giordano2019coordinated

Coordinated Control of Spacecraft’s Attitude and End-Effector for Space Robots

Authors: Giordano, Ott, Albu-Schäffer · Year: 2019 · Venue: IEEE Robotics and Automation Letters (RA-L)
Raw: md

Summary

This letter proposes a coordinated controller that simultaneously regulates a space robot’s spacecraft attitude, the system center-of-mass (CoM) position, and the end-effector pose, while deliberately leaving the spacecraft translation free. The key device is a “triangular” dynamics/actuation decomposition built on a circumcentroidal velocity coordinate that decouples the end-effector task from the spacecraft’s force (thruster) actuator. Because the base force is then needed only to relocate the CoM — not to execute end-effector or attitude tasks — contact-free maneuvers cost zero (or near-zero) translational fuel. The method is proven asymptotically stable via a cascade Lyapunov argument and validated on the DLR OOS-Sim hardware-in-the-loop facility.

Key Claims

• The system can be expressed in transformed coordinates (v_c, ω_b, ν_e^⊕) for which the CoM translational equation is inertially decoupled from the coupled base-attitude/end-effector subsystem: m v̇_c = f_c (22a), with the remaining 9-DOF block (22b) carrying the coupling perturbation C_c v_c.
• The actuation map (32) is block lower-triangular: the base force f_b is activated only to control the inertial CoM location; the end-effector and base-attitude tasks never command f_b. This decoupling is a specific property of the circumcentroidal velocity ν_e^⊕ and does not hold for the absolute end-effector velocity ν_e (whose actuation map (33) is full).
• Consequence: during any contact-free maneuver the CoM is conserved, so x̃_c = v_c = 0 and hence f_b = 0 — contact-free end-effector maneuvering requires no base force. If τ_b is supplied by momentum-exchange devices (reaction wheels) rather than thrusters, contact-free maneuvering consumes exactly zero fuel.
• The transformed attitude/end-effector inertia and Coriolis matrices satisfy the skew-symmetry passivity property [ω_b^T ν_e^⊕T](Ṁ̆ − 2C̆)[ω_b; ν_e^⊕] = 0 (23), which is what makes the Lyapunov derivative collapse to V̇ = −v̆^T D̆ v̆ ≤ 0.
• Fuel-consumption ranking (ideal simulation, eq. 39 model): full base control > partial base control (proposed) > floating-base control (exactly zero). Partial base control saves the translational fuel of full base control at the cost of an uncontrolled base-position drift after the maneuver.
• A nonredundant manipulator (n = 6) is assumed so that J_{ν_e}^⊕ is square and invertible; singularities of J_{ν_e}^⊕ are excluded from the stability region Ω via σ_min(J_{ν_e}^⊕) > 0.

Method

Regime — free-FLYING. The spacecraft is explicitly fully actuated: “external forces and torques are exerted on the spacecraft by means of the spacecraft actuators” (Sec. II.A). This is a free-flying base, not free-floating. However, the controller’s design intent is hybrid: it controls base attitude and the system CoM but deliberately leaves base translation free (called “partial base control”). The paper contrasts three regimes explicitly: full base control [9] (free-flying, position + attitude both rigidly controlled), the proposed partial base control, and floating-base control [5] (free-floating, base translation and rotation both uncontrolled, CoM and angular momentum conserved instead).

Model. Serial chain of n+1 bodies, n arm joints. Dynamics in base + joint coordinates (4):

• M(q) ∈ ℝ^{(6+n)×(6+n)}, C(q,v_b,ω_b,q̇), joint torques τ ∈ ℝ
• Inertia sub-blocks (5): M_t = mE, M_r = I_b, M_tr = −m[p_bc]^∧, M_tm = m J̄_v, etc.

Circumcentroidal decomposition. The end-effector velocity is split into CoM motion plus motion around the CoM (15):
ν_e = G_{v_c} v_c + ν_e^⊕, where the circumcentroidal velocity ν_e^⊕ ≜ G_{ω_b} ω_b + J_{ν_e}^⊕ q̇ (16) equals the body velocity ν_ce of E relative to the nonrotating CoM frame C. Crucially J_{ν_e}^⊕ = J_{ν_e} − [R_eb; 0] J̄_v (14) eliminates only the translational base motion — distinguishing it from the free-floating generalized Jacobian [3,5] which removes both translation and rotation.

Triangular dynamics. A coordinate transform Γ (19) maps (v_b, ω_b, q̇) → (v_c, ω_b, ν_e^⊕); forces transform as the dual (20). Inverting (for nonsingular J_{ν_e}^⊕) yields (21)→(22): CoM decoupled m v̇_c = f_c; coupled 9-DOF block with passivity (23).

Controllers (springs/dampers dual to the new coordinates): CoM f_c = −K_c x̃_c − D_c v_c (28); base τ_b^⊕ = −J_{x̃_b}^T K_b x̃_b − D_b ω_b (29); end-effector w_e^⊕ = −J_{x̃_e}^T K_e x̃_e − D_e ν_e (30). Errors use quaternion-vector coordinates; the representation Jacobian J_{x̃_e} is singularity-free (it cannot blow up). All K, D symmetric positive definite.

Stability. Cascade structure z = (z_1, z_2): z_1 = (x̃_c, ẋ̃_c) is linear and autonomous (34a) and globally asymptotically stable for PD gains; z_2 driven by z_1. With z_1 = 0, Lyapunov V = ½ v̆^T M̆ v̆ + ½ x̃^T K̆ x̃ (37) gives V̇ = −v̆^T D̆ v̆ ≤ 0 (38) using property (23); LaSalle closes the argument. Asymptotic stability holds on Ω excluding J_{ν_e}^⊕ singularities (Prop. IV.1). At singularity the algorithm does not fail numerically but loses actuation in the singular direction.

Relevance to thesis

This is a near-direct template for the nominal free-flying-manipulator control problem: a fully actuated 6-DOF base plus arm, with a principled choice of what to actuate rigidly vs. leave free. The circumcentroidal coordinate and the resulting block-triangular actuation map give a clean, provably stable way to (i) keep the end-effector task off the thruster channel and (ii) reduce thruster usage — directly relevant to a fuel/actuation cost layer and later risk-aware budgeting. The cascade Lyapunov proof and the passivity identity (23) are reusable scaffolding for our own stability arguments. The explicit three-way regime comparison (full / partial / floating) is a useful map of the design space and clarifies exactly where the free-flying assumption buys you attitude pointing at the price of fuel.

Connections

Topics: coordinated_control · circumcentroidal_motion · generalized_jacobian · free_flying_vs_free_floating

Key Equations / Quotes

“The control is based on a triangular actuation decomposition that decouples the end-effector task from the spacecraft’s force actuator, increasing fuel efficiency.” (Abstract)

CoM decoupling and triangular block (22):

m v̇_c = f_c (22a)
M̆ [ω̇_b; ν̇_e^⊕] + C̆ [ω_b; ν_e^⊕] + C_c v_c = [τ_b^⊕; w_e^⊕] (22b)

Passivity / skew-symmetry property (23):

[ω_b^T ν_e^⊕T] (Ṁ̆ − 2C̆) [ω_b; ν_e^⊕] = 0, ∀ ω_b, ν_e^⊕ ∈ ℝ^6

Triangular actuation map (32):

[f_b; τ_b; τ] = [[R_cb^T, 0, 0]; [[p_bc]^∧ R_cb^T, E, G_{ω_b}^T]; [J̄_v^T R_cb^T, 0, J_{ν_e}^⊕T]] [f_c; τ_b^⊕; w_e^⊕]

“all operations that do not involve contact will require no base force. The base force will be activated only when contact occurs … In designs in which the thrusters are used to actuate only f_b, but the actuation of τ_b is accomplished by momentum exchange devices …, the proposed control would have the remarkable advantage of consuming exactly zero fuel for contact-free end-effector maneuvering.” (Sec. IV.C)

“the proposed controller is subject to the singularities of the Jacobian J_e^⊕. At singularity the algorithm does not fail computationally but only results in loss of actuation in a singular direction.” (Sec. VI.A)

Open Questions

• The analysis assumes a nonredundant arm (n = 6) so J_{ν_e}^⊕ is square-invertible. How does the triangular decomposition and stability proof extend to redundant arms (where one would normally exploit a null space)?
• Singularities of J_{ν_e}^⊕ are merely excluded from Ω; the paper offers no singularity-robust modification. What happens to attitude/end-effector regulation near these dynamic singularities of the centroidal Jacobian?
• No orbital/environmental disturbances are modeled (assumed negligible vs. actuation). How does the CoM-conservation argument (and thus the zero-base-force claim) degrade under persistent disturbances?
• The fuel model (39) is a simplified L1-of-command integral with no thrust discretization or distribution envelope; real thruster minimum-impulse-bit and on/off discretization are left to future work.
• Base position drifts after maneuvers under partial control; is there a bounded-drift guarantee, and how is the workspace kept favorable w.r.t. the target over many maneuvers?