ness (hence invertibility) of the GIM; over what configuration set does
  positive-definiteness actually hold for a redundant ($N>6$) free-flying arm, and how does that set
  relate to the dynamic-singularity locus of the coupling block?