Free Floating Dynamics

Definition

Free-floating dynamics describe a space manipulator whose base spacecraft has its
attitude- and position-control system switched off: with no external forces or
torques, the base translates and rotates purely in reaction to arm motion, and total
linear and angular momentum are conserved (alali2024reinforcement,
nanos2015avoiding). This is the contrasting regime to
the free-flying system studied in this project, whose base is fully actuated; see
free_flying_vs_free_floating. Two consequences follow
that fixed-base manipulators never face: (i) the conserved-momentum relation acts as a
nonholonomic constraint coupling base motion to joint rates, so the end-effector pose
depends on the path taken in joint space, not just the final joint angles
(alali2024reinforcement); and (ii) the
end-effector velocity depends on base motion as well as joint rates, producing
dynamic singularities that are functions of the inertial properties and are invisible
to a purely kinematic analysis (nanos2015avoiding).

Key Equations

Symbols per notation.md.

With zero initial momentum, conservation of linear+angular momentum gives the constraint
relating base twist ${\dot{\mathbf{x}}}_b=[{\mathbf{v}}_0;{\mathbf{\omega }}_0]$ to joint
rates $\dot{\mathbf{\Phi }}$ (alali2024reinforcement, Eq. 8):

$$\left[\begin{array}{c}\mathbf{p}\\ \mathbf{L}\end{array}\right]={\mathbf{M}}_b{\dot{\mathbf{x}}}_b+{\mathbf{M}}_m\dot{\mathbf{\Phi }}=0,$$

where ${\mathbf{M}}_b$ is the base inertia and ${\mathbf{M}}_m$ the base–arm coupling
inertia. (Source symbols: ${\mathbf{M}}_m$ here is the rate-level coupling block of the
momentum map; it is the kinematic-momentum counterpart of the dynamics-level coupling
inertia ${\mathbf{M}}_{bm}$ in notation.md — flagged so the glyphs are
not conflated.) Solving the constraint for ${\dot{\mathbf{x}}}_b$ and substituting into
${\dot{\mathbf{x}}}_e={\mathbf{J}}_b{\dot{\mathbf{x}}}_b+{\mathbf{J}}_m\dot{\mathbf{\Phi }}$
folds the base reaction into the generalized Jacobian ${\mathbf{J}}_g$
(alali2024reinforcement, Eq. 9):

$${\dot{\mathbf{x}}}_e=({\mathbf{J}}_m-{\mathbf{J}}_b{\mathbf{M}}_b^{-1}{\mathbf{M}}_m)\dot{\mathbf{\Phi }}={\mathbf{J}}_g\dot{\mathbf{\Phi }}.$$

A dynamic singularity occurs where ${\mathbf{J}}_g$ loses rank; unlike a kinematic
singularity it depends on the mass/inertia properties and its location in the workspace is
path-dependent (nanos2015avoiding). The coupled
equations of motion carry the dynamic coupling in the off-diagonal block ${\mathbf{M}}_{bm}$
of the generalized inertia matrix (alali2024reinforcement, Eq. 10).

Source Support

• alali2024reinforcement — primary: cleanly splits space-robot research into free-flying (thruster-controlled base) vs free-floating (uncontrolled base reacting via momentum conservation); gives the momentum-conservation constraint, the ${\mathbf{J}}_g$ derivation, and the coupled inertia matrix with the dynamic-coupling block ${\mathbf{M}}_{bm}$.
• nanos2015avoiding — formal free-floating dynamics: angular-momentum conservation with possibly nonzero initial momentum, and the resulting path-dependent dynamic singularities of the generalized (Umetani–Yoshida) Jacobian.
• rybus2017control — supporting: free-floating satellite-manipulator model ($n+6$ generalized coordinates, nonholonomic); NMPC and Dynamic-Jacobian controllers that explicitly account for the free-floating base reaction.
• sze2024obstacle — supporting: free-floating mode defined as zero linear+angular momentum; gives the mass matrix and the generalized Jacobian in the spacecraft frame for an RL trajectory planner.

Related Topics

• ffsm_dynamics — the full coupled $(6+n)$-DOF equations of motion; free-floating dynamics is the regime obtained when the base wrench is set to zero.
• momentum_conservation — the conserved linear+angular momentum that is the defining constraint of this regime.
• angular_momentum_conservation — the angular part specifically; its non-integrability is what makes the regime nonholonomic and path-dependent.
• generalized_jacobian — ${\mathbf{J}}_g$, the map that folds the momentum constraint into the EE Jacobian; rank loss here is the dynamic singularity.
• free_flying_vs_free_floating — the regime contrast: our actuated-base (free-flying) system does not obey the momentum constraint above, so ${\mathbf{J}}_g$ is replaced by the circumcentroidal Jacobian.
• dynamic_singularity — the inertia-dependent, path-dependent rank loss of ${\mathbf{J}}_g$ that this regime introduces.

Open Questions

• All four sources assume an uncontrolled base (free-floating). Our system is free-flying (fully-actuated 6-DOF base): the momentum-conservation constraint and ${\mathbf{J}}_g$ no longer hold — which results carry over, and which (e.g. dynamic singularities of ${\mathbf{J}}_g$) simply vanish under base actuation?
• nanos2015avoiding admits nonzero initial angular momentum, while sze2024obstacle and alali2024reinforcement assume zero. For a free-flying base that can dump momentum with thrusters/wheels, does the nonzero-momentum machinery have any residual role?