zhang2020adaptive

Adaptive robust decoupling control of multi-arm space robots using time-delay estimation technique

Authors: Zhang, Liu, Gao, Ju · Year: 2020 · Venue: Nonlinear Dynamics (Springer)
Raw: md

Summary

The paper addresses the dynamic coupling between the manipulators and base of a multi-arm space robot and proposes a model-free decoupling controller combining time-delay estimation (TDE) with sliding-mode control (SMC). The full MIMO dynamics are derived via a composite rigid body algorithm (Lagrangian inertia matrix + Newton–Euler recursion for the bias force), then a constant diagonal gain matrix recasts the system so the lumped nonlinear term is estimated from one-step-delayed input/output rather than computed online. Global asymptotic stability (boundedness of the TDE error) is proven via a Lyapunov candidate and a discrete-time induction argument, and contrastive 2D dual-arm simulations against CTC and CTC-based SMC show the TDE-based SMC is simpler yet more effective under disturbances and uncertainties.

Key Claims

• For a multi-arm system the coupling appears not only between each arm and the base but also between arms (via the shared base acceleration ${\ddot{X}}_0$ in Eqs. 24a–24c), so the classical two-subsystem (base / manipulator) decoupling of Longman et al. is inadequate.
• TDE replaces online computation of $H$ and $C$ with the one-step estimate $\hat{N}\approx N(t-\delta )=F(t-\delta )-\bar{H}\ddot{q}(t-\delta )$ (Eqs. 37b, 38), making the controller intrinsically adaptive and cheaper than CTC, which matters when the model is high-dimensional.
• Stability requires the SMC switching gain to dominate the disturbance/uncertainty bounds: for CTC-based SMC $G_i>D_i+U_i$; for TDE-based SMC $G_i>E_i$, where $E_i$ bounds the TDE error $\epsilon =N-\hat{N}$.
• A discrete-time induction bound shows the estimation-error dynamics $\xi (k)$ are bounded by $\Vert \xi (k)\Vert \le \frac{n({\beta }_1+A_{\max }{\beta }_2)}{1-A_{\max }}$ with $0<A_{\max }<1$, where $A(t)=H^{-1}(t)\bar{H}-E$; this is the load-bearing condition tying stability to the choice of the constant gain $\bar{H}$.
• The boundary-layer saturation $\mathrm{sat}(s_i,{\psi }_i)$ (Eq. 35) replaces $\mathrm{sgn}$ to suppress chattering, trading robustness for smoothness as ${\psi }_i$ grows.

Method

Regime: free-FLYING. The paper explicitly contrasts the two regimes. It notes that the motion-planning literature (disturbance maps, RNS, etc.) targets free-floating robots whose base is uncontrolled to save fuel, requiring kinematic redundancy to null reactions. Here the base is actively controlled (station-keeping for grasping, communication), so the base wrench $F_B$ is a genuine control input alongside the joint torques ${\tau }_M^k$. The decoupling is achieved by control, not by null-space planning.

Dynamics (composite rigid body). The full system inertia is assembled from kinetic energy $T=\frac{1}{2}{\dot{q}}^{\mathrm{T}}H\dot{q}$ (Eq. 5), giving the block form (Eq. 4):
$\left[\begin{array}{ccc}{H}_B& H_{BM}^1& \dots \\ H_{BM}^{1\mathrm{T}}& H_M^1& \\ ⋮& & \ddots \end{array}\right]\ddot{q}+C=F,{\ddot{X}}_0=[{\dot{v}}_0^{\mathrm{T}},{\dot{\omega }}_0^{\mathrm{T}}{]}^{\mathrm{T}}.$
The bias force $C$ is obtained by Newton–Euler recursion with $\ddot{q}=0$ (Eq. 23), i.e. $C=f_{NE}(\ddot{q}=0)$. $H_{BM}^k$ is the base/arm-k coupling block.

Assumptions: rigid bodies; no gravity ($g=0$); bounded, known disturbance/uncertainty bounds. The perturbed model is $H\ddot{q}+C+f_D+f_U=F$ (Eq. 28), with $f_U=\Delta H\ddot{q}+\Delta C$.

TDE reformulation (Eqs. 36–40). Introduce constant diagonal $\bar{H}$: $\bar{H}\ddot{q}+N=F$, $N=(H-\bar{H})\ddot{q}+C+f_D+f_U$. Sliding surface $s=\dot{e}+\Lambda e$ (Eq. 29). Equivalent control $F_{TDEeq}=\bar{H}({\ddot{q}}_d+\Lambda \dot{e})+\hat{N}$ (Eq. 39); full law $F=F_{TDEeq}+G\mathrm{sgn}(s)$ (Eq. 40). Lyapunov $V=\frac{1}{2}{s}^{\mathrm{T}}s$ gives $\dot{V}<0$ for $G_i>E_i$.

Notation flag: the constant gain matrix is written $\bar{H}$ but in places the OCR renders it as plain $H$ (e.g. Eq. 38 transcribes $N(t-\delta )=F(t-\delta )-H\ddot{q}(t-\delta )$ where $\bar{H}$ is intended). The base generalized acceleration symbol ${\ddot{X}}_0$ also appears as $X_0$ in the Eq. 4 vector. The sliding-surface matrix is named $\Lambda$ in Eq. 29 but referred to as "$K$" in surrounding prose.

Relevance to thesis

Directly on-regime: a fully-actuated, controlled-base (free-flying) multi-arm space robot where dynamic coupling is the central nuisance. The composite-rigid-body block structure (separate $F_B$ for the base, ${\tau }_M^k$ per arm, explicit $H_{BM}^k$ coupling) is exactly the model class for our free-flyer, and the inter-arm coupling argument generalizes to base-plus-single-arm-plus-payload settings. The TDE approach is a candidate baseline for the nominal coordinated controller before adding a risk layer: it is model-light (good when inertial parameters are uncertain post-capture) but its guarantees rest on bounded, known disturbance bounds and a well-chosen $\bar{H}$ — assumptions the risk-aware layer would later have to interrogate. The discrete-time boundedness proof and the $G_i>E_i$ gain condition are useful templates for rigor.

Connections

Topics: dynamic_coupling · coordinated_control · free_flying_vs_free_floating · time_delay_estimation · sliding_mode_control

Key Equations / Quotes

“The most distinctive difference between a space robot and a base-fixed robot is its free-flying/floating base, which results in the dynamic coupling effect.” (Abstract, p. 1)

“Free-flying space robots can actively control their bases to solve this problem.” (Sect. 1, p. 2)

Dynamics (Eq. 4): $H\ddot{q}+C=F$ with base block $H_B$, arm blocks $H_M^k$, coupling $H_{BM}^k$.

TDE estimate (Eqs. 37b, 38): $\hat{N}\approx N(t-\delta )$, $N(t-\delta )=F(t-\delta )-\bar{H}\ddot{q}(t-\delta )$.

Control law (Eqs. 39–40): $F=\bar{H}({\ddot{q}}_d+\Lambda \dot{e})+\hat{N}+G\mathrm{sgn}(s)$, $s=\dot{e}+\Lambda e$.

Stability bound (Eq. 56): $\Vert \xi (k)\Vert \le \frac{n({\beta }_1+A_{\max }{\beta }_2)}{1-A_{\max }}$, $A(t)=H^{-1}(t)\bar{H}-E$, $0<A_{\max }<1$.

Gain condition (after Eq. 41): TDE error $\epsilon =N-\hat{N}$ bounded by $|{\epsilon }_i|\le E_i$; choose $G_i>E_i$.

Open Questions

• How is $\bar{H}$ chosen in practice to guarantee $0<A_{\max }<1$ for a real multi-arm system? The proof assumes it exists; the construction (beyond trial-and-error) is not given.
• The bounds $D_i,U_i,E_i$ are assumed known a priori. After capturing an unknown tumbling target the inertial parameters change abruptly — do the TDE-error bounds still hold, and how is $G_i>E_i$ maintained?
• All simulations are planar (2D dual-arm). How does the coupling analysis and TDE accuracy degrade in full 6-DOF base motion with attitude coupling?
• The sampling period $\delta =1$ ms is asserted “sufficiently small”; no quantitative link between $\delta$, $A_{\max }$, and the residual bound is given for hardware-realistic rates.