nenchev2013reaction

Reaction Null Space of a multibody system with applications in robotics

Authors: Nenchev · Year: 2013 · Venue: Mechanical Sciences (Mech. Sci.), 4, 97–112
Raw: md

Summary

A 25-year retrospective by the originator of the Reaction Null Space (RNS) formalism. The RNS is defined as the kernel of the coupling inertia matrix ${\mathbf{M}}_{bm}$ — the off-diagonal block of the system inertia matrix that couples the unfixed base to the actuated manipulator joints. Joint velocities/accelerations/torques drawn from this kernel produce reactionless motion: motion of the arm that imposes no momentum or spatial force on the base, hence complete dynamical decoupling of base from arm. The paper unifies four superficially different problems — free-floating space-robot base-attitude preservation, flexible-base vibration suppression, macro/mini manipulation, and humanoid balance — under one joint-space decomposition, and adds a recent result reinterpreting the manipulator end-link as the “unfixed base” to yield an alternative to Khatib’s Operational Space Formulation (OSF).

Key Claims

• Reactionless motion (complete dynamical decoupling of base from arm) is achievable iff the coupling momentum ${\mathcal{L}}_{bm}={\mathbf{M}}_{bm}\dot{\mathbf{\theta }}$ is conserved, i.e. ${\mathbf{M}}_{bm}\dot{\mathbf{\theta }}=0$ (Eq. 12).
• The RNS projector has $\mathrm{rank}{\mathbf{P}}_{bm}=n-6$; a 7-DOF arm yields a 1-D reactionless-motion manifold. Redefining the coupling inertia w.r.t. only base orientation (ignoring base translation) raises the rank to $n-3$, enlarging the reactionless path set — appropriate for free-floating space robots where base attitude (not position) matters.
• Using the Moore–Penrose pseudoinverse in the redundancy resolution makes the two joint-space components orthogonal and locally minimizes the coupling kinetic energy ${\mathcal{V}}_b^T{\mathbf{M}}_{bm}\dot{\mathbf{\theta }}$ (not the total kinetic energy as in OSF).
• The RNS end-link dynamics formulation (Eqs. 33–34) remains valid at kinematic singularities, where the OSF operational-space inertia ${\mathbf{M}}_e=(\mathbf{J}{\mathbf{M}}_m^{-1}{\mathbf{J}}^T{)}^{-1}$ (Eq. 32) fails because $\mathbf{J}$ is rank-deficient.
• Because RNS minimizes the well-behaved coupling kinetic energy rather than the highly nonlinear total kinetic energy, it avoids the spurious peak joint velocities/arm reconfiguration that OSF exhibits near singularities, while delivering equal end-link motion/force tracking performance (experimentally confirmed, Hara et al. 2012, Figs. 9–10).
• On-orbit validation: the RNS reactionless feedforward generator (Eq. 13) was flown on ETS-VII (~20 min experiment, external perturbations neglected), the first free-floating space robot.

Method

Notation. A tree-structured rigid MBS of one or more arms on an unfixed base, $n$ single-DOF joints $\mathbf{\theta }\in {\Re }^n$, base pose $X\in SE(3)$, generalized coordinates $q=(X,\theta )$. Spatial vectors are linear-then-angular: ${\mathcal{V}}_O=[{\mathbf{v}}_O^T{\mathbf{\omega }}^T{]}^T$, ${\mathcal{F}}_O=[f^T{\mathbf{n}}_O^T{]}^T$.

Momentum-based derivation (regime: free-FLOATING base — joints actuated, base NOT actuated). Treating the system as a composite rigid body (CRB), spatial momentum referenced to the base frame gives

${\mathbf{M}}_b{\mathcal{V}}_b+{\mathbf{M}}_{bm}\dot{\mathbf{\theta }}={\mathcal{L}}_b(4)$

with ${\mathbf{M}}_b$ the $6\times 6$ CRB (locked) inertia and ${\mathbf{M}}_{bm}\in {\Re }^{6\times n}$ the coupling inertia matrix. Under zero initial momentum (${\mathcal{L}}_b=0$), ${\mathbf{M}}_b{\mathcal{V}}_b=-{\mathbf{M}}_{bm}\dot{\mathbf{\theta }}$ — the reaction momentum equals minus the coupling momentum. For a redundant arm ($n>6$) the general solution is

$\dot{\mathbf{\theta }}=-{\mathbf{M}}_{bm}^{\#}{\mathbf{M}}_b{\mathcal{V}}_b+{\mathbf{P}}_{bm}{\dot{\mathbf{\theta }}}_a(11)$

where ${\mathbf{P}}_{bm}$ is the null-space projector. The kernel $\{{\dot{\mathbf{\theta }}}_{rl}={\mathbf{P}}_{bm}{\dot{\mathbf{\theta }}}_a\}$ is the RNS; integral curves of the reactionless vector field $n_{bm}(q)\in \ker {\mathbf{M}}_{bm}$ via $\dot{\mathbf{\theta }}=b{n}_{bm}$ (Eq. 13) trace reactionless paths.

Dynamics-based derivation. Full coupled dynamics (Eq. 14):
$\left[\begin{array}{cc}{\mathbf{M}}_b& {\mathbf{M}}_{bm}\\ {\mathbf{M}}_{bm}^T& {\mathbf{M}}_m\end{array}\right]\left[\begin{array}{c}{\dot{\mathcal{V}}}_b\\ \ddot{\mathbf{\theta }}\end{array}\right]+\left[\begin{array}{c}{C}_b\\ {\mathbf{c}}_m\end{array}\right]=\left[\begin{array}{c}{\mathcal{F}}_b\\ \mathbf{\tau }\end{array}\right]+\left[\begin{array}{c}{}^b{\mathbf{T}}_e^T\\ {\mathbf{J}}^T\end{array}\right]{\mathcal{F}}_e$
The CRB (upper) row gives ${\mathbf{M}}_b{\dot{\mathcal{V}}}_b+{\mathbf{M}}_{bm}\ddot{\mathbf{\theta }}+C_b={\mathcal{F}}_{\text{ext}}$ (Eq. 15), and the redundant joint-acceleration solution mirrors Eq. 11. Crucially for the thesis: the base-force term ${\mathcal{F}}_b$ is given a broad role to include “base constraint and/or actuator forces,” explicitly so the same equation models a free-FLYING space robot with attitude-controlled base (reaction/momentum wheels), a flexible-base manipulator, or a humanoid (p. 4, after Eq. 14).

Applications. (i) Single-body flexible base: elastic base force ${\mathcal{F}}_b=-{\mathbf{D}}_b{\mathcal{V}}_b-{\mathbf{K}}_b\Delta {\mathcal{X}}_b$ (Eq. 20); the dual-task law (Eq. 23) injects inertial damping via the pseudoinverse while the RNS term keeps motion reactionless, yielding mass–damper–spring closed-loop ${\mathbf{M}}_b{\dot{\mathcal{V}}}_b+({\mathbf{D}}_b+{\mathbf{G}}_b){\mathcal{V}}_b+{\mathbf{K}}_b\Delta {\mathcal{X}}_b=0$ (Eq. 24). (ii) Macro/mini (CRB flexible base, passive macro joints, Eqs. 25–27). (iii) Humanoid balance via foot as unfixed base (Eqs. 40–41). (iv) RNS end-link formulation: rename end-links A, B; treat end-link A as the reference “base,” yielding CRB dynamics in end-link coordinates (Eq. 34) with joint motion explicitly visible through $-{\mathbf{M}}_{Am}\ddot{\mathbf{\theta }}$ — usable at singularities, unlike OSF.

Relevance to thesis

This is the canonical RNS reference and the conceptual bridge between the free-floating literature and our free-FLYING problem. The free-floating derivation (zero base actuation) is the limiting case; the paper itself flags that an attitude-controlled base is captured by promoting ${\mathcal{F}}_b$ to an actuator wrench — exactly our regime. For a fully-actuated 6-DOF base, the RNS/coupling-inertia decomposition becomes a choice (decouple arm reactions from the base even though we could fight them with thrusters/wheels) rather than a necessity, and is directly useful for fuel/torque-economical coordinated guidance: feedforward reactionless arm motion plus pseudoinverse base regulation. The coupling-kinetic-energy minimization and singularity-robustness arguments connect to our dynamic-singularity and redundancy-resolution work, and the joint-space decomposition is a natural nominal layer to wrap a later risk-aware (CVaR/chance-constrained) base-disturbance budget around.

Connections

Topics: reaction_null_space, coupling_inertia_matrix, reactionless_motion, free_flying_vs_free_floating

Key Equations / Quotes

“reactionless motion (and hence, complete dynamical decoupling) can be achieved if and only if the coupling momentum is conserved” (p. 4, Eq. 12 context).

${\mathbf{M}}_{bm}\dot{\mathbf{\theta }}=0(12)$

“the base force ${\mathcal{F}}_b$ term could be assigned a broader role to include base constraint and/or actuator forces. This will allow us … to model other types of systems with the same equation, e.g. a free-flying space robot with attitude controlled base (i.e. using reaction/momentum wheels as actuators) …” (p. 4).

“differently from the OSF end-link dynamics (Eq. 31), the above equation can be applied even at kinematic singularities.” (p. 9, on Eq. 34).

OSF operational-space inertia that RNS sidesteps:
${\mathbf{M}}_e(\mathbf{\theta })=(\mathbf{J}(\mathbf{\theta }){\mathbf{M}}_m^{-1}(\mathbf{\theta }){\mathbf{J}}^T(\mathbf{\theta }){)}^{-1}\in {\Re }^{6\times 6}(32)$

“the coupling kinetic energy, minimized under the RNS formulation, is quite well behaved, even within a relatively small vicinity of ill-defined inertial coupling, i.e. where the coupling inertia matrix becomes rank deficient.” (p. 10).

Open Questions

• The paper notes the coupling inertia matrix ${\mathbf{M}}_{bm}$ can itself become rank-deficient (“ill-defined inertial coupling”) — these dynamic singularities of ${\mathbf{M}}_{bm}$ are mentioned but not characterized geometrically here. How do they relate to the kinematic singularities of $\mathbf{J}$, and how should redundancy resolution behave near them?
• The dual-task feedback controllers (humanoid CRB feedback, Sect. 6.1; flexible-base) are repeatedly deferred (“Results will be reported elsewhere”) — no stability proof for the combined reactionless-feedforward + pseudoinverse-feedback loop is given here.
• For an actuated base (free-flying), what is the optimal split between reactionless arm planning and active base actuation under fuel/torque cost — a question the paper enables but does not pose.