Kibern. vyčisl. teh., 2018, Issue 1 (195), pp.
Yefymenko M.V., PhD.,
associate professor of Zaporizhzhya National Technical University,
Scientific Production Enterprise “HARTRON-YUKOM”
Soborny аv., 166, Zaporozhye, 69035, Ukraine
SOLUTION OF THE PROBLEMS OF CONTROLLING THE MOTION OF A POINT ON A SPHERE
Introduction. There are a number of control objects, the movement of which in space can be interpreted as the movement of a point along a sphere of a given radius. As an example of such a motion, the angular motion of a spacecraft can be considered. Using the orientation quaternion and its derivative to describe the angular motion of a spacecraft, the angular motion can be represented as the motion of a point along a unit sphere in R4.
While controlling such objects, the methods for solving the basic problems of controlling the motion of a point along the unit sphere in the Rn space are of interest.
The purpose of the article is to build the following algorithms for controlling the motion of a point along the sphere:
– algorithm of a point motion stabilization with respect to program trajectory;
– algorithm of a point relocation from the current position to a specified position in minimum time;
– algorithm of a point relocation from the current position to a specified position in fixed time.
Results. The methods for solving the various problems of controlling the motion of a point along the sphere have been proposed.
Conclusion. On the basis of main properties of point along the sphere movement, the methods for solving the problems of controlling the motion of a point along the unit sphere in n-dimensional space have been proposed. Using the proposed methods, the solutions for the following control tasks have been found:
– problems of stabilizing the motion of a point along the sphere with respect to program trajectory;
– speed problems taking place when a point moves on along the sphere;
– problems of a point on the sphere relocation from the current position to a specified position in fixed time.
The efficiency of the proposed algorithms has been demonstrated on the example of spacecraft angular motion control. The results obtained can be applicable in the development of various control systems, the spacecraft angular motion control systems in particular.
Keywords: sphere, control, point projection, quaternion.
1 Kirichenko N.V., Matvienko V.T. Algorithms of asymptotic terminal and adaptive stabilization of the rotational motions of a rigid body. Problems of Control and Informatics. 2003. No 1. P. 5-15. (in Russian). https://doi.org/10.1615/JAutomatInfScien.v35.i1.10
2 Kirichenko N.F., Lepekha N.P. Perturbation of pseudoinverse and projection matrices and their application to the identification of linear and nonlinear dependencies. Problems of Control and Computer Science. 2001. N 1. P. 6-22.
3 Kirichenko N.F., Lepekha N.P. Pseudo-inversion in control and observation problems. Automation. 1993. N 5. P. 69-81.
4 Kirichenko N.F., Matvienko V.T. Optimal synthesis of structures for linear systems. Problems of Control and Informatics. 1996. N 1-2. P. 162-171.
5 Yefimenko N.V. Mathematical model of the angular motion of the spacecraft in the parameters of Rodrigues-Hamilton and its properties. Electronic modeling. 2018. Vol. 40. No 6. P. 21-36 (in Russian). https://doi.org/10.15407/emodel.40.06.021
6 Quakernaak X., Sivan R. Linear optimal control systems. Moscow: Mir, 1977. 650 p. (in Russian).
7 Yefimenko N.V. Synthesis of the space-optimal time-reversal of a spacecraft using the dynamic equation of the rotational motion of a rigid body in the Rodrig Hamilton parameters. Problems of Control and Computer Science. 2017. No 3. P. 109-128. (in Russian).
8 Yefimenko N.V. Synthesis of spacecraft reorientation control algorithms using the dynamic equations of the rotational motion of a rigid body in the Rodrig Hamilton parameters. Problems of Control and Computer Science. 2015. No 3. P. 145-155. (in Russian).