rybus2017control

Control System for Free-Floating Space Manipulator Based on Nonlinear Model Predictive Control (NMPC)

Authors: Rybus, Seweryn, Sasiadek · Year: 2017 · Venue: Journal of Intelligent & Robotic Systems (Springer; received 2015, published online 2016)
Raw: md

Summary

The paper proposes a two-module control system for an unmanned satellite-manipulator capturing a tumbling target: (1) an offline trajectory-planning module that minimizes a quadratic motor-power functional via calculus of variations / a boundary-value problem, and (2) an online Nonlinear Model Predictive Controller (NMPC) using the full nonlinear free-floating dynamics (no linearization) to track the planned end-effector trajectory. In planar 2-DOF simulations the NMPC is benchmarked against a Dynamic-Jacobian-inverse controller and Modified Simple Adaptive Control (MSAC), and shows smaller total end-effector tracking error under disturbances and parameter mismatch, at the cost of high computational expense.

Key Claims

• Trajectory optimization reduced the power-related cost functional L from 0.0719 (non-optimal) to 0.044 (optimal) in the capture scenario, a 38.8 % reduction.
• NMPC achieves smaller total end-effector position error than the Dynamic-Jacobian-inverse controller and MSAC across three test regimes: no disturbance, 30 % satellite/manipulator mass-inertia mismatch, and white-noise torque disturbance.
• NMPC’s advantage over the Dynamic-Jacobian-inverse controller is “clearly seen” specifically under parameter mismatch and torque noise; under noise MSAC degrades to roughly Dynamic-Jacobian-level performance.
• Because the reference trajectory is known in advance, NMPC anticipates square corners (initiates -x motion before the corner), reducing overshoot relative to the reactive controllers.
• High computational cost is the main drawback: the generalized-coordinate formulation [Junkins & Schaub quasivelocities] is flagged as having high computational complexity for manipulators with >4 DOF, making real-time NMPC on a real free-floating system “very difficult without any simplifications.”
• No instability was observed in any simulation, but the authors explicitly state a rigorous (numerical and analytical) stability analysis has not been performed.

Method

Regime — free-FLOATING (not free-flying). The satellite’s position/attitude controller is deliberately switched off during the capture maneuver (disturbances from manipulator motion deemed too large to fully compensate). The base therefore moves passively in reaction to arm motion; this is the explicit modeling assumption throughout. Contrast with the thesis’s free-FLYING system (fully actuated 6-DOF base).

Dynamics. Lagrangian formulation in the inertial frame, generalized coordinates q_p = [r_s, Θ_s, θ]^T (satellite CoM position, satellite orientation, joint angles). Equations of motion (Eq. 24):
Q = M(q_p) q̈_p + C(q̇_p, q_p) q̇_p, with Q = [F_s, H_s, u]^T the generalized forces (external satellite force/torque and joint torques u). Mass matrix M partitions into the kinetic-energy submatrices A,B,D,E,F,N (Eqs. 7-12, 25).

Non-zero, non-conserved momentum. A notable feature (from Seweryn & Banaszkiewicz): momentum and angular momentum are carried as time-dependent functions f_m = ∫ F_s dt, f_am = ∫ (H_s + r̃_s F_s) dt (Eqs. 16-17), so the system need not assume P = L = 0. The end-effector velocity (Eq. 18) is split into a momentum-driven term and a joint-driven term via H_2, H_3. The “dynamic” / generalized Jacobian appears as (J_M - J_s H_2^{-1} H_3).

Planning module. Minimize L = ½ uᵀu (Eq. 28) subject to the direct dynamics g (Eq. 29); Hamiltonian H = L + λ_vpᵀ g, stationarity ∂H/∂u = 0. Augmented state with multipliers (Eq. 32) yields a BVP solved by MATLAB bvp4c (Lobatto IIIa). Inverse kinematics at the velocity level uses the Generalized Jacobian Matrix.

NMPC module. Receding-horizon control on the full nonlinear plant (Eq. 39, q̈_p = M^{-1}(Q - C q̇_p)); the prediction model is treated as ideal while the simulated “real” plant carries parameter error / noise. State extended with r_ee (Eq. 40); least-squares objective with weighting W = eye([... 10 10]) penalizing only end-effector x,y. Horizon 1 s / 20 intervals; implemented in ACADO Toolkit. The authors elect not to feed forward u_ref since the NMPC already uses the full model.

Baselines. Dynamic-Jacobian-inverse law u_contr = G_θ̇ (J_dyn^{-1} G_ee e_p - θ̇) (Eq. 44), singular when J_dyn non-invertible (planned trajectory assumed far from dynamic singularities). MSAC uses the fixed-base transposed Jacobian u_contr = J_kinᵀ(K_p e_p + K_d e_v) with adaptively tuned gains (Eqs. 45-51).

Relevance to thesis

This is a direct methodological reference for the planning+tracking architecture, but it is the free-floating counterpart to our free-flying problem — useful precisely as a contrast. Their justification for free-floating (base controller off because reactions are too large) is the very disturbance the thesis’s actuated base is meant to handle, so the paper motivates why a free-flying formulation matters. The non-zero/non-conserved-momentum generalized Jacobian, the calculus-of-variations BVP planner, and the full-nonlinear NMPC tracker are all transferable to a 6-DOF actuated-base setting. Their honest flag that the quasivelocity formulation scales poorly past 4 DOF is a cautionary note for spatial extensions. The parameter-mismatch and torque-noise tests are a natural seam for the eventual risk-aware / uncertainty layer.

Connections

Topics: generalized_jacobian · free_floating_dynamics · nonlinear_mpc · trajectory_optimization

Key Equations / Quotes

“we assume that system responsible for controlling position and orientation of the satellite is switched off during the capture maneuver as disturbances caused by manipulator motion are too high to be fully compensated” (Sec. 1)

“In previous studies devoted to MPC for space manipulators the manipulator base was assumed to be fixed… In our approach we use a full nonlinear model of a free-floating system… we do not perform any linearization of the system.” (Sec. 3.3)

Equation of motion (Eq. 24): Q = M(q_p) q̈_p + C(q̇_p, q_p) q̇_p

Non-conserved momentum (Eqs. 13, 16-17): [P; L] = H_2 [v_s; ω_s] + H_3 θ̇ = [f_m; f_am], with f_m = ∫ F_s dt, f_am = ∫ (H_s + r̃_s F_s) dt.

“trajectory optimization allowed for reduction of the cost functional by 38.8 % from the non-optimal value.” (Sec. 4.4; L: 0.0719 → 0.044)

Open Questions

• Stability of the NMPC is unproven; only empirically observed. What guarantees (terminal cost/constraint, horizon length) would close this for the free-floating plant?
• Can real-time NMPC be made feasible for a spatial (>4-DOF) system given the flagged computational complexity of the quasivelocity formulation? What simplified/linearized model would suffice?
• Would a reaction-minimizing cost functional (raised in Future Work) materially reduce base attitude excursion versus the power-norm functional?
• All capture-maneuver results are disturbance-free with perfect parameter knowledge — how does the integrated planner+NMPC perform under the same mismatch/noise applied in the square-trajectory tests?