wilde2018equations

Equations of Motion of Free-Floating Spacecraft-Manipulator Systems: An Engineer’s Tutorial

Authors: Wilde, Kwok Choon, Grompone, Romano · Year: 2018 · Venue: Frontiers in Robotics and AI 5:41 (doi:10.3389/frobt.2018.00041)
Raw: md

Summary

A complete, self-contained tutorial derivation of the coupled equations of motion of a spacecraft-manipulator system (a single N-link manipulator on a 6-DOF base) via the Generalized Jacobian Matrix (GJM) approach of Yoshida & Umetani (1993), using the Lagrangian method. Its distinctive value is the full symbolic expression of every inertia-property matrix — the base-spacecraft, manipulator, and dynamic-coupling inertia matrices — together with their time derivatives and joint-angle derivatives, plus a generalized Jacobian for an arbitrary point on any link. A secondary contribution refines the coarse free-flying/free-floating dichotomy into five precisely defined maneuvering modes distinguished by which momenta (linear/angular) are conserved versus actively controlled.

Key Claims

• The N+6 coupled scalar equations of motion (eq. 39) for a floating system reduce, under conserved momentum, to N generalized scalar equations governed by the [N×N] generalized inertia matrix H* = H_m − H_0mᵀ H_0⁻¹ H_0m (eq. 45). Under zero initial momentum (M_0 = 0) this collapses to the fixed-base-like form Hq̈ + C(q,q̇) = τ (eq. 50).
• The [6×6] base-spacecraft inertia matrix H_0 admits a closed-form symbolic inverse via the Banachiewicz/Schur-complement formula (eq. 52), with the Schur complement S_U = H_S + m_tot r_0C^× r_0C^× being a symmetric [3×3] matrix invertible in closed form (eqs. 53–61). This avoids any numerical [6×6] inversion.
• The standard free-flying/free-floating pair is too coarse. The authors propose five modes (Table 1): floating (P, L conserved), rotation-floating (P, L conserved; attitude held by momentum-exchange devices), rotation-flying (P conserved, L controlled by external torque), translation-flying (P controlled, L conserved), and flying (both P and L controlled).
• A generalized Jacobian J_Xi = J_mXi − J_0Xi H_0⁻¹ H_0m (eq. 110) maps joint rates to the spatial velocity of an arbitrary point X_i on link i; specializing to the end-effector recovers Yoshida & Umetani’s J, and inverse kinematics follows as q̇ = J*⁻¹ [v_E; ω_E]ᵀ (eq. 111).
• Two simulation case studies (planar 4-link; spatial 6-link from Yoshida & Umetani 1993) verify the symbolic model under Computed-Torque/PD control; with zero internal-torque actuation, base reaction motion is confirmed equal-and-opposite to the manipulator, and the system center-of-mass stays fixed.

Method

Lagrangian derivation in inertial frame 𝒥, generalized coordinates = base pose 𝒥x_0 = [v_0ᵀ ω_0ᵀ]ᵀ and joint angles q. Custom Denavit-Hartenberg convention (base = link 0, joint 0 at base CM; link frame ℒ_i aligned with 𝒥_{i+1}, origin at CM_i). Kinematics given in both DCM/homogeneous-transform form and unit-quaternion form. Kinetic energy is partitioned (eq. 27) into base inertia H_0 [6×6], manipulator inertia H_m [N×N], and dynamic-coupling inertia H_0m [6×N]; H_0m = [J_TS; H_Sq] (eq. 31) is precisely the term encoding base/manipulator coupling. The generalized-Jacobian reduction uses the conserved-momentum relation [P;L] = H_0 ẋ_0 + H_0m q̇ = M_0 (eq. 42) solved for ẋ_0 = −H_0⁻¹ H_0m q̇ (eq. 109, for M_0 = 0).

Regime — this is the key caveat for our thesis. Despite the five-mode taxonomy, the actual EOM derivation is performed only for the floating mode (the strictest free-floating case: base fully uncontrolled, both P and L conserved, potential energy zero). The reduction to N equations and the generalized Jacobian both rely on momentum conservation (eqs. 42–43) and the M_0 = 0 assumption (eq. 46). The authors note rotation-floating can be reached by adding momentum-exchange-device torques to the RHS of eq. 37 (per Wie 2008), and non-zero L is treated in Nanos & Papadopoulos (2011), but neither is carried through here. A genuinely free-FLYING base (our system) with external forces/torques on all 6 base DOF breaks the momentum-conservation premise; the H*/J* reduction does not directly apply without re-deriving with the base wrench retained on the RHS.

Relevance to thesis

This is the cleanest available reference for the full symbolic inertia bookkeeping (H_0, H_m, H_0m and all their q- and t-derivatives) that any model-based controller or planner for our free-flying manipulator needs — even though our base is actuated. It makes the dynamic-coupling matrix H_0m explicit, which is the object our coordinated base+arm control must contend with. Critically, it is a worked example of the free-floating limit; the paper’s own taxonomy lets us state precisely how our flying regime differs (both P and L controlled), so we can document exactly which of their equations survive and which must be re-derived with the base control wrench retained. The computed-torque section (eqs. 112–115) even flags inertia-parameter uncertainty (fuel slosh / tank settling) as the practical reason H̃*, C̃* differ from truth — a direct hook for our later risk layer.

Connections

Topics: generalized_jacobian · generalized_inertia_matrix · dynamic_coupling · free_flying_vs_free_floating

Key Equations / Quotes

“The General Jacobian Matrix approach describes the motion of the end-effector of an underactuated manipulator system solely by the manipulator joint rotations, with the attitude and position of the base-spacecraft resulting from the manipulator motion.” (Abstract, p. 1)

Generalized inertia matrix (eq. 45):
$H^{\star }=H_m-H_{0m}^T{H}_0^{-1}{H}_{0m}.$

Reduced generalized EOM, M_0 = 0 (eq. 50):
$H^{\star }\ddot{q}+C^{\star }(q,\dot{q})=\tau .$

Closed-form base-inertia inverse via Schur complement (eqs. 52–53):
$H_0^{-1}=\left[\begin{array}{cc}\frac{1}{m_{tot}}{\mathbb{I}}_{3,3}-r_{0C}^{\times }{S}_U^{-1}{r}_{0C}^{\times }& r_{0C}^{\times }{S}_U^{-1}\\ S_U^{-1}{r}_{0C}^{\times }& S_U^{-1}\end{array}\right],S_U=H_S+m_{tot}{r}_{0C}^{\times }{r}_{0C}^{\times }.$

Generalized Jacobian for arbitrary point X_i (eq. 110) and base-velocity elimination (eq. 109):
$J_{X_i}^{\star }=J_{m_{X_i}}-J_{0_{X_i}}{H}_0^{-1}{H}_{0m},{\dot{x}}_0=-H_0^{-1}{H}_{0m}\dot{q}.$

“the two commonly used categories (free-flying and free-floating) are expanded by the introduction of five categories (namely floating, rotation-floating, rotation-flying, translation-flying, and flying).” (Introduction, p. 2; Table 1, p. 4)

Open Questions

• The derivation is restricted to the floating mode with M_0 = 0. What is the explicit form of the generalized EOM and J* when the base wrench (forces and torques on all 6 DOF) is actively commanded — i.e. our flying regime where neither P nor L is conserved? The paper points to Nanos & Papadopoulos (2011) for non-zero L only.
• Inverse kinematics via q̇ = J*⁻¹ [v_E; ω_E]ᵀ (eq. 111) presumes J* is square and invertible; the paper does not treat dynamic singularities of J* or redundancy resolution, which are central to robust planning.
• Computed-torque control assumes perfect inertia knowledge (H̃* = H*); the authors name fuel-slosh/tank-settling uncertainty but do not quantify its effect on tracking — an opening for the risk-aware layer.

Flags

OCR note: several transcribed equations in the raw md are mangled (the expanded Euler-angle DCMs eqs. 7–8, the J_Ti/J_Ri index ranges, and zero-block sizes “0_{3,N-1}” vs “0_{3,N-i}”). Equations cited above were cross-checked against context and the GJM literature; treat raw-md matrix entries with caution and prefer the published PDF for exact symbolic terms.