umetani1989resolved

Resolved Motion Rate Control of Space Robotic Manipulators with Generalized Jacobian Matrix

Authors: Umetani, Yoshida · Year: 1989 · Venue: Journal of the Robotics Society of Japan (日本ロボット学会誌); also presented as the foundational Generalized Jacobian work later in IEEE T-RA 1989
Raw: md

Note: the body text is Japanese with an English abstract. OCR of the prose is rough but the displayed equations and table transcribe cleanly. See flags.

Summary

The paper derives the Generalized Jacobian Matrix (GJM), ${\mathbf{J}}^{\ast }$, a new Jacobian for a manipulator mounted on a base satellite whose position and attitude are not actively controlled (free-floating). By folding the system’s linear- and angular-momentum conservation laws into the kinematic formulation, the end-effector velocity is related to the joint velocities alone, eliminating the base degrees of freedom while still accounting for the reaction motion they undergo. This enables Resolved Motion Rate Control (RMRC) of a space manipulator in the inertial frame, and the same formulation is shown to subsume the attitude-controlled (reaction-wheel) case. Validity is demonstrated by 3-D simulations of a realistic 3-DOF robot-satellite capture scenario.

Key Claims

• For a free-floating $n+1$-link system with no external forces, the conventional fixed-base Jacobian fails because the end-effector pose depends on the entire history of arm motion; an analytic inverse-kinematic solution at the position/joint-angle level does not exist (Eq. (10) is non-integrable in space).
• At the velocity level, however, momentum/angular-momentum conservation gives a history-independent linear map. Combining the manipulator characteristic equation (14) with momentum conservation (16) and eliminating the base rates ${\dot{\mathbf{\Phi }}}_S$ yields $\dot{\mathbf{P}}={\mathbf{J}}^{\ast }{\dot{\mathbf{\Phi }}}_M+{\dot{\mathbf{P}}}_0$ with ${\mathbf{J}}^{\ast }={\hat{\mathbf{J}}}_M-{\hat{\mathbf{J}}}_S{\hat{\mathbf{I}}}_S^{-1}{\hat{\mathbf{I}}}_M$ (Eqs. (18)–(19)).
• ${\mathbf{J}}^{\ast }$ is $6\times n$; when $n=6$ (non-redundant) it is square, and away from singularities the inverse kinematics is solved analytically: ${\dot{\mathbf{\Phi }}}_M=[{\mathbf{J}}^{\ast }{]}^{-1}(\dot{\mathbf{P}}-{\dot{\mathbf{P}}}_0)$ (Eq. (20)).
• The GJM generalizes the terrestrial Jacobian: as the base inertia $\to \infty$, ${\hat{\mathbf{I}}}_S^{-1}{\hat{\mathbf{I}}}_M\to 0$ and ${\hat{\mathbf{J}}}_M\to \mathbf{J}$, recovering $\dot{\mathbf{P}}=\mathbf{J}{\dot{\mathbf{\Phi }}}_M$ — “a ground-fixed manipulator is a link system attached to the very large inertia of the Earth.”
• A cyclic (“non-holonomic”) arm maneuver returns the joints to their start but changes the base attitude — confirming history dependence at the position level; conversely, retracing the same joint path at any speed yields the same reaction-induced pose change (speed-independent).
• The same machinery handles the attitude-controlled case: imposing ${\dot{\mathbf{\Phi }}}_S=0$ gives the required reaction momentum ${\mathbf{L}}_c=-{\hat{\mathbf{I}}}_M{\hat{\mathbf{J}}}_M^{-1}(\dot{\mathbf{P}}-{\dot{\mathbf{P}}}_0)$ (Eq. (22)) for sizing reaction wheels; partial (2-axis) attitude control is also demonstrated.
• Simulation (Table 1: 2000 kg base, 20/50/50 kg links): the capture maneuver induces base attitude excursions of roughly $-23^{\circ }$ (roll) and $+14^{\circ }$ (pitch) plus translation, yet the end-effector tracks the target path exactly — reaction effects are “by no means negligible.”

Method

Model assumptions (Sec. 2.2): (1) the robot satellite carries one $n$-DOF revolute-joint manipulator and all bodies are rigid; (2) the system floats freely in inertial space with no base position/attitude control, so no external forces act and the conservation laws hold; (3) full state and target motion are observable in the inertial frame.

This is the canonical free-FLOATING regime (uncontrolled base) — distinct from our thesis’s free-FLYING (fully-actuated 6-DOF base). The paper explicitly contrasts the two stances (Sec. 1): the “first position” actively fixes the base (cited Yamada et al., Lindberg et al.), which the authors deem to demand high base-control authority and be a special case; they instead adopt the “second position,” letting the base move and compensating via arm control — and then show in Sec. 4.3 that the attitude-controlled case is recovered as a special instance of the general formulation.

Notation (Sec. 2.3): $r_i$ link CoM positions (inertial), $r_g$ system CoM, $p_n$ end-effector position, $\alpha ,\beta ,\gamma$ base yaw/pitch/roll, ${\varphi }_j$ joint angles. Base rates ${\dot{\mathbf{\Phi }}}_S=(\dot{\alpha },\dot{\beta },\dot{\gamma }{)}^t$ (3), manipulator rates ${\dot{\mathbf{\Phi }}}_M=({\dot{\varphi }}_1,\dots ,{\dot{\varphi }}_n{)}^t$ ($n$).

Governing relations:

• System CoM: ${\sum }_{i=0}^n{m}_i{r}_i=r_g{\sum }_{i=0}^n{m}_i$ (1)
• Linear momentum: $\sum m_i{\dot{r}}_i=$ const (2); angular momentum: $\sum (I_i{\mathbf{\omega }}_i+m_i{r}_i\times {\dot{r}}_i)=$ const (3)
• Joint constraint $r_i-r_{i-1}=a_i+b_{i-1}$ (4); end-effector $p_n=r_0+b_0+\sum l_i$ (5)
• Solving (1)+(4) gives $r_i={\sum }_j{K}_{ij}(\cdots )+r_g$ (6) with mass-ratio coefficients $K_{ij}$ (7), so ${\dot{r}}_i$ and ${\omega }_i$ are linear in $(\dot{\alpha },\dot{\beta },\dot{\gamma },{\dot{\varphi }}_j)$ (8)–(9).

Differentiating (5) yields the space-manipulator characteristic equation $\dot{\mathbf{P}}=[{\dot{p}}_n;{\omega }_n]=\hat{\mathbf{J}}(\mathbf{\Phi })\dot{\mathbf{\Phi }}+V_0$, partitioned as $\dot{\mathbf{P}}={\hat{\mathbf{J}}}_S{\dot{\mathbf{\Phi }}}_S+{\hat{\mathbf{J}}}_M{\dot{\mathbf{\Phi }}}_M+{\mathbf{V}}_0$ (14) — note this has 6 equations for $n+3$ unknowns. Momentum conservation supplies the closing 3 equations: ${\hat{\mathbf{I}}}_S{\dot{\mathbf{\Phi }}}_S+{\hat{\mathbf{I}}}_M{\dot{\mathbf{\Phi }}}_M={\mathbf{L}}_0$ (16). Eliminating ${\dot{\mathbf{\Phi }}}_S$ (17) into (14) produces the GJM (18)–(19). The appendix defines the roll-pitch-yaw transform $A_0=E^{k\gamma }{E}^{j\beta }{E}^{i\alpha }$ (App. 1) and the base angular-velocity / Euler-rate map (App. 6).

Relevance to thesis

This is the foundational GJM paper that the entire free-floating space-robotics literature builds on, and it is the natural baseline against which our free-FLYING formulation must be positioned. Key contrast: their GJM exists because the base is uncontrolled and momentum is conserved — the conservation laws are what reduce $n+3$ unknowns to a solvable system. For our fully-actuated 6-DOF base, momentum is not conserved (base thrusters/torquers inject external wrench), so ${\mathbf{J}}^{\ast }$ in this exact form does not apply; instead the base DOFs are independent controls. The paper’s Sec. 4.3 attitude-control treatment (impose ${\dot{\mathbf{\Phi }}}_S=0$, solve for required ${\mathbf{L}}_c$) is the conceptual bridge to the actuated-base case and worth studying for how it embeds base actuation as a constraint. The dynamic-singularity insight (the GJM has configuration-dependent singularities distinct from the kinematic Jacobian) is directly relevant to our singularity/redundancy analysis.

Connections

Topics: generalized_jacobian · free_flying_vs_free_floating · resolved_motion_rate_control · momentum_conservation

Key Equations / Quotes

“the authors investigate the kinematics of free-flying multi-jointed link systems by introducing the momentum conservation law into the formulation and derive a new Jacobian matrix in generalized form for space robotic arms.” (Abstract, p. 1) — note: the abstract says “free-flying,” but assumption (2) makes the regime free-floating (no base control); the term usage is pre-standardization.

Generalized Jacobian (Eqs. (18)–(19)):
$\dot{\mathbf{P}}=({\hat{\mathbf{J}}}_M-{\hat{\mathbf{J}}}_S{\hat{\mathbf{I}}}_S^{-1}{\hat{\mathbf{I}}}_M){\dot{\mathbf{\Phi }}}_M+{\dot{\mathbf{P}}}_0\equiv {\mathbf{J}}^{\ast }{\dot{\mathbf{\Phi }}}_M+{\dot{\mathbf{P}}}_0,{\dot{\mathbf{P}}}_0={\mathbf{V}}_0+{\hat{\mathbf{J}}}_S{\hat{\mathbf{I}}}_S^{-1}{\mathbf{L}}_0$

Inverse kinematics (non-redundant, $n=6$, away from singularity) (Eq. (20)):
${\dot{\mathbf{\Phi }}}_M=[{\mathbf{J}}^{\ast }{]}^{-1}(\dot{\mathbf{P}}-{\dot{\mathbf{P}}}_0)$

Large-base-inertia limit (Eqs. (18)→): ${\hat{\mathbf{I}}}_S^{-1}{\hat{\mathbf{I}}}_M\to 0,{\hat{\mathbf{J}}}_M\to \mathbf{J}\Rightarrow \dot{\mathbf{P}}=\mathbf{J}{\dot{\mathbf{\Phi }}}_M$.

Reaction momentum for attitude hold (Eq. (22)): ${\mathbf{L}}_c=-{\hat{\mathbf{I}}}_M{\hat{\mathbf{J}}}_M^{-1}(\dot{\mathbf{P}}-{\dot{\mathbf{P}}}_0)$.

Open Questions

• Singularity structure of ${\mathbf{J}}^{\ast }$: the paper notes “away from singularities” $[{\mathbf{J}}^{\ast }{]}^{-1}$ exists but does not characterize the dynamic singularities (configuration- and mass-distribution-dependent) — later literature (Papadopoulos & Dubowsky) addresses this.
• Redundancy ($n>6$): only the square non-redundant inverse (20) is given; no pseudoinverse / null-space resolution is developed here.
• All control is off-line/open-loop RMRC; no feedback or disturbance-rejection treatment, and no uncertainty in the mass/inertia parameters that $K_{ij}$ and ${\mathbf{J}}^{\ast }$ depend on.