Resolved Motion Rate Control

Definition

Resolved Motion Rate Control (RMRC) is velocity-level inverse kinematics: given a desired
end-effector task-space velocity, solve the differential kinematics $\dot{\mathbf{P}}=\mathbf{J}\dot{\mathbf{\Phi }}$
for the joint rates $\dot{\mathbf{\Phi }}$ that realize it, then integrate/command those rates
(Whitney 1969). umetani1989resolved carries the scheme to a space
manipulator by replacing the fixed-base Jacobian with the generalized Jacobian
${\mathbf{J}}_g$ (written ${\mathbf{J}}^{\ast }$ in the source), which folds the system’s linear/angular
momentum conservation into the kinematics so that base reaction to arm motion is accounted for.
Regime caveat: Umetani’s derivation assumes momentum is conserved (no external wrench, base
reacts passively) — the free-floating regime — even though its English abstract says “free-flying.”
For our free-flying base (fully actuated, 6-DOF), momentum is not conserved, ${\mathbf{J}}_g$ does
not apply, and the RMRC map is built on the circumcentroidal Jacobian
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ instead (see Open Questions).

Key Equations

Symbols per notation.md.

RMRC is the inverse of the differential kinematics. With the manipulator Jacobian (here the source’s
generalized Jacobian ${\mathbf{J}}_g$, eq. 19) mapping joint rates to the task-space EE velocity,

$$\dot{\mathbf{P}}={\mathbf{J}}_g{\dot{\mathbf{\Phi }}}_M+{\dot{\mathbf{P}}}_0,$$

the square non-redundant ($n=6$) inverse away from singularities is (Umetani eq. 20)

$${\dot{\mathbf{\Phi }}}_M={\mathbf{J}}_g^{-1}(\dot{\mathbf{P}}-{\dot{\mathbf{P}}}_0).$$

Here ${\dot{\mathbf{P}}}_0={\mathbf{V}}_0+{\hat{\mathbf{J}}}_S{\hat{\mathbf{I}}}_S^{-1}{\mathbf{L}}_0$
(Umetani eq. 18) is the drift term set by the initial conditions: ${\mathbf{V}}_0$ from the initial
system-CoM velocity (linear momentum) and the second term from the initial angular momentum
${\mathbf{L}}_0$. It vanishes only when both initial momenta are zero, as assumed in the paper’s
simulations. For a redundant or near-singular Jacobian the plain inverse is replaced by the
pseudoinverse / damped least-squares solution.

Notation flag. notation.md reserves ${\mathbf{J}}^{\ast }$ for Papadopoulos’s free-floating EE Jacobian
and uses ${\mathbf{J}}_g$ for the Umetani–Yoshida GJM; the source itself writes the GJM as ${\mathbf{J}}^{\ast }$.
This page uses the canonical ${\mathbf{J}}_g$.

Source Support

• umetani1989resolved — origin of the space-manipulator RMRC: applies the classical resolved-rate concept (eq. 11) on the generalized Jacobian (eqs. 18–20), validated in capture-operation simulations of a base + 3-DOF arm.

Related Topics

• pseudoinverse_jacobian — the generalized inverse used when ${\mathbf{J}}_g$ is non-square (redundant) or ill-conditioned, replacing the literal ${\mathbf{J}}_g^{-1}$ in the RMRC step.
• generalized_jacobian — the space-manipulator Jacobian ${\mathbf{J}}_g$ that RMRC inverts here; note its free-floating (momentum-conservation) assumption.
• kinematic_redundancy_resolution — how RMRC exploits extra DOF (null-space terms) when the arm has more joints than task constraints.
• trajectory_tracking — RMRC is the velocity-level inner step that drives an EE along a commanded Cartesian path.
• closed_loop_inverse_kinematics — adds a pose-error feedback term to open-loop RMRC to arrest numerical drift from integrating joint rates.

Open Questions

• The source assumes a free-floating base (momentum conserved); the inverse (eq. 20) absorbs base reaction through ${\dot{\mathbf{P}}}_0$ and ${\mathbf{J}}_g$. For our free-flying, fully-actuated base, momentum is not conserved — does the analogous RMRC map built on ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ recover the same form, or does base actuation add an independent input term?
• Umetani’s eq. 20 uses a literal matrix inverse and is silent on singularity handling beyond “away from singular points.” How does the conditioning of ${\mathbf{J}}_g$ (vs. ${\sigma }_G$ for $\mathbf{\Gamma }$) govern RMRC stability near a singularity in the free-flying case?