Coupling Inertia Matrix

Definition

The coupling inertia matrix is the off-diagonal block of the full coupled inertia
$\mathbf{M}$ (notation.md) that maps joint motion onto base-frame momentum
(and vice versa) — i.e. the inertial cross-coupling between the spacecraft base and the actuated
arm. In the partitioned dynamics it is the block linking the base twist
$[{\mathbf{v}}_b;{\mathbf{\omega }}_b]$ to the joint rates $\dot{\mathbf{q}}$; Giordano
writes its translational and manipulator pieces as ${\mathbf{M}}_{tr}=-m[{\mathbf{p}}_{bc}{]}^{\wedge }$
and ${\mathbf{M}}_{tm}=m{\bar{\mathbf{J}}}_v$ (current_sota eq 1.4 / Giordano eqs 5a–5d),
Nenchev as ${\mathbf{M}}_{bm}\in {\mathbb{R}}^{6\times n}$, and Wilde as $H_{0m}\in {\mathbb{R}}^{6\times N}$.
It is the physical object whose properties drive two phenomena central to this thesis: its kernel
is the reaction null space (joint motions producing zero base reaction), and its rank loss is a
dynamic singularity of the inertially-coupled velocity map.

Distinct from the generalized inertia matrix

This page is the coupling block ${\mathbf{M}}_{tr}/{\mathbf{M}}_{tm}$ (${\mathbf{M}}_{bm}$,
$H_{0m}$) — an off-diagonal cross term, ${\mathbb{R}}^{6\times n}$. The
generalized_inertia_matrix is the reduced $n\times n$
matrix $H^{\star }=H_m-H_{0m}^T{H}_0^{-1}{H}_{0m}$ (Wilde eq 45) obtained by the Schur complement that
folds the coupling block out. They are different objects; do not conflate them.

Inconsistency — three names, no canonical symbol

Giordano’s sub-block names (${\mathbf{M}}_{tr},{\mathbf{M}}_{tm}$), Nenchev’s ${\mathbf{M}}_{bm}$,
and Wilde’s $H_{0m}$ all denote the base-arm coupling block, but notation.md
registers none of them (nor the RNS projector ${\mathbf{P}}_{bm}$). The closest canonical
forms are the current_sota sub-block names ${\mathbf{M}}_{tr},{\mathbf{M}}_{tm}$ used below; a
central symbol row is requested in this page’s returned warnings.

Key Equations

Coupling block inside the full coupled dynamics. Partitioning
$\mathbf{M}\in {\mathbb{R}}^{(6+n)\times (6+n)}$ along $[{\mathbf{v}}_b;{\mathbf{\omega }}_b|\dot{\mathbf{q}}]$,
the off-diagonal sub-blocks are the coupling inertia:

$$\mathbf{M}=\left[\begin{array}{cc}{\mathbf{M}}_b& {\mathbf{M}}_c\\ {\mathbf{M}}_c^T& {\mathbf{M}}_m\end{array}\right],{\mathbf{M}}_c=\left[\begin{array}{c}{\mathbf{M}}_{tm}\\ {\mathbf{M}}_{rm}\end{array}\right],{\mathbf{M}}_{tr}=-m[{\mathbf{p}}_{bc}{]}^{\wedge },{\mathbf{M}}_{tm}=m{\bar{\mathbf{J}}}_v.$$

(current_sota eq 1.4 / Giordano eqs 5a–5d) — ${\mathbf{M}}_b$ is the $6\times 6$ locked (composite-rigid-body)
base inertia, ${\mathbf{M}}_m$ the $n\times n$ arm inertia, and ${\mathbf{M}}_c$ the base-arm coupling block.
The mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v=\frac{1}{m}{\sum }_j{m}^{(j)}{\mathbf{R}}_{jb}^T{\mathbf{J}}_{v_j}$
(current_sota eq 1.6 / Giordano eq 7) is what makes the coupling inertia-distribution-dependent.

Coupling momentum and the reaction null space. With the base referenced to its own frame and the
coupling block written ${\mathbf{M}}_{bm}\in {\mathbb{R}}^{6\times n}$ (Nenchev’s symbol), the
base-momentum balance is

$${\mathbf{M}}_b{\mathbf{\nu }}_b+{\mathbf{M}}_{bm}\dot{\mathbf{q}}={\mathbf{L}}_b,$$

(nenchev2013reaction eq 4) so that the coupling momentum is ${\mathbf{M}}_{bm}\dot{\mathbf{q}}$.
Reactionless motion — and hence complete dynamical decoupling of base from arm — holds iff

$${\mathbf{M}}_{bm}\dot{\mathbf{q}}=0⟺\dot{\mathbf{q}}\in \ker {\mathbf{M}}_{bm}=\text{RNS},$$

(nenchev2013reaction eq 12) the kernel being the reaction null space. For a redundant arm the
general resolution is $\dot{\mathbf{q}}=-{\mathbf{M}}_{bm}^{\#}{\mathbf{M}}_b{\mathbf{\nu }}_b+{\mathbf{P}}_{bm}{\dot{\mathbf{q}}}_a$
with projector ${\mathbf{P}}_{bm}$, $\mathrm{rank}{\mathbf{P}}_{bm}=n-6$
(nenchev2013reaction eq 11).

Rank loss ⇒ dynamic singularity. Where the coupling block becomes rank-deficient the inertial
coupling is “ill-defined”: the Schur reduction below cannot be formed and the inertia-weighted
velocity map loses rank — a dynamic_singularity:

$$\mathrm{rank}{\mathbf{M}}_{bm}<6⟹\text{dynamic singularity (ill-defined inertial coupling).}$$

(nenchev2013reaction, p. 10) — Nenchev flags these but does not characterise them geometrically;
their relation to kinematic singularities of ${\mathbf{J}}_{{\nu }_e}$ is an open question.

Folded out by the Schur complement (the link to the generalized inertia matrix). Eliminating the
base velocity under momentum conservation removes the coupling block via a Schur complement,
producing the reduced generalized inertia $H^{\star }$:

$$H^{\star }=H_m-H_{0m}^T{H}_0^{-1}{H}_{0m},H_{0m}\in {\mathbb{R}}^{6\times N}\text{(the coupling block)}.$$

(wilde2018equations eqs 45, 31) — $H_{0m}$ here is Wilde’s name for the same coupling block; the
generalized inertia matrix is what survives its elimination.

Free-flying vs free-floating (load-bearing)

nenchev2013reaction and wilde2018equations both derive the coupling block in the free-FLOATING
regime (base uncontrolled; momentum conserved, ${\mathbf{L}}_b=0$). Our system is
free-FLYING: the base wrench is commanded, so ${\mathbf{M}}_{bm}\dot{\mathbf{q}}=0$
is a design choice (decouple arm reactions even though thrusters/wheels could absorb them), not a
conservation law. Nenchev explicitly notes the same equation models an attitude-controlled base by
promoting ${\mathbf{F}}_b$ to an actuator wrench — exactly our regime
(free_flying_vs_free_floating). In Giordano’s
circumcentroidal formulation the coupling appears as ${\mathbf{M}}_{tm}=m{\bar{\mathbf{J}}}_v$
inside $\mathbf{\Gamma }$, and the residual conditioning hazard migrates to
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (see gamma_closed_form_inverse).

Source Support

• nenchev2013reaction — defines the coupling inertia ${\mathbf{M}}_{bm}$, its kernel as the RNS (eqs 4, 11, 12), and flags its rank deficiency as ill-defined inertial coupling (FREE-FLOATING; notes the free-flying extension via an actuated base wrench).
• sone2016reactionless — reactionless-motion planning built on the coupling-inertia kernel.
• wilde2018equations — full symbolic coupling block $H_{0m}=[J_{TS};H_{Sq}]$ and its Schur elimination into the generalized inertia $H^{\star }$ (eqs 31, 45; FREE-FLOATING, floating mode, $M_0=0$).
• giordano2019coordinated — the circumcentroidal sub-blocks ${\mathbf{M}}_{tr}=-m[{\mathbf{p}}_{bc}{]}^{\wedge }$, ${\mathbf{M}}_{tm}=m{\bar{\mathbf{J}}}_v$ (eqs 5a–5d) for the actuated-base (FREE-FLYING) system.

Related Topics

• generalized_inertia_matrix — the reduced $n\times n$ $H^{\star }$ obtained by Schur-eliminating this coupling block; the two are complementary, not the same matrix.
• reaction_null_space — defined exactly as the kernel of the coupling inertia matrix, $\ker {\mathbf{M}}_{bm}$.
• dynamic_singularity — occurs where the coupling block (and hence the inertia-weighted Jacobian) loses rank.
• dynamic_coupling — the broad phenomenon of which the coupling inertia matrix is the exact algebraic carrier in the inertia term.
• reactionless_motion — the motion class produced by joint rates lying in $\ker {\mathbf{M}}_{bm}$.

Open Questions

• Geometric characterisation of where ${\mathbf{M}}_{bm}$ becomes rank-deficient, and its relation to the kinematic singularities of ${\mathbf{J}}_{{\nu }_e}$ and the circumcentroidal ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (nenchev2013reaction leaves this open).
• For a fully-actuated base, what is the optimal split between reactionless arm planning ($\dot{\mathbf{q}}\in \ker {\mathbf{M}}_{bm}$) and active base actuation under a fuel/torque cost — the free-flying generalisation of the RNS resolution.
• Whether to register a canonical symbol for the coupling block in notation.md (and which: ${\mathbf{M}}_{bm}$ vs the Giordano sub-block names), since the clash across sources currently forces a per-equation choice.