Dynamic Modeling and Control for a Double-State Microgravity Vibration Isolation System

Abstract

The microgravity vibration isolation systems (MVIS) are characterized by system coupling, uncertainties, nonlinearity, variable mass and momentum of inertia. These characteristics lead to a challenging control system design problem. In this paper, a double-stage microgravity vibration isolation system is introduced. The relative motion dynamics and absolute motion dynamics of the system are established via Newton's law, thereafter the linearized and decoupled model is derived for control design. A robust backstepping controller is proposed to suppress the accelerations transmitted to the payload, as well as to address the challenges of system uncertainties and nonlinearity. The validity of the relative motion dynamics of the system is demonstrated by comparing the experimental data with the simulation results. The Lyapunov theory is used for proving the stability of the proposed controller. Simulations show that the robust backstepping controller is capable of suppressing microgravity acceleration disturbances from 1 to 100 Hz at around 19 dB and tracking the displacement to zero position fast, which performs well in the time domain and frequency domain.

Appendices

Appendix

To express Eqs. (32) and (33) of motion in concise form, the following definitions are introduced.

$\begin{array}{c}{\mathbf{C}}_1=\left[\begin{array}{cc}c_{11}&c_{12}\\c_{21}&c_{22}\end{array}\right],\;{\mathbf{K}}_1=\left[\begin{array}{cc}k_{11}&k_{12}\\k_{21}&k_{22}\end{array}\right],\;{\mathbf{D}}_1=\left[\begin{array}{c}d_1\\d_2\end{array}\right]\\{\mathbf{C}}_2=\left[\begin{array}{cc}c_{31}&c_{32}\\c_{41}&c_{42}\end{array}\right],\;{\mathbf{K}}_2=\left[\begin{array}{cc}k_{31}&k_{32}\\k_{41}&k_{42}\end{array}\right],\;{\mathbf{D}}_2=\left[\begin{array}{c}d_3\\d_4\end{array}\right]\end{array}$  

with

$$\begin{array}{l}{c}_{11}={2}^{I}{{\varvec{\upomega}}}^{C\times }+{\mathbf{C}}_{tt}/{m}_{S}\\ {c}_{12}=\left({\mathbf{C}}_{tr}-{\mathbf{C}}_{tt}{\mathbf{r}}_{{u}_{0}u}^{\times }\right)/{m}_{S}\\ {c}_{21}=\left({\mathbf{C}}_{rt}+{\mathbf{r}}_{{u}_{0}u}^{\times }{\mathbf{C}}_{tt}\right)\\ {c}_{22}={\mathbf{J}}_{S}^{-1}[{{\mathbf{J}}_{S}}^{I}{{\varvec{\upomega}}}^{C\times }{+}^{I}{{\varvec{\upomega}}}^{C\times }{\mathbf{J}}_{S}-{({{\mathbf{J}}_{S}}^{I}{{\varvec{\upomega}}}^{C})}^{\times }]\\ {k}_{11}=\frac{{\mu }_{E}}{{r}_{{I}_{0}{B}_{0}}^{3}}{\mathbf{A}}_{r}{-}^{I}{\alpha }^{C\times }{-}^{I}{{\varvec{\upomega}}}^{C\times }{}^{I}{{\varvec{\upomega}}}^{C\times }+{\mathbf{K}}_{tt}/{m}_{S}{k}_{12}=\left({\mathbf{K}}_{tr}-{\mathbf{K}}_{tt}{\mathbf{r}}_{{u}_{0}u}^{\times }\right)/{m}_{S}\\ {k}_{21}=-({\mathbf{C}}_{rt}{\mathbf{r}}_{{u}_{0}u}^{\times }-{\mathbf{C}}_{rr}-{\mathbf{r}}_{{u}_{0}u}^{\times }{\mathbf{C}}_{tr}+{\mathbf{r}}_{{u}_{0}u}^{\times }{\mathbf{C}}_{tt}\cdot {\mathbf{r}}_{{u}_{0}u}^{\times })\\ {k}_{22}=-({\mathbf{K}}_{rt}{\mathbf{r}}_{{u}_{0}u}^{\times }-{\mathbf{K}}_{rr}-{\mathbf{r}}_{{u}_{0}u}^{\times }{\mathbf{K}}_{tr}+{\mathbf{r}}_{{u}_{0}u}^{\times }{\mathbf{K}}_{tt}{\mathbf{r}}_{{u}_{0}u}^{\times })\\ +{\mathbf{J}}_{S}^{-1}\left[{{\mathbf{J}}_{S}}^{S}{{\varvec{\upalpha}}}^{C\times }+{\mathbf{J}}_{S}{({{(}^{C}{\mathbf{Q}}^{B}{}^{I}{{\varvec{\upomega}}}^{B})}^{\times }{+}^{B}{{\varvec{\upomega}}}^{C})}^{\times }\right.\left.-{{\mathbf{J}}_{S}}^{I}{{\varvec{\upomega}}}^{C\times }{}^{I}{{\varvec{\upomega}}}^{C\times }{+}^{I}{{\varvec{\upomega}}}^{C\times }{{\mathbf{J}}_{S}}^{I}{{\varvec{\upomega}}}^{C\times }\right]\\ {d}_{1}=({\mathbf{F}}_{\Delta S}-{\mathbf{F}}_{cF})/{m}_{S}-\left({\mathbf{F}}_{\Delta B}/{m}_{B}+{\mathbf{F}}_{\Delta C}/{m}_{C}\right)\\ -\left(\frac{{\mu }_{E}}{{r}_{{I}_{0}{B}_{0}}^{3}}{\mathbf{A}}_{r}{-}^{I}{\alpha }^{B\times }{-}^{I}{{\varvec{\upomega}}}^{B\times I}{{\varvec{\upomega}}}^{B\times }\right){\mathbf{r}}_{{B}_{0}{C}_{0}}+{\mathbf{F}}_{\Delta u}\\ {d}_{2}={\mathbf{J}}_{S}^{-1}(-{\mathbf{M}}_{cF}+{\mathbf{M}}_{\Delta S}){-}^{I}{{\varvec{\upalpha}}}^{C}\\ -{({}^{C}{\mathrm{Q}}^{BI}{{\varvec{\upomega}}}^{B})}^{\times B}{{\varvec{\upomega}}}^{C}-{\mathbf{J}}_{S}^{-1}{}^{I}{{\varvec{\upomega}}}^{C\times }{{\mathbf{J}}_{S}}^{I}{{\varvec{\upomega}}}^{C}\end{array}$$

Robust to Noise Microgravity

$$\begin{array}{l}{c}_{31}={2}^{I}{{\varvec{\upomega}}}^{S\times }\\ {c}_{32}={0}_{3\times 3}\\ {c}_{41}={0}_{3\times 3}\\ {c}_{42}={\mathbf{J}}_{F}^{-1}\left[{{\mathbf{J}}_{F}}{}^{I}{\varvec{\upomega}}^{S\times }{+}^{I}{{\varvec{\upomega}}}^{S\times }{\mathbf{J}}_{F}-{({{\mathbf{J}}_{F}}{}^{I}{\omega }^{S})}^{\times }\right]\\ {k}_{31}=\frac{{\mu }_{E}}{{r}_{{I}_{0}{B}_{0}}^{3}}{\mathbf{A}}_{r}{-}^{I}{{\varvec{\upalpha}}}^{S\times }{-}^{I}{{\varvec{\upomega}}}^{S\times }{}^{I}{{\varvec{\upomega}}}^{S\times }\\ {k}_{32}={0}_{3\times 3}\\ {k}_{41}={0}_{3\times 3}\\ {k}_{42}={\mathbf{J}}_{F}^{-1}\left[{{\mathbf{J}}_{F}}{}^{I}{\alpha }^{S\times }-{({{\mathbf{J}}_{F}}{}^{I}{{\varvec{\upomega}}}^{S})}^{\times }{}^{I}{{\varvec{\upomega}}}^{S\times }{+}^{I}{{\varvec{\upomega}}}^{S\times }{{\mathbf{J}}_{F}}{}^{I}{{\varvec{\upomega}}}^{S\times }\right]\\ {d}_{3}={\mathbf{F}}_{\Delta F}/{m}_{F}-{\mathbf{F}}_{\Delta B}/{m}_{B}\\ -\left(\frac{{\mu }_{E}}{{r}_{{I}_{0}{B}_{0}}^{3}}{\mathbf{A}}_{r}{-}^{I}{{\varvec{\upalpha}}}^{B\times }{-}^{I}{{\varvec{\upomega}}}^{B\times I}{{\varvec{\upomega}}}^{B\times }\right)({\mathbf{r}}_{{B}_{0}{C}_{0}}+{\mathbf{r}}_{{C}_{0}{S}_{0}})\\ {d}_{4}={\mathbf{J}}_{F}^{-1}{\mathbf{M}}_{\Delta F}{-}^{I}{{\varvec{\upalpha}}}^{S}-{\mathbf{J}}_{F}^{-1}{}^{I}{{\varvec{\upomega}}}^{S\times }{\mathbf{J}}_{F}{}^{I}{{\varvec{\upomega}}}^{S}\end{array}$$