FFSM Dynamics

Definition

The coupled rigid-body equations of motion of a free-flying space manipulator (FFSM): a
fully-actuated 6-DOF base spacecraft carrying an $n$-joint serial arm, written over the
generalized velocity $[{\mathbf{v}}_b;{\mathbf{\omega }}_b;\dot{\mathbf{q}}]\in {\mathbb{R}}^{6+n}$
(base linear/angular velocity stacked on joint rates). Because base and arm exchange momentum
through a non-block-diagonal inertia matrix $\mathbf{M}(\mathbf{q})$, the system is
dynamically coupled: arm motion induces base motion and vice versa. The defining contrast with the
more common free-floating regime is that here all six base degrees of freedom are actuated
(${\mathbf{f}}_b,{\mathbf{\tau }}_b\ne 0$), so linear and angular momentum are not
conserved and the generalized-Jacobian / momentum-elimination reductions of the floating literature do
not apply unmodified.

Key Equations

All symbols are rendered from notation.md. The canonical form is Giordano’s
free-flying model; the floating-regime reductions are noted as such.

Full coupled equations of motion. With $\mathbf{M},\mathbf{C}\in {\mathbb{R}}^{(6+n)\times (6+n)}$
the full coupled inertia and Coriolis matrices, and the actuation stacked as base force ${\mathbf{f}}_b$,
base torque ${\mathbf{\tau }}_b$, joint torques $\mathbf{\tau }$:

$$\mathbf{M}(\mathbf{q})\left[\begin{array}{c}{\dot{\mathbf{v}}}_b\\ {\dot{\mathbf{\omega }}}_b\\ \ddot{\mathbf{q}}\end{array}\right]+\mathbf{C}\left[\begin{array}{c}{\mathbf{v}}_b\\ {\mathbf{\omega }}_b\\ \dot{\mathbf{q}}\end{array}\right]=\left[\begin{array}{c}{\mathbf{f}}_b\\ {\mathbf{\tau }}_b\\ \mathbf{\tau }\end{array}\right].$$

(Giordano eq 4 / current_sota eq 1.4)

The free-flying regime is precisely the statement that the entire right-hand side is commandable; the
free-floating limit sets ${\mathbf{f}}_b={\mathbf{\tau }}_b=0$, recovering momentum
conservation (see free_flying_vs_free_floating).

Inertia sub-blocks — where the coupling lives. The leading rows of $\mathbf{M}$ make the
base-arm coupling explicit. With total mass $m={\sum }_{j=0}^n{m}^{(j)}$, the base-to-CoM vector
${\mathbf{p}}_{bc}$, and the mass-averaged linear Jacobian ${\bar{\mathbf{J}}}_v$:

$${\mathbf{M}}_t=m\mathbf{E},{\mathbf{M}}_{tr}=-m[{\mathbf{p}}_{bc}{]}^{\wedge },{\mathbf{M}}_{tm}=m{\bar{\mathbf{J}}}_v.$$

(Giordano eqs 5a–5d / current_sota eq 1.4 sub-blocks)

$${\mathbf{p}}_{bc}=\frac{1}{m}\sum _{j=0}^n{m}^{(j)}{\mathbf{p}}_{bj},{\bar{\mathbf{J}}}_v=\frac{1}{m}\sum _{j=0}^n{m}^{(j)}{\mathbf{R}}_{jb}^T{\mathbf{J}}_{v_j}.$$

(Giordano eqs 6, 7 / current_sota eqs 1.5, 1.6)

Here ${\mathbf{M}}_t$ is the translational block (decouples to pure total mass), while the
off-diagonal ${\mathbf{M}}_{tr}$ (translation–rotation, through ${\mathbf{p}}_{bc}$) and
${\mathbf{M}}_{tm}$ (translation–manipulator, through ${\bar{\mathbf{J}}}_v$) are the inertial
coupling terms — they are nonzero precisely because the CoM moves as the arm articulates
(see coupling_inertia_matrix, dynamic_coupling).

Reduced (circumcentroidal) equations of motion. Pushing the full dynamics through the coordinate
transform $\mathbf{\Gamma }$ (which maps $[{\mathbf{v}}_b;{\mathbf{\omega }}_b;\dot{\mathbf{q}}]\to [{\mathbf{v}}_c;{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$) by congruence yields the block-diagonal
transformed inertia $\hat{\mathbf{M}}={\mathbf{\Gamma }}^{-T}\mathbf{M}{\mathbf{\Gamma }}^{-1}$
and the $\dot{\mathbf{\Gamma }}$-corrected $\hat{\mathbf{C}}$:

$$\hat{\mathbf{M}}=\left[\begin{array}{cc}m\mathbf{E}& 0\\ 0& \breve{\mathbf{M}}\end{array}\right],\hat{\mathbf{C}}=\left[\begin{array}{cc}0& -{\mathbf{C}}_c^T\\ {\mathbf{C}}_c& \breve{\mathbf{C}}\end{array}\right].$$

(Giordano eq 21 / current_sota eqs 3.1, 3.2)

The whole-system CoM then obeys a decoupled translational law while the attitude and end-effector
form a single coupled $9\times 9$ block ($\breve{\mathbf{M}},\breve{\mathbf{C}}\in {\mathbb{R}}^{9\times 9}$),
forced by the residual CoM-coupling Coriolis ${\mathbf{C}}_c$:

$$m{\dot{\mathbf{v}}}_c={\mathbf{f}}_c,$$ $$\breve{\mathbf{M}}\left[\begin{array}{c}{\dot{\mathbf{\omega }}}_b\\ {\dot{\mathbf{\nu }}}_e^{\oplus }\end{array}\right]+\breve{\mathbf{C}}\left[\begin{array}{c}{\mathbf{\omega }}_b\\ {\mathbf{\nu }}_e^{\oplus }\end{array}\right]+{\mathbf{C}}_c{\mathbf{v}}_c=\left[\begin{array}{c}{\mathbf{\tau }}_b^{\oplus }\\ {\mathbf{w}}_e^{\oplus }\end{array}\right],{\mathbf{C}}_c=[{\mathbf{C}}_{bc};{\mathbf{C}}_{ec}].$$

(Giordano eqs 22a, 22b / current_sota eqs 3.3, 3.4)

This decoupling — and the passivity of the coupled block — is the dynamical foundation the coordinated
controller exploits; see circumcentroidal_decoupling and
coordinated_control_passivity.

Regime caveat

The reduction above is for a free-flying base with $n=6$ (so ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is
square and $\mathbf{\Gamma }$ invertible). It is not the free-floating generalized-inertia
reduction ${\mathbf{M}}^{\star }={\mathbf{M}}_m-{\mathbf{M}}_{0m}^T{\mathbf{M}}_0^{-1}{\mathbf{M}}_{0m}$
(wilde2018equations eq 45), which presumes conserved momentum (${\mathbf{M}}_0=0$).

Source Support

• sukhanov2015dynamic — (seeded) treats dynamic coupling in a
  space-manipulator setting; original stub citation.
• giordano2019coordinated — canonical free-flying model.
  Full coupled EOM (eq 4) with sub-blocks (eqs 5–7), the circumcentroidal coordinate transform
  $\mathbf{\Gamma }$ (eq 19), and the reduced/decoupled EOM (eqs 21–23). Base is fully actuated
  (free-flying); base translation deliberately left free (“partial base control”).
• virgili-llop2016spacecraft — DeNOC/Newton-Euler
  derivation supporting both flying and floating bases; explicit block partition of the inertia
  into base, manipulator, and base-manipulator coupling blocks; O(n) recursive construction.
• wilde2018equations — tutorial Lagrangian derivation of the
  coupled EOM (eq 39) and the coupling inertia ${\mathbf{H}}_{0m}$; reduction performed for the
  free-floating mode only (${\mathbf{M}}_0=0$). Useful for the full symbolic inertia
  bookkeeping and the five-mode flying/floating taxonomy.

Related Topics

• dynamic_coupling — the phenomenon carried by the off-diagonal inertia blocks
  ${\mathbf{M}}_{tr},{\mathbf{M}}_{tm}$; FFSM dynamics is the equation set that exhibits it.
• coupling_inertia_matrix — the base-manipulator coupling block of
  $\mathbf{M}$ (${\mathbf{M}}_{0m}$ / ${\mathbf{H}}_{0m}$) named and analyzed in isolation.
• generalized_inertia_matrix — the full $\mathbf{M}(\mathbf{q})$
  and its free-floating Schur-complement reduction ${\mathbf{M}}^{\star }$.
• circumcentroidal_motion — the $\oplus$ coordinate split that turns the
  full coupled EOM into the reduced, partially-decoupled form used by the controller.
• coordinated_control — the control law built directly on the reduced EOM
  (§3) of this page.
• free_flying_vs_free_floating — the regime distinction that
  determines whether momentum is conserved and which reduction is valid.

Open Questions

• The reduced EOM (Giordano eqs 21–22) assume a nonredundant arm ($n=6$). What is the explicit form of
  $\breve{\mathbf{M}},\breve{\mathbf{C}}$ and the decoupling for a redundant arm ($n>6$), where
  ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is no longer square?
• wilde2018equations derives the floating reduction with ${\mathbf{M}}_0=0$; what is the
  closed-form free-flying EOM when the base wrench is retained on the RHS — i.e. how do
  ${\mathbf{M}}^{\star },{\mathbf{C}}^{\star }$ change when neither $\mathbf{p}$ nor $\mathbf{L}$
  is conserved?
• Notation clash to resolve at verification: $\mathbf{M}$ (this wiki) vs $\mathbf{H}$
  (wilde2018, virgili-llop2016) for the inertia matrix — confirm the cross-walk is recorded so the
  examiner sees one symbol authority.