Angular Momentum Conservation

Definition

In a space manipulator with no external forces or torques acting on it, the total angular
momentum about the system center of mass is conserved. This holds in the free-floating regime,
where the base actuators (thrusters, reaction wheels) are off and the base is left to translate and
rotate in reaction to arm motion (nanos2011use). The conservation law
algebraically couples the base angular velocity to the joint rates, and because it is not
integrable, the free-floating system is nonholonomic: base attitude depends on the path the
joints take, not just their final values. For a free-flying base — our regime — the base
actuators supply external torque on purpose, so angular momentum is not conserved (it is exchanged
with the actuators); this conservation constraint is what free-flying control deliberately breaks.

Key Equations

Symbols per notation.md.

Conservation of angular momentum for the rigid free-floating system, no external torques
(nanos2011use Eq. 1; symbols mapped to canonical: $\mathbf{L}$ = conserved angular momentum,
${\mathbf{\omega }}_b$ = base angular velocity in the base frame, $\dot{\mathbf{q}}$ = joint
rates, ${\mathbf{R}}_{bt}$ = base→inertial rotation, so ${\mathbf{R}}_{bt}^{\top }$ maps
inertial→base — matching the source’s ${\mathbf{R}}_0^{\top }$, per notation.md
${\mathbf{R}}_{xy}:\mathcal{X}\to \mathcal{Y}$ with $\mathcal{T}$ the inertial frame):

$${}^b\mathbf{D}(\mathbf{q}){\mathbf{\omega }}_b+{}^b{\mathbf{D}}_q(\mathbf{q})\dot{\mathbf{q}}={\mathbf{R}}_{bt}^{\top }\mathbf{L}$$

where ${}^b\mathbf{D}(\mathbf{q})$ and ${}^b{\mathbf{D}}_q(\mathbf{q})$ are
configuration-dependent inertia-type coupling matrices (the source’s ${}^0\mathbf{D}$,
${}^0{\mathbf{D}}_q$; not in notation.md — they play the role of the base self-inertia and the
base–arm coupling inertia ${\mathbf{M}}_{bm}$). With zero initial momentum
($\mathbf{L}=0$) this reduces to the homogeneous reaction constraint
${}^b\mathbf{D}{\mathbf{\omega }}_b=-{}^b{\mathbf{D}}_q\dot{\mathbf{q}}$, the
basis for reactionless_motion and the
generalized_jacobian. With nonzero $\mathbf{L}$, the right-hand
side acts as a drift term and the end effector can no longer be held fixed indefinitely
(nanos2011use).

Source Support

• nanos2011use — primary; derives the angular-momentum conservation
  law for a free-floating manipulator with nonzero initial momentum, shows it makes the system
  nonholonomic, and identifies workspace subsets where the end effector can remain fixed despite the
  accumulated momentum. Notes that Umetani & Yoshida’s generalized Jacobian “reflects both momentum
  conservation laws and kinematic relations.”

Related Topics

• momentum_conservation — parent topic carrying both the conserved
  linear ($\mathbf{p}$) and angular ($\mathbf{L}$) momentum; this page is the angular-momentum
  specialization.
• free_floating_dynamics — the regime in which this conservation holds
  (base actuators off); the equations of motion are derived under exactly this assumption.
• reaction_null_space — the homogeneous (zero-momentum) form of the
  conservation law defines the joint motions that produce no base reaction.
• generalized_jacobian — folds the momentum-conservation constraint into
  the end-effector Jacobian for the free-floating regime (Umetani & Yoshida).
• reactionless_motion — arm trajectories that satisfy the zero-momentum
  conservation constraint so the base attitude is undisturbed.

Open Questions

• nanos2011use assumes a free-floating base (actuators off). For our free-flying base the
  base torque is an external input, so $\mathbf{L}$ is not conserved — how does the “fixed-point
  workspace” analysis change once the base can actively reject the momentum drift?
• The nonzero-momentum drift makes joint rates proportional to $\mathbf{L}$ but torques
  super-proportional (the Coriolis terms ${\mathbf{C}}_h$ are nonlinear in $\dot{\mathbf{q}}$);
  what is the corresponding torque budget when an actuated base shares the load?