# Detection of the Rotational Motion of the *AIST-2D* Small Spacecraft by Magnetic Measurements

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### **Abstract**—

The paper presents the reconstruction results of rotational motion of the *AIST-2D* small spacecraft by onboard measurements of vectors of angular velocity and the strength of Earth’s magnetic field obtained in summer 2016. The reconstruction method is based on the reconstruction of kinematic equations of the rotational motion of a solid body. According to the method, measurement data of both types collected on a certain time interval are processed together. Measurements of the angular velocity are interpolated by piecewise-linear functions, which are replaced in kinematic differential equations for a quaternion that defines the transformation from the satellite instrument coordinate system to the inertial coordinate system. The obtained equations represent the kinematic model of the rotational motion of a satellite. A solution to these equations that approximates the actual motion is derived from the condition of the best (in the sense of the least squares method) match between the measurement data of the strength vector of Earth’s magnetic field and its calculated values. The initial conditions of the approximating solution, constant bias in angular velocity measurements, and angles specifying the matrices of transformation from magnetometer intrinsic coordinate systems to the instrument coordinate system of the satellite (measurements of the angular velocity are specified in it) are refined. The described method makes it possible to reconstruct the actual rotational motion of a satellite using one solution of kinematic equations over time intervals longer than 10 h.

## Notes

### ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 17-01-00143.

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