ye2019research

Research on Adaptive Reaction Null Space Planning and Control Strategy Based on VFF-RLS and SSADE-ELM Algorithm for Free-Floating Space Robot

Authors: Ye, Dong, Huang · Year: 2019 · Venue: Electronics (MDPI) 8(10):1111
Raw: md

Summary

After a space robot captures an unknown (non-cooperative) target, the resulting “complex” system has uncertain, time-varying inertial parameters that invalidate standard Reaction Null Space (RNS) reactionless planning. The paper proposes (i) an Adaptive RNS (ARNS) joint-velocity planner that online-identifies the unknown coupling parameters via a Variable Forgetting Factor Recursive Least Squares (VFF-RLS) regression — avoiding the data-saturation of plain RLS for time-varying systems — and (ii) an adaptive-robust joint tracking controller built on an Extreme Learning Machine (ELM) whose input weights/biases are optimized by a staged Strategy Self-Adaptation Differential Evolution (SSADE) scheme, with a Lyapunov stability proof and a sliding-mode-style robust term. Claims are supported by 3-DOF planar Matlab/Simulink simulation and an air-bearing semi-physical experiment.

Key Claims

• For a redundant manipulator, the reactionless joint motion lives in the null space of the base-arm angular-momentum coupling matrix ${\mathbf{H}}_{\omega \varphi }$; the RNS planner is ${\dot{\mathbf{q}}}_{d|RNS}={\mathbf{H}}_{\omega \varphi }^{+}(\mathbf{L}-{\mathbf{r}}_g\times \mathbf{P})+(\mathbf{E}-{\mathbf{H}}_{\omega \varphi }^{+}{\mathbf{H}}_{\omega \varphi })\dot{\mathbf{\xi }}$ (Eq. 19).
• Plain RLS exhibits a data-saturation phenomenon and fails to track the time-varying parameters introduced by an unknown captured target; an exponentially-relaxing forgetting factor $\lambda (k)={\lambda }_{max}-{\sigma }_1{e}^{-{\sigma }_{2}k}$ (Eq. 26) restores tracking while preserving late-phase stability. Tuned values: ${\sigma }_1=0.1$, ${\sigma }_2=0.04$, ${\lambda }_{max}=0.99$, ${\mathbf{Q}}_{(0)}=\delta \mathbf{E}$ with $\delta =1.4$.
• The unknown coupling enters as a regression ${\mathbf{y}}^T=\mathbf{\Phi }\mathbf{W}$ with regressor $\mathbf{\Phi }=[1{\mathbf{\omega }}_0\dot{\mathbf{\xi }}{]}^T$ and parameter block $\mathbf{W}=[{\mathbf{K}}_1{\mathbf{K}}_2{\mathbf{K}}_3{]}^T$ (Eqs. 21-24); ${\mathbf{K}}_1,{\mathbf{K}}_2,{\mathbf{K}}_3$ are estimation errors of the pseudo-inverse/null-space products.
• The SSADE-ELM tracking controller is the most accurate of the three compared (SSADE-ELM, plain ELM, PSO-ELM): simulation average joint-tracking error $-3.0\times 10^{-7}$ vs ELM $-4.5\times 10^{-7}$ and PSO-ELM $-4.2\times 10^{-7}$ (Table 3); experiment $-5.3\times 10^{-6}$ vs $-7.6\times 10^{-6}$ and $-6.9\times 10^{-6}$ (Table 4).
• Trade-off: plain ELM is fastest (no parameter optimization) but least accurate; SSADE-ELM pays optimization overhead for best precision.
• Lyapunov candidate $V=\frac{1}{2}{\mathbf{r}}^T\mathbf{Mr}+\frac{1}{2}{\sum }_{i,j}{\mu }_{ij}^{-1}{\tilde{\beta }}_{ij}^2$ (Eq. 48) yields $\dot{V}\le -{\mathbf{r}}^T\mathbf{Kr}-{\sum }_i(E_{\tau ai}^{max}-E_{\tau ai})|r_i|\le 0$ (Eq. 49), provided the bound $E_{\tau ai}^{max}$ dominates the ELM approximation error.

Method

Regime — free-FLOATING. The paper explicitly states the base is uncontrolled during capture and stabilization: the complex maintains linear and angular momentum conservation (“the space robot is in a free-floating state,” p. 5; the planner exploits $\mathbf{P},\mathbf{L}$ conservation). This is the opposite assumption from our free-flying (actuated 6-DOF base) platform — the entire RNS mechanism here exists because the base cannot be commanded, so reactionless arm motion is the only lever on base attitude.

Dynamics (Eq. 1-2): post-capture complex obeys ${\mathbf{M}}_t\ddot{\mathbf{q}}+{\mathbf{C}}_t\dot{\mathbf{q}}+{\mathbf{\tau }}_k={\mathbf{\tau }}_e$, with $\Delta \mathbf{M}={\mathbf{M}}_t-\mathbf{M}$, $\Delta \mathbf{C}={\mathbf{C}}_t-\mathbf{C}$ lumped into a composite disturbance $\mathbf{d}={\mathbf{\tau }}_e-\Delta \mathbf{M}\ddot{\mathbf{q}}-\Delta \mathbf{C}\dot{\mathbf{q}}$. Crucially the paper assumes $\mathbf{M},\Delta \mathbf{M}$ are constant diagonal matrices (Eq. following Eq. 2), which decouples Eq. 3 joint-wise into ${\ddot{q}}_i=d_i^{\prime }-M_{ii}^{-1}{\tau }_{ki}$ (Eq. 4). Sliding variable $\mathbf{r}=\dot{\mathbf{e}}+\mathbf{\Lambda }\mathbf{e}$ (Eq. 6), $\mathbf{e}=\mathbf{q}-{\mathbf{q}}_d$.

Momentum model (Eq. 9-18): linear/angular momentum assembled from per-link Jacobians ${\mathbf{J}}_{Ti},{\mathbf{J}}_{Ri}$ into the coupled form $[\mathbf{P};\mathbf{L}]=[[\cdots ]][{\mathbf{v}}_0;{\mathbf{\omega }}_0]+[{\mathbf{J}}_{T\omega };{\mathbf{H}}_{\omega \varphi }]\dot{\mathbf{q}}+[0;{\mathbf{r}}_g\times \mathbf{P}]$ (Eq. 17). ${\mathbf{H}}_{\omega \varphi }$ is the arm-to-base angular-momentum coupling (the object the null space is taken over).

Control law (Eq. 28): ${\tau }_{ki}=M_{ii}({\tau }_{ai}+{\tau }_{ci}+{\tau }_{PDi})$ — adaptive (ELM) + robust + PD. ELM hidden layer ${\mathbf{h}}_i=g_i({\mathbf{A}}_i{\mathbf{x}}_i+{\mathbf{b}}_i)$ with sigmoid $g$, input ${\mathbf{x}}_i=[{\ddot{q}}_{di}{\dot{q}}_{di}{q}_{di}{e}_i{\dot{e}}_i]$; output weights via ${\dot{\hat{\beta }}}_i={\mu }_i{\mathbf{h}}_i{r}_i$ (Eq. 34). Robust term ${\hat{\tau }}_{Ri}=E_{\tau ai}^{max}\mathrm{sign}(r_i)$ (Eq. 41) bounds the ELM residual.

Notation clashes to flag: $\mathbf{M}$ is overloaded — total system mass in the momentum block (Eq. 13/17) vs the diagonal joint-inertia matrix in the dynamics (Eq. 1-4). ${\mathbf{\omega }}_0$ denotes base angular velocity in the body but is listed as an initial condition = 0 in Table 1. The text says “Bringing Eq. (7) into Equation (1)” and “according to Equation (15)/(20)” where the referenced numbers are inconsistent with the displayed equations (likely an MDPI numbering/OCR drift); the regression update is Eq. 24-26.

Relevance to thesis

This is a useful contrast paper. Its core assumption — free-floating, uncontrolled base, momentum-conserving — is exactly what our free-flying platform relaxes. For us the base is fully actuated, so we do not need RNS to protect base attitude; we can instead coordinate base thrust/torque with arm motion. The paper is nonetheless relevant for (1) the online inertial-parameter identification problem after grasping an unknown target (VFF-RLS regression is transferable to our coordinated-control parameter estimation), and (2) the adaptive-robust + Lyapunov template for tracking under lumped uncertainty, which maps onto our risk-aware control layer. The constant-diagonal-$\mathbf{M}$ assumption is too strong for a fully-coupled 6-DOF base + redundant arm and should not be carried over uncritically.

Connections

Topics: reaction_null_space · dynamic_coupling · parameter_estimation · trajectory_tracking

Key Equations / Quotes

“The space robot in this paper is in a free-floating state, and its manipulator has redundancy.” (Sec. 3.1)

RNS planner (Eq. 19):
${\dot{\mathbf{q}}}_{d|RNS}={\mathbf{H}}_{\omega \varphi }^{+}(\mathbf{L}-{\mathbf{r}}_g\times \mathbf{P})+(\mathbf{E}-{\mathbf{H}}_{\omega \varphi }^{+}{\mathbf{H}}_{\omega \varphi })\dot{\mathbf{\xi }}$

Variable forgetting factor (Eq. 26):
$\lambda (k)={\lambda }_{max}-{\sigma }_1{e}^{-{\sigma }_{2}k}$

Lyapunov derivative bound (Eq. 49):
$\dot{V}\le -{\mathbf{r}}^T\mathbf{Kr}-{\sum }_{i=1}^n\left(E_{\tau ai}^{max}-E_{\tau ai}\right)|r_i|\le 0$

“The proposed algorithm does not require accurate system dynamics modeling, nor does it require information about unknown time-varying disturbance, enabling dynamic reactionless path tracking control.” (Conclusions)

Open Questions

• The Lyapunov bound requires $E_{\tau ai}^{max}\ge |E_{\tau ai}|$ at all times, but $E_{\tau ai}^{max}=1$ is fixed in Table 2 — how is this bound guaranteed for an arbitrary unknown target rather than assumed?
• The constant-diagonal joint inertia $\mathbf{M}$ assumption decouples the dynamics; how badly does it degrade once configuration-dependent off-diagonal inertia coupling is restored?
• Only a planar 3-DOF case is demonstrated; does the staged SSADE crowding metric (${\bar{d}}_{g,ave}$ thresholds $c=0.85$) scale to the higher-dimensional weight space of a spatial 6+ DOF arm?
• Conclusion notes “slow execution speed” from the evolutionary inner loop — is SSADE-ELM real-time feasible on flight hardware, or only offline weight tuning?