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\begin{document}

%% article title
\title{Supplementary materials for}
\titleitem{Learning-based data-driven control for micro-nano free-floating space robots in Cartesian space}

\doi{\\https://doi.org/10.1631/ENG.ITEE.2026.0054}
\longauthor{Renhao MAO, Tao MENG, Kun WANG, Zhonglin ZUO, Hang ZHOU, Shujian SUN, }



\publishyear{2026}
\vol{27}
\issue{7}
\articlenumber{260054}
\thispagestart{1}
\pagestart{1}
\pageend{4}


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%\articleType can be `Science Letters:', `Review:', `Comment:', etc.

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\section*{Proof of Theorem 1}
 \newtheorem{assumption}{Assumption}
  %\begin{assumption}
  \noindent \textbf{Assumption S1}\ \ \ \ The TFO variable $\boldsymbol{s}[\boldsymbol{z}(k)]$ is Lipschitz continuous with a Lipschitz constant $L_{\rm s}$.
  \label{assumption1}
  %\end{assumption}  
  Since the uncertainty vector $\boldsymbol{D}(k)$ is bounded, over a finite prediction horizon, each term $\Vert \boldsymbol{D}(k+i|k)\Vert _2\ (i=0,1,\cdots, N_{\rm p})$ satisfies $\Vert\boldsymbol{D}(k+i|k) \Vert _2\leq \varUpsilon $.\par
  
  Combined with Assumption \ref{assumption1}, the norm of the TFO prediction errors between the nominal and true systems is bounded as follows:
%  \begin{equation}
%      \begin{aligned}
%          &\Vert \boldsymbol{s}[\hat{\boldsymbol{z}}(k+1|k)]-\boldsymbol{s}[\boldsymbol{z}(k+1)] \Vert _2 \leq L_{\rm s} \Vert\boldsymbol{D} \Vert _2\leq L_{\rm s} \varUpsilon, \\
%          &\vdots \\
%          &\Vert \boldsymbol{s}[\hat{\boldsymbol{z}}(k+i|k)]-\boldsymbol{s}[\boldsymbol{z}(k+i)] \Vert _2 \leq \frac{1 - \Vert\boldsymbol{A}\Vert^i}{1 - \Vert\boldsymbol{A}\Vert} L_{\rm s} \varUpsilon.
%      \end{aligned}
%   \end{equation}\par
   \begin{equation}
   	\begin{cases}
   		\Vert \boldsymbol{s}[\hat{\boldsymbol{z}}(k+1|k)]-\boldsymbol{s}[\boldsymbol{z}(k+1)] \Vert _2 \leq L_{\rm s} \Vert\boldsymbol{D} \Vert _2\leq L_{\rm s} \varUpsilon, \\
   		\vdots \\
   		\Vert \boldsymbol{s}[\hat{\boldsymbol{z}}(k+i|k)]-\boldsymbol{s}[\boldsymbol{z}(k+i)] \Vert _2 \leq \frac{1 - \Vert\boldsymbol{A}\Vert^i}{1 - \Vert\boldsymbol{A}\Vert} L_{\rm s} \varUpsilon.
   	\end{cases}
   \end{equation}
  This completes the proof of Theorem 1.

\section*{Proof of Theorem 2}
%\begin{lemma} \citep{marruedo2002input}
  \noindent \textbf{Lemma S1} \citep{marruedo2002input}\ \ \ \ Consider the system $\boldsymbol{x}(k+1)=\boldsymbol{f}(\boldsymbol{x}(k),\boldsymbol{D}(k))$, if there exists a positive definite function $V(\boldsymbol{x}(k),\boldsymbol{D}(k))$, and $\mathcal{K}_{\infty}$ functions $a_3$, $a_4$, $a_5$, $a_6$, such that
  $a_3\ \left(\Vert \boldsymbol{x}(k) \Vert _2\right)\leq V(\boldsymbol{x}(k),\ \boldsymbol{D}(k))\leq a_4\left(\Vert \boldsymbol{x}(k) \Vert _2\right)$ and furthermore $V(\boldsymbol{x}(k+1),\boldsymbol{D}(k+1))- V(\boldsymbol{x}(k),\ \boldsymbol{D}(k))\leq -a_5\left(\Vert \boldsymbol{x}(k) \Vert _2\right)+a_6\left(\Vert \boldsymbol{w}(k) \Vert _2\right)$, then the system is input-to-state stable (ISS).
  \label{lemma1}
%\end{lemma}\par

%\begin{assumption}
    \noindent \textbf{Assumption S2}\ \ \ \ The stage cost function $L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+i|k)], \boldsymbol{u}(k+i|k)\}$ satisfies Lipschitz continuity with constant $L_{\rm a}$. Additionally, there exists a $\mathcal{K}_{\infty}$ function $a_1$, such that $L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+i|k)],\  \boldsymbol{u}(k+i|k)\}\geq a_1\left(\Vert\boldsymbol{s}[\hat{\boldsymbol{z}}(k+i|k)] \Vert _2\right)$.
%  \label{assumption2}
%\end{assumption}

%\begin{assumption}
      \noindent \textbf{Assumption S3}\ \ \ \ The terminal cost function $L_{\rm f}\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k)]\}$ is Lipschitz continuous with constant $L_{\rm b}$. There exists a $\mathcal{K}_{\infty}$ function $a_2$, such that $L_{\rm f}\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k)]\}\geq a_2\left(\Vert\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k)] \Vert _2\right)$.
%  \label{assumption3}
%\end{assumption}

%\begin{assumption}
      \noindent \textbf{Assumption S4}\ \ \ \ There exists a local stabilizing controller $\boldsymbol{\mu}(k) $ for the terminal set $\mathcal{X}_{\rm f}$, ensuring that $L_{\rm f}\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+1)]\}-L_{\rm f}\{\boldsymbol{s}[\boldsymbol{z}(k)]\} \leq L\{\boldsymbol{s}[\boldsymbol{z}(k)],\boldsymbol{\mu}(k)\}$.
%  \label{assumption4}
%\end{assumption}
Consider the following Lyapunov function definition:
\begin{equation}
    V(k)=J^*(k),
    \label{eqs2}
\end{equation}
where $J^*(k)$ represents the optimal cost function for the objective.\par
Let $\bar{J}(k+1)$ denote the cost function associated with a feasible trajectory at the subsequent time step $k+1$.\par
The discrepancy between $\bar{J}(k+1)$ and $J^*(k)$ can be expressed as
\begin{equation}
    \begin{aligned}
       {\rm \Delta} J(k)&=\bar{J}(k+1)-J^*(k) \\
        &=\sum_{i = 0}^{N_{\rm p}-2}\left(L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+i+1|k+1)], \boldsymbol{u}(k+i+1|k+1)\}-L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+i+1|k)],\boldsymbol{u}^*(k+i+1|k)\}\right) \\ 
        &\quad+ L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k+1)], \boldsymbol{u}(k+N_{\rm p}|k+1)\}- L\{\boldsymbol{s}[\boldsymbol{z}(k)], \boldsymbol{u}^*(k)\}+ L_{\rm f}\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}+1|k+1)]\} \\
        &\quad- L_{\rm f}\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k)]\}+ L_f\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k+1)]\}-L_{\rm f}\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k+1)]\}. 
    \end{aligned}
\end{equation}\par
Invoking Assumptions S2 and S3, we have
\begin{equation}
    \begin{aligned}
    &L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+i+1|k+1)], \boldsymbol{u}(k+i+1|k+1)\}- L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+i+1|k)], \boldsymbol{u}^*(k+i+1|k)\}\leq L_{\rm a} L_{\rm s} \Vert\boldsymbol{A}\Vert^i\varUpsilon \\
    &L_{\rm f}\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k+1)]\}- L_{\rm f}\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k+N_{\rm p}|k)]\}\leq L_{\rm b} L_{\rm s} \Vert\boldsymbol{A}\Vert^{N_{\rm p}-1}\varUpsilon.
    \end{aligned}
\end{equation}\par
Invoking Assumption S4, we have
\begin{equation}
    \begin{aligned}
    {\rm \Delta} J(k)&\leq L_{\rm c} \varUpsilon -L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k)], \boldsymbol{u}^*(k)\},
    \end{aligned}
\end{equation}
where $L_{\rm c}=\left(L_{\rm b}\Vert\boldsymbol{A}\Vert^{N_{\rm p}-1} +L_{\rm a}\frac{1 - \Vert\boldsymbol{A}\Vert^{N_{\rm p}-1}}{1- \Vert\boldsymbol{A}\Vert}\right) L_{\rm s}$.\par
Invoking Assumption S2, $L\{\boldsymbol{s}[\hat{\boldsymbol{z}}(k)],\boldsymbol{u}^*(k)\}\geq a_1\Vert\boldsymbol{s}[\boldsymbol{z}(k)]\Vert _2$, the first difference of \eqref{eqs2} is
\begin{equation}
    {\rm \Delta} V(k)=J^*(k+1)-J^*(k)\leq L_{\rm c}\varUpsilon -a_1\Vert \boldsymbol{s}[\boldsymbol{z}(k)] \Vert _2.
\end{equation}\par
Invoking Lemma S1, the TFO variable $\boldsymbol{s}[\boldsymbol{z}(k)]$ is ISS.\par
  This completes the proof of Theorem 2.

\section*{Proof of Theorem 3}
%\begin{lemma} \citep{li2022nonlinear}
      \noindent \textbf{Lemma S2} \citep{li2022nonlinear}\ \ \ \ The relation between $\binom{\alpha}{j}$ and $\Gamma(\chi )$ can be established by
    \begin{equation}
      \sum_{j = 0}^{L} (-1)^j \binom{\alpha}{j} = \frac{\Gamma (L+1-\alpha)}{\Gamma (1-\alpha)\Gamma(L+1)},
     \end{equation}
    where $\Gamma(\chi )=\int_{0}^{\infty }  \tau ^{\chi-1}{\rm e}^{-\tau }{\rm d} \tau$. \par
    Considering the bounded sequence $y(k), k=0,1,\cdots $ with $\max \{ y(k-1) \}\leq \rho_{\rm y} $, the corresponding fractional-order difference is bounded by
    \begin{equation}
      \frac{1}{({\rm \Delta} T)^{\alpha }} \sum_{j = 1}^{L} (-1)^j \binom{\alpha}{j} y(k-j) \leq \frac{(\mathcal{N} -1)\rho_{\rm y}}{({\rm \Delta} T)^{\alpha }},
     \end{equation}
    where $\mathcal{N} = \frac{\Gamma (L+1-\alpha)}{\Gamma (1-\alpha)\Gamma(L+1)} \geq 1$.
    \label{lemma2}
%\end{lemma}

%\begin{lemma} \citep{li2013discrete}
      \noindent \textbf{Lemma S3} \citep{li2013discrete}\ \ \ \   Considering a dynamical system $y(k+1) - y(k) + \xi \lceil y(k) \rfloor ^{\lambda} = g(k)$ with $\vert g(k) \vert \leq \gamma $, there is a finite step to guarantee
  \begin{equation}
    \vert y(k) \vert \leq \Lambda \cdot \max \left\{ \left( \frac{\gamma}{\xi} \right)^{\frac{1}{\lambda}}, \xi^{\frac{1}{1 - \lambda}} \right\},
  \end{equation}
where $\Lambda = 1 + \lambda ^{\frac{\lambda}{1-\lambda}} - \lambda ^{\frac{1}{1-\lambda}}$.
  \label{lemma3}
%\end{lemma}\par

%\begin{lemma} \citep{toth2015convergence}
      \noindent \textbf{Lemma S4} \citep{toth2015convergence}\ \ \ \     For a sequence of vectors $\{\boldsymbol{x}_k\}$ in the vector space $\mathbb{C}^n$ and a vector $\boldsymbol{x}_0$, where $\boldsymbol{x}_k = \begin{bmatrix} a_1^{(k)}, \cdots, a_n^{(k)} \end{bmatrix}^{\rm T}$ and $\boldsymbol{x}_0 = \begin{bmatrix} a_1, \cdots, a_n \end{bmatrix}^{\rm T}$, 
  if the condition $\lim_{k \to \infty} a_j^{(k)} = a_j$ holds $ \forall 1 \leq j \leq n$, then the sequence $\{\boldsymbol{x}_k\}$ converges to $\boldsymbol{x}_0$ with respect to the vector norm.
  \label{lemma4}
%\end{lemma}

The first difference of $\boldsymbol{s}[\boldsymbol{z}(k)]$ is
\begin{equation}
  \boldsymbol{s}[\boldsymbol{z}(k+1)] - \boldsymbol{s}[\boldsymbol{z}(k)] = \boldsymbol{e}(k+1) - \boldsymbol{e}(k) + {\rm \Delta} T \boldsymbol{\xi} \boldsymbol{\mathcal{D}} ^{\alpha} \lceil \boldsymbol{e}(k) \rfloor ^{\lambda}.
  \label{eqs10}
\end{equation}\par
By invoking the Grünwald-Letnikov fractional-order definition and setting $\boldsymbol{\varrho }(k) = \boldsymbol{s}[\boldsymbol{z}(k+1)] - \boldsymbol{s}[\boldsymbol{z}(k)]$, \eqref{eqs10} can be reformulated as
\begin{equation}
  \begin{aligned}
  \boldsymbol{\varrho }(k) = \boldsymbol{e}(k+1) - \boldsymbol{e}(k) + \frac{\boldsymbol{\xi}}{({\rm \Delta} T)^{\alpha -1}} \lceil \boldsymbol{e}(k) \rfloor ^{\lambda} + \frac{\boldsymbol{\xi}}{({\rm \Delta} T)^{\alpha -1}} \sum_{j = 1}^{L} (-1)^j \binom{\alpha}{j}\lceil \boldsymbol{e}(k-j) \rfloor ^{\lambda}. 
   \end{aligned}
\end{equation}\par
Without loss of generality, for each element $e_i(k)\ (i = 1,\cdots,n)$ in $\boldsymbol{e}(k)$, there exists
\begin{equation}
  \begin{aligned}
  \varrho_i(k) =  e_i(k+1) - e_i(k) + \frac{\xi_i}{({\rm \Delta} T)^{\alpha -1}} \lceil e_i(k) \rfloor ^{\lambda} + \frac{\xi_i}{({\rm \Delta} T)^{\alpha -1}} \sum_{j = 1}^{L} (-1)^j \binom{\alpha}{j}\lceil e_i(k-j) \rfloor ^{\lambda} .
  \label{eqs12}
  \end{aligned}
\end{equation}\par
By applying Lemma S2\ref{lemma2} and considering the bounded sequence $e_i(k)$ with $\max \{ e_i(k-1) \}\leq \rho_i $, we obtain
\begin{equation}
  \frac{1}{({\rm \Delta} T)^{\alpha -1}} \sum_{j = 1}^{L} (-1)^j \binom{\alpha}{j}\lceil e_i(k-j) \rfloor ^{\lambda} \leq \frac{(\mathcal{N}_i -1)\rho_i}{({\rm \Delta} T)^{\alpha -1}},
 \end{equation}
where $\mathcal{N}_i = \frac{\Gamma (L+1-\alpha)}{\Gamma (1-\alpha)\Gamma(L+1)} \geq 1$.\par
When $\boldsymbol{s}[\boldsymbol{z}(k)]$ converges, $\boldsymbol{\varrho}(k)$ also converges. For each element $\varrho_i(k)$ in $\boldsymbol{\varrho}(k)$, there exists a positive constant $\kappa_i$ such that $ \vert \varrho_i(k) \vert \leq \kappa_i$. Consequently, \eqref{eqs12} can be reformulated as
\begin{equation}
  e_i(k+1) - e_i(k) + \frac{\xi_i}{({\rm \Delta} T)^{\alpha -1}} \lceil e_i(k) \rfloor ^{\lambda} \leq \kappa_i - \frac{(\mathcal{N}_i -1)\xi_i \rho_i}{({\rm \Delta} T)^{\alpha -1}}.
 \end{equation}\par
By applying Lemma S3\ref{lemma3}, $e_i(k)$ converges within a finite number of steps:
\begin{equation}
  \begin{aligned}
  \vert e_i(k) \vert \leq \Lambda \cdot \max \left\{ \left[ \frac{({\rm \Delta} T)^{\alpha -1} \kappa_i - (\mathcal{N}_i -1)\xi_i \rho_i}{\xi_i} \right]^{\frac{1}{\lambda}}, \left( ({\rm \Delta} T)^{1-\alpha} \xi_i \right)^{\frac{1}{1 - \lambda}} \right\}.
  \label{eqb6}
  \end{aligned}
\end{equation}
where $\Lambda = 1 + \lambda ^{\frac{\lambda}{1-\lambda}} - \lambda ^{\frac{1}{1-\lambda}}$.\par
By applying Lemma S4\ref{lemma4}, if the condition that $e_i(k)$ converges holds $\forall 1 \leq i \leq n$, then the error vector $\boldsymbol{e}(k)$ also converges. Therefore, $\boldsymbol{e}(k)$ converges whenever $\boldsymbol{s}[\boldsymbol{z}(k)]$ converges.\par
This completes the proof of Theorem 3.
\section*{Monte Carlo simulations}
We conducte two sets of Monte Carlo simulations with different target-point ranges, each comprising 200 runs. During the simulations, the base satellite and each manipulator component are subject to $20\%$ uncertainties in mass and inertia. In addition, each joint is subjected to an unknown disturbance torque ${\Delta}_{\rm m}^i(t) = 0.01 \times (\sin(20t) + \cos(10t) - {\rm e}^{-0.1t})$ $\rm{N\cdot m}$  ($i=1,2,\ldots,6$).\par
We define the DEN fitting error during the control process by
\begin{equation}
  \boldsymbol{r}(k) = \boldsymbol{p}_{\rm e}(k+1) - \boldsymbol{p}_{\rm e}(k) - \boldsymbol{\Psi}(k) \big(\theta(k+1) - \theta(k)\big) .
\end{equation}\par
During the Monte Carlo simulations, we record the terminal end-effector tracking error $\Vert \boldsymbol{p}_{{\rm ref}} - \boldsymbol{p}_{\rm e}(T) \Vert_2$, as well as the maximum DEN fitting error over the simulation, defined by ${r}_{\max} = \max \Vert \boldsymbol{r}(k) \Vert_2$.\par
In the first set of Monte Carlo simulations, the target point is specified as
\begin{equation}
  \boldsymbol{p}_{\mathrm{ref}}=\begin{bmatrix}0.5\\ 0\\ 0.5\end{bmatrix}\mathrm{m}+
  \begin{bmatrix}{\delta} x\\ { \delta} y\\ { \delta} z\end{bmatrix}\mathrm{m},
\end{equation}
where ${ \delta} x$, ${ \delta} y$, and ${ \delta} z$ are independently sampled from a uniform distribution over $(-0.1,\,0.1)$.\par
In the second set of Monte Carlo simulations, the target point is defined in the same manner, except that ${ \delta} x$, ${ \delta} y$, and ${ \delta} z$ are independently sampled from a uniform distribution over $(-0.2,\,0.2)$.\par

Figs. \ref{monte_1} and \ref{monte_2} present the results of the two sets of Monte Carlo simulations. Defining the success criterion as a terminal end-effector tracking error of less than $2 \times 10^{-3}$ m, set 1 achieves a $100\%$ success rate, whereas set 2 achieves $93.5\%$. 
An analysis of the maximum DEN fitting error reveals that substantial fitting inaccuracies during the simulation are the primary cause of failure. This issue stems from two main factors. First, the kinematic model of the FFSR is influenced by both its geometric configuration and dynamic factors, resulting in deviations when the system is subjected to external disturbances and model uncertainties. Second, in scenarios involving widely dispersed target points, the system is more likely to approach kinematic singularities, which degrades the DEN's fitting performance in out-of-distribution regions.
\begin{figure}[H]
  \centering
  \subfloat[Terminal end-effector tracking error]{
  \includegraphics[height=4cm]{case1_end.eps}
  }
  \subfloat[Maximum DEN fitting error]{
  \includegraphics[height=4cm]{case1_max_r.eps}
  }
  \caption{Results of Monte Carlo simulation (set 1)}
  \label{monte_1}
\end{figure}\par
\begin{figure}[H]
  \centering
  \subfloat[Terminal end-effector tracking error]{
  \includegraphics[height=4cm]{case2_end.eps}
  }
  \subfloat[Maximum DEN fitting error]{
  \includegraphics[height=4cm]{case2_max_r.eps}
  }
  \caption{Results of Monte Carlo simulation (set 2)}
  \label{monte_2}
\end{figure}\par



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