The Magneto-Torquer is a coil wound around with a conductor wire (usually copper) over some kind of support; this wire is coated with varnish to provide electrical insulation. This way of generating a magnetic field is known as an electromagnet, and the density generated due to the magnetic field may differ proportionally from the number of turns the coil might have and the quantity of current flowing through.

The coil is formed by fitting the core in the solenoid ( Figure 1 ). Given that the iron has more permeability than the air, the lines of force can flow easily, and the magnetic field lines generated by the coil are concentrated in the core, producing a flow density increase inside the coil [5] .

The moment of inertia of a solid is a scalar magnitude defined as [6] :

$I={\displaystyle \underset{i=1}{\overset{N}{\sum }}{m}_i{r}_i^2},$

$r_i$ represents the distance from the axis of rotation and $m_i$ as the masses.

Having said this, it can be deduced that the moment of inertia of a solid depends on the axis of rotation, and as mentioned before, when it is about a solid, the addition of the formula can be substituted by the following integral

$I=p{\displaystyle \int \text{d}m{r}^2},$

where, $\text{d}m$ represents the mass element of the solid directly related to the solid density $p$ and $r$ is the distance from the axis of rotation.

Given the mass element of the solid $\text{d}m$ and the solid density $p$ besides if this is homogeneous, it is possible to calculate the density of the integral when substituting $\text{d}m$ in the previous formula

$\text{d}m=p\text{d}V$

$I=p{\displaystyle \int r^2\text{d}V}$

where, $\text{d}V$ represents a solid volume and, to calculate a homogeneous solid’s inertia moment, it is necessary to solve the previous integral.

Taking the situation above into account, the variation per unit time is the linear velocity factor $\stackrel{\dot{}}{r}=v$ and in case motion may take place along a radius circumference $r$ the magnitude of the linear velocity will be the following [7] ,

$v=r\frac{\text{d}\theta }{\text{d}t}=rw$ (1)

where, $v$ is the linear velocity, $w$ is the angular velocity.

Therefore, the angular moment $L$ applied to a rigid solid is obtained

$L={\displaystyle \int r\cdot mwr\text{d}m}={\displaystyle \int w{r}^2\text{d}m}=I\cdot w,$ (2)

it is known that a Magneto-Torquer operation is based on the ability to create a magnetic dipole; when a current flows through a coil, the equation of angular moment is obtained [8] ,

$L=I\cdot w.$ (3)

Remembering that the magnetic torque $\tau$ applied to a coil exposed to an external magnetic field may be expressed as the vector product of dipole moment and external magnetic field [5] [9] ,

$\tau =\mu \times B,$ (4)

where, $\mu$ is the dipole moment produced by the Magneto-torque, $B$ is the external magnetic field in such a way that the modulus of the magnetic torque can be expressed [9] ,

$\tau =\mu \cdot B\cdot sen\left(\theta \right),$ (5)

where $\theta$ is the angle that forms the axis of action of the magnetic force produced from the Magneto-Torquer and the desired angle. By saying this, it is possible to find the equation of motion of the system. To carry this on, it is necessary to apply the theorem of conservation of angular momentum. The law of conservation of angular momentum states if the net external torque is zero (it does not imply the external forces are zero, but it is an isolated system), the total angular momentum is conserved in other words, it is constant [6] .

${\displaystyle \sum \text{Τ}}=I\cdot \stackrel{\dot{}}{w},$ (6)

with $\text{Τ}$, it is the torque applied to the system.

However, the magnetic torque produced by the magneto-torque, Equation (5), is not the only one that interacts with the system; there is also a resistive torque produced by different factors. From Equation (6), it can be deduced that when applying a torque to a solid, it can create an angular acceleration about the axis around which the object rotates, leading to the motion equation for the Magneto-Torquer

$\tau -{\text{Τ}}_r=I\cdot \omega ,$ (7)

where ${\text{Τ}}_r$, represents the resistive torque which opposes the motion. Besides, the operating principle of a Magneto-Torquer is based on the torque produced by the magnetic dipole moment. Thus, it can possibly be related to the current intensity flowing through the coils along the satellite movement.

And so, relating the Equations (5), (7) it may bring

$\mu \cdot B\cdot sen\left(\theta \right)-{\text{Τ}}_r=I\cdot \omega ,$ (8)

the dipole moment produced by a coil is calculated as Equation (7) shows. The angular velocity function $\omega$ is obtained as:

$\omega =\frac{\mu B}{I}\cdot sen\left(\theta \right)-\frac{{\text{Τ}}_r}{I},$ (9)

where, $I$ represents the moment of inertia, $\mu$ is the dipole moment, $B$ is the external magnetic field, ${\text{Τ}}_r$ is a resistive torque, $\theta$ is the angle that forms the axis of action of the magnetic force produced by the Magneto-Torque, and the reference angle.

The inertia matrix is defined as a symmetric matrix whose elements are formed from the moment of inertia of three perpendicular axes and three inertia products. If $I\in {ℝ}^{3\times 3}$ respect to an origin $O$, it may bring up [10] [11] ,

$I=\left[\begin{array}{ccc}{I}_x& -I_{xy}& -I_{xz}\\ -I_{xy}& I_y& -I_{yz}\\ -I_{xz}& I_{yz}& I_z\end{array}\right]$ (10)

where, $I=I^{\text{T}}$.

Well, if the body’s coordinate system axes coincide with the main axes of inertia, that is, the principal frame of the satellite is chosen where the inertia matrix $I$ is diagonal, just like it happens with most of the satellites, the matrix is reduced as [10] [11] ,

$I=\left[\begin{array}{ccc}{I}_x& 0& 0\\ 0& I_y& 0\\ 0& 0& I_z\end{array}\right]$. (11)

Therefore, the satellite’s Attitude dynamics are expressed by applying Euler’s equation:

$I_x{\omega }_x=-\left(I_z-I_y\right){\omega }_y{\omega }_z+{\tau }_x,$ (12)

$I_y{\omega }_y=-\left(I_x-I_z\right){\omega }_x{\omega }_z+{\tau }_y,$ (13)

$I_z{\omega }_z=-\left(I_y-I_x\right){\omega }_y{\omega }_x+{\tau }_z,$ (14)

where, $\omega =\left({\omega }_x,{\omega }_y,{\omega }_z\right)\in {ℝ}^3$ is the angular velocity vector of the satellite and $\tau =\left({\tau }_x,{\tau }_y,{\tau }_z\right)\in {ℝ}^3$ is the torque vector performing on the system.

The geomagnetic field is the magnetic field extending from the Earth’s core. The Earth’s magnetic field may be approximated as a dipole inclined to 15 degrees to the axis of rotation of the Earth creates a magnetic field; its magnitude on the Earth’s surface has an average from 25 to 65 μT (microteslas) being stronger at the poles and weaker at the magnetic equator.

Table 1 shows the characteristics of the Earth’s geomagnetic field according to [12] .

where Re, defines the radius of the Earth’s model as 6.378 km. The dipole intensity is the dipolar portion intensity of the planetarium magnetic field outside the planet. In Gauss-Re³, Re is in units of the planet’s radius. (Dividing by the distance Re³ yields the field in Gauss).

The Earth’s atmosphere is the gaseous layer of the Earth, being the most outermost and the least dense of the planet. This layer is composed of several gases that vary as the pressure at different altitudes. This mixture of gases that compose the atmosphere is genuinely called air. 75% of the atmospheric mass is at the first 11 km of altitude above the sea level. The main gases that compose it are Oxygen (21%) and nitrogen (78%), followed by argon, carbon dioxide, and water vapor.

Table 2 and Table 3 point out the characteristics and composition of the Earth’s atmosphere according to [12] .

The CubeSats are cube-shaped satellites with the following specifications:

1) The coordinate system of a CubeSat 1U must coincide with the P-POD coordinate system (Poly Picosatellite Orbital Deployer), while integrated into the P-POD. The origin of the CubeSat system coordinates is located in the geometric center of the CubeSat.

2) The maximum mass of a CubeSat 1U is 1.33 kg.

3) The gravity center must be located no more than 2 cm. from the geometric origin in the X, Y & Z direction.

1) The deployment switch must be operated all the time while it is integrated into the P-POD (Poly Picosatellite Orbital Deployer).

2) The CubeSat must have an RBF pin (Remove Before Flight).

3) The CubeSats will incorporate protection in the battery circuit to charge and discharge, thus avoiding unbalanced cell conditions.

Stability can be tested with the Lyapunov function criterion [15] . Considering the following positive define function (Lyapunov candidate function),

$V\left(x_1,x_2\right)=\frac{1}{2}{x}_1^{\text{T}}{K}_p{x}_1+\frac{1}{2}{x}_2^{\text{T}}M\left(q\right)x_2,$ (24)

the representation in terms of position and velocity errors is obtained by substituting the state variables in Equation (24), and the result is a positively defined function

$V\left(\tilde{q},\stackrel{\dot{}}{q}\right)=\frac{1}{2}{\tilde{q}}^{\text{T}}{K}_p\tilde{q}+\frac{1}{2}{\stackrel{\dot{}}{q}}^{\text{T}}M\left(q\right)\stackrel{\dot{}}{q},$ (25)

when deriving Equation (25):

$\stackrel{\dot{}}{V}\left(\tilde{q},\stackrel{\dot{}}{q}\right)={\tilde{q}}^{\text{T}}{K}_q\stackrel{\dot{}}{\tilde{q}}+{\tilde{q}}^{\text{T}}M\left(q\right)\stackrel{\dot{}}{\tilde{q}}+\frac{1}{2}{\stackrel{\dot{}}{q}}^{\text{T}}\stackrel{\dot{}}{M}\left(q\right)\stackrel{\dot{}}{q},$ (26)

The state variable $x_2$ in Equation (21) will give:

$M\left(q\right)\ddot{q}=K_p\tilde{q}-K_v\stackrel{\dot{}}{q}-C\left(q,\stackrel{\dot{}}{q}\right)\ddot{q},$ (27)

when substituting Equations (26), (27):

$\stackrel{\dot{}}{V}\left(\tilde{q},\stackrel{\dot{}}{q}\right)={\tilde{q}}^{\text{T}}{K}_p\stackrel{\dot{}}{\tilde{q}}+{\stackrel{\dot{}}{q}}^{\text{T}}{K}_p\tilde{q}-{\stackrel{\dot{}}{q}}^{\text{T}}{K}_v\stackrel{\dot{}}{q}+\frac{1}{2}{\stackrel{\dot{}}{q}}^{\text{T}}\left[\stackrel{\dot{}}{M}\left(q\right)-2C\left(q,\stackrel{\dot{}}{q}\right)\right]\stackrel{\dot{}}{q}$. (28)

By applying the antisymmetric matrix properties in Equation (28), we get:

$\stackrel{\dot{}}{V}\left(\tilde{q},\stackrel{\dot{}}{q}\right)={\tilde{q}}^{\text{T}}{K}_p\stackrel{\dot{}}{\tilde{q}}+{\stackrel{\dot{}}{q}}^{\text{T}}{K}_p\tilde{q}-{\stackrel{\dot{}}{q}}^{\text{T}}{K}_v\stackrel{\dot{}}{q}.$ (29)

When using the state variable ${\stackrel{\dot{}}{x}}_1$, Equation (19) will give:

${\stackrel{\dot{}}{x}}_1=\stackrel{\dot{}}{\tilde{q}}=\{\begin{array}{l}\stackrel{\dot{}}{q}-{\stackrel{\dot{}}{q}}_d=\stackrel{\dot{}}{q}\\ {\stackrel{\dot{}}{q}}_d-\stackrel{\dot{}}{q}=-\stackrel{\dot{}}{q}\end{array}$ (30)

And applying the properties of the transpose

${\left(xyz\right)}^{\text{T}}=x^{\text{T}}{y}^{\text{T}}{z}^{\text{T}},$ (31)

${\left(x^{\text{T}}\right)}^{\text{T}}=x,$ (32)

considering that $K_p$ is symmetric, as, $K_p=K_p^{\text{T}}$ and with the described properties in Equations (31), (32), (30) can be modified as follows

$\stackrel{\dot{}}{V}\left(\tilde{q},\stackrel{\dot{}}{q}\right)=-{\stackrel{\dot{}}{q}}^{\text{T}}{K}_v\stackrel{\dot{}}{q}.$ (33)

Finally, the derivative of the function of Lyapunov is less or equal to zero; this is because when the error of the position is zero, this term $-{\stackrel{\dot{}}{q}}^{\text{T}}{K}_v\stackrel{\dot{}}{q}$ becomes zero. In other words, it is indeed less or equal to zero. Concluding that the equilibrium point $x_1=x_2=x^{*}\left(0,0\right)$ located at the origin of the state space, is stable.

For this case, the face X of the satellite is oriented in order to calculate the Attitude matrix between the satellite’s frame and the orbital system based on the Attitude vectors function of the Sun and the geomagnetic field measured by the sensors. In addition, a specific orbit over the State of Puebla was chosen.

Three sets of orthogonal magnetic torsion coils are used for the Attitude control, and for this case, it is used a NCTR-M002 Magneto-Torquer (see Table 4 ) whose characteristics make it particularly useful to be implemented for the CubeSat due to its low use of energy and its lightweight. Table 5 shows the simulation parameters used.

It was selected as a 1U CubeSat model as shown in Figure 2 .

With a mass equal to M = 1.2 kg and with an inertia matrix

$I=\left[\begin{array}{ccc}2.166\text{e}-3& 0& 0\\ 0& 2.166\text{e}-3& 0\\ 0& 0& 2.166\text{e}.3\end{array}\right]$ (34)

The next Keplerian elements were taken into consideration for the simulation, such as the Earth’s radius, orbit, eccentricity, and the orbit’s inclination. For this case, a polar orbit was selected, and the rest of the elements, such as the perigee argument, average anomaly, and ascending node longitude, were selected.

The variation of the Keplerian elements is calculated according to the Earth’s shape. An initial Attitude (initial position) for the CubeSat in radians is proposed, and an angular velocity is given in radians per second.

$eulxyz={\left[30,50,10\right]}^{\prime }\frac{\pi }{180};$

$w_{ang}={\left[0.1,0,0.5\right]}^{\prime };$

The inertia matrix is symmetric as it is with most of the satellites, the coordinate system axes coincide with the principal inertia axes. The matrix values must be given in Kg.

Three magnetic torsion coils are used for the Attitude control, one for each axis, with a magnetic moment of 0.2 Am².

The satellites usually use sensors to find out their location and orientation in orbit (Attitude). These sensors work based on the direction of the Earth, Sun, Moon, stars, the geomagnetic field, and among others. This is why it is important to keep the sensors mentioned during the simulation in mind. Propat offers distinct models that allow simulating the Sun’s position in the geocentric inertial system, the Earth’s orientation in a specific moment (Greenwich Sidereal Time), and the Earth’s magnetic field. The simulation code shows exactly the model used to test the sensors at a specific time.

The time propagation of the simulation is given in seconds. The time, angles, angular velocity, and the orbit vector start initializing. The control torque and the current of the magneto torques start initializing.

As a numeric integration is estimated, the precision is defined. With this, it is now possible to calculate the Orbit and Attitude propagation. Also, the propagation of the orbit and conversion of the Keplerian elements to a vector of state.

In this case, perturbations will not be considered. Therefore, the Attitude vector will be defined and initialized.

It is necessary to have knowledge of the direction and the intensity of the geomagnetic field to orient a Satellite’s face towards the Earth.

It is also recommended that the ODE45 numerical integrator be used for the propagation of the Attitude.

Additionally, it is necessary to know not only the position vector but also the Sun’s position vector in the Satellite’s frame.

The Gaussian was added to the Sun’s position vector and to the geomagnetic field since the data of the sensors is corrupted by the line noise before being processed, thus limiting the transmission distance.

The axes are adjusted so that the X axis aims at the Earth, the Y axis is orthogonal to the X axis in the orbital plane, and the Z axis is normal to the orbital plane.

Considering two reference systems, one belonging to the Earth and the other belonging to the satellite, the Attitude matrix is calculated for two pairs of vectors known in two different reference systems. If the two coordinate systems are the same, the matrix gets its identity back, as shown in Figure 3 . At last, the orbital angular velocity is calculated. As was mentioned previously, the control is valid only if the satellite’s angular velocity is a minimum of four times the revolutions per orbit.

The current supplied to the Magneto-Torquer is calculated considering the number of spins and the maximum current they can hold.

Current in the coils

EB = teta’.*earth_field;

K = 1e5;

MI = −k*(cross(EB, (w_ang-wo_b)));

N = I max;

xyzC = MI/(710*0.04);

xcurrent = xyzC(1);

ycurrent = xyzC(2);

zcurrent = xyzC(3);

Finally, the data is saved for plotting.

time = cat(2, time, t);

omeg = cat(2, omeg, w_ang);

euler = cat(2, euler, teta*180/pi);

corx = cat(2, corx, xcurrent);

cory = cat(2, cory, ycurrent);

corz = cat(2, corz, zcurrent).

According to the next expression of the control PD, we can say:

$u=K_p\theta a+K_d\left({\omega }_{orb}-{\omega }_{ob}\right),$ (35)

$\theta$ and $a$ are Euler’s angle and axis, while ${\omega }_{orb}$ is the orbital angular velocity with respect to the orbital system and ${\omega }_{ob}$ is the orbital angular velocity. Meanwhile, the selection of the gain $K_p=-0.00025$ and $K_d=-0.002$ was done based on the criteria of $K_p<K_d$ since this is the way to prioritize the equilibrium point, however, if the value of the gains is reversed, the system reduces its angular velocity, making the system reach its equilibrium point in a longer period [16] .

% Vangular-Vorbital

cont_torq = −0.00025*teta − 0.002*(w_ang-wo_b)

Considering the gains of $K_p$ & $K_d$ and the characteristics of the satellite the following results are obtained, which represent the orientation measured in Euler angles, the angular velocity measured in rad/s, and finally, the variation of the current supplied to the Magneto-Torquer.

As Figure 4 shows, it can be observed that the satellite reached the desired position even after the position error was zero due to the line noise.

In Figure 5 , it can be clearly observed that when the satellite reaches the desired position, the angular velocity is equal to zero.

As it is shown in Figure 6 , the current supplied does not exceed 0.03 A because the maximum current the Magneto-Torquer can handle is 0.04 A.