caccavale2001kinematic

Kinematic Control of Redundant Free-Floating Robotic Systems

Authors: Caccavale, Siciliano · Year: 2001 · Venue: Advanced Robotics, Vol. 15, No. 4, pp. 429–448
Raw: md

Summary

The paper develops a family of closed-loop inverse kinematics (CLIK) algorithms for a redundant manipulator mounted on a free-floating (uncontrolled, momentum-conserving) spacecraft. Reaction coupling is captured by the generalized Jacobian of Umetani and Yoshida, and the central methodological contribution is replacing minimal (Euler-angle) orientation descriptions with the unit quaternion to obtain a globally singularity-free orientation error for both the spacecraft attitude and the end-effector pose. Redundancy with respect to the combined spacecraft/end-effector task is resolved by task-space augmentation with a task-priority projection; a notable special case recovers the fixed-attitude-restricted (FAR) Jacobian when the spacecraft attitude is to be held constant.

Key Claims

• The generalized Jacobian $J_G$ folds the reaction of manipulator motion on the free-floating base into a single $(6\times n)$ map $v_e=J_G\dot{q}$; as spacecraft inertia grows, $J_G\to {\bar{J}}_e$ (the manipulator Jacobian), recovering the fixed-base limit.
• A quaternion-based CLIK (Eq. 19) uses the geometric Jacobian rather than the analytical Jacobian, so representation singularities cannot occur in the orientation error dynamics; only the vector part $e_{de}$ of the relative quaternion enters the error.
• The Euler-angle CLIK (Eq. 14) employs the analytical Jacobian $J_A$, which is singular at representation singularities; the case studies demonstrate that crossing such a singularity corrupts both spacecraft and end-effector motion, whereas the quaternion algorithm tracks cleanly through the same trajectory.
• Task-priority redundancy resolution (Eq. 21) projects the lower-priority (constraint) task onto $\mathcal{N}(J_G)$ via $(I_n-J_G^{†}{J}_G)$ and uses the transpose of the constraint Jacobian with a feedback correction, avoiding a second pseudoinversion and improving robustness to algorithmic singularities from conflicting tasks.
• For the constant-attitude task (${\omega }_{sd}=0$), algebraic manipulation collapses the projected solution exactly to the transpose of the fixed-attitude-restricted (FAR) Jacobian: $J_G(I_n-J_C^{†}{J}_C)={\bar{J}}_e(I_n-M_e^{†}{M}_e)$ (Eq. 26), giving end-effector trajectories that do not disturb the base attitude.

Method

Regime: explicitly free-FLOATING. The base has no actuation — “there are no devices intended to change spacecraft attitude, e.g. reaction wheels or thrusters” (p. 433). Linear and angular momentum are conserved and assumed initially null, which is exactly what makes the generalized Jacobian valid. This contrasts with our free-FLYING platform (fully actuated 6-DOF base), where the base velocity $v_s$ is an independent control input rather than a quantity eliminated via momentum conservation.

Kinematics. End-effector pose relative to the inertial frame (Eqs. 1–2):
$r_e=r_s+R_s{r}_{es}^s(q),R_e=R_s{R}_e^s(q),$
where $r_{es}^s(q)$ and $R_e^s(q)$ are the ordinary direct kinematics in the base frame ${\Sigma }_s$. Differentiating gives the generalized velocity map (Eq. 3) coupling base velocity $v_s$ and joint rates $\dot{q}$.

Momentum elimination. With initial CoM velocity null, translational and rotational momentum conservation give (Eq. 9) $M_s{\omega }_s+M_e\dot{q}=0$. Since $M_s$ (spacecraft rotational inertia) is non-singular, ${\omega }_s$ is solved and substituted, eliminating dependence on base attitude changes and yielding (Eqs. 10–11):
$v_e=J_G\dot{q},J_G={\bar{J}}_e-{\bar{J}}_s{M}_s^{-1}{M}_e.$

CLIK algorithms. Position error $\Delta r_{de}=r_d-r_e(q)$ (Eq. 12). Two orientation formulations:

• Euler-angle error $\Delta {\varphi }_{de}={\varphi }_d-{\varphi }_e(q)$ with solution $\dot{q}=J_A^{-1}({\dot{x}}_d+K_P\Delta x_{de})$ (Eq. 14), $J_A=\mathrm{blockdiag}(I_3,S^{-1}(\varphi ))J_G$ (Eq. 15). Yields exponentially stable error (Eqs. 16–17) only away from representation singularities.
• Quaternion error $e_{de}=R_e{e}_{de}^e$ (Eq. 18), using only the vector part of the relative Euler parameters (App. Eqs. A.11–A.12). Solution $\dot{q}=J_G^{-1}(v_d+K_P{e}_{de})$ (Eq. 19) gives non-homogeneous error dynamics $\Delta {\omega }_{de}+k_{Po}{e}_{de}=0$ (Eq. 20), stability via a Lyapunov argument.

Redundancy ($n>m_s+m_e$). Priority projection (Eq. 21):
$\dot{q}=J_G^{†}(v_d+K_P{e}_{de})+(I_n-J_G^{†}{J}_G)J_C^T{K}_C{e}_C.$
When $n-m_e=m_s$, the constraint Jacobian for the base-attitude task derives from Eq. 9 as $J_C=-M_s^{-1}{M}_e$ (Eq. 22). Switching priority to keep attitude constant gives Eq. 24, which simplifies (Eqs. 25–27) to the FAR-Jacobian transpose form.

Notation flag. The OCR renders the imaginary/angular-velocity symbol $\omega$ as “x”/"$\Omega$" inconsistently (e.g. "${\omega }_e$" appears as "$x_e$"), the quaternion scalar $\eta$ appears as ”´”, and various accents are mangled. Equation numbers and matrix structure are intact; symbol glyphs should be read with care.

Relevance to thesis

This is a foundational reference for the kinematic layer of coordinated spacecraft/manipulator control and the canonical demonstration that the unit quaternion removes orientation-representation singularities from CLIK. Two cautions for our free-flying work: (i) the entire generalized-Jacobian machinery rests on momentum conservation with no base actuation — for our fully actuated base, ${\omega }_s$ is not eliminated but commanded, so $J_G$ is replaced by the direct $(v_s,\dot{q})$ map of Eq. 3, and the coupling becomes a design freedom rather than a constraint. (ii) The task-priority/null-space and quaternion-error constructs, however, transfer directly and are arguably more natural for us, since base attitude can be a controlled task variable instead of an emergent reaction. The FAR-Jacobian result is the free-floating analogue of “command the arm without disturbing the base,” which for a free-flyer is replaced by explicit reaction compensation through the actuated base.

Connections

Topics: generalized_jacobian · unit_quaternion · closed_loop_inverse_kinematics · task_priority_redundancy

Key Equations / Quotes

“the reaction effects of the manipulator motion on the spacecraft can be taken into account by proper kinematic and dynamic modeling of the whole system. To the purpose, a general description of the reaction effects is the generalized Jacobian concept” (p. 430)

“there are no devices intended to change spacecraft attitude, e.g. reaction wheels or thrusters. In this case, momentum conservation dictates that …” (p. 433)

Generalized Jacobian (Eqs. 10–11):
$v_e=J_G\dot{q},J_G={\bar{J}}_e-{\bar{J}}_s{M}_s^{-1}{M}_e.$

Quaternion-based CLIK (Eq. 19):
$\dot{q}=J_G^{-1}(q)(v_d+K_P{e}_{de}),e_{de}=[\Delta r_{de}^T{e}_{de}^T{]}^T.$

Task-priority redundancy (Eq. 21):
$\dot{q}=J_G^{†}(v_d+K_P{e}_{de})+(I_n-J_G^{†}{J}_G)J_C^T{K}_C{e}_C.$

FAR-Jacobian identity (Eq. 26):
$J_G(I_n-J_C^{†}{J}_C)={\bar{J}}_e(I_n-M_e^{†}{M}_e).$

“differently from the previous Euler angles-based algorithm (14), the geometric Jacobian appears in lieu of the analytical Jacobian and thus representation singularities cannot occur” (p. 435)

Open Questions

• The pseudoinverse-based priority scheme (Eq. 21) “assumes $J_G$ is non-singular” — the paper does not characterize dynamic/algorithmic singularities of $J_G$ itself (distinct from representation singularities, which the quaternion removes). How robust is the scheme near a singularity of the generalized Jacobian?
• Case study with Eq. 24 exhibits a residual steady-state end-effector orientation error from an algorithmic singularity caused by conflict between the end-effector-orientation constraint and the spacecraft-orientation task; the paper notes but does not resolve this conflict-handling limitation.
• Extensions flagged but not pursued: use of redundancy to minimize impact effects in microgravity, and multi-arm systems.