nanos2011use

On the use of free-floating space robots in the presence of angular momentum

Authors: Nanos, Papadopoulos · Year: 2011 · Venue: Multibody System Dynamics (Springer; Special Issue, received 2010, published online Dec 2010)
Raw: md

Summary

The paper relaxes the near-universal assumption of zero initial angular momentum for free-floating space manipulators and studies how a non-zero, conserved system angular momentum h_CM affects the ability to hold the end effector fixed in inertial space. With momentum present, the end effector generally cannot remain at a point indefinitely; the authors derive the kinematic and dynamic conditions under which a fixed end-effector location is sustainable, and they identify subsets of the reachable workspace (rings in 2D, spherical shells in 3D) that are “immune” to momentum accumulation. The system continually moves base and joints to absorb the momentum while the end effector stays put, and they show this requires a manipulator of at least 3 DOF (spatial) or 2 DOF (planar) for position, and at least 6 DOF for position-plus-orientation fixity.

Key Claims

• With non-zero conserved angular momentum, keeping the end-effector position fixed requires solving the stacked momentum + zero-end-effector-velocity system Eq. (13); this has a solution only if the manipulator has N ≥ 3 DOF (spatial). Planar case requires N ≥ 2 (Eq. 37). Fixing both position and orientation requires N ≥ 6 (spatial, Eq. 23–24) or N ≥ 3 (planar).
• A feasible “fixed end-effector” configuration exists iff the dynamic-singularity-type determinant condition Eq. (16) holds: det(−⁰J₁₁ ⁰D⁻¹ ⁰D_q + ⁰J₁₂) ≠ 0. This depends only on configuration q, not on the momentum magnitude.
• The end-effector distance from the system CM, R = ‖⁰r_E(q)‖, is independent of base attitude (Eq. 17). Hence singular configurations map to Cartesian circles (2D) / spheres (3D) that must be removed from the reachable workspace, leaving the momentum-tolerant region.
• Crucially, the very condition Eq. (16) is the dynamic-singularity condition of Papadopoulos & Dubowsky [5]. The system can tolerate non-zero momentum exactly in the workspace subsets where dynamic singularities cannot occur. If the end effector sits where dynamic singularities can appear, it cannot be held fixed under momentum.
• The required base angular velocity ⁰ω₀ (Eq. 19/27) and joint rates q̇ (Eq. 20/26) are linearly proportional to h_CM and vanish when h_CM = 0 (recovering the classical result that a zero-momentum free-floater can sit still).
• Planar 2-DOF example (Table 1): a = 0.4255 m, b = 1.7872 m, c = 0.9681 m give a feasible ring 1.2447 m ≤ r_E ≤ 2.3298 m. The motion is arbitrarily long-lived (t_f = 2000 s shown), torques small and smooth.
• Joint rates scale linearly with h_CM but torques scale super-linearly (faster than doubling when h_CM doubles), because the Coriolis/centrifugal term C_h is nonlinear in q̇.
• Spatial 3-DOF example (Table 2): |α| = 0.4444 m, c = 0.9222 m, d = 0.9778 m give shell 0.5 m ≤ ‖r_E‖ ≤ 1.4556 m, further intersected with the Eq. (16) region.

Method

Regime: free-FLOATING — explicitly a satellite base with all actuators OFF, so the base translates/rotates passively in reaction to arm motion. This is the underactuated, nonholonomic case driven by angular-momentum conservation, NOT our free-flying (fully-actuated 6-DOF base) setting. The opening section is careful: “free-flying” denotes the thruster/reaction-wheel-equipped base; turning everything off yields the “free-floating” mode studied here.

Assumptions: rigid open-chain manipulator with N revolute joints (N+6 total DOF); no external forces/torques; zero initial linear momentum so the system CM is fixed and chosen as inertial origin; non-zero initial angular momentum h_CM (from small collisions or on-off attitude controller error).

Conservation of angular momentum (Eq. 1):
⁰D(q) ⁰ω₀ + ⁰D_q(q) q̇ = R₀ᵀ(ε,n) h_CM

End-effector linear velocity (Eq. 3): ṙ_E = ṙ_CM + R₀(⁰J₁₁ ⁰ω₀ + ⁰J₁₂ q̇); angular velocity (Eq. 4): ω_E = R₀(⁰ω₀ + ⁰J₂₂ q̇).

Reduced equations of motion (Eq. 9): H(q) q̈ + c_h(ε, n, h_CM, q, q̇) = τ, with H = ⁰D_qq − ⁰D_qᵀ ⁰D⁻¹ ⁰D_q (Eq. C1). In 3D, c_h depends on base attitude.

Fixed-end-effector condition: set ṙ_E = 0 ⟹ ⁰J₁₁ ⁰ω₀ + ⁰J₁₂ q̇ = 0 (Eq. 12), stacked with Eq. (1) into A[⁰ω₀; q̇] = [R₀ᵀh_CM; 0] (Eq. 13), A the 6×(N+3) matrix of Eq. (14). Solvability hinges on Eq. (16). For N>3 a least-squares pseudoinverse solution (Eq. 21) is used.

Planar specialization exploits that the base attitude θ₀ is a cyclic (ignorable) variable: a Routhian R(q,q̇) (Eq. 34) is built by eliminating θ₀ via Eq. (29), giving reduced dynamics (Eq. 35) τ = H q̈ + C_h(h_CM,q,q̇)q̇ + g_h(q,h_CM), where the momentum produces a gyroscopic-like C_h correction (Eq. 36a) and a configuration-only “potential” term g_h = ½ h_CM² ∂(D⁻¹)/∂q (Eq. 36b) that does not vanish at q̇ = 0.

Relevance to thesis

Our base is fully actuated, so we are NOT bound by the nonholonomic momentum constraint this paper lives under. But the work is directly relevant in two ways. (1) It gives the exact dynamic-singularity condition Eq. (16) and the clean fact that it is configuration-only (attitude-independent), with the elegant mapping to Cartesian circles/spheres via R = ‖⁰r_E(q)‖ — a template for characterizing where coupling pathologies live in the workspace. (2) It quantifies the cost of passively absorbing accumulated momentum (continuous base+joint motion, super-linear torque growth). For a free-flyer, momentum could instead be dumped by the base actuators; this paper is the baseline that justifies why a fully-actuated base is worth the propellant/wheel budget, and what the failure mode looks like if one chose to coast. The Routhian reduction for the cyclic base attitude is also a useful modeling pattern even in the actuated case.

Connections

Topics: generalized_jacobian · dynamic_singularity · angular_momentum_conservation · free_flying_vs_free_floating

Key Equations / Quotes

“in the presence of momentum, the manipulator end effector in general cannot remain at a given location for indefinite time.” (Abstract, p. 1)

“Equation (16) shows that the system can tolerate non-zero angular momentum at the same workspace subsets in which dynamic singularities cannot exist.” (Sect. 3)

Conservation of angular momentum (Eq. 1):

⁰D(q) ⁰ω₀ + ⁰D_q(q) q̇ = R₀ᵀ(ε, n) h_CM

Solvability / dynamic-singularity condition (Eq. 16):

det(−⁰J₁₁ ⁰D⁻¹ ⁰D_q + ⁰J₁₂) ≠ 0

Attitude-independence of CM distance (Eq. 17):

R = ‖r_E(ε,n,q)‖ = ‖R₀(ε,n) ⁰r_E(q)‖ = ‖⁰r_E(q)‖

Planar 2-DOF singularity function & rates (Eq. 48–49):

S(q₁,q₂) = ab D₂ s₁ + bc D₀ s₂ − ac D₁ s₁₂ ≠ 0
θ̇₀ = (bc s₂ / S) h_CM, q̇₁ = −((ac s₁₂ + bc s₂)/S) h_CM, q̇₂ = ((ab s₁ + ac s₁₂)/S) h_CM

“When A is singular (i.e., S(q₁,q₂)=0), the joint rates … increase to infinity and the end effector is displaced from its desired location.” (Sect. 4A)

Open Questions

• The feasible region is found by subtracting singular circles/spheres; the paper does not characterize the topology/connectedness of the remaining momentum-tolerant set, nor how it shrinks as ‖h_CM‖ grows toward joint-rate/torque limits.
• Joint angles can grow without bound (the planar example’s q₁ increases monotonically); the paper notes joint limits eventually cap the hold time but gives no design rule for max sustainable hold time vs. momentum magnitude.
• Position-and-orientation fixity needs N ≥ 6; no analysis of redundancy resolution or singularity structure of the augmented A* (Eq. 24) is given.
• For N > 3 the least-squares pseudoinverse solution (Eq. 21) is stated but its behavior near rank deficiency of A is not studied.