sukhanov2015dynamic

Dynamic Equations of Free-flying Space Robot for Feedback Control Tasks

Authors: Sukhanov, Silaev, Glumov · Year: 2015 · Venue: Automation and Remote Control, vol. 76, no. 8 (DOI 10.1134/S0005117915080093)
Raw: md

Summary

The paper derives a planar dynamic model for a free-flying space manipulative robot (SMR): a three-link manipulator carrying a load on a fully controlled carrying body (control forces F_x, F_y and orientation moment M_ϑ). Its central contribution is a coordinate transformation of the rigid-body Lagrangian model so that the controlled end-point-to-target deviation coordinates (X_ε, Y_ε) appear explicitly as states, yielding a “modified model” amenable to classical feedback control. It also (i) gives a matrix-algebra device for modeling self-stopping (deadening) joint drives, and (ii) establishes a quantitative “ϑ-dynamics” error criterion under which the full nonlinear model may be replaced by a simplified decoupled model when joint speeds are small (“creeping”).

Key Claims

• For the planar SMR-L the 6-DOF Lagrangian equations (3 base coordinates q⁰=(X₀,Y₀,ϑ), 3 joint angles qᵅ=(α₁,α₂,α₃)) take block form A(q)q̈ = M + f(q,q̇) with A₁₂=A₂₁ being the base/manipulator coupling block — i.e. base and arm motions are dynamically coupled and cannot be treated separately.
• By substituting the doubly-differentiated kinematic relation (8) between (X₀,Y₀) and (X_ε,Y_ε), the model can be rewritten in the controlled deviation coordinates q̃=(qᵋ,qᵅ) with q^ε=(X_ε,Y_ε,ϑ), preserving the same block structure (Eq. 9).
• Self-stopping joint deadening (α̇_i=0, M_i^α=0) is represented by left/right multiplying the inertia/control system by I_{i+3} (a 6×6 identity with the (i+3)-th diagonal element zeroed); the general case uses I=diag(1,1,1,δ₁,δ₂,δ₃) with δ_i∈{0,1} (Eq. 10).
• The full model (11) may be reduced to the decoupled model (12) — A₁₁q̈⁰+A₁₂q̈ᵅ=M⁰, A₂₂ᵈq̈ᵅ=Mᵅ — when three conditions hold: powerful/self-stopping drives so Mᵅ≫A₂₁q̈⁰, technical-controllability so A₂₂→A₂₂ᵈ=diag A₂₂, and small joint speeds so the Coriolis/centrifugal terms f⁰,f^α (quadratic in q̇) are negligible.
• Numerical example: for the test configuration, a 3% ϑ-dynamics error bound (Eq. 16) corresponds to a “creep” joint-speed boundary of about ᾱ_{r,max}≈0.07 rad/s. Within this regime the reduced and full models match closely (Δϑ≤0.07 rad, ΔX_ε,ΔY_ε≤0.1 m; joint-angle mismatch Δα_r≤2×10⁻⁶ rad).

Method

The SMR is a planar mechanical system: a carrying body plus a hinged three-link manipulator holding a load. Generalized coordinates q=(q⁰,qᵅ): base position/attitude (X₀,Y₀,ϑ)≐(q₁,q₂,q₃) in the inertial CXY frame, and interlink angles (α₁,α₂,α₃)≐(q₄,q₅,q₆) in the body frame oxy. The Lagrangian equations of motion (Eq. 1, from ref. [3]) are written componentwise with configuration-dependent inertia coefficients a_{ij}(q) and quadratic velocity terms f_i(q,q̇)=Σ_{j,k} b^i_{jk}(q) q̇_j q̇_k.

Vector-matrix block form (Eqs. 2–3):
A(q)q̈ = M + f(q,q̇), partitioned as A₁₁ (base 3×3, symmetric), A₂₂ (manipulator 3×3, symmetric, including drive inertias J_{pi}i²_{pi}), and A₁₂=A₂₁ (3×3 coupling). The drive moment vector entries are M_i^α = i_{pi}(k_{ui}u_i − M_{Ti}) − k_{α̇i}α̇_i − J_{pi}i²_{pi}α̈_i.

The end-point-to-target deviation is ρ_aA=(X_ε,Y_ε) with X_ε=X_A−X_a, Y_ε=Y_A−Y_a. After shifting the inertial origin onto the target A, X_ε=−X_a, Y_ε=−Y_a, giving the kinematic coupling (Eqs. 4–7) and its second derivative (Eq. 8). Substituting yields the modified model (Eq. 9) in q̃=(q

Regime — free-FLYING (not free-floating): The base is fully actuated; the equations carry explicit base control force/torque M⁰=(F_x,F_y,M_ϑ). The simplification analysis explicitly sets M⁰=0 only as a test condition to probe attitude (ϑ) drift driven by joint motion (“free-flying SMR (M⁰=0)” — note the authors call this state free-flying, whereas in much of the Western literature a base with zero base control over the maneuver is termed free-floating). The cited dynamic-singularity reference [2] (Papadopoulos & Dubowsky) is a free-floating result; here the base actuation is available, so the modeling target differs.

Model-reduction validity is judged by comparing the “ϑ-dynamics” of full (11) vs reduced (12) models under a common test moment M_T^α, via integral indices J_N=∫|ϑ_N|dt, J_R=∫|ϑ_R|dt and the relative error ε_ϑ(q̇)=|1 − J_R/J_N(q̇)| ≤ ε̄_ϑ (Eqs. 14–16).

Relevance to thesis

Directly on-target: this is an explicitly free-flying (fully-actuated base) SMR model, matching our 6-DOF actuated-base platform rather than the free-floating literature. Two ideas are useful: (1) re-coordinatizing the dynamics so the regulated end-point/target deviation is a state (a guidance-friendly error formulation for trajectory tracking and feedback design); and (2) a principled, quantitative criterion (the ϑ-dynamics error bound) for when coupling and Coriolis/centrifugal nonlinearities may be dropped — useful both for nominal controller synthesis and, later, for bounding model-reduction error in the risk-aware layer. The self-stopping-drive deadening device (diagonal selector matrix) is a clean way to model joints that lock, relevant to staged/sequential manipulation.

Connections

Topics: ffsm_dynamics · dynamic_coupling · free_flying_vs_free_floating · task_space_error_dynamics

Key Equations / Quotes

“the impact of manipulator motion on the motion of the carrying body makes it impossible to consider these motions separately” (Sec. 1)

Block dynamics (Eq. 3):
$\left[\begin{array}{cc}{A}_{11}(q)& A_{12}(q)\\ A_{21}(q)& A_{22}(q)\end{array}\right]\left[\begin{array}{c}{\ddot{q}}^0\\ {\ddot{q}}^{\alpha }\end{array}\right]=\left[\begin{array}{c}{M}^0\\ M^{\alpha }(u)\end{array}\right]+\left[\begin{array}{c}{f}^0(q,\dot{q})\\ f^{\alpha }(q,\dot{q})\end{array}\right]$

Reduced (simplified) model (Eq. 12):
$A_{11}{\ddot{q}}^0+A_{12}{\ddot{q}}^{\alpha }=M^0,A_{22}^d{\ddot{q}}^{\alpha }=M^{\alpha }$

ϑ-dynamics reduction error (Eq. 15):
${\epsilon }_{\vartheta }(\dot{q})=\left|1-\frac{J_R}{J_N(\dot{q})}\right|,{\epsilon }_{\vartheta }(\dot{q})\le {\bar{\epsilon }}_{\vartheta }(\text{Eq. 16})$

“to the left of which the full model (11) can be replaced in a first approximation by the simplified model (12)” — with ε̄_ϑ=3% giving ᾱ_{r,max}≈0.07 rad/s (Sec. 5)

Open Questions

• The coefficients ã_{ij} and b^i_{jk} of both the original and modified models are deferred to ref. [3]; the paper itself does not transcribe them, so the model is not self-contained.
• The ϑ-dynamics error criterion is validated on a single planar example; no general analytical bound relating ε̄_ϑ to a guaranteed joint-speed limit (independent of configuration) is given.
• Extension beyond planar (3-DOF base) to the full spatial 6-DOF free-flying case is not addressed.
• The “technical controllability” conditions [6] that justify diagonalizing A₂₂ are invoked but not stated here.