A Direct Parametric Approach to Spacecraft Attitude Tracking Control

Xiao-Yi Wang Guang-Ren Duan

Xiao-Yi Wang and Guang-Ren Duan. A Direct Parametric Approach to Spacecraft Attitude Tracking Control. International Journal of Automation and Computing, vol. 14, no. 5, pp. 626-636, 2017 doi:  10.1007/s11633-017-1089-4
Citation: Xiao-Yi Wang and Guang-Ren Duan. A Direct Parametric Approach to Spacecraft Attitude Tracking Control. International Journal of Automation and Computing, vol. 14, no. 5, pp. 626-636, 2017 doi:  10.1007/s11633-017-1089-4

doi: 10.1007/s11633-017-1089-4
基金项目: 

National Natural Science Foundation of China 61321062

A Direct Parametric Approach to Spacecraft Attitude Tracking Control

Funds: 

National Natural Science Foundation of China 61321062

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    Author Bio:

    Guan-Ren Duan received the B. Sc. degree in applied mathematics from Yanshan University, China in 1983, the M. Sc. degree in control systems theory from Harbin Engineering University, China in 1986, and received the Ph. D. degree in control systems theory from Harbin Institute of Technology, China in 1989. From 1989 to 1991, he was a post-doctoral researcher at Harbin Institute of Technology, where he became a professor of control systems theory in 1991. He visited the University of Hull, UK, and the University of Sheffield, UK from December 1996 to October 1998, and worked at the Queens University of Belfast, UK from October 1998 to October 2002. Since August 2000, he has been elected specially employed professor at Harbin Institute of Technology sponsored by the Cheung Kong Scholars Program of the Chinese government. He is currently the director of the Centre for Control Systems and Guidance Technology at Harbin Institute of Technology. He is the author and co-author of over 300 publications. He is a chartered engineer in the UK, a senior member of IEEE and a fellow of IEE.
         His research interests include robust control, eigenstructure assignment, descriptor systems, missile autopilot control and magnetic bearing control.
         E-mail:g.r.duan@hit.edu.cn

    Corresponding author: Xiao-Yi Wang received the B. Sc. degree from Department of Control Science and Engineering, Harbin Institute of Technology, China in 2014. She received the M. Sc. degree in Centre for Control Theory and Guidance Technology, Harbin Institute of technology, China in 2017.
         Her research interests include nonlinear robust control and spacecraft attitude control.
         E-mail:xiaoyiwang51@outlook.com (Corresponding author)
         ORCID iD:0000-0002-8511-1273
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出版历程
  • 收稿日期:  2016-08-19
  • 录用日期:  2017-03-29
  • 网络出版日期:  2017-07-03
  • 刊出日期:  2017-10-01

A Direct Parametric Approach to Spacecraft Attitude Tracking Control

doi: 10.1007/s11633-017-1089-4
    基金项目:

    National Natural Science Foundation of China 61321062

    作者简介:

    Guan-Ren Duan received the B. Sc. degree in applied mathematics from Yanshan University, China in 1983, the M. Sc. degree in control systems theory from Harbin Engineering University, China in 1986, and received the Ph. D. degree in control systems theory from Harbin Institute of Technology, China in 1989. From 1989 to 1991, he was a post-doctoral researcher at Harbin Institute of Technology, where he became a professor of control systems theory in 1991. He visited the University of Hull, UK, and the University of Sheffield, UK from December 1996 to October 1998, and worked at the Queens University of Belfast, UK from October 1998 to October 2002. Since August 2000, he has been elected specially employed professor at Harbin Institute of Technology sponsored by the Cheung Kong Scholars Program of the Chinese government. He is currently the director of the Centre for Control Systems and Guidance Technology at Harbin Institute of Technology. He is the author and co-author of over 300 publications. He is a chartered engineer in the UK, a senior member of IEEE and a fellow of IEE.
         His research interests include robust control, eigenstructure assignment, descriptor systems, missile autopilot control and magnetic bearing control.
         E-mail:g.r.duan@hit.edu.cn

    通讯作者: Xiao-Yi Wang received the B. Sc. degree from Department of Control Science and Engineering, Harbin Institute of Technology, China in 2014. She received the M. Sc. degree in Centre for Control Theory and Guidance Technology, Harbin Institute of technology, China in 2017.
         Her research interests include nonlinear robust control and spacecraft attitude control.
         E-mail:xiaoyiwang51@outlook.com (Corresponding author)
         ORCID iD:0000-0002-8511-1273

English Abstract

Xiao-Yi Wang and Guang-Ren Duan. A Direct Parametric Approach to Spacecraft Attitude Tracking Control. International Journal of Automation and Computing, vol. 14, no. 5, pp. 626-636, 2017 doi:  10.1007/s11633-017-1089-4
Citation: Xiao-Yi Wang and Guang-Ren Duan. A Direct Parametric Approach to Spacecraft Attitude Tracking Control. International Journal of Automation and Computing, vol. 14, no. 5, pp. 626-636, 2017 doi:  10.1007/s11633-017-1089-4
    • Spacecraft attitude control has been a hot problem for several years as a significant part of spacecraft navigation control. Many different methods are applied on the problem of spacecraft and aircraft attitude control [1-20]. For instance, Parlos and Sunkel [4] used linear quadratic Gaussian (LQG) control to solve a type of large spacecraft attitude angular maneuver. Bang et al. [5] designed a new sliding model control method which can deal with flexible spacecraft attitude maneuver problem. Singh and Zhang [6] utilized adaptive output feedback control method to track the flexible spacecraft attitude. Additionally, Guan [1, 2, 21] raised a direct parametric approach to stabilize the nonlinear second-order spacecraft attitude model, and the controller can turn the original model into a stable linear constant system.

      According to the research background of spacecraft attitude control, spacecraft attitude tracking needs further research and has more profound applications. Compared with Euler representation, quaternion representation can describe large-angle attitude tracking without singular points which should appear when the changing of Euler angle is bigger than 90°. Moreover, this paper raises a fully-actuated second-order system based on the spacecraft attitude dynamical and kinematical differential equations. Although most of methods deal with the spacecraft attitude control problem by the first-order models, this paper takes advantage of the fully-actuated second-order model and makes the process of controller design easier and clearer than methods based on the first-order models. Finally, this paper builds up a control model for the spacecraft attitude tracking through the error equation between controlled object and the mobile target. According to the direct parametric approach provided by Duan in [1, 2, 22-24], a controller is designed to stabilize the control model, which means the controlled spacecraft successfully tracks the target. To design the controller, the first step is to compensate the nonlinear term ${\xi}$ of the non-approximated control model by controller ${{\mathit{\boldsymbol{u}}}_{c}}$ then obtain the quasi-linear system. The next step is to convert the stabilization problem of the quasi-linear system into the problem of finding the solution for relative Sylvester equation, and yield the expression of the state feedback controller ${u_{f}}$ . Moreover, the controller ${u_{f}}$ can be optimized by the free parameters ${F}_{2n\times2n}$ and ${Z}_{2n\times n}$ . All the elements in $F$ and $Z$ can be arbitrary, thus the direct parametric approach could provide all degrees of freedom of the prospective controller. With the useful advantage of the approach, the controller can be easily designed to satisfy the requirement of performance of closed-loop system. By the adopted way, this method can turn the original highly nonlinear model into a linear closed-loop constant system. Finally, the closed-loop system is simulated in Matlab to verify the availability and robustness of the controller when there is certain external disturbance torque applied on the control model.

    • In this section, a second-order spacecraft attitude control model is built by attitude dynamical equations and kinematical equations. Additionally, that model is described by quaternion and without any approximation. The problem of quaternion-based spacecraft attitude tracking is raised based on the attitude control model.

    • According to the attitude dynamics and kinematics of a rigid spacecraft in the inertial frame, the error attitude dynamics of a rigid spacecraft relative to a mobile tracking target in the inertial frame is given by

      $ \begin{align} & J{{{\dot{\omega }}}_{e}}=-\omega \times J\omega + \\ & \quad \quad \quad J\left( \omega \times C\left( {{q}_{e}} \right){{\omega }_{r}}-C\left( {{q}_{e}} \right){{{\dot{\omega }}}_{r}} \right)+u+d \\ \end{align} $

      (1)

      where

      $ {J} = {\rm diag}\left( {{J_x}, {J_y}, {J_z}} \right) $

      (2)

      is the rotating inertia matrix of the spacecraft, ${\omega}$ is the angular rate vector, $q_{e}$ is the error attitude quaternion between controlled spacecraft and mobile target, ${\omega_e}$ is the error angular rate vector relative to the mobile tracking target, ${\omega_r}$ is the angular rate vector for target, ${u}$ is the input control torque vector, and $d$ is the disturbance torque vector. Besides, ${\omega}$ , ${\omega_e}$ , ${\omega_r}$ , ${u}$ and $d$ belong to ${\bf R}^{1\times 3}$ .

      In addition, $C(q_{e})$ is the transition matrix and given by [19, 20]

      $ {C( {{q_e}})} = \left[{{C_1}( {{q_e}} )}~~~{{C_2}( {{q_e}} )}~~~{{C_3}( {{q_e}} )} \right] $

      (3)

      where

      $ {{C_1}\left( {{q_e}} \right)}= \left[{\begin{array}{*{20}{c}} {q_{e1}^2-q_{e2}^2-q_{e3}^2 + q_{e0}^2}\\ {2\left( {{q_{e1}}{q_{e2}}-{q_{e3}}{q_{e0}}} \right)}\\ {2\left( {{q_{e1}}{q_{e3}} + {q_{e2}}{q_{e0}}} \right)} \end{array}} \right]\ $

      (4)

      $ {{C_2}\left( {{q_e}} \right)} = \left[{\begin{array}{*{20}{c}} {2\left( {{q_{e1}}{q_{e2}} + {q_{e3}}{q_{e0}}} \right)}\\ {-q_{e1}^2 + q_{e2}^2-q_{e3}^2 + q_{e0}^2}\\ {2\left( {{q_{e2}}{q_{e3}}-{q_{e1}}{q_{e0}}} \right)} \end{array}} \right] $

      (5)

      $ {{C_3}\left( {{q_e}} \right)} = \left[{\begin{array}{*{20}{c}} {2\left( {{q_{e1}}{q_{e3}}-{q_{e2}}{q_{e0}}} \right)}\\ {2\left( {{q_{e2}}{q_{e3}} + {q_{e1}}{q_{e0}}} \right)}\\ {-q_{e1}^2-q_{e2}^2 + q_{e3}^2 + q_{e0}^2} \end{array}} \right] $

      (6)

      $ {{q_e}} = {\left[{\begin{array}{*{20}{c}} {{q_{e0}}} & {{q_{e1}}} & {{q_{e2}}} & {{q_{e3}}} \end{array}} \right]^{\rm T}}. $

      (7)

      The variable ${\omega_e}$ can be expressed by

      $ {{\omega }_{e}}=\omega -C\left( {{q}_{e}} \right){{\omega }_{r}}. $

      (8)

      Then, the kinematics of a rigid spacecraft in the inertial frame based on error quaternion is given by

      $ \left[{\begin{array}{*{20}{c}} {{{\dot q}_{e0}}}\\ {{{\dot q}_{e1}}}\\ {{{\dot q}_{e2}}}\\ {{{\dot q}_{e3}}} \end{array}}\right]=\frac{1}{2}\left[{\begin{array}{*{20}{c}} {-{q_{e1}}} & {-{q_{e2}}} & {-{q_{e3}}}\\ {{q_{e0}}} & { - {q_{e3}}} & {{q_{e2}}}\\ {{q_{e3}}} & {{q_{e0}}} & { - {q_{e1}}}\\ { - {q_{e2}}} & {{q_{e1}}} & {{q_{e0}}} \end{array}}\right]\left[{\begin{array}{*{20}{c}} {{\omega _{ex}}}\\ {{\omega _{ey}}}\\ {{\omega _{ez}}} \end{array}}\right]\ $

      (9)

      where ${\omega_e}$ is the error angular rate vector relative to the mobile tracking target, and $q_{e}$ is the error quaternion relative to the target.

      When the system is stable, this means

      $ \left\{ {\begin{array}{*{20}{l}} {\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {\omega _e} = 0,\;\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {q_e}_0 = 1}\\ {\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {q_{e1}} = 0,\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {q_{e2}} = 0,\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {q_{e3}} = 0.} \end{array}} \right. $

      (10)

      To stabilize the controlled system, a suitable controller ${u}$ is given by

      $ {u} = f\left( {{{\omega _e}}, {q_{e0}}, {q_{e1}}, {q_{e2}}, {q_{e3}}, t} \right). $

      (11)

      When the controlled system is stable, the spacecraft error attitude $q_{e}$ will successfully achieve the stable equilibrium point [1, 0, 0, 0]T, which means that the controlled spacecraft successfully tracks on the mobile target.

    • Generally, the error quaternion $q_{e}$ has the constraint which is

      ${q_{e0}}^2 + q_{e1}^2 + q_{e2}^2 + q_{e3}^2 = 1\ $

      (12)

      ${e}={\left[{\begin{array}{*{20}{c}} {{e_1}} & {{e_2}} & {{e_3}} \end{array}} \right]^{\rm T}}={\left[{\begin{array}{*{20}{c}} {{q_{e1}}} & {{q_{e2}}} & {{q_{e3}}} \end{array}} \right]^{\rm T}}.\ $

      (13)

      Additionally, $q_{e}$ $_{0}$ can be denoted as

      ${q_{e0}} = {e_0} = \sqrt {1 - \left( {e_1^2 + e_2^2 + e_3^2} \right)}. $

      (14)

      With the expression of $e$ , we get from (9) that

      $\dot e = \frac{1}{2}T\left( e \right){\omega _e} = \frac{1}{2}\left[{\begin{array}{*{20}{c}} {{e_0}} & {-{e_3}} & {{e_2}}\\ {{e_3}} & {{e_0}} & {-{e_1}}\\ {-{e_2}} & {{e_1}} & {{e_0}} \end{array}} \right]{\omega _e}. $

      (15)

      Taking derivative of (15) with respect to time t gives

      $\ddot{ {e}}= \frac{1}{2}\frac{{{\rm d}{{T}}\left( {e} \right)}}{{{\rm d}t}}{{\omega _e}} + \frac{1}{2}{ {T}}\left( { {e}} \right){\dot {{\omega _e}}}. $

      (16)

      The expression for ${\omega_e}$ can be derived from (15) as

      $ \begin{array}{*{35}{l}} {{\omega }_{e}}= & 2\left[ -\dot{e}\left( e \right){{T}^{\text{T}}}\left( e \right) \right]{{\left[ -\frac{1}{2}{{e}^{\text{T}}}\omega \dot{e} \right]}^{\text{T}}}= \\ {} & 2\left( \frac{1}{2}e{{e}^{\text{T}}}{{\omega }_{e}}+{{T}^{\text{T}}}\left( e \right)\dot{e} \right)= \\ {} & e{{e}^{\text{T}}}{{\omega }_{e}}+2{{T}^{\text{T}}}\left( e \right)\dot{e}= \\ {} & 2{{\left( {{I}_{3}}-e{{e}^{\text{T}}} \right)}^{-1}}{{T}^{\text{T}}}\left( e \right)\dot{e}= \\ {} & 2\Phi \left( e \right){{T}^{\text{T}}}\left( e \right)\dot{e} \\ \end{array} $

      (17)

      where

      $ {\Phi} \left( {e} \right) = {\left( {{{I}_3} - {e{e^{\rm T}}}} \right)^{ - 1}} = \left[{\begin{array}{*{20}{c}} {{\Phi _1}\left( e \right)}\\ {{\Phi _2}\left( e \right)}\\ {{\Phi _3}\left( e \right)} \end{array}} \right]. $

      (18)

      Thus,

      $ \left\{ \begin{array}{*{35}{l}} {{\omega }_{ex}}=2{{\Phi }_{1}}\left( e \right){{T}^{\text{T}}}\left( e \right)\dot{e} \\ {{\omega }_{ex}}=2{{\Phi }_{2}}\left( e \right){{T}^{\text{T}}}\left( e \right)\dot{e} \\ {{\omega }_{ex}}=2{{\Phi }_{3}}\left( e \right){{T}^{\text{T}}}\left( e \right)\dot{e}. \\ \end{array} \right. $

      (19)

      The next step is to summarize the second-order spacecraft control model through (1) and (16). First of all, because of

      ${\omega } = {\omega _e} - {C}\left( {e} \right){\omega _r}. $

      (20)

      Equation (1) can be written as

      $ \begin{array}{*{35}{l}} J{{{\dot{\omega }}}_{e}}= & -\left( {{\omega }_{e}}-C\left( e \right){{\omega }_{r}} \right)\times J\left( {{\omega }_{e}}-C\left( e \right){{\omega }_{r}} \right)+ \\ {} & J\left( \left( {{\omega }_{e}}-C\left( e \right){{\omega }_{r}} \right)\times C\left( {{q}_{e}} \right){{\omega }_{r}}-C\left( {{q}_{e}} \right){{{\dot{\omega }}}_{r}} \right)+ \\ {} & u+d. \\ \end{array} $

      (21)

      Then, the expression of $ \dot{\omega _e} $ can be derived by (21) as

      $ \begin{array}{*{35}{l}} {{{\dot{\omega }}}_{e}}= & {{J}^{-1}}\left[ {{\omega }_{e}}\times \left( J{{\omega }_{e}} \right) \right]- \\ {} & {{J}^{-1}}\left\{ {{\omega }_{e}}\times \left[ JC\left( e \right){{\omega }_{r}} \right] \right\}- \\ {} & {{J}^{-1}}\left\{ \left[ C\left( e \right){{\omega }_{r}} \right]\times \left[ J{{\omega }_{e}} \right] \right\}- \\ {} & {{J}^{-1}}\left\{ \left[ C\left( e \right){{\omega }_{r}} \right]\times \left[ JC\left( e \right){{\omega }_{r}} \right] \right\}+ \\ {} & \left[ {{\omega }_{e}}\times \left( C\left( e \right){{\omega }_{r}} \right) \right]-C\left( e \right){{{\dot{\omega }}}_{r}}+ \\ {} & {{J}^{-1}}\left( u+d \right). \\ \end{array} $

      (22)

      Denote

      ${\Gamma \left( {J, e, \dot e} \right)} = \left[{\begin{array}{*{20}{c}} 0 & 0 & {{\Gamma _c}}\\ {{\Gamma _a}} & 0 & 0\\ 0 & {{\Gamma _b}} & 0 \end{array}} \right]\ $

      (23)

      $\left\{ \begin{array}{l} {\Gamma _a} = \dfrac{{{J_z} - {J_x}}}{{{J_y}}}{{\Phi _3}}\left( {e} \right){{T^{\rm T}}}\left( {e} \right)\dot {{e}}\\ {\Gamma _b} = \dfrac{{{J_x} - {J_y}}}{{{J_z}}}{{\Phi _1}}\left( {e} \right){{T^{\rm T}}}\left( {e} \right)\dot {{e}}\\ {\Gamma _c} = \dfrac{{{J_y} - {J_z}}}{{{J_x}}}{{\Phi _2}}\left( {e} \right){{T^{\rm T}}}\left( {e} \right)\dot {{e}} \end{array} \right.\nonumber\\[-5mm] $

      (24)

      $G = \left[{\begin{array}{*{20}{c}} {{G_1}}\\ {{G_2}}\\ {{G_3}} \end{array}} \right] = \left[{\begin{array}{*{20}{c}} {{{\left[{JC\left( e \right){\omega _r}} \right]}_x}}\\ {{{\left[{JC\left( e \right){\omega _r}} \right]}_y}}\\ {{{\left[{JC\left( e \right){\omega _r}} \right]}_z}} \end{array}} \right]\ $

      (25)

      and

      ${H} = \left[{\begin{array}{*{20}{c}} {{H_1}}\\ {{H_2}}\\ {{H_3}} \end{array}} \right] = {\left[{\begin{array}{*{20}{c}} {{{\left[{C\left( e \right){\omega _r}} \right]}_x}}\\ {{{\left[{C\left( e \right){\omega _r}} \right]}_y}}\\ {{{\left[{C\left( e \right){\omega _r}} \right]}_z}} \end{array}} \right]}. $

      (26)

      The expression for ${\dot{\omega_e}}$ (22) can be expanded as

      $ \begin{array}{*{35}{l}} {{{\dot{\omega }}}_{e}}= & 4\Gamma \left( J,e,\dot{e} \right)\Phi \left( e \right){{T}^{\text{T}}}\left( e \right)\dot{e}- \\ {} & {{J}^{-1}}\left[ \begin{matrix} {{G}_{3}}{{\Phi }_{2}}-{{G}_{2}}{{\Phi }_{3}} \\ {{G}_{1}}{{\Phi }_{3}}-{{G}_{3}}{{\Phi }_{1}} \\ {{G}_{2}}{{\Phi }_{1}}-{{G}_{1}}{{\Phi }_{2}} \\ \end{matrix} \right]2{{T}^{\text{T}}}\dot{e}- \\ {} & {{J}^{\text{-}1}}\left[ \begin{matrix} {{H}_{2}}{{J}_{z}}{{\Phi }_{3}}-{{H}_{3}}{{J}_{y}}{{\Phi }_{2}} \\ {{H}_{3}}{{J}_{x}}{{\Phi }_{1}}-{{H}_{1}}{{J}_{z}}{{\Phi }_{3}} \\ {{H}_{1}}{{J}_{y}}{{\Phi }_{2}}-{{H}_{2}}{{J}_{x}}{{\Phi }_{1}} \\ \end{matrix} \right]2{{T}^{\text{T}}}\dot{e}+ \\ {} & \left[ \begin{matrix} {{H}_{2}}{{\Phi }_{3}}-{{H}_{3}}{{\Phi }_{2}} \\ {{H}_{3}}{{\Phi }_{1}}-{{H}_{1}}{{\Phi }_{3}} \\ {{H}_{1}}{{\Phi }_{2}}-{{H}_{2}}{{\Phi }_{1}} \\ \end{matrix} \right]2{{T}^{\text{T}}}\dot{e}- \\ {} & {{J}^{-1}}\left[ H\times \left( JH \right) \right]-C\left( e \right){{{\dot{\omega }}}_{r}}+ \\ {} & {{J}^{-1}}\left( u+d \right). \\ \end{array} $

      (27)

      Substituting (17), (22) and (26) into (16) gives

      $ \begin{array}{*{35}{l}} \ddot{e}= & \frac{1}{2}\frac{\text{d}T\left( e \right)}{\text{d}t}{{\omega }_{e}}+\frac{1}{2}T\left( e \right){{{\dot{\omega }}}_{e}}= \\ {} & \frac{1}{2}\frac{\text{d}T\left( e \right)}{\text{d}t}\left[ 2\Phi \left( e \right){{T}^{\text{T}}}\left( e \right)\dot{e} \right]= \\ {} & \left( \begin{array}{*{35}{l}} \frac{\text{d}T}{\text{d}t}\Phi \left( e \right)+2T\Gamma \left( J,e,\dot{e} \right)\Phi \left( e \right)+ \\ {{J}^{-1}}T\left[ \begin{matrix} {{G}_{3}}{{\Phi }_{2}}-{{G}_{2}}{{\Phi }_{3}} \\ {{G}_{1}}{{\Phi }_{3}}-{{G}_{3}}{{\Phi }_{1}} \\ {{G}_{2}}{{\Phi }_{1}}-{{G}_{1}}{{\Phi }_{2}} \\ \end{matrix} \right]+ \\ {{J}^{\text{-}1}}T\left[ \begin{matrix} {{H}_{2}}{{J}_{z}}{{\Phi }_{3}}-{{H}_{3}}{{J}_{y}}{{\Phi }_{2}} \\ {{H}_{3}}{{J}_{x}}{{\Phi }_{1}}-{{H}_{1}}{{J}_{z}}{{\Phi }_{3}} \\ {{H}_{1}}{{J}_{y}}{{\Phi }_{2}}-{{H}_{2}}{{J}_{x}}{{\Phi }_{1}} \\ \end{matrix} \right]- \\ T\left[ \begin{matrix} {{H}_{2}}{{\Phi }_{3}}-{{H}_{3}}{{\Phi }_{2}} \\ {{H}_{3}}{{\Phi }_{1}}-{{H}_{1}}{{\Phi }_{3}} \\ {{H}_{1}}{{\Phi }_{2}}-{{H}_{2}}{{\Phi }_{1}} \\ \end{matrix} \right] \\ \end{array} \right){{T}^{\text{T}}}\left( e \right)\dot{e}- \\ {} & \frac{1}{2}T\left\{ {{J}^{-1}}\left[ H\times \left( JH \right) \right]-C\left( e \right){{{\dot{\omega }}}_{r}} \right\}+ \\ {} & \frac{1}{2}T{{J}^{-1}}\left( u+d \right). \\ \end{array} $

      (28)

      In addition, the second-order spacecraft attitude tracking control model (28) can be written in the second-order form as

      $ \left\{ \begin{array}{l} {{A_2}\left( {\theta, e, \dot e} \right)\ddot e} + {{A_1}\left( {\theta, e, \dot e} \right)\dot e} +\\ {{A_0}\left( {\theta, e, \dot e} \right)e} + {\xi} \left( {{\theta}, {e}, {\dot{ e}}, t} \right) \end{array} \right\} = {B\left( {\theta, e, \dot e} \right)u} $

      (29)

      where

      $ {{A_2}\left( {\theta, e, \dot e} \right)} = {I_3} $

      (30)

      $ \begin{align} & {{A}_{1}}\left( \theta ,e,\dot{e} \right)= \\ & \quad \quad \quad \left( \begin{array}{*{35}{l}} \frac{\text{d}T}{\text{d}t}\Phi \left( e \right)+2T\Gamma \left( J,e,\dot{e} \right)\Phi \left( e \right)+ \\ {{J}^{-1}}T\left[ \begin{matrix} {{G}_{3}}{{\Phi }_{2}}-{{G}_{2}}{{\Phi }_{3}} \\ {{G}_{1}}{{\Phi }_{3}}-{{G}_{3}}{{\Phi }_{1}} \\ {{G}_{2}}{{\Phi }_{1}}-{{G}_{1}}{{\Phi }_{2}} \\ \end{matrix} \right]+ \\ {{J}^{\text{-}1}}T\left[ \begin{matrix} {{H}_{2}}{{J}_{z}}{{\Phi }_{3}}-{{H}_{3}}{{J}_{y}}{{\Phi }_{2}} \\ {{H}_{3}}{{J}_{x}}{{\Phi }_{1}}-{{H}_{1}}{{J}_{z}}{{\Phi }_{3}} \\ {{H}_{1}}{{J}_{y}}{{\Phi }_{2}}-{{H}_{2}}{{J}_{x}}{{\Phi }_{1}} \\ \end{matrix} \right]- \\ T\left[ \begin{matrix} {{H}_{2}}{{\Phi }_{3}}-{{H}_{3}}{{\Phi }_{2}} \\ {{H}_{3}}{{\Phi }_{1}}-{{H}_{1}}{{\Phi }_{3}} \\ {{H}_{1}}{{\Phi }_{2}}-{{H}_{2}}{{\Phi }_{1}} \\ \end{matrix} \right] \\ \end{array} \right)\times {{T}^{\text{T}}}\left( e \right)\dot{e} \\ \end{align} $

      (31)

      $ {{A_0}\left( {\theta, e, \dot e} \right)} = {0_3} $

      (32)

      $ {B\left( {\theta, e, \dot e} \right)} = \frac{1}{2}{T\left( e \right){J^{ - 1}}} $

      (33)

      $ {\xi} \left( {{\theta}, {e}, {\dot e}, t} \right) = \frac{1}{2}{T}\left\{ \begin{array}{l} {{J^{ - 1}}\left[{H \times \left( {JH} \right)} \right]}- \\ {C\left( e \right){{\dot \omega }_r} - {J^{ - 1}}d} \end{array} \right\} $

      (34)

      where $A_{0}$ , $A_{1}$ , $A_{2}$ are the coefficient matrices, and ${B}$ is the input matrix, $ {\xi}$ is the nonlinear term, ${u}$ is the controller, ${d}$ is the external disturbance.

      When the second-order system is stable, which means

      $ \left\{ {\begin{array}{*{20}{l}} {\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {e_1} = 0,\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {e_2} = 0,\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {e_3} = 0}\\ {\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {{\dot e}_1} = 0,\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {{\dot e}_2} = 0,\mathop {\lim }\limits_{t \to \infty } {\mkern 1mu} {{\dot e}_3} = 0.} \end{array}} \right. $

      (35)

      To stabilize the controlled system (28), a suitable controller ${u}$ is given by

      $ u = f\left( {{e_1}, {e_2}, {e_3}, t} \right). $

      (36)

      When the controlled system is stable, the spacecraft error attitude ${e}$ will successfully achieve the stable equilibrium point [0, 0, 0]T, this means that the controlled spacecraft successfully tracks the mobile target.

    • To use the direct parametric approach provided by Duan in [1] to design controller for the control model, there are three assumptions needed to satisfy.

      Assumption 1. The values of the system parameter $\theta =\theta (t)\in \Omega \subset {{\bf{R}}^{l}}$ .

      Define the parameter $ \theta $ as

      $ {\theta} \left( t \right) = \left[{\begin{array}{*{20}{c}} {{J_x}\left( t \right)} & {{J_y}\left( t \right)} & {{J_z}\left( t \right)} \end{array}} \right] $

      (37)

      where ${{\text{J}}_{i}}(i=x,y,z)$ is the rotational inertia of one axis.

      Particularly, the case of space capturing is considered. It is supposed that the two spacecrafts are joined together at the time t1, then they departed from each other at time t2. The rotational inertia can be defined as

      $ {{J}_{i}}\left( t \right)=\left\{ \begin{array}{*{35}{l}} \begin{matrix} {{J}_{i1}}, & t<{{t}_{1}} \\ \end{matrix} \\ \begin{matrix} {{J}_{i2}}, & {{t}_{1}}\le t<{{t}_{2}} \\ \end{matrix} \\ \begin{matrix} {{J}_{i3}}, & t\le {{t}_{2}} \\ \end{matrix} \\ \end{array} \right.i=x,y,z $

      (38)

      where $J_{xi} $ , $J_{yi} $ and $J_{zi} $ (i = 1, 2) are constant positive scalars. Thus, parameter $\theta (t)$ satisfies the Assumption 1.

      Assumption 2. det( ${{A} _2 }) \neq $ 0, $ \forall {{e}}, \space{{\dot{e} }}$ and ${{ \theta }} ({t}) \in \space{\Omega}.$

      According to (29), det ( ${A} _2 $ ) = 1. So the system should satisfy Assumption 2.

      Assumption 3. ${B}$ is uniformly bounded and det $({{B}}) \neq $ 0, $ \forall {{e}}$ , $\space{{ \dot{e} }}$ and ${{ \theta }} ({t}) \in \space{ \Omega}.$

      According to

      $ {B} = \frac{1}{2}{T\left( e \right){J^{ - 1}}} $

      (39)

      $ \det \left( {B} \right) = \frac{1}{{8{J_x}{J_y}{J_z}}}{e_0}. $

      (40)

      Assumption 3 is equivalent to $ {{e}_{0}}\ne 0 $ , because it has

      $ {e_0} = \sqrt {1 - \left( {e_1^2 + e_2^2 + e_3^2} \right)} $

      (41)

      where $ {{e}_{0}}\ne 0 $ is equal to

      $ e_1^2 + e_2^2 + e_3^2 < 1. $

      (42)

      Although this condition cannot be satisfied all the time, it can be met when the initial values of ${e}$ are chosen carefully. When this condition is met, Assumption 3 is satisfied.

      A controller ${u}$ is designed for the spacecraft attitude tracking model (29), which is made up of two parts as

      $ {u} = {u_c} + {u_f} $

      (43)

      where $u _c $ is the compensating controller for the term $ \xi $ , and its expression is

      $ \quad {u_c} = {\left( {\frac{1}{2}{T}\left( {e} \right){{{J}}^{ - 1}}} \right)^{ - 1}}\frac{1}{2}{T}\left\{ \begin{array}{l} {{J}^{ - 1}}\left[{H \times \left( {JH} \right)} \right] +\\ {C}\left( {e} \right){{\dot \omega }_r} - {{J^{ - 1}}d} \end{array} \right\}.\quad $

      (44)

      In addition, ${u} _f $ is the state feedback controller for the compensated control system and given by

      $ {u_f} = {K_0}\left( {\theta, e, \dot e} \right){e} + {K_1}\left( {\theta, e, \dot e} \right)\dot {{e}} + {v} $

      (45)

      where ${K} _0 $ and $ {K}_1 \in {\bf R}^{3\times 3} $ are the continuous state feedback gains, and ${v}$ is an external signal.

      Then, a parametric controller ${u} _f $ is going to be designed by a direct parametric approach.

      Define

      $ {F} = \left\{ \begin{array}{l} \left. {F} \right|{F} \in {{\bf R}^{2n \times 2n}}, {\rm and}~\exists {Z} \in {{\bf R}^{2n \times 2n}}\\ {\rm s.t.}~~\det \left[ {\begin{array}{*{20}{c}} {Z}\\ {ZF} \end{array}} \right] \ne 0 \end{array} \right\} $

      (46)

      where ${F}$ and ${Z}$ are parametric matrices, and $n$ is the dimension of the second-order system (29).

      Step 1. Find the Sylvester equation for the control model (29).

      Denote

      $ {{V^{ - 1}}{A_c}\left( {\theta, e, \dot e} \right)V} = {F} $

      (47)

      $ {{V}_{2n \times 2n}} = \left[{\begin{array}{*{20}{c}} {{{\left( {{V_0}} \right)}_{n \times 2n}}}\\ {{{\left( {{V_1}} \right)}_{n \times 2n}}} \end{array}} \right] = \left[{\begin{array}{*{20}{c}} {{V_0}}\\ {{V_0}F} \end{array}} \right] $

      (48)

      and

      $ \det \left( {V} \right) \ne 0 $

      (49)

      where ${A} _c $ is the closed-loop system matrix, and ${{V}} \in \textbf{R}^{2n\times2n} $ is a parametric matrix.

      Through the direct parametric approach, the relative Sylvester equation for the system (29) can be given by

      $ \left\{ \begin{array}{l} {{A_2}\left( {\theta, e, \dot e} \right){V_0}{F^2}} +\\ {{A_1}\left( {\theta, e, \dot e} \right){V_0}F} +\\ {{A_0}\left( {\theta, e, \dot e} \right){V_0}} \end{array} \right\} = {B\left( {\theta, e, \dot e} \right)W} $

      (50)

      where ${{W}} \in \textbf{R}^{n\times 2n} $ is the solution for (50).

      Step 2. Solve the Sylvester equation and gain the expression for feedback controller ${{u}}_f$ .

      It is suggested by the direct parametric approach that the solution for the Sylvester equation is

      $ \left\{ \begin{array}{*{35}{l}} V=\left[ \begin{array}{*{35}{l}} {{N}_{0}}Z \\ {{N}_{0}}ZF \\ \end{array} \right]=\left[ \begin{matrix} Z \\ ZF \\ \end{matrix} \right] \\ W={{B}^{-1}}\left( {{A}_{2}}Z{{F}^{2}}+{{A}_{1}}ZF+{{A}_{0}}Z \right) \\ \end{array} \right. $

      (51)

      and ${V}$ satisfies the condition which is det $({{V}}) \neq $ 0, then the controller ${{u} _f }$ can be written as

      $ {u_f} = {{K_0}\left( {\theta, e, \dot e} \right)e} + {{K_1}\left( {\theta, e, \dot e} \right)\dot e} + {v} $

      (52)

      where

      $ {K_F} = \left[{\begin{array}{*{20}{c}} {{K_0}} & {{K_1}} \end{array}} \right] = {W}{{V}^{ - 1}}. $

      (53)

      Denote

      $ {X} = \left[{\begin{array}{*{20}{c}} {e}\\ {\dot e} \end{array}} \right]. $

      (54)

      Then, the closed-loop system can be converted into the first-order form:

      $ {\dot X} = {{A_c}X} + {{B_c}v} $

      (55)

      where

      $ {A_c} = \left[{\begin{array}{*{20}{c}} {0} & {{I_n}}\\ {{-{A_2}^{-1}A_0^c}} & {{-{A_2}^{ - 1}A_1^c}} \end{array}} \right] $

      (56)

      $ {B_c} = \left[{\begin{array}{*{20}{c}} {0}\\ {{{A_2}^{-1}\left( {\theta, e, \dot e} \right)B\left( {\theta, e, \dot e} \right)}} \end{array}} \right] $

      (57)

      and

      $ \qquad \left\{ \begin{array}{l} {A_0^c\left( {\theta, e, \dot e} \right)} = {{A_0}\left( {\theta, e, \dot e} \right) - B\left( {\theta, e, \dot e} \right){K_0}\left( {\theta, e, \dot e} \right)}\\ {A_1^c\left( {\theta, e, \dot e} \right)} ={ {A_1}\left( {\theta, e, \dot e} \right) - B\left( {\theta, e, \dot e} \right){K_1}\left( {\theta, e, \dot e} \right)}. \end{array}\qquad \right. $

      (58)

      Step 3. Optimize the controller ${{u} _f }$ by the optimizing index ${{J}} _{opt} $ .

      To maintain the robustness of performance and robustness of stability for a controlled system with uncertain parameters, the beneficial method is to minimize the closed-loop eigenvalues sensitivities given by [21]. As the closed-loop system should be a constant stable linear system, the optimizing index could be chosen as

      $ \begin{matrix} {{J}_{opt}}={{J}_{opt}}\left( F,Z \right)=\left\| V \right\|\left\| {{V}^{-1}} \right\|= \\ \qquad \left\| \left[ \begin{matrix} Z \\ ZF \\ \end{matrix} \right] \right\|\left\| {{\left[ \begin{matrix} Z \\ ZF \\ \end{matrix} \right]}^{-1}} \right\|. \\ \end{matrix} $

      (59)

      Furthermore, the constrained nonlinear programming problem can be described as

      $ \begin{array}{*{35}{l}} \min {{J}_{opt}}\left( F,Z \right) \\ \rm{s}.\rm{t}.\left\{ \begin{array}{*{35}{l}} \det \left( \begin{matrix} Z \\ ZF \\ \end{matrix} \right)\ne 0,~Z\in {{\bf{R}}^{n\times 2n}} \\ {{\lambda }_{i}}\in {{C}^{-}},~i=1,2,\cdots ,2n. \\ \end{array} \right. \\ \end{array} $

      (60)

      Besides, to simplify the optimizing process, the parameter F can be considered as

      $ {F} ={\rm diag} \left( {{\lambda _1},{\lambda _2}, \cdots ,{\lambda _{2n}}} \right) $

      (61)

      where $\lambda_{i}$ is the eigenvalue of the closed-loop system.

      Step 4. Simulation.

      Put the controller $u$ into the control model, then utilize Matlab to acquire the response for closed-loop system. Finally, analyze the performance of the closed-loop system and make the conclusion.

    • A set of practical spacecraft attitude system control data example is provided for the simulation [3].

      The rotational inertia of the spacecraft is

      $ {J} = \left[{\begin{array}{*{20}{c}} {{J_x}} & 0 & 0\\ 0 & {{J_y}} & 0\\ 0 & 0 & {{J_z}} \end{array}} \right] = \left[{\begin{array}{*{20}{c}} {18} & 0 & 0\\ 0 & {21} & 0\\ 0 & 0 & {24} \end{array}} \right]{\rm kg\cdot{m^2}} $

      The disturbance torque $d$ is

      ${d} = 1.0 \times {10^{ - 3}}\left[{\begin{array}{*{20}{c}} {\cos \left( {0.01t} \right)-0.3}\\ {0.3\cos (0.02t) + 0.6}\\ {0.5\sin (0.02t)} \end{array}} \right]{\rm{N}} \cdot {\rm{m}}. $

      The initial attitude quaternion of controlled spacecraft ${{q}} _d $ (0) is

      ${q_d}\left( 0 \right) = {\left[{\begin{array}{*{20}{c}} 1 & 0 & 0 & 0 \end{array}} \right]^{\rm T}}. $

      The initial attitude quaternion of target spacecraft ${{q}} _r $ (0) is

      $ {q_r}\left( 0 \right) = {\left[{\begin{array}{*{20}{c}} {0.668\, 3} & {-0.554\, 6} & {0.399\, 9} & {0.293\, 1} \end{array}} \right]^{\rm T}}. $

      As $ q_e $ (0) = $ q_{r}^{\ast} $ (0) $\, \circ\, {{q}}_{d}$ (0), the initial error attitude quaternion ${{q}} _{e}$ (0) is

      $ {q_e}\left( 0 \right) = {\left[{\begin{array}{*{20}{c}} {0.668\, 3} & {{\rm{0}}{\rm{.554\, 6}}} & {{\rm{-0}}{\rm{.399\, 9}}} & {{\rm{-0}}{\rm{.293\, 1}}} \end{array}} \right]^{\rm T}}. $

      Thus, the initial state ${e}$ (0) and $ \dot{e} $ (0) are

      $ {e}\left( 0 \right) = {\left[{\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.554\, 6}}} & {{\rm{-0}}{\rm{.399\, 9}}} & {{\rm{-0}}{\rm{.293\, 1}}} \end{array}} \right]^{\rm T}} $

      and

      $ {\dot e}\left( 0 \right) = \left[{\begin{array}{*{20}{c}} {{\rm{0}}{\rm{.071\, 5}}} & {{\rm{0}}{\rm{.104\, 7}}} & {{\rm{0}}{\rm{.060\, 9}}} \end{array}} \right]. $

      In addition, the target spacecraft is mobile and its angular rate $ \omega_r $ is

      $ {\omega _r} = 0.01\left[{\begin{array}{*{20}{c}} {\sin \left( {0.02t} \right)}\\ {-2\sin \left( {0.02t} \right)}\\ {\sin \left( {0.02t} \right)} \end{array}} \right]{\rm rad/s}. $

      First of all, the controller ${u}$ is designed by the approach in section 4 as

      $ \begin{align} & u={{u}_{c}}+{{u}_{f}}= \\ & \quad \ \ {{B}^{-1}}\left( \theta ,e,\dot{e} \right)\xi \left( \theta ,e,\dot{e},t \right)+ \\ & \quad \ \ {{K}_{0}}\left( \theta ,e,\dot{e} \right)e+{{K}_{1}}\left( \theta ,e,\dot{e} \right)\dot{e}+v. \\ \end{align} $

      (62)

      Moreover, the controller should be optimized by the ${{J} _{opt} }$ as

      $ \begin{array}{*{35}{l}} \min {{J}_{opt}}\left( F,Z \right) \\ \rm{s}.\rm{t}.\left\{ \begin{array}{*{35}{l}} \det \left( \begin{matrix} Z \\ ZF \\ \end{matrix} \right)\ne 0,~~Z\in {{\bf{R}}^{n\times 2n}} \\ {{\lambda }_{i}}\in {{C}^{-}},~~i=1,2,\cdots ,2n. \\ \end{array} \right. \\ \end{array} $

      (63)

      Then, to simplify the optimization, the parameter ${F}$ is considered as

      $ \begin{align} & F=\text{diag}\left( {{\lambda }_{1}},{{\lambda }_{2}},\cdots ,{{\lambda }_{6}} \right)= \\ & \quad \ \ \text{diag}\left( \text{-0}.\text{1},\text{-0}.\text{15},\text{-0}.\text{2},\text{-0}.\text{25},\text{-0}.\text{3},\text{-0}.\text{35} \right). \\ \end{align} $

      The initial ${Z}$ (0) is chosen by

      $ {Z}\left( 0 \right) = \left[{\begin{array}{*{20}{c}} 2 & 0 & 0 & 1 & 0 & 0\\ 0 & 2 & 0 & 0 & 1 & 0\\ 0 & 0 & 2 & 0 & 0 & 1 \end{array}} \right]. $

      Additionally, initial ${{J} _{opt}}$ (0) is 17.551 4.

      Thus, the final optimal parameter ${Z}$ is

      $ {Z} = {\left[{\begin{array}{*{20}{c}} {1.385\, 6} & {-0.001\, 1} & {0.000\, 3}\\ {0.001\, 3} & {1.474\, 2} & {-0.000\, 1}\\ {-0.000\, 6} & {0.000\, 0} & {1.335\, 9}\\ {1.199\, 1} & { - 0.000\, 9} & {0.001\, 2}\\ { - 0.000\, 4} & {1.163\, 1} & {0.000\, 0}\\ { - 0.000\, 2} & {0.000\, 1} & {1.285\, 9} \end{array}} \right]^{\rm{T}}}. $

      Additionally, the final ${{J} _{opt}}$ is 14.336 4.

      Note that

      $ {Z{F^2}{V^{ - 1}}} = {\left[{\begin{array}{*{20}{c}} {{\rm{-}}0.025\, 0} & {{\rm{-}}0.000\, 0} & {0.000\, 0}\\ {{\rm{-}}0.000\, 0} & {{\rm{ - }}0.045\, 0} & {{\rm{ - }}0.000\, 0}\\ {0.000\, 0} & {{\rm{ - }}0.000\, 0} & {{\rm{ - }}0.070\, 0}\\ {{\rm{ - }}0.350\, 0} & {{\rm{ - }}0.0001} & {0.000\, 1}\\ {{\rm{ - }}0.000\, 0} & {{\rm{ - }}0.450\, 0} & {{\rm{ - }}0.000\, 0}\\ {0.000\, 0} & {{\rm{ - }}0.000\, 0} & {{\rm{ - }}0.550\, 0} \end{array}} \right]^{\rm{T}}}. $

      Because of the expression

      $ {V{V^{ - 1}}} = \left[{\begin{array}{*{20}{c}} {Z}\\ {ZF} \end{array}} \right]{V^{ - 1}} = \left[{\begin{array}{*{20}{c}} {Z{V^{-1}}}\\ {ZF{V^{-1}}} \end{array}} \right] = {I_6} $

      (64)

      these equations can be derived as

      $ {{A_1}\left( {\theta, e, \dot e} \right)ZF{V^{ - 1}}} = \left[{\begin{array}{*{20}{c}} {{0_3}} & {{A_1}\left( {\theta, e, \dot e} \right)} \end{array}} \right] $

      (65)

      $ {ZF{V^{ - 1}}} = \left[{\begin{array}{*{20}{c}} {{0_3}} & {{I_3}} \end{array}} \right] $

      (66)

      $ \begin{align} & {{K}_{F}}=\left[ {{K}_{0}}~~~~{{K}_{1}} \right]=W{{V}^{-1}}= \\ & \quad \quad B{{\left( \theta ,e,\dot{e} \right)}^{-1}}\left( Z{{F}^{2}}+{{A}_{1}}\left( \theta ,e,\dot{e} \right)ZF \right){{V}^{-1}}= \\ & \quad \quad B{{\left( \theta ,e,\dot{e} \right)}^{-1}}\left( Z{{F}^{2}}{{V}^{-1}}+{{A}_{1}}\left( \theta ,e,\dot{e} \right)ZF{{V}^{-1}} \right)= \\ & \quad \quad B{{\left( \theta ,e,\dot{e} \right)}^{-1}}\left( Z{{F}^{2}}{{V}^{-1}}+\left[ {{0}_{3}}~~~~{{A}_{1}}\left( \theta ,e,\dot{e} \right) \right] \right). \\ \end{align} $

      (67)

      Furthermore, with the controller $u$ , the response of error attitude quaternion ${{q}_{e}} = [{{q}_{e0}}~~ {{q}_{e1}}~~ {{q}_{e2}}~~ {{q}_{e3}}]^{\rm T}$ and the state variable ${e}$ can be plotted in Matlab as Figs. 1 and 2.

      Figure 1.  Error attitude quaternion ${q} _e $

      Figure 2.  Response curve of variable ${e}$ and $\dot{e} $ of controlled system

      Based on Figs. 1 and 2, the variable ${e}$ and $\dot{e}$ are successfully made stable in time = 70 s, which reach the stable balance point [0, 0, 0]T. Thus, the controller ${u}$ is successful for the attitude stabilization. According to (19), the error angular rate $ \omega_e $ also reached the zero point, which means the spacecraft successfully tracks the target with the angular rate ${{ \omega_r }} = 0.01[\sin (0.02{t}), -2\sin (0.02{t}), \sin (0.02{t})] ^{\rm T} $ . According to (13) and (14), when the system is disturbed by external torque $d$ , the error attitude quaternion $q_e $ also reaches the stable equilibrium point [1, 0, 0, 0]T. Thus, the controlled spacecraft successfully tracks the mobile target. Moreover, the controller needs to be tested for availability and stability.

      Additionally, to justify the relative attitude quaternion between $q_r $ for target spacecraft and $q_d $ for controlled spacecraft, the tracking process is plotted by Matlab. Fig. 3 shows that when the simulation time gets near to 70 s, the controlled spacecraft attitude $q_d $ successfully tracks on the target spacecraft attitude $q_r $ .

      Figure 3.  Comparison between controlled spacecraft attitude and target spacecraft attitude

      According to Fig. 4, the controller is stable which is pointed out by the stable curve, and it can be utilized for the control model. Furthermore, there is a limitation of 2 N $\cdot $ m for each axis based on that controlled spacecraft [3]. In Fig. 4, each axis control torque satisfies that requirement. Hence, the controller ${u}$ is useful, practical, and has robustness eliminating the effects of the external disturbance torque.

      Figure 4.  Controller ${u}$ with respect to time curve

      We verify whether the closed-loop system is a linear constant system as

      $ {\dot X }= {{A_c}X + {B_c}v} $

      (68)

      where

      $ {A_c} = \left[{\begin{array}{*{20}{c}} 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ {{\rm{-}}0.025\, 0} & {{\rm{-}}0.000\, 0} & {0.000\, 0} & {{\rm{-}}0.350\, 0} & {{\rm{ - }}0.000\, 0} & {0.000\, 0}\\ {{\rm{ - }}0.000\, 0} & {{\rm{ - }}0.045\, 0} & {{\rm{ - }}0.000\, 0} & {{\rm{ - }}0.000\, 1} & {{\rm{ - }}0.450\, 0} & {{\rm{ - }}0.000\, 0}\\ {0.000\, 0} & {{\rm{ - }}0.000\, 0} & {{\rm{ - }}0.070\, 0} & {0.000\, 1} & {{\rm{ - }}0.000\, 0} & {{\rm{ - }}0.550\, 0} \end{array}} \right] $

      $ {B_c} = \left[{\begin{array}{*{20}{c}} 0\\ {\dfrac{1}{2}{T}\left( {e} \right){J^{-1}}} \end{array}} \right]. $

      The state variable response curve of that linear constant system is plotted in Matlab by Fig. 5.

      Figure 5.  Response curve of variable ${e}$ and $\dot{e} $ of controlled system

      This linear constant system is stable as Fig. 5 describes. Although the state response of closed-loop linear system is similar to the stated response of variable ${e}$ and ${ \dot{e}, }$ the state error between practical closed-loop and theoretical closed-loop linear constant systems need further test. Furthermore, the state error between practical closed-loop system and theoretical closed-loop linear constant system can be defined as

      $ state\_error\left( i \right) = {\left\| {{{\left[{\begin{array}{*{20}{c}} {e\left( i \right)} & {\dot e\left( i \right)} \end{array}} \right]}^{\rm T}} - {{\left[{X\left( i \right)} \right]}^{\rm T}}} \right\|_2}. $

      (69)

      The response of the sate error is plotted in Fig. 6.

      Figure 6.  Curve of the state error between practical system and theoretical system

      According to Fig. 6, the state error between practical system and theoretical system is quite small which is of 10-14 level. Therefore, the controller ${u}$ has successfully turned the original system into a linear constant stable system.

    • A controller is designed by the direct parametric approach and applied on a practical spacecraft attitude system. In this paper, the nonlinear and non-approximated spacecraft attitude control model is turned into a stable closed-loop linear constant system by the controller ${u}$ . Even though many other control strategies prefer to simplify the spacecraft attitude system with approximation, a fully-actuated second-order control model is built up without approximation. Furthermore, the values of the closed-loop eigenstructure are undetermined parameters. The user of the controller with all degrees of freedom ${F}$ and ${Z}$ can easily design the controller with desired closed-loop eigenstructure. Moreover, the degrees of freedom through the parameters ${F}$ and ${Z}$ are beneficial for the optimization of the controller to satisfy the requirement of performance of the controlled system, because the parameters can be uncertain before the simulation experiment. Additionally, the simulation by Matlab with a set of practical data for spacecraft attitude tracking on a mobile target verifies the availability and robust stability of the controller when there is the external disturbance torque applied on the control model. Finally, compared with the theoretical closed-loop linear constant system, the controller actually can convert the original system into a stable linear constant system, which means the controller is advantageous in easy placement of the desired stable closed-loop poles.

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