nanos2015avoiding

Avoiding Dynamic Singularities in Cartesian Motions of Free-Floating Manipulators

Authors: Nanos, Papadopoulos · Year: 2015 · Venue: IEEE Transactions on Aerospace and Electronic Systems, 51(3), pp. 2305–2318
Raw: md

Summary

For a free-floating space manipulator (spacecraft attitude control OFF, no thrusters), Cartesian end-effector motion is corrupted by dynamic singularities (DSs): configurations where the generalized Jacobian loses rank and whose location in the workspace is path-dependent (nonholonomic). The paper presents a trajectory-planning method that, for an arbitrary prescribed end-effector path and orientation, computes the initial spacecraft attitude (and corresponding initial configuration) that guarantees DS avoidance throughout the motion, thereby unlocking the full workspace including the path-dependent region (PDW). The method handles nonzero initial angular momentum, applies to planar and spatial systems, and is nonredundant (no extra DOF needed).

Key Claims

• A spatial free-floating manipulator needs a minimum of N = 6 revolute joints to follow a general 6-DOF end-effector trajectory (position + attitude); in the planar case the minimum is N = 3 (from the row-count of the combined momentum/velocity system, Eq. (4), requiring N ≥ 6).
• DS avoidance is enforced not by reaching S = det(S*) = 0 but by keeping S bounded away from 0 by a chosen margin S₀: require S_min > S₀ > 0 (or S_max < S₀ < 0). Values of S near 0 already cause large accelerations/torques and are undesirable.
• S_min is shown to be a continuous function of the initial spacecraft attitude (Eq. 13a; planar form S_min = S(θ₀,in), Eq. 13b), which makes the search for feasible initial attitudes well-posed.
• For planar systems the method yields the exact feasible RANGE of initial spacecraft attitudes (Eqs. 29a/29b), not just isolated solutions; Appendix C proves the number of boundary attitudes θ₀,in satisfying S_min = S₀ in [0,2π) is always even.
• Reversal property (Appendix A): running the path backwards (reversed end-effector velocity AND reversed-sign angular momentum) over the same duration returns the system to its exact starting configuration (x₃ = x₁, Eq. 47). This lets a known attitude at one path point be propagated to feasible attitudes elsewhere.
• With nonzero angular momentum the singularity-avoidance solution depends on the motion’s final time t_f (Eq. 27) — a dependence absent in the zero-momentum case.
• For a 3-DOF planar arm with specified end-effector orientation, the 3-DOF singularity problem reduces to a 2-DOF problem in (q₁, q₂): det(S*) is independent of q₃ (Eq. 21a). Analogously, a spatial 3-DOF-plus-spherical-wrist arm has S = S(q₁,q₂,q₃) independent of wrist angles (Eq. 37).
• For an arm mounted on a spacecraft principal axis with the two transverse moments of inertia equal, S* factors as ⁰R₁(q₁)·S_n(q₂,q₃), so S = det(S_n(q₂,q₃)) is independent of q₁ (Eqs. 39–40); singularity/margin surfaces are then surfaces of revolution with planar-like cross-sections.

Method

Regime: free-FLOATING (spacecraft attitude controller off; spacecraft drifts in reaction to arm motion via dynamic coupling). The paper does NOT control the base — base attitude is an outcome, and the design lever is the initial base attitude. This is the key contrast with our free-FLYING vehicle, whose fully actuated 6-DOF base would never exhibit these momentum-induced DSs in the first place; here the singularity is purely a consequence of the uncontrolled base.

Model (N-DOF arm, N+6 total DOF). Zero external force ⇒ fixed system CM at inertial origin O. Angular-momentum conservation (Eq. 1):

⁰D(q)⁰ω₀ + ⁰D_q(q) q̇ = R₀ᵀ(ε,n) h_CM

End-effector twist (Eq. 2): v_E = [ṙ_E; ω_E] = C(ε,n)[⁰J*₁₁ ⁰J*₁₂][⁰ω₀; q̇], with C = blkdiag(R₀, R₀).

Stacking momentum (1) and twist (2) gives the 9×(N+3) system (Eq. 4) with matrix A (Eq. 5). For N = 6, a unique solution exists iff det(A) = det(⁰D)·det(S*) ≠ 0, hence the DS condition is

S(q) = det(S*(q)) ≠ 0, S* = ⁰J*₁₁ ⁰D⁻¹ ⁰D_q + J*₁₂ (Eqs. 6–7)

S* is the generalized Jacobian of Umetani & Yoshida. Solving (4) gives base rate ⁰ω₀ (Eq. 8a) and joint rates q̇ (Eq. 8b), with quaternion kinematics (Eq. 8c) integrated alongside. The workspace splits into the PIW (path-independent, DS-free) and PDW (path-dependent, DS may occur depending on path) per Papadopoulos & Dubowsky.

Planning logic (Section IV–V):

• A margin curve/surface is the level set S(q) = S₀ ≠ 0; a singularity curve/surface is S₀ = 0. A motion curve is the joint-space trace of (8) for a given path; a distance curve is R(q) = R_i (Eq. 16, 28).
• Case I (interior minimum): require the motion curve tangent to the margin curve — common tangent condition λ₁ = λ₂ (Eqs. 23a/23b, 27), where λ₁ = −(∂S/∂q₁)/(∂S/∂q₂) and λ₂ = q̇₂/q̇₁.
• Case II (endpoint minimum): the path endpoint just touches the margin curve; touch point = intersection of margin curve with the distance curve R(q) = R_i.
• The closed-form margin-curve solutions for the planar 2-DOF-reduced problem (Eqs. 22a/22b) come from writing S = k₀(q₁) + k₁(q₁)s(q₂) + k₂(q₁)c(q₂) (coefficients in Appendix B).
• Feasible ranges for a single PDW area: Eqs. 29a/29b. Multi-PDW paths are split at common points lying in the PIW into L segments; feasible ranges per segment are intersected (Θ¹₁&₂ = Θ¹₁ ∩ Θ¹₂) and propagated by the reversal property (flowchart, Fig. 5).

Path parameterized by quintic arclength s(τ) = 10τ³ − 15τ⁴ + 6τ⁵, τ = t/t_f (Eqs. 18b, 19), giving zero endpoint velocity/acceleration.

Examples. Ex. 1: planar 3-DOF, h₀ = 0.5 N·m·s, parabolic path A→B with end-link normal to path, t_f = 100 s; method yields feasible initial range Θ^A_AB = [0°,63°] ∪ [161°,360°]; an attitude at 140° (outside) goes singular, 200° (inside) succeeds. Ex. 2: spatial 6-DOF (3-DOF arm + spherical wrist), h_CM = [0.5,0.3,0.1]ᵀ N·m·s, straight line A→B, S₀ = −5; a feasible initial quaternion is found that follows the path DS-free, whereas a naive initial quaternion hits a DS at point C.

Relevance to thesis

This is the canonical statement of the dynamic singularity problem and the generalized-Jacobian formalism we must reference. Crucially it is a free-floating result: every DS here is induced by the uncontrolled base reacting through angular-momentum conservation. For our free-flying (fully actuated 6-DOF base) manipulator the base attitude is a controlled input, so these momentum-coupling DSs do not arise — but the paper sharply defines what we are buying with base actuation, and the generalized-Jacobian / dynamic-coupling machinery (Eqs. 1–8) is the same backbone we use. The continuity-of-S_min argument, the margin-vs-singularity level-set geometry, and the quintic path parameterization are directly reusable. The reversal/time-dependence results are a useful cautionary tale for any planner that relies on momentum, relevant if we ever operate our base in a degraded (thrusters-off) contingency mode.

Connections

Topics: dynamic_singularity, generalized_jacobian, free_floating_dynamics, cartesian_path_planning

Key Equations / Quotes

“At DSs, the manipulator is unable to move its end-effector in some inertial direction, while their location in the workspace is path dependent due to the nonholonomic nature of the Cartesian motion.” (Introduction, p. 2305)

“DS avoidance then requires that S_min > S₀ > 0 or S_max < S₀ < 0, where S₀ a constant.” (Section IV)

Generalized Jacobian and DS condition (Eqs. 6–7):

det(A) = det(⁰D) det(S*(q)) ≠ 0 ⇒ S(q) = det(S*) ≠ 0, S* = ⁰J*₁₁ ⁰D⁻¹ ⁰D_q + J*₁₂

Time dependence under nonzero momentum (Eq. 27):

λ₁(q₁ₜ,q₂ₜ) = λ₂(θ₀ₜ, qₜ, τₜ, h_CM, t_f)

Reversal property (Eq. 47): x₂ − x₁ = −(x₃ − x₂) ⇒ x₃ = x₁.

Open Questions

• Computing the full feasible attitude ranges for spatial systems (3 independent attitude variables) is stated as complex and left as future work; only isolated feasible attitudes are obtained in Ex. 2.
• The method assumes accurately known mass/inertia parameters (barycentric vectors a,b,c,d depend on mass distribution); robustness to parameter uncertainty / identification is acknowledged as out of scope.
• Selecting the margin S₀ trades workspace coverage against acceleration/torque headroom; no systematic rule is given (Ex. 1 uses S₀ ≈ 3.3% of S_max heuristically).