bruschi2025singularity

Singularity-Free Task-Priority Design for Trajectory Tracking in Space Robots

Authors: Bruschi et al. · Year: 2025 · Venue: IEEE conference (Denver, CO, 08-10 July 2025; added to IEEE Xplore 21 Aug 2025)
Raw: md

Summary

The paper proposes a hierarchical, singularity-free task-priority controller for trajectory tracking of a fully-actuated (free-flying) space manipulator during the pre-capture phase of an in-orbit servicing / debris-removal mission. End-effector pose tracking (the capture) is given strict priority over base pose tracking (safe distance / attitude), with the secondary task projected into the null space of the end-effector Jacobian. The key insight is that, because the 6-DOF base is actuated, the full end-effector Jacobian J_N(q) = [J_b J_m(q)] is full rank for all q even when the manipulator Jacobian J_m(q) is singular, so the primary task never loses rank. The design uses a 6D matrix (SR(6)) representation of rigid motion, yielding a control law that is globally valid across the state space and invariant to inertial-frame changes.

Key Claims

• The end-effector Jacobian J_N(q) = [J_b J_m(q)] has rank(J_N(q)) = 6 for all q ∈ T^N, because the base block J_b = ^N X_0 is always full rank; hence the outer-loop control law (12) is non-singular on the entire configuration manifold M = SR(6) × T^N (Property 1). This is the central enabling fact that distinguishes a free-FLYING base from a free-floating one.
• The controller achieves almost-global asymptotic tracking of any smooth end-effector trajectory (primary task), lim_{t→∞} X̃_N(t) = I_6 for almost all initial conditions (Theorem 1).
• The secondary (base) task converges, lim X̃_0(t) = I_6 and lim q̇(t) = 0, only when the desired trajectories are compatible (Definition 2) AND the manipulator stays non-singular, i.e. det(J_m(q) J_m(q)^⊤) ≠ 0 ∀ t > t_0. Lemma 1: when J_m(q) is full rank, J_0 N_N(q) J_0^⊤ is positive definite.
• The hierarchical design yields a cascade closed-loop structure: the base/joint subsystem (15)-(16) is perturbed by the end-effector subsystem (14); convergence follows from (almost) ISS arguments plus a GAS inner loop.
• The SR(6) formulation avoids gimbal-lock of minimal parameterizations and the double-cover of quaternions, and is invariant to inertial-frame change X' = ^{T'}X_T (Property 2) — useful when only relative state is measured.
• It distinguishes feasibility (Def. 1) from compatibility (Def. 2): feasibility is necessary for ground robots but NOT for space robots if only the end-effector trajectory matters, since the actuated base can place the end-effector with joints frozen.

Method

Regime: free-FLYING. The base is fully actuated with 6 DOF: the control input u := (τ_0, f_0, τ_q) ∈ R^{6+N} explicitly includes base torque AND base force, and the base pose X_0 is part of the state and a performance output. This is the load-bearing departure from free-floating literature, where the base is uncontrolled and momentum coupling constrains the reachable set; here the authors lean on full/over-actuation throughout.

Configuration: (X_0, q) ∈ M := SR(6) × T^N, a (6+N)-dimensional nonlinear manifold. Kinematics (1):

• Ẋ_0 = X_0 ν_0^×, q̇ = Ω.

Dynamics (2): H(q) v̇ + C(v,q) v = u, with v := (ν_0, Ω), exploiting skew-symmetry of ½ Ḣ − C.

End-effector pose by spatial product (3): X_N = X_0 ∏_1^N {}^{i-1}X_i X_{J_i}^{-1}(q_i).

Task Jacobians (4)-(5): J_0 = [I_6 0_{6×N}], J_N(q) = [J_b J_m(q)] with J_b := ^N X_0 and J_m(q) := [^N X_1 X_{J_1}(q_1) s_r ⋯ X_{J_N}(q_N) s_r], s_r := [0 0 1 0 0 0]^⊤ the revolute-joint motion constraint.

Modular inner-outer control (6): u := γ^v(v − v^c) + C(v,q) v^c + H(q) v̇^c. Outer loop sets the virtual velocity (12):
v^c := J_N(q)^† w(X̃_N, t) + N_N(q) J_0^⊤ w(X̃_0, t), with N_N(q) := I_{6+N} − J_N(q)^† J_N(q) the null-space projector and w(X̃_k, t) := γ^X(X̃_k) + X̃_k^{-1} ν̄_k(t) (feedback + feedforward).

Error coordinates (9): X̃_k := X̄_k^{-1} X_k, so R̃_k = R̄_k^⊤ R_k, p̃_k = R̄_k^⊤(p_k − p̄_k); convergence ⇔ X̃_k → I_6. Closed-loop primary error (14): Ẋ̃_N = X̃_N (γ^X(X̃_N))^×. Rigid-body stabilizer (18) from ref [4]: γ^X(X̃_k) = −(S^{-1}(skew(K_k^R R̃_k), R̃_k^⊤ K_k^p p̃_k)), with K_k^R symmetric, distinct eigenvalues, trace(K_k^R) I_3 − K_k^R > 0, K_k^p > 0.

Inner loop (20): proportional γ^v(ṽ) = −K^v ṽ, giving H(q) ṽ̇ = −C(v,q) ṽ − K^v ṽ; ṽ = 0 is GAS (Proposition 1).

Notation flag: the OCR mangles several symbols — v̇^c appears as {v}c in (6); det(...) = 0 in Lemma 1 should read ≠ 0 (non-singular case); equation (18) has an unbalanced parenthesis. Transcriptions above are corrected to the evident intent.

Relevance to thesis

This is directly on-target for a free-flying space manipulator. The core result — that an actuated 6-DOF base renders the combined end-effector Jacobian full rank everywhere, decoupling primary-task feasibility from manipulator kinematic singularities — is a clean nominal-algorithm building block. The strict task-priority / null-space projection cleanly separates capture (primary) from base safety/comms pose (secondary), and the cascade stability proof is a template for a rigorous nominal-then-risk development. The SR(6) frame-invariant formulation matches the relative-only-measurement setting typical of capture. The feasibility-vs-compatibility distinction (Defs. 1-2) is the natural place to later inject uncertainty / chance constraints on whether the base trajectory remains compatible.

Connections

Topics: kinematic_singularity · generalized_jacobian · task_priority_control · free_flying_vs_free_floating

Key Equations / Quotes

“While J_0 and J_b are always full rank by construction, the manipulator Jacobian J_m(q) can lose rank at certain joint angles configurations, which are known as kinematic singularities.”

“By observing that the end-effector can be placed wherever desired just by moving the base with the joint angles kept fixed, we propose a priority-oriented design that ensures tracking of the end-effector trajectory even when passing through singular configurations of the manipulator.”

Outer-loop law (12): v^c := J_N(q)^† w(X̃_N, t) + N_N(q) J_0^⊤ w(X̃_0, t).

Property 1 / Theorem 1: rank(J_N(q)) = 6 ∀ q ∈ T^N; lim_{t→∞} X̃_N(t) = I_6 for almost all initial conditions; and under compatibility with det(J_m J_m^⊤) ≠ 0, lim X̃_0 = I_6, lim q̇ = 0.

“Feasibility is a fundamental requirement for trajectory tracking in ground robots … However, this property is not strictly necessary for space robots if only the end-effector trajectory is of interest: controlling the base motion while keeping the joint angles fixed is sufficient to accomplish the task.”

Open Questions

• The inner loop is a deliberately simple proportional stabilizer; the authors defer sloshing, parametric uncertainty, and more robust inner-loop designs to future work — how does the cascade guarantee degrade under model error in H(q), C(v,q)?
• Convergence of the secondary task explicitly excludes manipulator-singular solutions (det(J_m J_m^⊤) ≠ 0); behavior of the base task exactly at / through singular q is not characterized.
• “Almost all initial conditions” — the excluded measure-zero set (the unstable rigid-body equilibria from the antipodal attitude error) is standard, but no explicit basin/region-of-attraction estimate is given.
• How is compatibility (Def. 2) guaranteed or recovered online when the target is tumbling and the estimated grasping-point trajectory drifts? (Natural entry point for the risk layer.)