yao2021adaptive

Adaptive trajectory tracking control of a free-flying space manipulator with guaranteed prescribed performance and actuator saturation

Authors: Yao · Year: 2021 · Venue: Acta Astronautica
Raw: md

Summary

Yao proposes adaptive trajectory-tracking controllers for an attitude-controlled free-flying space manipulator subject to parametric uncertainty, external disturbance, and actuator saturation. A model-based backstepping controller built on an asymmetric time-varying barrier Lyapunov function (BLF) enforces prescribed performance bounds on the position tracking errors; an RBF neural network then makes the scheme model-free by approximating the lumped uncertainty; finally a high-gain observer removes the velocity-measurement requirement, yielding an output-feedback controller. The whole closed loop is proven semi-globally uniformly ultimately bounded (SGUUB) while the position errors provably never leave the prescribed envelope.

Key Claims

• The asymmetric time-varying BLF (Eq. 16) keeps each position error component strictly inside the time-varying envelope -k_a(t) <= e_1(t) <= k_b(t), with the envelope shaped by an exponential performance function delta_i(t) = (e_{1i,0} - e_{1i,inf}) e^{-gamma_i t} + e_{1i,inf} (Eq. 15), giving prescribed convergence rate and maximum overshoot, provided -k_a(0) <= e_1(0) <= k_b(0).
• The RBFNN with sigma-modification adaptation (Eq. 29) makes the full-state controller model-independent: M(q), C(q,q̇), the saturation residual Δτ, and disturbance d may all be completely unknown.
• A high-gain observer (Eq. 45, Lemma 3) reconstructs q̇ from position only; the velocity estimation error ẽ_2 = ξ_2 satisfies ||ξ_2|| <= κ h_2 for all t >= t*, with κ a small gain.
• Both controllers are proven SGUUB (Theorems 1, 2). The full-state gain condition is λ_min(K_2 - (1/2) I_n) > 0 (Eq. 41); the output-feedback condition tightens to λ_min(K_2 - (3/2) I_n) > 0 (Eq. 63), reflecting the extra observer cross-terms.
• The error ultimate bound is e_{1i}^2 <= k_{·i}^2 (1 - e^{-R}) with R = 2(V(0) + N/ρ) (Eqs. 30, 44), so the BLF bound is never violated even though convergence is only to a neighborhood of zero.
• Numerical study: planar two-link manipulator post-capturing a non-cooperative target (n = 3 DOF: base attitude + 2 joints), τ_m = 20 Nm saturation, with full-state outperforming output-feedback on settling time and energy E_τ = (1/2) ∫ ||τ|| ds because velocity estimation costs transient time and effort.

Method

The system is an attitude-controlled free-FLYING space manipulator. The author’s three-way taxonomy (Sec. 1) is worth transcribing: type 1 = base free in rotation and translation (free-floating); type 2 = rotation actively controlled, translation free; type 3 = both rotation and translation controlled (arbitrary flight). Types 2 and 3 are both labelled “free-flying.” This paper treats the attitude-controlled case and writes the generalized coordinate q ∈ R^n as “the attitude of the spacecraft base and the joint angles” — i.e. it does NOT include base translation in the simulated model (n = 3 for a planar 2-link arm: one base attitude + two joints). So this is a controlled-base manipulator dynamics, not the fully 6-DOF actuated-translation case; the regime distinction matters for our thesis (see below).

Dynamics (Eq. 1, Euler–Lagrange form with saturation and disturbance):

M(q) q̈ + C(q,q̇) q̇ = sat(τ) + d

Saturation is factored multiplicatively as sat(τ) = Θ(τ) τ (Eq. 2-3), giving M q̈ + C q̇ = τ + Δτ + d with residual Δτ = (Θ(τ) - I_n) τ (Eq. 4). Standard properties are assumed: bounded inertia m I_n <= M(q) <= m̄ I_n, ||C|| <= c̄ ||q̇|| (Property 1), skew-symmetry of Ṁ - 2C (Property 2), and bounded disturbance ||d|| < d̄ (Assumption 1).

Backstepping with errors e_1 = q - q_d (Eq. 11) and e_2 = q̇ - μ (Eq. 12). The asymmetric BLF V_1 (Eq. 16) uses indicator p(i) to switch between the upper/lower barriers depending on the sign of e_1i. The lumped uncertainty -M μ̇ - C μ + Δτ + d is approximated by an RBFNN Ŵ^T S(Z) with regressor Z = [q^T, q̇^T, μ^T, μ̇^T]^T (Eq. 27). Output feedback replaces e_2 and Z by observer-based estimates ê_2, Ẑ (Eq. 48-49); Lemma 4 handles the regressor mismatch S(Ẑ) = S(Z) + ς S_l.

Regime note: the model is the attitude-controlled free-flying (controlled-base) case. Despite the title saying “free-flying,” base linear translation is not an actuated DOF in the worked model; the strong base/arm dynamic coupling is folded entirely into the uncertain M, C matrices and compensated by the NN rather than exploited structurally.

Relevance to thesis

This is a directly relevant point of comparison for our nominal free-flying controller. It offers: (i) a clean BLF route to hard prescribed-performance envelopes on tracking error — a candidate for our risk/safety layer, since the envelope can encode collision-avoidance margins between base and arm; (ii) a model-free RBFNN compensation of the lumped base/arm coupling and unknown captured-target inertia, relevant to our uncertainty handling; (iii) an output-feedback construction via high-gain observer if we ever face missing velocity measurements. Caveat for our pencil-and-paper judge: the paper’s “free-flying” model omits actuated base translation, so the dynamic-coupling structure is hidden inside uncertain M, C rather than derived (no generalized Jacobian, no reaction-coupling analysis). Our fully-actuated 6-DOF base demands that the coupling be modelled explicitly, so this paper is a control-architecture reference, not a dynamics reference.

Connections

Topics: trajectory_tracking · barrier_lyapunov_function · prescribed_performance_control · high_gain_observer

Key Equations / Quotes

“The first type is classified as the free-floating mode, while the latter two types are classified as the free-flying mode.” (p. 2)

“The attitude-controlled free-flying mode is the most commonly used when the space manipulator approaches or grasps a target since it significantly enhances the dexterity and flexibility of end-effector.” (p. 2)

Dynamics (Eq. 1): M(q) q̈ + C(q,q̇) q̇ = sat(τ) + d

Asymmetric time-varying BLF (Eq. 16):
V_1 = (1/2) Σ_i [ p(i) ln( k_bi^2 / (k_bi^2 - e_1i^2) ) + (1 - p(i)) ln( k_ai^2 / (k_ai^2 - e_1i^2) ) ]

Performance function (Eq. 15): δ_i(t) = (e_{1i,0} - e_{1i,inf}) e^{-γ_i t} + e_{1i,inf}

RBFNN lumped-uncertainty model (Eq. 27): -M μ̇ - C μ + Δτ + d = W*^T S(Z) + ε(Z)

Sigma-modified adaptation (Eq. 29): Ŵ̇_i = Γ_i S_i(Z_i) e_2i - η_i Γ_i Ŵ_i

High-gain observer error (Eq. 46-47): ξ_2 = ζ_2/κ - q̇, ẽ_2 = ζ_2/κ - μ - e_2 = ξ_2, with ||ξ_2|| <= κ h_2.

Convergence (Eq. 38 / 60): V̇ <= -ρ V + N, giving V <= V(0) + N/ρ and e_{1i}^2 <= k_{·i}^2 (1 - e^{-R}).

Open Questions

• The proof of Theorem 2 (output feedback) is stated “similar with that of Theorem 1 and thus is omitted.” The peaking phenomenon inherent to high-gain observers near t = 0 is not analyzed against the BLF — could observer peaking transiently violate |e_1i| < k_·i and blow up the BLF before t*? The author asserts the envelope is never violated but does not bound the interaction during the initial peaking window.
• The RBFNN regressor Z includes q̇ and μ̇; in output feedback these become estimates Ẑ, yet the text writes Ẑ = [q^T, q̇^T, μ^T, μ̇^T]^T (Eq. 48) identical to the full-state Z — likely an OCR/typo carry-over; the intended Ẑ should use observer estimates. Flagged.
• Extensions named as future work: finite-time / fixed-time convergence, and distributed coordinated tracking of a manipulator formation under a communication topology.