Using the principle of gravitational stabilization and orientation in the design of small spacecraft

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Introduction

Currently, a number of world space powers are conducting research and design developments to create small spacecraft (SS). Reducing the size and mass of the SS is a great advantage in the field of space instrumentation, since this results in a significant reduction in financial costs for launching a spacecraft into orbit and designing it [1–2].

An important component system of spacecraft is the orientation and stabilization system (OSS), the tasks of which are to orient the apparatus or any of its individual elements in a given direction, as well as to counteract disturbing forces, both external and internal, that tend to change orientation and position of the device in space. The efficiency of the useful payload (UP) of the SS depends on the high-quality application of the OSS.

A correctly oriented SS allows the use of directional antennas (instead of omnidirectional ones) as UPs, which significantly improves the power and quality of the supplied signal. In addition, the strategy of orienting solar panels towards the Sun makes it possible to maximize the generated electrical power on the spacecraft’s onboard equipment, and the orientation of the radiation panels is set in such a way as to minimize their illumination by the Sun, which helps to ensure thermal balance. In this way, providing the survivability of the spacecraft is achieved when energy consumption is minimized while maintaining guaranteed level of electricity generation. Since the generation of electricity is associated with the orientation of solar panels towards the Sun, the mode for ensuring the survivability of the SS is a set of measures to exclude programmable elements from the orientation algorithm and the orientation is carried out directly according to the presence of the Sun sensors [3].

Various types of radio receivers and radio transmitters most often play the role of UP with directional antennas. For narrower frequency ranges on which antennas transmit information, a more precise specified orientation is required, and the error in the case of stabilization (keeping the satellite in one position) should be minimal. For spacecraft whose UP includes optical means (photo and video telescopes), the task of changing orientation with a given minimum speed is added to the OSS tasks.

The spacecraft is given a certain angular motion relative to given landmarks by rotating around the center of mass. Visible celestial and terrestrial objects (stars, the Sun, horizon line) or directions in space (local vertical, geomagnetic field strength vector, oncoming air flow velocity vector), which can be measured with instruments, are used as landmarks [3–5].

The devices of the orientation and stabilization systems of the SS are subject to certain requirements caused by the main restrictions on the mass, size of the device, its energy and computing resources. The process of orientation in a given direction of the spacecraft is based on a change from an unoriented position to an oriented position. The task of stabilization includes restoring the original position, which was disturbed due to the influence of any disturbing factors.

Active and passive systems are used to orient the SS. Strict requirements for mass and dimensions are imposed on the SS, but the use of an active attitude control system is not always advisable. In the SS designs, it is often difficult or even impossible to install correction jet engines due to limitations in the weight and size of the SS design itself; moreover, the absence of working gas on board the spacecraft allows a significant increase in the useful mass of the SS [3–5].

Passive stabilization methods are also distinguished by the fact that they do not require large reserves of additional energy on board the SS.

Due to the small size of the satellite, the influence of aerodynamic forces and solar pressure is negligible and cannot act as an orientation system. The optimal option is a gravitational orientation system based on the use of gravitational moment.

Gravity orientation and stabilization system

The principles of gravitational orientation of the SS are based on the influence of gravitational and centrifugal forces arising due to differences in the moments of inertia of the SS along its axes and leading to the alignment of the axis of the smallest moment of inertia with the radius vector of the orbit, and the axes of the maximum moment of inertia are with the binormal to the orbit [6].

The magnitude of the stabilizing gravitational moments when the SS moves in a circular orbit and for small deflection angles is determined by the equations in the plane of pitch, roll and yaw, respectively:

 

$M_z=3{\omega }_{\text{КА}}^2\cdot (I_Y-I_X)\cdot {\Theta }_C,$ (1)

$M_Y=4{\omega }_{\text{КА}}^2\cdot (I_Z-I_X)\cdot {\phi }_C,$ (2)

$M_X={\omega }_{\text{КА}}^2\cdot (I_Z-I_Y)\cdot {\Psi }_C,$ (3)

where ${\omega }_{\text{КА}}$ is the orbital angular velocity of the SS; $I_X,I_Y,I_Z$ – moments of inertia of the SS relative to the axes of the associated coordinate system; ${\phi }_c,{\Theta }_c,{\Psi }_c$ – roll, pitch and yaw angles, respectively (angles that determine the relative position of the orbital and associated coordinate systems (ACS)).

The magnitude of gravitational moments depends on the difference between the moments of inertia of the SS and its angular velocity, which is affected by the orbital altitude. Consequently, gravitational OSS is more effective in low circular orbits. The efficiency of the OSS operation depends on the magnitude of the moments of inertia; the greater they are (they depend on the distance between the elements of the SS located in the direction of one of the axes of the ACS and the center of mass of the satellite), the stronger the device resists a change in position in space, i.e. at high moments of inertia, it is difficult for the SS to rotate around the center of mass [7; 8].

The gravitational moment has a restoring effect on the deviation of the axes of the SS from the base coordinate system in the case of the direction of the axis of the smallest moment of inertia of the SS along the radius vector, and the highest moment of inertia of the SS I_z – along the binormal orbit. It is possible for a three-axis orientation $I_X<I_Y<I_Z,$ however, this is provided when implementing the design of the SS in the form of a dumbbell.

The angular motion of the satellite under the influence of gravitational restoring moments at small deflection angles is an independent oscillatory motion along the planes of roll and yaw.

Assuming that the satellite is axisymmetric $I_Y=I_Z,$ i.e., a spacecraft with a uniaxial orientation, then the position of only one axis directed to the center of the Earth is controlled. In the absence of damping, the SS will perform undamped harmonic oscillations near the equilibrium direction - the local vertical, coinciding with the radius vector of the orbit with a pitch oscillation frequency ${\omega }_{\Theta }$:

${\omega }_{\Theta }={\omega }_{КА}\sqrt[]{\frac{3(I_X-I_Y)}{I_X}}.$ (4)

Gravitational stabilization becomes possible when angular rotation is limited to a level that ensures transition to oscillatory motion, i.e., capture conditions:

 

${\left(\frac{{\dot{\Theta }}_{Н.З}}{{\omega }_{\Theta }}\right)}^2<\mathrm{0,5}(1+\cos 2{\Theta }_{\text{Н.З}}),$ (5)

where ${\Theta }_{\text{Н.З}}, {\dot{\Theta }}_{\text{Н.З}}$ are the initial values of the pitch angular velocity at which capture is ensured.

The absence of an external damping medium in outer space forces the creation of special devices for energy dissipation in order to dampen vibrations around the equilibrium position. Such devices in the form of mechanical or magnetic dampers convert the energy of vibrational motion into heat. Advantage, in this case, can be given to magnetic dampers, which are relatively compact and reliable in operation.

Most often, when designing satellites with magneto-gravitational OSS, while ensuring uniaxial orientation of the small spacecraft with an error of 5–10°, it is necessary to create a maximum ratio of the moments of inertia of the SS after its insertion into orbit at a level of 15–20, as well as gravitational control torque over disturbing moments by 10–15 times.

Thus, the uniaxial orientation of the SS to the Earth is ensured by the choice of the appropriate composition: actuators of the gravitational device, a device for energy dissipation and guidance sensors.

Application of gravitational device in the SS

We propose a project of a small CubeSat spacecraft of 3U size with a gravitational orientation system (Fig. 1), where the gravitational device is shown conventionally. The design of such a satellite requires the presence of a gravitational stabilization device, which is necessary for the deployment of the SS after its separation from the launch vehicle, as well as for creating a righting moment. The gravity device is supposed to be placed between the rigidly fastened 2U SS and the third U SS. In its transport form, the gravity device is in the assembled position, the dimensions of which should not exceed the dimensions of the SS in the sum of the dimensions of the shipping container.

 

Рис. 1. Внешний вид МКА CubeSat размера 3U

Fig. 1. Appearance of CubeSat SS size 3U

 

According to the CubeSat design specification, the maximum mass of a 3U CubeSat should not exceed 4 kg. The estimated rib size of each single module is 10 cm [9; 10].

The gravity device is designed to create the required ratio moments of inertia of the spacecraft relative to the oriented axes $I_X<I_Y=I_Z.$ If we take the inertial model in the form of a dumbbell with a massive SS and a load (in our case, one of the CubeSats) moving relative to the center of mass of the spacecraft from the initial position $X_0$ to the final position $X_B$, then the moments of inertia $I_Y$ of such a spacecraft after extending the load can be determined by the formula

$I_Y=I_{Y_0}+m_{\text{и.г}}\cdot X_B^2\cdot \left(1-\frac{m_{\text{и.г}}}{M_{КА}}\right),$ (6)

where $I_{Y_0}$ is the moment of inertia of the spacecraft along the Y axis in the initial position (we will consider the moment of inertia for a rectangular section); $m_{\text{и.г}}$ – mass of inertial load; $M_{\mathrm{KA}}$ – spacecraft mass.

Equation (6) allows to calculate the required length of the gravitating device $l_{ш}$ for given requirements for the ratio of the moments of inertia of the spacecraft after the load is extended:

 

$X_B=l_{\text{ш}}+X_0=\sqrt{\frac{k_{YX}\cdot I_{X_0^{}}-I_{Y_0}}{1-\frac{m_{\text{и.г}}}{M_{\text{КА}}}}},$ (7)

where $k_{YX}$ – moment of inertia ratio coefficient (15–20), $k_{YX}=\frac{I_Y}{I_{X_0}}\ge 15$; $I_{X_0}$ – moment of inertia of the spacecraft along the X axis in the initial position.

Then the equation for finding the length of the gravitating device takes the form:

 

$l_{\text{ш}}=\sqrt{\frac{k_{YX}\cdot I_{X_0}-I_{Y_0}}{1-\frac{m_{\text{и.г}}}{M_{\text{КА}}}}}-X_0.$ (8)

However, this calculation does not consider the presence of a single rod or a system of several rods. Clarification is required on the estimated mass of the inertial load, which is a satellite with a balancing rod (a system of gravitational devices). To do this, it is necessary to know the type and material of the gravitating object [11–13].

Two variants for the design of a gravitational orientation system were considered [14; 15]. The first and most often used option is the use of a gravity telescopic rod (GTR) (Fig. 2, a). Such a device has the shape of a rod that removes parts of the satellite from each other by a certain distance. Due to the fact that the force of attraction of each part of the satellite depends on the distance to the center of the Earth, a moment is created that tends to align the bar along a straight line directed towards the center of the Earth (local vertical). The operating principle of the gravity orientation system is as follows. Initially, the GTR is assembled due to the tension of the wire on the coil of the stabilizing motor shaft. When the spacecraft is launched into orbit, the stabilizing engine, together with the coil, the GTR is set in motion. By lengthening the wire that was wound on the reel, the main shaft moves into the working position. As a result, the interaction of the permanent magnet in the GTR with the Earth's magnetic field is enhanced, thereby creating the necessary control torque. Thus, the GTR orients the satellite towards the center of the Earth. The effect of the gravitational moment can be reduced. To do this, it is necessary to rotate the shaft of the stabilizing motor in the other direction, while the wire is wound onto a reel, the length of the rod decreases, and therefore the effect of the gravitational moment decreases.

 

Рис. 2. Устройства гравитационной стабилизации: а – гравитационная телескопическая штанга; б – актуатор на основе желобчатой ленты

Fig. 2. Gravitational stabilization devices: а – gravity telescopic rod; b –the groved belt actuator

 

The second option is an actuator based on a grooved tape (Fig. 2, b). Due to the cross-sectional shape, the tape is flexible enough to be wound onto a drum and has the necessary margin of stability in the deployed state. This design consists of three tapes, two of which are auxiliary and smaller in size, in order to prevent deformation of the grooved tape from uneven thermal heating by solar rays [3].

The grooved belt actuator mounted on the satellite for gravity orientation works on the principle of using gravitational forces to control the orientation of the satellite. The operating principle of the actuator is as follows. Under the influence of gravitational force, the different mass of the grooved tape creates an uneven distribution of the gravitational moment relative to the axis of rotation of the satellite. This uneven gravitational moment causes the satellite to rotate until the gravitational forces become balanced. Thus, the grooved belt actuator uses gravitational forces to control the orientation of the satellite and ensures its stable and accurate orientation in space.

Gravity rod and grooved belt actuator have their own advantages and disadvantages, and the choice between them depends on the specific requirements and application conditions. Here are a few factors to consider when comparing these two actuators.

Simplicity of design. A gravity bar is typically simpler to construct and has fewer moving parts, which can make it easier to manufacture and maintain. Actuator grooved tape may be more complex in design and require more precise calibration.

Flexibility and precision. An actuator based on a grooved belt makes it possible to achieve more accurate and flexible adjustment of the satellite orientation, since the masses in different sections of the belt can be changed over a wide range. The gravity rod may have limited fine-tuning capabilities.

Efficiency. The gravity rod can consume less energy from the SS for its operation, since its operation is based on the use of gravitational forces. An actuator based on a grooved belt can consume more energy from the SS due to the need for belt movement and changes in distributed mass.

Reliability. Both actuators can be reliable enough, but a gravity rod may have fewer moving parts and therefore less damage and wear.

At the initial stages of design, we will use a gravity rod as a gravitating device due to its versatility and ease of manufacture.

In the folded position, the rod is attached to the upper and lower frames of the product so that the axis of the rod is parallel to the axis of the spacecraft. In the working position, the rod rotates perpendicular to the spacecraft axis around the axis of rotation of the hinge unit and is fixed in this position.

After the stage of separation from the launch vehicle, any spacecraft must carry out the process of calming, i.e., damping the initial angular velocities. As mentioned earlier, the role of a calming device is a permanent magnet (magnetic damper), which is automatically captured by the Earth’s magnetic field. The process of extending the bar should take place in the regions of the North Pole after ensuring the orientation of the longitudinal axis of the spacecraft by a gravitating object to the Earth.

Conclusion

When designing the SS, it is necessary to take into account the advantages of the proposed OSS design, since it allows to place more payload on the SS without overloading it with various instruments for the orientation and stabilization system. However, when designing, it is necessary to take into account the features of the proposed design of the rod (the mechanism for attaching two main modular cubes with the third one), its mass and dimensional characteristics, as well as the possibility of deformation of the rod during operation.

Sobre autores

Polina Еsina

Reshetnev Siberian State University of Science and Technology

Autor responsável pela correspondência
Email: polina_alex13@mail.ru

student

Rússia, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037

Vladimir Kornev

Reshetnev Siberian State University of Science and Technology

Email: 2604775@mail.ru

Cand. Sc., Associate Professor; Associate Professor of the Department of technical mechanics

Rússia, 31, Krasnoyarskii rabochii prospekt, Krasnoyarsk, 660037

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