giordano2018workspace

Workspace fixation for free-floating space robot operations

Authors: Giordano, Calzolari, Albu-Schäffer · Year: 2018 · Venue: IEEE conference paper (DLR / TU Munich)
Raw: md

Summary

The paper addresses workspace shift: when a space robot contacts a target, momentum exchange drifts the whole system so the target eventually leaves the reachable workspace (which is centered at the system center of mass, CoM). The authors extend their earlier momentum-dumping work into a single unified controller that simultaneously (i) regulates the end-effector pose to an inertially-fixed target, (ii) dumps accumulated angular momentum, and (iii) regulates the system CoM to a desired inertial location — thereby fixing the workspace without rigidly position-controlling the base. The base is otherwise left to float freely, so base actuators consume fuel only to arrest contact-induced drift. Validated on the DLR OOS-Sim hardware-in-the-loop facility (7-DOF KUKA arm on a satellite-simulator base).

Key Claims

• The maximum reachable workspace is centered at the whole-system CoM (citing Umetani & Yoshida); hence regulating the CoM position in the inertial frame keeps the (stationary) target inside the reachable workspace during operations.
• Coordinate transformation to the task space v_new = [v_c^T h_r^T ν_{e,int}^T]^T yields a triangular CoM / angular-momentum / end-effector dynamics, enabling cascade controller design.
• In the new task coordinates the Coriolis/centrifugal terms for the centroid and angular-momentum equations vanish identically, reducing the system to m v̇_c = f_c, ḣ_r = τ_c, plus the decoupled internal end-effector equation.
• A passivity / skew-symmetry property holds for the transformed end-effector dynamics: ν_{e,int}^T (Ṁ_e^* − 2C_e^*) ν_{e,int} = 0, used in the Lyapunov proof.
• The equilibrium set (p̃_c = ṗ̃_c = h_r = 0, x̃_e = ν_{e,int} = 0) is asymptotically stable, proven via cascade theorems for compact invariant sets plus a LaSalle argument, valid in region Ω where the generalized Jacobian J_m^* is nonsingular.
• The actuator distribution (Eq. 27) is triangular: base force f_b serves only CoM control; base torque τ_b realizes the centroidal torque plus compensates the lever-arm coupling of f_c; joint torques realize the internal end-effector wrench. No end-effector force is counterbalanced by external (base) actuators — the fuel-saving advantage of using the internal wrench w_{e,int} rather than the absolute wrench w_e.
• In the contact-free nominal case the controller collapses to f_b = 0, τ_b = 0, and the classical transposed-Jacobian free-floating law τ = −J_m^{*T}(J^T K_e x̃_e + D_e ν_e) — i.e. zero fuel when no contact occurs.
• The controller inherits the dynamic singularity behavior of the transposed-Jacobian free-floating controller: it does not fail computationally at a singularity of J_m^* but loses actuation along the singular direction.
• Experimental contrast: pure momentum-dumping drives the manipulator to a practically singular configuration (~t = 140 s) after repeated contacts, losing end-effector control; CoM control keeps manipulability favorable, restoring end-effector pose to ~1.5 cm (static-friction limited) after each contact.

Method

A serial-link space robot of n+1 bodies with n arm joints. Base generalized force w_b ∈ ℝ^6 is exerted by satellite actuators; orbital/environmental disturbances neglected. The full dynamics (Eq. 2) is partitioned into base-translation / base-rotation / manipulator blocks with mass matrix M(q) and Coriolis matrix C. Frames: B (base CoM, body frame), E (end-effector), T (inertially-fixed target), C (nonrotating frame at whole-system CoM). The Adjoint transformation (Eq. 1) maps twists/wrenches between frames.

Key constructs:

• Generalized Jacobian J_m^* = J_m − A_{eb} M_b^{-1} M_{bm} ∈ ℝ^{6×n} (Eq. after 6).
• Internal end-effector velocity ν_{e,int} = J_m^* q̇ (Eq. 6); absolute velocity ν_e = ν_{e,int} + A_{eb} M_b^{-1} A_{cb}^T h (Eq. 7). Note h ≠ 0 here, unlike the usual literature assumption h = 0.
• Total momentum around C: h = A_{cb}^{-T} h_b; v_c = (1/m) h_t (Eq. 9).
• Transformation Γ ∈ ℝ^{12×(6+n)} (Eq. 10) maps [v_b ω_b q̇] to [v_c h_r ν_{e,int}]; dual force transform (Eq. 11). Inputs f_c (centroidal force), a_r/τ_c (net angular acceleration/torque about C), w_{e,int} (internal end-effector wrench).
• Reduced triangular dynamics (Eq. 13): m v̇_c = f_c; ḣ_r = τ_c; M_e^* ν̇_{e,int} + C_e^* ν_{e,int} + C_{ec} v_c + C_{er} h_r = w_{e,int} with M_e^* = (J_m^* M_m^{*-1} J_m^{*T})^{-1}.

Control laws: end-effector impedance w_{e,int} = −J_{x̃_e ν_e}^T K_e x̃_e − D_e ν_e (Eq. 18) with quaternion-based pose error (Eq. 15); CoM regulation f_c = −K_c p̃_c − D_c v_c (Eq. 19); momentum dumping τ_c = −D_{hr} h_r (Eq. 20). Closed loop (Eq. 22) decomposes into a linear, robot-independent “whole” subsystem and the arm subsystem, proven asymptotically stable via cascade + Lyapunov V = ½ ν_{e,int}^T M_e^* ν_{e,int} + ½ x̃_e^T K_e x̃_e with V̇ = −ν_{e,int}^T D_e ν_{e,int} ≤ 0.

Regime. The title and base controller heritage say “free-floating”, but the paper’s actual contribution is fundamentally a free-flying one in this wiki’s sense: the base is fully actuated (w_b = [f_b; τ_b] ∈ ℝ^6 from satellite thrusters/actuators) and those actuators are used during contact to regulate CoM and dump momentum. The distinction here is subtle and deliberate: the system behaves free-floating (actuators off, fuel-free) during contact-free operation, but switches to active 6-DOF base actuation to fix the workspace when contact occurs — all within one controller, without controller switching. So it is an actuated-base (free-flying-capable) controller engineered to recover the fuel economy of free-floating whenever possible. Base attitude is not regulated (it converges to unknown values per impulse), which is a flag for free-flying applications that need attitude pointing.

Relevance to thesis

Directly relevant to a fully-actuated 6-DOF base manipulator. It shows how to exploit base actuation sparingly: a triangular/cascade decomposition lets the base actuators handle only the “outer” rigid-body motion (CoM position, net angular momentum) while joint torques handle the end-effector, with no wasteful base-vs-arm force fighting. The CoM-regulation-as-workspace-fixation idea is a clean nominal-guidance objective, and the manipulability collapse under pure momentum dumping is a concrete motivation for the risk layer (workspace shift as a hazard). The singularity region Ω and dynamic-singularity caveat tie into singularity-aware planning.

Connections

Topics: generalized_jacobian · center_of_mass_regulation · dynamic_singularity · momentum_dumping

Key Equations / Quotes

“Since the maximum reachable workspace is centered at the CoM of the whole system [7], controlling this point around a suitable position would guarantee that the target object always remains in the reachable workspace while the manipulator operates.” (p. 1)

Generalized Jacobian and internal velocity:

J_m^* = J_m − A_{eb} M_b^{-1} M_{bm}, ν_{e,int} = J_m^* q̇ (Eq. 6)

Reduced triangular dynamics (Eq. 13a–13c):

m v̇_c = f_c
ḣ_r = τ_c
M_e^* ν̇_{e,int} + C_e^* ν_{e,int} + C_{ec} v_c + C_{er} h_r = w_{e,int}

Passivity property (Eq. 14):

ν_{e,int}^T (Ṁ_e^* − 2 C_e^*) ν_{e,int} = 0 ∀ ν_{e,int} ∈ ℝ^6

Nominal (contact-free) reduction to the classical free-floating law (Eq. 28):

f_b = 0, τ_b = 0, τ = −J_m^{*T}(J_{x̃_e ν_e}^T K_e x̃_e + D_e ν_e)

“the proposed controller (27) shares singularity properties similar to those of the transposed Jacobian free-floating controller: a) it is subject to dynamic singularities [15], b) when a singularity of J_m^* is encountered, it does not fail computationally but only results in loss of actuation in a singular direction.” (p. 4)

Open Questions

• Base attitude is uncontrolled and drifts to unknown values per impulse, which “may cancel or accumulate”; the authors note unfavorable attitudes can still yield zero manipulability. How should a reorientation task be superimposed (using the stated remaining 3-DOF redundancy) without breaking the triangular decoupling?
• The nonredundant n = 6 assumption is used “without loss of generality”; how does the triangular form and the generalized-Jacobian inverse J_m^{*-1} (Eq. 22e) generalize cleanly to the redundant 7-DOF arm actually used in the experiments?
• Feedback of p̃_c, v_c, h_r relies on the inertia model and kinematics; what is the sensitivity of CoM regulation to inertial-parameter and state-estimation error (camera/LIDAR/IMU fusion suggested but not analyzed)?
• The end-effector steady-state error is static-friction limited (~1.5 cm); how does this interact with the singularity-proximity actuation loss in a risk/uncertainty setting?