Reactionless motion is manipulator motion that imposes zero coupling momentum on the base, so the
arm is completely dynamically decoupled from the base: the base experiences no reaction and holds its
state while the arm moves. Following Nenchev (2013), it is generated from the kernel of the coupling
inertia matrixMbm — the cross block of the system inertia that maps joint rates to base
momentum — projected by the reaction null-space projector. The derivation in nenchev2013reaction assumes a free-FLOATING robot (actuated arm, unactuated base) under conserved spatial momentum, where the property is most valuable because the base
cannot otherwise resist the reaction. It requires kinematic redundancy (n>6, or n>3 when only base attitude is protected): rank PRNS=n−6.
With zero initial momentum, the base momentum equation Mbνb+Mbmq˙=0
admits the redundant (n>6) general solution
\qquad \boldsymbol P_{\mathrm{RNS}}=\boldsymbol E-\boldsymbol M_{bm}^{+}\boldsymbol M_{bm}.$$
The reactionless set is the second term alone: any $\dot{\boldsymbol q}_{\mathrm{rl}}=\boldsymbol P_{\mathrm{RNS}}\dot{\boldsymbol q}_a$
satisfies the homogeneous **coupling-momentum-conservation** condition
$$\boldsymbol M_{bm}\,\dot{\boldsymbol q} = \boldsymbol 0,$$
i.e. zero coupling momentum $\boldsymbol{\mathcal L}_{bm}\equiv\boldsymbol M_{bm}\dot{\boldsymbol q}=\boldsymbol 0$. Reactionless motion
$\Leftrightarrow$ coupling momentum conserved. These rates trace the *reactionless motion manifold* in joint
space (dimension $n-6$); the same construction lifts to acceleration level ($\boldsymbol P_{\mathrm{RNS}}\ddot{\boldsymbol q}_a$)
for feedforward torque control.
> Note: the source writes joint rates as $\dot{\boldsymbol\theta}$ and the projector as $\boldsymbol P_{bm}$; both are
> rendered here in canonical notation ($\dot{\boldsymbol q}$, $\boldsymbol P_{\mathrm{RNS}}$). $\boldsymbol{\mathcal L}_{bm}$ for coupling
> momentum is not in [notation.md](../notation.md) ($\boldsymbol p,\boldsymbol L$ are the conserved linear/angular momentum);
> used here as a local label, not a conflicting glyph.
## Source Support
- [nenchev2013reaction](../sources/nenchev2013reaction.md) — primary; defines reactionless motion as the
kernel of $\boldsymbol M_{bm}$, derives it from momentum conservation, and shows the joint-space
decomposition (RNS feedforward + pseudoinverse feedback) for free-floating space robots, flexible-base,
macro/mini, and humanoid systems. On-orbit verification on ETS-VII.
## Related Topics
- [reaction_null_space](reaction_null_space.md) — the kernel $\ker\boldsymbol M_{bm}$ itself; reactionless
motion is exactly motion drawn from that null space.
- [dynamic_coupling](dynamic_coupling.md) — reactionless motion is the operating point where this
arm $\leftrightarrow$ base coupling is nulled ($\boldsymbol M_{bm}\dot{\boldsymbol q}=\boldsymbol 0$).
- [base_disturbance_minimization](base_disturbance_minimization.md) — reactionless motion is the exact
($=0$) case; minimization is the relaxed objective when redundancy is insufficient for a true null.
- [momentum_conservation](momentum_conservation.md) — the conserved-momentum premise that makes the
homogeneous condition $\boldsymbol M_{bm}\dot{\boldsymbol q}=\boldsymbol 0$ equivalent to reactionlessness.
- [null_space_projection](null_space_projection.md) — the projector mechanism $\boldsymbol P_{\mathrm{RNS}}$
that selects reactionless rates; here applied to $\boldsymbol M_{bm}$ rather than a task Jacobian.
Also related: [kinematic_redundancy](kinematic_redundancy.md) (redundancy $n>6$ is what makes
$\ker\boldsymbol M_{bm}$ nontrivial) and the [generalized_jacobian](generalized_jacobian.md) (the companion
free-floating EE-mapping construct).
## Open Questions
- The derivation assumes a free-**floating** base whose value proposition is "the base can't resist, so
null the reaction." For our free-**flying** (fully-actuated) base the actuators *can* hold pose against
reaction — is reactionless motion then a propellant/fuel-economy objective rather than a necessity, and
how does it trade against the [circumcentroidal_motion](circumcentroidal_motion.md) coordinated-control
formulation we actually use?
- Reactionless decoupling holds only as far as the inertial model is exact; the source notes models
"cannot be perfect" and pairs the RNS feedforward with a pseudoinverse feedback term. How sensitive is
the null to inertial-parameter error, and does that motivate a risk-aware treatment?
- The reactionless manifold has dimension $n-6$ (or $n-3$ for attitude-only). For our redundant arm, is
the manifold large enough to *also* satisfy task and singularity-avoidance constraints, or do these
objectives compete for the same null space?