International Journal of Aerospace Engineering

Volume 2018, Article ID 9768475, 12 pages

https://doi.org/10.1155/2018/9768475

## Design and Numerical Validation of an Algorithm for the Detumbling and Angular Rate Determination of a CubeSat Using Only Three-Axis Magnetometer Data

School of Aerospace Engineering, Sapienza University of Rome, Via Salaria 851, 00138 Rome, Italy

Correspondence should be addressed to Stefano Carletta; ti.1amorinu@attelrac.onafets

Received 4 November 2017; Revised 2 March 2018; Accepted 16 April 2018; Published 2 May 2018

Academic Editor: Vaios Lappas

Copyright © 2018 Stefano Carletta and Paolo Teofilatto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A detumbling algorithm is developed to yield three-axis magnetic stabilization of a CubeSat deployed with unknown RAAN, orbit phase angle, inclination, attitude, and angular rate. Data from a three-axis magnetometer are the only input to determine both the control torque and the angular rate of the spacecraft. The algorithm is designed to produce a magnetic dipole moment which is constantly orthogonal to the geomagnetic field vector, independently of both the attitude and the angular rate of the rigid spacecraft. The angular rates are calculated in real time from magnetometer data, and the use of a second-order low-pass filter allows to rapidly reduce the measurement error within ±0.2 deg/sec. Numerical validation of the algorithm is performed, and a variety of feasible scenarios is simulated assuming the CubeSat to operate in low Earth orbit. The robustness of the algorithm, with respect to unknown deployment conditions, different sampling rates, and uncertainties on the moments of inertia of the CubeSat, is verified.

#### 1. Introduction

The stabilization of a spacecraft after deployment is never a trivial issue, and several satellites have been lost due to anomalies or dysfunctions occurred during this phase. Typically, the angular rates at the deployment are much higher than those desired for attitude maneuvering and the satellite is said to be tumbling. Therefore, a specific *detumbling* control must be designed to stabilize the spacecraft within the minimum time compatible with the mission requirements. After the spin motion of the spacecraft has been damped to the desired level, the control policy can be switched to fine pointing or attitude maneuvering and mission operations can start.

Additional issues often arise when implementing such a detumbling control on a CubeSat, mainly because of the limited capabilities, when compared to bigger satellites, of the Attitude Determination and Control Systems (ADCS) commonly used. Nevertheless, CubeSat technology has grown dramatically fast over the last ten years and is nowadays opening to scientific missions. Therefore, improving their reliability, and in particular by increasing the efficiency of their ADCS, is for sure a main target for the years to come.

In this work, we propose a solution for the contingent scenario in which the ADCS sensor addressed to angular rate measurement (i.e., a rate gyro) is not capable of providing any suitable information. This can happen after a failure of the sensor itself or if the CubeSat angular rates exceed the measuring range of the sensor, which is therefore not suitable to produce accurate measurements.

Focusing on CubeSats operating in low Earth orbit (LEO), an algorithm producing both the detumbling and angular rate determination using only three orthogonal magnetorquers and the measurements by a three-axis magnetometer is developed. It is worth to outline that these two results are independent one from each other. The algorithm is specifically designed to be implemented on a low computationally efficient CubeSat onboard computer. Additional constraints are represented by the limited energy budget, the strict limit on the peak current, and the low sampling rate.

The detumbling module of the algorithm is based on the popular B-dot control, here rearranged to explicitly generate a magnetic dipole moment constantly orthogonal to the geomagnetic field vector. The stability of the control was verified numerically, simulating the detumbling of the 3 U CubeSat Tigrisat. Numerical simulations also proved that an accurate selection of the proportional control parameter results in a commanded magnetic dipole moment which never exceeds a maximum value here fixed to 0.3 Am^{2}.

The problem of magnetometer-only attitude determination has been extensively examined through the last thirty years, with relevant results obtained first by Natanson et al., for spacecraft rotating with constant angular rate and later extended to the case with no a priori knowledge of the spacecraft state with uses of a Kalman filter [1, 2]. As reviewed by Hajiyev and Guler [3], several algorithms include a single- or multiple-step extended Kalman filter [4–9] or an unscented Kalman filter [10]. Other approaches are based on deterministic two vector methods [11–13] and are typically preferred when computational efficiency of the onboard computer is limited and lower measurement accuracy is acceptable.

In the proposed algorithm, measurements of the angular rates are based on the geometric properties relating three consecutive samples of the geomagnetic field vector, processed in real time through a second-order low-pass filter. Simulations proved that the algorithm is suitable for measuring the angular rates within a steady state error of ±0.2 deg/sec and robust with respect to unknown deployment conditions and uncertainties on the inertial properties of the CubeSat.

In Sections 3 and 4, the design for the detumbling control and the angular rate determination algorithm are presented and numerically validated, simulating the detumbling of the 3 U CubeSat Tigrisat. In Section 5, a variety of detumbling scenarios, including unknown deployment parameters and uncertainties, is simulated to evaluate the performance of the algorithm and the results are discussed.

#### 2. Mathematical Model

Some preliminary considerations are worth being outlined to clearly define the framework in which the detumbling problem is here studied. The algorithm is developed for a CubeSat deployed in LEO, without a priori knowledge about its RAAN (), orbit phase angle (), inclination (*i*), attitude, and angular rate. The ADCS only includes one three-axis magnetometer and three magnetorquers mutually orthogonal and aligned with the principal axes of inertia, which define the body fixed reference frame ().

Attitude dynamics is modeled assuming the spacecraft as a rigid body on which only the magnetic dipole torque acts. This is the torque due to the interaction of the magnetic dipole moment produced by magnetorquers with the geomagnetic field vector. The nonlinear angular rate dynamics can be expressed by the following equation [14]: where is the angular rate vector, is the tensor of inertia, is the magnetic dipole moment, and is the geomagnetic field vector in . Both and its rate represent the angular velocity and acceleration of with respect to , namely, the Geo Centric Inertial (GCI) reference frame [15]. The time derivatives of the geomagnetic field vector in the two frames are related by the following equation:

Numerical simulations, whose results are discussed in Sections 3, 4, and 5, were run integrating the full nonlinear spacecraft attitude dynamics equations [16], using the fixed step solver ODE8. The values of during the time of the simulation are obtained from the International Geomagnetic Reference Field (IGRF) model.

#### 3. Detumbling Control

The proposed detumbling control represents a variation of the classical B-dot, in which the variable is defined to be constantly orthogonal to [17]. Such a result is obtained assuming that is negligible with respect to the other variations. This approximation is reasonably accurate only if the angular rates are higher than the maximum rate of change of the geomagnetic field vector, equal to twice the orbital rate () of the spacecraft [14]. The scenarios investigated in this work verify the mentioned condition (see Section 5), and thus (2) can be rearranged as follows: where is the projection of in the direction orthogonal to .

The cross-product does not allow the determination of by inverting (3). Consequently, a guessed expression for is considered as follows:

The suitability of (4) follows from the following:

Recalling that in LEO , the rhs of (5) is approximately equal to , as it is required by (3).

The magnetic dipole moment is now expressed as follows: where is a constant parameter, namely, the control gain, to be defined in accord to the specifics of the mission. Equation (6) clearly shows that is constantly orthogonal to .

The control algorithm was validated through numerical analysis in MATLAB. Simulation parameters are those of the 3 U CubeSat Tigrisat, launched by the School of Aerospace Engineering in 2014, reported in Table 1.