Inertial-Space Disturbance Rejection for Space-Based Manipulators
Authors: Holt, Desrochers · Year: 1992 · Venue: NASA Conference Publication / NASA Conf. Space Telerobotics (NASA report N94-26281)
Raw: md
Summary
The paper presents a kinematic, end-point-feedback control law that uses a 6-DOF PUMA manipulator to reject unknown disturbances in the joints of the 3-DOF platform on which it is mounted, keeping the end-effector tracking a desired pose in inertial space. The central technical device is an “approximate pseudoinverse Jacobian” J‡ that exploits the block-triangular structure of the PUMA Jacobian, replacing one 6×6 inverse/SVD with two 3×3 SVDs (≈4× faster) and remaining well-defined through kinematic singularities. Experimental results on real hardware are reported for step (10° and 30°), sinusoidal, and random disturbances in the platform rotational axis, including behavior near the Arm Fully Stretched singularity.
Key Claims
- The approximate pseudoinverse J‡ equals J⁻¹ when J is nonsingular (property 1), but does not satisfy the Moore–Penrose conditions when J is singular (property 2).
- Computing J‡ costs about one quarter of the pseudoinverse: two 3×3 SVDs are O(2(N/2)³) vs. one O(N³) 6×6 SVD. Table 1 reports ≈6 ms (J‡) vs. ≈25 ms (J†) vs. ≈1 ms (J⁻¹, where defined).
- A threshold σ_min on the smallest nonzero singular value trades tracking accuracy against control-signal norm: larger σ_min widens the singular “well” (smaller, smoother control near singularities); smaller σ_min shrinks the well but yields a large, highly discontinuous norm.
- Closed-loop stability is governed by the matrix M_{k,k-1}; with the scalar gain choice K_c = k_c I, 0 ≤ k_c ≤ 2, the eigenvalues of M lie on a circle of radius k_c (or at zero when J is singular).
- The J‡ controller behaves like a high-pass filter on the disturbance: low-frequency disturbance components are attenuated most; relative stability depends on disturbance amplitude, relative performance on disturbance frequency.
- Near a singularity the control becomes very weak in certain (“forbidden”) directions, implying unavoidable tracking error there; with σ_min = 0.1 the control along the workspace boundary weakens ~30° before the Arm Fully Stretched point, preventing the arm from crossing it; too-small σ_min can cause “chattering” between flex/noflex configurations.
Method
A 6-DOF PUMA is mounted on a 3-DOF platform; the goal is to hold the inertial end-effector pose ⁰u_{0,E} against arbitrary, unknown platform-joint disturbances. Regime note: despite the title’s “space-based,” the platform here is actively disturbed but the base motion is unknown to the controller — this is closer to a free-flying base (fully actuated platform DOF, treated as an external disturbance) than to a free-floating (momentum-conserving, uncontrolled) base. The paper does not use generalized-Jacobian / reaction-dynamics machinery; the platform enters purely kinematically as an unknown joint displacement δ, and end-point sensing is assumed available because the disturbance is not known exactly (only nominal platform pose η_o and bound δ̄ are known).
The Jacobian maps joint to Cartesian rates, du = J(q) dq (1). In frame 6 the PUMA Jacobian is block lower-triangular,
⁶J_{3,9} = [[B, 0], [D, E]] (3),
so the exact inverse is [[B⁻¹, 0], [−E⁻¹DB⁻¹, E⁻¹]] (4). The approximate pseudoinverse replaces the inverses by 3×3 pseudoinverses:
J‡ ≜ [[B†, 0], [−E†DB†, E†]] (6).
When J is singular it returns the minimum-norm solution as if dp (linear) and dφ (angular) were decoupled, i.e. it separately minimizes ‖B dq₁ − dp‖₂ and ‖E dq₂ − dφ‖₂ (property 3, eq. 7). The approximation error is measured by ‖JJ‡J − J‖₂ (8)–(9), bounded by ‖J‖₂ when both B,E singular; by ‖E‖₂ if only B nonsingular (11); by ‖B‖₂ if only E nonsingular (12); and 0 when both nonsingular (13).
The disturbance-rejection model isolates platform dependence by a frame transform:
⁰du_{0,E} = ⁰J_{3,E}(η_o+δ, θ) dθ + dv = ⁰₃R(η_o+δ) ³J_{3,E}(θ) dθ + dv (14).
Discretizing (Δu_k = u_k − u_{k-1}, etc.) gives (17). The proposed J‡ control law uses the nominal platform rotation ⁰₃R(η_o) (the disturbance δ is not used in the controller):
Δθ_d = ³J‡_{3,E}(θ_k) ³₀R(η_o) K_c (⁰u_d − ⁰u_k) (19).
With a one-period actuation delay (20) and M_{k,k-1} ≜ ³₀R(η_o+δ_k)³J_{3,E}(θ_k)³J‡{3,E}(θ{k-1})³₀R(η_o)K_c (22), the closed loop is linear time-varying:
⁰u_k = (I − M_{k,k-1}) ⁰u_{k-1} + M_{k,k-1} ⁰u_d + Δv_k (23).
Gain design: K_c = k_c I (24); if δ̄ is known, α ≈ sup_i arg(λ̄) of ³₀R(η_o+δ̄)³₀R(η_o) (25) and k_c = 2/√(tan²α + 1) (26).
Relevance to thesis
This is an early, hardware-validated attempt to use a manipulator to compensate for base motion — the kinematic dual of the coordinated free-flying control problem. For our fully-actuated 6-DOF base it is a cautionary baseline: Holt treats the base as an unknown disturbance with end-point feedback rather than exploiting known base kinematics/dynamics (the generalized-Jacobian approach). The σ_min thresholding and the explicit “weak control in forbidden directions near singularities” result are directly relevant to singularity-robust inverse-kinematics layers in our guidance stack, and the high-pass-filter characterization plus the amplitude/frequency dependence of stability are useful priors when we add a risk-aware layer over base-motion uncertainty.
Connections
Topics: singularity_robust_inverse, damped_least_squares, inertial_space_tracking, base_disturbance_rejection
Key Equations / Quotes
“The usual method of dealing with singularities of the Jacobian is to avoid them. This approach is not applicable to the disturbance rejection problem since a sufficiently large disturbance could force the manipulator into a singular configuration.” (Sec. 2)
“If J is singular, the approximate pseudoinverse finds the minimum norm solution as if dp and dφ were decoupled” (Sec. 2, property 3)
“the J‡ controller is like a high-pass filter; the lowest frequency components of the disturbance signal are attenuated the most.” (Sec. 9)
“The control in certain directions becomes very weak near singularities. This implies that there may be an unavoidable tracking error in the ‘forbidden’ directions when the arm is at or near a singularity.” (Sec. 9)
Control law (19): Δθ_d = ³J‡_{3,E}(θ_k) ³₀R(η_o) K_c (⁰u_d − ⁰u_k)
Approximate pseudoinverse (6): J‡ = [[B†, 0], [−E†DB†, E†]]
Gain (26): k_c = 2 / √(tan²α + 1)
Open Questions
- The approximate pseudoinverse abandons the Moore–Penrose conditions and the linear/angular decoupling is heuristic; what is lost in optimality relative to a damped least squares (Levenberg–Marquardt) inverse, which the paper does not compare against?
- Stability is argued only via the spectral radius of a “slowly time-varying” M; no rigorous LTV/robust-stability proof is given for fast or large disturbances. Under what δ̄ does the slowly-varying assumption break?
- Block-triangular structure is specific to the PUMA frame-6 Jacobian; does J‡ generalize to a redundant or differently-structured space manipulator Jacobian?
- Results are limited to disturbances in the platform rotational axis; how does the method perform under coupled translational+rotational base disturbances?