Finite-time integrated target tracking for spacecraft with two-dimensional turntable under performance constraints

Abstract

The finite-time integrated tracking control scheme for spacecraft with two-dimensional turntable is investigated under the performance constraints. First, an integrated attitude error model of the spacecraft platform and turntable is constructed based on the D’Alembert principle, and the expected angular acceleration is treated as a disturbance in the controller design. Next, a predefined-time observer is applied to handle external disturbance and expected angular acceleration. Furthermore, a novel performance function is designed to constrain the dynamic response process of the attitude-tracking error. Based on the performance function, the sliding surface is designed, and a finite-time tracking controller is developed, which can ensure fast convergence of tracking errors. To prevent the tracking errors breaking the performance function, the controller gain is adjusted by the performance function. At last, numerical simulation is performed to verify the feasibility and effectiveness of the proposed algorithm.

Appendix A: Expressions for parameters

${{\varvec{J}}_{g\_o}}$ denotes the rotational inertia of the whole spacecraft relative to $\Re _G$. ${\varvec{J}_{a\_a0}}$ denotes the rotational inertia of turntable part A around the connection point $O_{a0}$. ${\varvec{J}_{b\_b0}}$ denotes the rotational inertia of turntable part B around the connection point $O_{b0}$.

$$\begin{aligned} \varvec{I}_1^g= & {} \left( {{m_A} + {m_B}} \right) \cdot \left[ {{\varvec{r}}_{o\_a0}}^\mathrm{{T}}\left( {{{\varvec{r}}_{o\_a0}} + {{\varvec{T}}_{ga}}{{\varvec{\rho }}_\Sigma }} \right) \mathbf{{I}} \right. \\{} & {} \quad \left. - {{\varvec{r}}_{o\_a0}}{{\left( {{{\varvec{r}}_{o\_a0}} + {{\varvec{T}}_{ga}}{{\varvec{\rho }}_\Sigma }} \right) }^\mathrm{{T}}} \right] \\ {\varvec{I}}_1^a= & {} {m_B} \cdot {{\varvec{T}}_{ga}}\left[ {{\varvec{\rho }}_{a0\_b0}}^\mathrm{{T}}\left( {{{\varvec{\rho }}_{a0\_b0}}\mathrm{{ + }}{{\varvec{T}}_{ab}}{\varvec{l}_b}} \right) \mathbf{{I}} \right. \\{} & {} \quad \left. - {{\varvec{\rho }}_{a0\_b0}}{{\left( {{{\varvec{\rho }}_{a0\_b0}} + {{\varvec{T}}_{ab}}{\varvec{l}_b}} \right) }^\mathrm{{T}}} \right] {{\varvec{T}}_{ag}} \\ {\varvec{I}}_2^a= & {} {\left[ {{\varvec{T}}_{ga}}\left( {{m_A}{{\varvec{\rho }}_a} + {m_B}{{\varvec{\rho }}_{a0\_b0}}} \right) \right] ^\mathrm{{T}}}{{\varvec{r}}_{o\_a0}}{} \mathbf{{I}} \\{} & {} \quad - {{\varvec{T}}_{ga}}\left( {{m_A}{{\varvec{\rho }}_a} + {m_B}{{\varvec{\rho }}_{a0\_b0}}} \right) {{\varvec{r}}_{o\_a0}}^\mathrm{{T}} \\ {\varvec{I}}_1^b= & {} {\left( {{{\varvec{T}}_{ga}}{{\varvec{T}}_{ab}}{{\varvec{l}}_b}} \right) ^\mathrm{{T}}}\left( {{{\varvec{r}}_{o\_a0}} + {{\varvec{T}}_{ga}}{\varvec{\rho }_{a0\_b0}}} \right) \mathbf{{I}} \\{} & {} \quad - {{\varvec{T}}_{ga}}{{\varvec{T}}_{ab}}{{\varvec{l}}_b}{\left( {{{\varvec{r}}_{o\_a0}} + {{\varvec{T}}_{ga}}{\varvec{\rho }_{a0\_b0}}} \right) ^\mathrm{{T}}}{m_B} \\ {{\varvec{I}}_\Sigma }= & {} \left( {{m_A} + {m_B}} \right) \left[ {{\varvec{r}}_a^\mathrm{{T}}\left( {{{\varvec{T}}_{ga}}{{\varvec{\rho }}_\Sigma }} \right) \mathbf{{I}} - {{\varvec{r}}_a}{{\left( {{{\varvec{T}}_{ga}}{{\varvec{\rho }}_\Sigma }} \right) }^\mathrm{{T}}}} \right] \\ {\varvec{I}}_2^b= & {} {{\varvec{T}}_{ga}}\left[ {{{\left( {{{\varvec{T}}_{ab}}{{\varvec{l}}_b}} \right) }^\mathrm{{T}}}{\varvec{\rho }_{a0\_b0}}{} \mathbf{{I}} - {{\varvec{T}}_{ab}}{{\varvec{l}}_b}{\varvec{\rho }_{a0\_b0}}^\mathrm{{T}}{m_b}} \right] {{\varvec{T}}_{ag}} \\ {\varvec{I}}_3^g= & {} {\varvec{r}_{o\_a0}}^\mathrm{{T}}{{\varvec{T}}_{ga}}{{\varvec{T}}_{ab}}{{\varvec{l}}_b}{} \mathbf{{I}} - {\varvec{r}_{o\_a0}}{\left( {{{\varvec{T}}_{ga}}{{\varvec{T}}_{ab}}{{\varvec{l}}_b}} \right) ^\mathrm{{T}}}{m_b} \\ {\varvec{D}}_1^g= & {} \left( {{m_A} + {m_B}} \right) {{\varvec{r}}_{o\_a0}} \times \left( {{{\varvec{r}}_{o\_a0}} + {{\varvec{T}}_{ab}}{{\varvec{\rho }}_\Sigma }} \right) \\ {\varvec{D}}_1^a= & {} {m_B}\left( {{{\varvec{T}}_{ga}}{{\varvec{\rho }}_{a0\_b0}}} \right) \times \left( {{{\varvec{T}}_{ga}}{{\varvec{\rho }}_{a0\_b0}} + {{\varvec{T}}_{ga}}{{\varvec{T}}_{ab}}{\varvec{l}_b}} \right) \\ {\varvec{D}}_2^a= & {} \left( {{m_A}{{\varvec{T}}_{ga}}{{\varvec{\rho }}_a}\mathrm{{ + }}{m_B}{{\varvec{T}}_{ga}}{{\varvec{\rho }}_{a0\_b0}}} \right) \times {{\varvec{r}}_{o\_a0}} \\ {\varvec{D}}_1^b= & {} {m_B}\left( {{{\varvec{T}}_{ga}}{{\varvec{T}}_{ab}}{{\varvec{l}}_b}} \right) \times \left( {{{\varvec{r}}_{o\_a0}} + {{\varvec{T}}_{ga}}{\varvec{\rho }_{a0\_b0}}} \right) \\ {{\varvec{D}}_\Sigma }= & {} \left( {{m_A} + {m_B}} \right) {{\varvec{r}}_a} \times \left( {{{\varvec{T}}_{ga}}{{\varvec{\rho }}_\Sigma }} \right) \\ {\varvec{D}}_2^b= & {} {m_B}\left( {{{\varvec{T}}_{ga}}{{\varvec{T}}_{ab}}{{\varvec{l}}_b}} \right) \times \left( {{{\varvec{T}}_{ga}}{\varvec{\rho }_{a0\_b0}}} \right) \\ {\varvec{D}}_3^g= & {} {m_B}{\varvec{r}_{o\_a0}} \times \left( {{{\varvec{T}}_{ga}}{{\varvec{T}}_{ab}}{{\varvec{l}}_b}} \right) \\ \end{aligned}$$

${{\varvec{\rho }}_\Sigma }$ denotes the position vector from the connection point $O_{a0}$ to the mass center of turntable. ${\varvec{\rho }_a}$ represents the vector from the connection point $O_{a0}$ to the mass center of the turntable part A. ${\varvec{l}_b}$ represents the vector from the connection point $O_{b0}$ to the mass center of turntable part B. The remaining $\varvec{r}$, $\varvec{\rho }$, and $\varvec{l}$ denote the vectors under the coordinate systems $\Re _G$, $\Re _A$, and $\Re _B$, respectively, and their points are given by their subscripts.