Optimization Issues in the Problem of Small Satellite Attitude Determination and Control

  • Zaure RakishevaEmail author
  • Anna Sukhenko
  • Nazgul Kaliyeva
Part of the Springer Optimization and Its Applications book series (SOIA, volume 144)


The problems of synthesis of attitude determination and control system of small satellites regarding the influence of external perturbations due to their small mass and restrictions in using high-precision actuators, due to the limitations of energy budget and construction, are considered. The solutions to this problem are proposed involving the development of high-accuracy algorithms for satellite attitude determination and control, using the minimum set of sensors, various types of actuators, and optimization principles.



Positive constant

\( \overrightarrow{\omega_{bi}^b} \)

Angular velocity of the small satellite in the body coordinate system

\( \overrightarrow{\omega_{oi}^o} \)

Angular velocity of the orbital coordinate system relative to the inertial coordinate system

Δω1, Δω2, Δω3

Components of the vector of deviation of the small satellite angular velocity w.r.t. the required angular velocity

\( \overrightarrow{\omega_{\mathrm{r}}} \)

Required angular velocity

Multiplication operator for quaternions


System matrix


Control matrix

\( \overrightarrow{B} \)

Geomagnetic induction vector

\( \overrightarrow{B^o}={\big[}{B}_x^o,{B}_y^o,{B}_z^o{\big]} \)

Geomagnetic induction vector in the orbital coordinate system


Misalignment of the angular position of the small satellite

\( \overrightarrow{G_b} \)

Angular momentum of satellite

\( \overrightarrow{h} \)

Vector of conjugate variables

\( \overrightarrow{h_a^b} \)

Angular momentum of the reaction wheels


Inertia moments of the small satellite

\( {\tilde{J}}_x \), \( {\tilde{J}}_{\mathrm{y}} \), \( {\tilde{J}}_z \)

Nominal values of the inertia moments

ΔJx, ΔJy, ΔJz

Value of deviations of the satellite moment of inertia

Kp, Kd

Proportional gain and derivative gain


Linear matrix inequalities

\( \overrightarrow{M_e^b} \)

Moment of the external forces in the body coordinate system

\( \overrightarrow{M_c^b} \)

Control moment of the actuators in the body coordinate system

\( \overrightarrow{m} \)

Magnetic moment of the coils


Matrix of system noise


Initial covariance matrix


Proportional-derivative controller

\( \Delta \overrightarrow{\mathrm{Q}} \)

Difference between the current angular position and the required angular position

\( \overrightarrow{{\mathrm{Q}}_{bi}} \)

Quaternion that sets the current angular position of the satellite in the inertial coordinate system

\( {\overrightarrow{Q}}_{bo} \)

Quaternion that sets the current angular position of the satellite in the orbital coordinate system

\( \overrightarrow{{Q_{b\mathrm{o}}}^{\ast }} \)

Quaternion that is the inverse of \( {\overrightarrow{Q}}_{bo} \)

\( \overrightarrow{{\mathrm{Q}}_{\mathrm{r}}} \)

Desired angular position

Δq1, Δq2, Δq3

Components of the vector part of the quaternion which describe the deviation of the small satellite current orientation regarding the desired orientation

\( {R}_b^o \)

Direction cosine matrix representing the rotation of the orbital coordinate system axes regarding the axes of the body coordinate system

\( \overrightarrow{u} \)

Control vector

W ≥ 0, P > 0

Constant matrices

\( \overrightarrow{X} \)

State vector of the dynamical system

\( \overrightarrow{\Delta X} \)

Deviation of the current state vector of the dynamic system w.r.t. the desired state vector

δx, δy, δz

Normalized parametric uncertainties of the satellite inertia moments


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Zaure Rakisheva
    • 1
    Email author
  • Anna Sukhenko
    • 1
  • Nazgul Kaliyeva
    • 1
  1. 1.Department of MechanicsAl-Farabi Kazakh National UniversityAlmatyKazakhstan

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