Momentum Conservation

Definition

For a space manipulator system subject to no external forces or torques, total linear and
angular momentum are integrals of the motion: arm motion is compensated by reactive base motion so
that the system center of mass does not accelerate and total angular momentum is fixed. This is the
defining property of the free-floating regime (uncontrolled base): Umetani & Yoshida introduce
the linear- and angular-momentum conservation laws into the velocity kinematics to build the
generalized Jacobian, and Nanos & Papadopoulos write the same angular-momentum
law to expose the nonholonomic drift it induces. The premise does not hold for our free-flying
system: a fully-actuated 6-DOF base applies external base wrenches, so momentum is exchanged with the
actuators rather than conserved — see “Open Questions”.

Key Equations

Symbols per notation.md.

Linear- and angular-momentum conservation for an $n+1$-body chain with no external loads
(Umetani & Yoshida 1989, Eqs. 2–3; symbols below follow that source — $m^{(j)}$ body masses,
${\mathbf{I}}^{(j)}$ body inertias, ${\mathbf{r}}^{(j)}$ inertial body-CoM positions, ${\mathbf{\omega }}^{(j)}$ body angular rates):

$$\mathbf{p}=\sum _{j=0}^n{m}^{(j)}{\dot{\mathbf{r}}}^{(j)}=\text{const.},\mathbf{L}=\sum _{j=0}^n\left({\mathbf{I}}^{(j)}{\mathbf{\omega }}^{(j)}+m^{(j)}{\mathbf{r}}^{(j)}\times {\dot{\mathbf{r}}}^{(j)}\right)=\text{const.}$$

With zero initial linear momentum the system CoM is fixed, and the angular-momentum law reduces to a
configuration-dependent relation between the base angular velocity and the joint rates. Nanos &
Papadopoulos (Eq. 1) state it for a possibly non-zero initial momentum $\mathbf{L}={\mathbf{h}}_{\mathrm{CM}}$
(notation: ${\mathbf{h}}_{\mathrm{CM}}$ is theirs; ${}^0\mathbf{D},{}^0{\mathbf{D}}_q$ are configuration-dependent inertia blocks):

$${}^0\mathbf{D}(\mathbf{q}){}^0{\mathbf{\omega }}_0+{}^0{\mathbf{D}}_q(\mathbf{q})\dot{\mathbf{q}}={\mathbf{R}}_0^{\top }(\mathbf{ϵ},\eta ){\mathbf{h}}_{\mathrm{CM}}.$$

The right-hand side is non-integrable in the base attitude, which is the source of the nonholonomic
behavior of free-floating systems.

Source Support

• umetani1989resolved — primary: folds linear (Eq. 2) and angular (Eq. 3) momentum conservation into the velocity kinematics to derive the Generalized Jacobian Matrix for RMRC; states the conservation premise explicitly (Assumption 2: base position/attitude uncontrolled, no external loads).
• nanos2011use — support: writes the angular-momentum law (Eq. 1) for non-zero initial momentum ${\mathbf{h}}_{\mathrm{CM}}$ and shows it drives a drift that, in general, prevents the end-effector from holding a fixed inertial point; merged here from angular_momentum_conservation.

Related Topics

• angular_momentum_conservation — the rotational specialization of this law; the component that becomes nonholonomic and drives base-attitude drift.
• ffsm_dynamics — these conservation laws are the kinematic constraints that the free-floating equations of motion must satisfy.
• dynamic_coupling — conservation is the mechanism of coupling: a momentum budget forces base reaction whenever the arm moves.
• free_floating_dynamics — the regime in which momentum is actually conserved (no base actuation / external loads).
• generalized_jacobian — the map obtained by substituting the conservation relation into the EE velocity kinematics (Umetani & Yoshida); valid only while momentum is conserved.

Open Questions

• Both sources assume an uncontrolled base (free-floating). With a fully-actuated 6-DOF base, external base wrenches make $\mathbf{p},\mathbf{L}$ non-constant — does any reduced conservation structure survive, or must the generalized-Jacobian machinery be replaced entirely (e.g. by the circumcentroidal split)?
• Nanos & Papadopoulos show non-zero ${\mathbf{h}}_{\mathrm{CM}}$ forces continual joint motion to hold an inertial point; for our actuated base this drift is absorbable by base control — at what residual momentum does relying on base authority become preferable to RNS-style reactionless joint motion?