#  SCIENTIA SINICA Informationis, Volume 50 , Issue 4 : 588-602(2020) https://doi.org/10.1360/N112019-00049

## Differential game learning approach for multiple microsatellites takeover of the attitude movement of failed spacecraft More info
• ReceivedFeb 28, 2019
• AcceptedJun 5, 2019
• PublishedApr 8, 2020
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Appendix

\begin{align*}&\Psi_{1}=\left[\begin{matrix}-\textstyle\theta_{1m}^{2} \cdots -\textstyle\theta_{Nm}^{2}\end{matrix}\right], \Psi_{21}=\begin{bmatrix} \frac{\lambda_{DM11}^{2}}{2\psi_{121}^{2}}&\cdots &\frac{\lambda_{DM1N}^{2}}{2\psi_{12N}^{2}} \\ \vdots& &\vdots \\ \frac{\lambda_{DMN1}^{2}}{2\psi_{N21}^{2}}&\cdots &\frac{\lambda_{DMNN}^{2}}{2\psi_{N2N}^{2}}\end{bmatrix}, \Psi_{22}=\begin{bmatrix}\textstyle{\sum_{j=1}^{N} \frac{\psi_{12j}^{2}\theta_{1M}^{2}}{2}} \cdots \textstyle{\sum_{j=1}^{N} \frac{\psi_{N2j}^{2}\theta_{NM}^{2}}{2}}\end{bmatrix}, \\ &\Psi_{41}=\begin{bmatrix} &\textstyle{\sum_{j=1}^{N} \frac{\lambda_{EMj1}^{2}}{32\psi_{j41}^{2}}}& & \\ & &\ddots & \\ & & &\textstyle{\sum_{j=1}^{N} \frac{\lambda_{EMjN}^{2}}{32\psi_{j4N}^{2}}}\end{bmatrix}, \Psi_{42}=\begin{bmatrix}\textstyle{\sum_{j=1}^{N} \frac{\psi_{14j}^{2}b_{1M}^{2}}{2}} \cdots \textstyle{\sum_{j=1}^{N} \frac{\psi_{N4j}^{2}b_{NM}^{2}}{2}}\end{bmatrix}, \\ &\Psi_{5}=\begin{bmatrix}\textstyle{\sum_{j=1}^{N} \frac{\psi_{15j}^{2}b_{1M}^{2}}{2}} \cdots \textstyle{\sum_{j=1}^{N} \frac{\psi_{N5j}^{2}b_{NM}^{2}}{2}}\end{bmatrix}+\begin{bmatrix}\textstyle{\sum_{j=1}^{N} \frac{b_{Dj1}^{2}}{8\psi_{j51}^{2}}} \cdots \textstyle{\sum_{j=1}^{N} \frac{b_{DjN}^{2}}{8\psi_{j5N}^{2}}}\end{bmatrix}, \\ &\Psi_{6}=\begin{bmatrix}\textstyle{\sum_{j=1}^{N} \frac{\psi_{16j}^{2}b_{1M}^{2}}{2}} \cdots \textstyle{\sum_{j=1}^{N} \frac{\psi_{N6j}^{2}b_{NM}^{2}}{2}}\end{bmatrix}+\begin{bmatrix}\textstyle{\sum_{j=1}^{N} \frac{b_{Ej1}^{2}}{8\psi_{j61}^{2}}} \cdots \textstyle{\sum_{j=1}^{N} \frac{b_{EjN}^{2}}{8\psi_{j6N}^{2}}}\end{bmatrix}, \\ &\Psi_{71}=\begin{bmatrix} \frac{\psi_{17}^{2}b_{1M}^{2}}{2} \cdots \frac{\psi_{N7}^{2}b_{NM}^{2}}{2}\end{bmatrix}, \Psi_{72}=\sum_{i=1}^{N} \frac{b_{e_{Hi}}^{2}}{2\psi_{i7}^{2}}, \end{align*} 其中$\Psi_{21}$, $\Psi_{41}\in\mathbb{R}^{{N}\times{N}}$, $\Psi_{1}$, $\Psi_{22}$, $\Psi_{42}$, $\Psi_{5}$, $\Psi_{6}$, $\Psi_{71}\in\mathbb{R}^{{1}\times{N}}$, $\Psi_{72}\in\mathbb{R}$.

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• Figure 2

(Color online) Value function trajectories of microsatellites

• Figure 3

(Color online) Convergence of critic NN weight vector estimations of microsatellites. (a) Microsatellite 1; protectłinebreak (b) Microsatellite 2; (c) Microsatellite 3

• Figure 4

(Color online) Attitude MRPs trajectories of combination

• Figure 5

(Color online) Attitude angular velocity trajectories of combination

• Figure 6

(Color online) Control torques trajectories of microsatellites. (a) Microsatellite 1; (b) Microsatellite 2; protectłinebreak (c) Microsatellite 3