calzolari2020singularity

Singularity Maps of Space Robots and their Application to Gradient-based Trajectory Planning

Authors: Calzolari, Lampariello, Giordano · Year: 2020 · Venue: Robotics: Science and Systems (RSS)
Raw: md

Summary

The paper presents a formal numerical method to compute singularity maps (S-Maps) in the configuration space of free-floating space robots, i.e. conservative enclosures of the configurations where the generalized Jacobian loses rank, with the complementary space guaranteed singularity-free. Two formal criteria are compared — Interval Arithmetic (IA) regularity testing of the generalized Jacobian, and Taylor-model evaluation of a determinant — within a branch-and-bound search, aided by heuristic pruning from a manipulability point cloud. The authors then build a C¹ signed-distance function to the S-Map and use it as a collision-avoidance-style inequality constraint inside gradient-based trajectory planning, demonstrated on a planar 3-DOF and a spatial 6-DOF (arm) free-floating robot.

Key Claims

• For spatial free-floating robots, the Path-Independent / Path-Dependent Workspace concepts of Papadopoulos & Dubowsky lose usefulness: “the entire workspace can become PDW,” so singularities should be analyzed/planned in configuration space, where the singularity surfaces are fixed (J*_m depends only on q).
• Using the inverse-chain factorization (11), all dynamic singularities arise solely from the degeneracy of M_em(q): det(J_m) = -det(M_ee⁻¹) det(M_em), and det(M_ee⁻¹) is never zero since M_ee is an inertia. This makes Taylor-model determinant evaluation on the smaller, simpler M_em tractable where it is infeasible on J_m itself.
• Singularities do not depend on the last link’s inertial properties nor on the last joint coordinate; the search space can therefore be reduced by one dimension.
• Taylor models drastically reduce conservativeness vs IA: for the planar 3-DOF case the Taylor-model map took 225 s vs 3300 s for IA at comparable precision (<7% of the time), because dependency tracking avoids the IA “wrapping effect.”
• IA’s regularity criterion (15) extends straightforwardly to non-square (redundant) Jacobians; the Taylor determinant criterion does not (it requires a square matrix), trading off the two methods’ applicability.
• The spatial 6-DOF S-Map (≈200k boxes, ~18 MB uncompressed) supports fast online queries: distance evaluation ~3 ms, segment-intersection test ~30 ms.
• An S-Map gives global knowledge of singularity location, so a planner can keep a guaranteed minimum distance Δ from singularities; a purely local manipulability-maximizing constraint can fail to detect an infeasible step that crosses a singularity (the determinant changes sign).

Method

Regime: free-FLOATING. The base spacecraft is unactuated; the analysis hinges on conservation of momentum to eliminate the base twist. The paper notes (Sec. VI) that the resulting S-Maps may also apply to base-actuated (free-flying) feedback controllers “whose singularity is determined by J*_m” — relevant to our setting but secondary in the paper.

Equation of motion, zero external base wrench (3):

$\left[\begin{array}{cc}{M}_b& M_{bm}\\ M_{bm}^T& M_m\end{array}\right]\left[\begin{array}{c}{\dot{\nu }}_b\\ \ddot{q}\end{array}\right]+c=\left[\begin{array}{c}0\\ \tau \end{array}\right]$

End-effector twist (5): ν_e = J_b ν_b + J_m q̇. With zero momentum h_b = 0, conservation (6) M_b ν_b = -M_bm q̇ is substituted to give the generalized Jacobian (8):

$J_m^{\ast }(q)=J_m-J_b{M}_b^{-1}{M}_{bm},{\nu }_e=J_m^{\ast }(q)\dot{q}.$

The inverse-chain reformulation (Abiko et al.) treats the end-effector as the base, yielding the alternative factorization (11): J_m = -M_ee⁻¹ M_em, hence the key determinant identity (13). Dynamic singularities are det(J_m(q)) = 0 (12) for the non-redundant case.

Two regularity criteria over an interval box q

• Criterion 1 (IA, eqn 14–15): enclose J*_m^I via IA (matrix inverse enclosed by Rohn–Farhadsefat). Strong regularity follows from Rex & Rohn: if ρ(|mid(A^I)⁻¹| rad(A^I)) < 1 then A^I is regular.
• Criterion 2 (Taylor models, eqn 16): build a 3rd-order Taylor model T_det of det(M_em(q^tm)); if 0 ∉ B(T_det) (an interval bound on the Taylor model) the set is regular.

A branch-and-bound search bisects the configuration box (initial 2^N boxes), testing each; boxes that cannot be proven regular at precision ε are stored as the S-Map. Heuristic pruning generates a point cloud S_q of near-singular configs by gradient descent on manipulability m(q) = |det(J*_m)| (17), then declares small boxes containing an S_q point as singular to avoid full bisection.

Utilization: each S-Map box is enclosed by a sphere (center = box midpoint, radius = norm of interval radius vector). The signed distance (18)–(19) d_s(q) = min_i (‖q^i_sm − q‖₂ − r^i_sm) is C¹. A segment-intersection test (20)–(22) gives a necessary-and-sufficient condition for a segment q_ab to pierce the map. Trajectory planning solves min_q γ(q) s.t. d_s(q) > Δ (23) via projected-gradient steps with line search guaranteeing the step does not cross the S-Map.

Relevance to thesis

Directly addresses dynamic (generalized-Jacobian) singularity avoidance for space manipulators — a core obstacle for our free-flying guidance/planning. Although the analysis assumes a free-floating base (momentum conservation eliminates ν_b), the authors explicitly flag that the same J*_m-based singularity structure governs certain base-actuated controllers (Giordano et al. workspace fixation / momentum dumping), which is precisely our fully-actuated regime. The C¹ signed-distance S-Map reframes singularity avoidance as collision avoidance, providing a global, guaranteed minimum-distance constraint — a clean primitive for our nominal planner and a natural anchor point for the later risk-aware layer (the margin Δ could absorb tracking/modeling error, hinting at a chance-constrained reading of singularity proximity).

Connections

Topics: generalized_jacobian, dynamic_singularity, interval_arithmetic, trajectory_optimization

Key Equations / Quotes

“det(J*_m(q)) = -det(M_ee⁻¹(q)) det(M_em(q))” — eqn (13); singularities arise solely from degeneracy of M_em.

“a representation of singularities in the workspace is not meaningful for spatial free-floating robots since, practically, the whole Cartesian space would appear singular … This does not happen in the configuration space, where the singularity surfaces are fixed.” (Sec. II-B)

Rex–Rohn regularity: ρ(|mid(A^I)⁻¹| rad(A^I)) < 1 ⇒ A^I strongly regular — eqn (15).

Signed distance: d_s(q) = min_i (‖q^i_sm − q‖₂ − r^i_sm) — eqns (18)–(19), C¹ continuous.

“the application of Taylor models for the detection of singularities is a novel approach.” (Contribution 1)

Open Questions

• How to extend the method to kinematically redundant arms? (IA criterion handles non-square Jacobians, but the Taylor determinant criterion does not; named as primary future work.)
• The sphere enclosure yields only C¹ distance with cusps; can a smooth C² mesh-based enclosure (per Luo et al.) improve optimizer convergence?
• The 6-DOF map took 170 hours on 8 cores for a restricted joint range; what implementation/parallelization is needed for practical, full-range coverage?
• How should Δ be chosen to guarantee joint-velocity feasibility near singularities rather than merely positional distance?
• Does the J*_m-based map transfer rigorously to base-actuated (free-flying) controllers, or only heuristically?