Stereographic Projection Method in Problems of Orientation of a Moving Object

Study of new kinematic parameters for strapdown orientation systems is a relevant task, since it allows increasing the accuracy and reliability of the object orientation determining under conditions of complex motion and an influence of external factors. Traditional kinematic parameters, such as Euler angles, have known limitations, for example, the problem of "gimbal lock", while new approaches to describing object rotations can provide more efficient determination of the object orientation. The aim of the work was to analyze and evaluate the applicability of the parameters (w, z) in the algorithms of strapdown orientation systems, as well as to develop and study a new scheme for combining gyroscopic and accelerometric measurements with data integration by the parameter w. Kinematic parameters (w, z) were relatively recently introduced into the theory of finite rotation, which specify the position and orientation of an object around a fixed point through two successive finite rotations. One of these rotations (angle z) characterizes the rotation of the object around one of the axes of the fixed coordinate system. The second rotation is described using a stereographic projection of the axis of the moving object onto the complex plane w = u + jv. This allows us to represent the orientation of the object as a point on this plane. Kinematic equations for the parameters (w, z) are given. Efficiency of parameters (w, z) using is shown as applied to gyroscopic strapdown orientation systems. It is shown that the kinematic equations for the function (w, z) arguments can be integrated independently of the angle z, and the general third order of the system is one unit lower than the kinematic equations in quaternions. Numerical experiments on integration of kinematic equations under the condition of constant rotation of the object with a given angular velocity of yaw and harmonic oscillations in pitch and roll angles are carried out. The modeling results are illustrated in functions of time, on the Riemann sphere and on the complex plane. Relationships are given that allow calculation of the function w(u, v) arguments using accelerometers. Stereographic projections of parameters obtained on the basis of measurements of gyroscopes and accelerometers containing instrumental errors are illustrated. A scheme for integrating gyroscopic and accelerometric data is given. This scheme differs from traditional methods because integration is performed not by pitch and roll angles but using arguments of the function w.

1. Chelnokov Yu. N. Quaternion Methods and Regular Models of Celestial and Spaceflight Mechanics: Using Euler (Rodrigues–Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Constrained Three-Body Problem. Bulletin of the Russian Academy of Sciences. Solid Body Mechanics. 2023;(1):3 -32. DOI: 10.31857/S0572329922600293

2. Avgustov L. I., Babichenko A. V., Orekhov M. I., Sukhorukov S. Ya., Shkred V. K. [Navigation of aircraft in nearearth space. Moscow, Nauchtekhlitizdat. 2015;592 p.

3. Pobegaylo A.P. Application of quaternions in computer geometry and graphics. Minsk, BSU. 2019;223 p.

4. Gordeyev V. N. Quaternions and biquaternions with applications in geometry and mechanics. Kiyev, Stal. 2016;316 p.

5. Chelnokov Yu. N. Quaternion models and methods of dynamics, navigation and motion control: monograph. Moscow, FIZMATLIT. 2011;554 p.

6. Fourati H., Belkhiat D.E.C. Multisensor Attitude Estimation: Fundamental Concepts and Applications: Devices, Circuits, and Systems, CRC Press. 2016; 606 p.

7. Ishlinsky A. Yu. Orientation, gyroscopes and inertial navigation. Moscow, Nauka. 1976;670 p.

8. Tsiotras P., Corless M., and Longuski J.M. A novel approach to the attitude control of axisymmetric spacecraft. Automatica. 1995;31(8):1099-1112.

9. Meng Z., Dimarogonas D., Johansson K. Attitude Coordinated Control of Multiple Underactuated Axisymmetric Spacecraft. IEEE Transactions on Control of Network Systems. 2016;1-1 p. DOI: 10.1109/TCNS.2016.2562503

10. Markley L., Crassidis J. Fundamentals of Spacecraft Attitude Determination and Control. New York, Springer. 2014;486 p.

11. Jerry R., Muir Jr. Complex Analysis: A Modern First Course in Function Theory, First Edition. John Wiley & Sons, Inc. 2015;277 p.

12. Aboelmagd N., Tashfeen B. K., Jacques G. Fundamentals of Inertial Navigation, Satellite-based Positioning and their Integration. Springer Science & Business Media 2012;314 p.