A generic model of a nanosatellite attitude control and stabilization system was developed on the basis of magnetorquers and reaction wheels, which are controlled by PID controllers with selectable gains. This approach allows using the same architectures of control algorithms (and software) for several satellites and adjusting them to a particular mission by parameter variation. The approach is illustrated by controlling a satellite attitude in three modes of operation: detumbling after separation from the launcher, nominal operation when the satellite attitude is subjected to small or moderate disturbances, and momentum unloading after any reaction wheel saturation. The generic control algorithms adjusted to each mode of operation were implemented in a complete attitude control system. The control system model was embedded into a comprehensive simulation model of satellite flight. The simulation results proved the efficiency of the proposed approach.

The advancements in miniaturization of sensors and mechanical devices and computing hardware and software lead to enhanced performance of small satellites. The novel concepts of cooperating satellites flying in a formation give prospects for a further increase of the possibility of providing services similar to those offered now by large and expensive spacecraft and also of offering new services or functionalities [

Satellite control torques may be generated in a passive or active way. Passive attitude control does not consume power but may be applied only for dumping of unwanted satellite motion resulting from external disturbances [

After separation from a launcher, the satellite may tumble, i.e., spin with a high angular rate, which [

The nominal control laws provide stabilization counteracting external disturbances, which, even when small (like gravity-gradient) but acting for a long time, may drift a satellite away from its required attitude. Another control task in the nominal mode is satellite rotational maneuvers. As primary actuators, reaction wheels are commonly used in the nominal mode, as they may assure a high attitude accuracy and force a relatively quick response (although slower than thrusters). To control reaction wheels, proportional-derivative (PD) controllers are used [

In such a case, reaction wheels require diminishing their momentum. For small satellites in low Earth orbits (LEO), magnetorquers are used to unload momentum, as cheap, reliable, and effective torque-generating devices [

For each mode of operation, the signals from the control law block should be passed as input to particular actuators. It may be achieved in various ways, for instance, using one type of actuators only (reaction wheels in the nominal mode or magnetorquers in the detumbling mode) and then minimizing the control effort [

The main objective of the study was to develop an attitude control system to be implemented on board CubeSats for prospective formation flying. The ease of applicability to similar satellite sizes was the main requirement to diminish the cost of control system development and tuning. This explains why the traditional PID approach was implemented with focus on flexibility of implementation to similar satellites and scalability for a number of actuators. To evaluate feasibility of the developed concept, a comprehensive WUT simulation model of a satellite flight was supplemented by a control system model, allowing evaluation of the control quality. The paper will describe the satellite simulation model, and then next, the generic structure of the control system will be presented. The simulation results will confirm the efficiency of the proposed control system and algorithms.

The satellite simulation software developed at the Warsaw University of Technology (Figure

Satellite simulation model architecture.

A satellite bus is modeled as a rigid body with six degrees of freedom and time-varying mass and inertia. The dynamic equations of motion are derived in the Body Frame Fixed (BFF) coordinate system (Figure

Coordinate system applied.

The rigid spacecraft dynamic equations of motion are described as

The inertia matrix

The

The satellite kinematic equations for position and attitude have the form

Matrix

It is assumed that on a satellite,

A reaction wheel is modelled as a flywheel fixed to the satellite body spinning with angular velocity

A torque acting on the satellite due to a single reaction wheel (in the BFF frame) is calculated as

Angular velocity

The reaction wheel is driven by an electric motor, the dynamics of which in general is described by combined electromechanical equations. As control signals, COTS (commercial off-the-shelf) reaction wheels use current, torque, or angular velocity, and their control electronics converts them to the input voltage of the motor. In this study, it is assumed that the armature current

The torque from a magnetorquer is a crossproduct of the magnetic dipole generated by a coil and the Earth magnetic field vector:

A magnetic dipole is calculated as

The general architecture of a satellite attitude control system investigated in this study is shown in Figure

General architecture of the attitude control system.

The differences between the required and current states are inputs to the satellite attitude control system (ACS), composed of two blocks denominated in Figure

For both control blocks, full PID control is envisaged and implemented in the algorithm.

The applied PID control is illustrated in Figure

Attitude control system modes of operation.

The detumbling mode is activated after separation of the satellite from the launch vehicle or at any time when the satellite’s angular velocity is too large. The satellite angular velocity is reduced by the attitude controller, and the momentum management controller is inactive. In the detumbling mode, only magnetorquers are used for spacecraft stabilization, and PID is reduced to differential control related to the actual value of satellite angular velocity

The gain matrix

When the specified low angular velocity is achieved, the attitude control system is switched from the detumbling mode to the more precise nominal mode.

The nominal mode is used to maintain or change satellite attitude as required with adequate accuracy, despite external disturbances. The control is performed by reaction wheels in a full PID form

In (

The derivative of attitude error

The gain matrices

The momentum unloading mode is used to prevent reaction wheel saturation by reducing their angular velocities. In the momentum unloading mode, reaction wheels are used for satellite attitude control, and simultaneously, magnetorquers are switched on for reaction wheel desaturation. The mode is activated when the velocity of at least one reaction wheel exceeds its design limit. However, for better mission performance, the momentum unloading system may be switched on from time to time to reduce accumulating wheel rotating velocity.

In the momentum unloading mode, the control of reaction wheels should assure satellite attitude control and diminish the spin of reaction wheels, so the total reaction wheel control

As the integral term in PID might slow down the control reaction, for satellite attitude control

To decelerate angular velocities of reaction wheels to a specified value

The magnetorquers are also controlled by the D controller:

Proper selection of control gains

To investigate the efficiency of the developed attitude control system, a simulation study was done for the generic satellite mass data given in Table

Satellite mass properties.

Description | Value | Unit |
---|---|---|

Mass | 6.2 | [kg] |

Moments of inertia: | 0.0756, 0.0763, 0.0209 | [kg·m^{2}] |

Products of inertia: | -0.0002, 0.0020, -0.0019 | [kg·m^{2}] |

Location of center of mass relative to geometrical center ( | 0.0034, -0.0036, -0.0007 | [m] |

Dimensions | [m] |

The initial satellite orbit elements are provided in Table

Initial satellite orbital parameters.

Description | Value | Unit |
---|---|---|

Semimajor axis | 6371 000 | [m] |

Eccentricity | 0 | [-] |

Longitude of ascending node | 331.7739 | [°] |

Orbit inclination | 51.6393 | [°] |

Argument of perigee | 0 | [°] |

True anomaly | 20.5285 | [°] |

Orbital period | 5500 | s |

Three configurations of reaction wheels: orthogonal, pyramidal, and tetrahedral (Figure

Reaction wheel configurations: (a) orthogonal, (b) pyramidal, (c) tetrahedral, and (d) skew.

The reaction wheel parameters used in simulations are based on Sinclair RW-0.03 data [

RW data.

Description | Value | Unit |
---|---|---|

Moment of inertia about spin axis | 51.16 | [kg·mm^{2}] |

Armature current | 0.35 | [A] |

Viscous friction | 3.837 | [kg·mm^{2}/s] |

Motor torque constant | 51.6393 | [°] |

The major parts of simulations were performed for the orthogonal configuration of reaction wheels as the most commonly used one [

The matrices

The set of magnetorquers contains three devices in an orthogonal configuration with nominal dipoles as in [

For the detumbling mode, the constant part of the gain matrix was calculated purely based on the satellite inertia and orbital parameters as per equation (

Initial attitudes for all simulated cases of yaw, pitch, and roll angle values were zero, i.e.,

The simulation rationale for all cases was to check the applicability of the algorithms developed. In particular, for the detumbling mode, the ability to suppress the satellite rotation was verified. For the normal operating mode, the ability to maintain or change the satellite attitude was investigated, for various angular velocities and configurations of reaction wheels. For the momentum unloading case, the ability to desaturate high rates of reaction wheels was verified.

The detumbling mode of the ACS operation was simulated to verify the effectiveness of reducing the satellite’s high value of 10°/s per axis of initial angular velocity, which may appear after separation from the launcher. The magnetorquers were actuators used in this mode. The time profiles of the satellite angular velocity module and magnetorquer dipole moments module are presented in Figures

(a) Satellite rate reduction in the detumbling mode; (b) closeup.

The satellite angular velocity was reduced below 0.5°/s after 3200 s and below 0.2°/s after 6000 s, i.e., in time about 9% longer than the orbital period which was 5500 s. The result is satisfying, in terms, for instance, of the requirement formulated in [

As indicated by other simulations, using the detumbling mode controller did not reduce the satellite rate below 0.2°/s. To cross this threshold, the ACS nominal operation mode should be used.

In the simulations of the nominal mode, only reaction wheels were active. The quantities monitored for this mode are satellite Euler angles:

Nominal mode simulation cases.

Case | Initial satellite rates | Commanded attitude | Initial RW spin | RW configuration |
---|---|---|---|---|

1 | 0.5°/s | 0° | 0 | Orthogonal 3 wheels |

2 | 0 | 0°, 0°, 0° | 0, 2000, 4000 rpm | Orthogonal 3 wheels |

3 | 0 | 0°, 0°, 0° | 0 rpm | Orthogonal 3 wheels |

4 | 0 | 120°, 60°, 150° | 0 rpm | Pyramid 4 wheels |

Tetrahedron 4 wheels |

The simulation of case 1 illustrates the transition from detumbling to nominal mode. The initial angular velocity vector is assumed to be 0.5°/s per axis, which is the residual of the detumbling mode. The requirement was to keep the zero Euler angles, i.e.,

At the beginning of the control, there was an increase in the reaction wheel angular rate, but after 10 seconds (Figure

(a) Stabilization in the nominal mode, case 1; (b) closeup.

The simulation case 2 was performed to investigate attitude control system performance for different initial reaction wheel rates. In this case, the initial attitude was

In case 2 for initial RW rates 0 and 2000 rpm, the performance is acceptable. The high initial reaction wheel rate such as 4000 rpm does not allow achieving a large commanded angle of rotation, which may be the result of reaction wheel saturation, as the second reaction wheel reached the maximal angular velocity of 6200 rpm (Figure

Nominal mode, case 2.

The case 3 simulation was performed to check the ACS performance for the cases similar for a time-varying nadir commanded attitude. In the first 100 seconds, the commanded attitude was

(a) Nominal mode, case 3; (b) closeup.

In this case, the control system response is similar to that in case 2. The system has a better performance for lower initial reaction wheel rates, as the commanded attitude was reached more quickly and with better accuracy. For initial rates of 4000 rpm, the commanded attitude is not achieved and a significant pointing error occurs. It may be a symptom of reaction wheel saturation, as a result of summing up the angular velocities of reaction wheels and satellite, which makes the controller ineffective.

The simulation case 4 was performed to verify whether the developed control laws may be also used for reaction wheel configurations other than the orthogonal one. Initial satellite angular velocity vector was 0°/s per axis. The pyramid and tetrahedron configurations of four reaction wheels were investigated. In control gain calculations, the configurations differ by matrices

(a) Nominal mode, case 4; (b) closeup.

For both configurations, the maneuver was completed after about 15 seconds. The satellite motion was different for each configuration, but the final results were similar. The overshoot for pyramid configuration was larger on the pitch axis than for the tetrahedron configuration, but on the roll axis, an opposite tendency occurred.

The systems using 4 reaction wheels reach the commanded attitude significantly earlier with smaller overshoot than the one with 3 reaction wheels (case no. 3), as using more reaction wheels allows to generate a higher torque on the satellite. The performance of reaction wheel arrays is different; however, in both cases, the maximal speed achieved by any reaction wheel was similar (above 2000 rpm for the 1^{st} wheel of the pyramid configuration and for the 2^{nd} wheel of the tetrahedron configuration).

The results presented in Figure

In this mode of ACS operation, both reaction wheels and magnetorquers are active. To verify the effectiveness of diminishing reaction wheel angular velocities, a desaturation process was performed for initial velocities of reaction wheels equal to 6000 rpm which is near their maximal value of operation—6200 rpm. Although it is a rather unrealistic value in practice, this case was studied to check the efficiency of the control. Simulations were done for orthogonal and pyramid configurations of reaction wheels.

For the orthogonal configuration of three reaction wheels (Figures

(a) Momentum unloading mode, reaction wheels in orthogonal configuration. (b) Closeup.

To verify whether it is possible to apply control laws developed during this study to more than three reaction wheels, the simulation for a set of 4 reaction wheels in the pyramid configuration was performed.

The desaturation process was successful for this case too (Figures

(a) Momentum unloading mode, reaction wheels in pyramid configuration. (b) Closeup.

The objective of the simulation study was to verify whether the attitude control system developed on the basis of linear control may be applied to the nonlinear satellite model. Three modes of operations were simulated, and the results of the control are satisfactory.

In the detumbling mode, the satellite angular velocity was reduced even for very high initial angular velocities, such as 10°/s per axis. The other simulation results indicated that using only magnetorquers for high accuracy pointing in the presence of external torque acting on the satellite may not be effective, which suggests that reaction wheels should be used in the final tuning of satellite attitude.

The simulations for the nominal mode of operation indicated that the control algorithm, applied in this mode, provides sufficient attitude control. The control algorithm developed in this study may be applied to various configurations of reaction wheels, which was illustrated by simulation results for orthogonal, pyramid, and tetrahedron reaction wheel configurations. The study revealed limits of using reaction wheels at high angular rates. As varying external loads acting on a satellite results in increase of reaction wheel angular velocities, the momentum unloading mode is necessary to maintain the required performance of reaction wheel actuation during a mission.

The simulations of the momentum unloading mode confirmed that the developed control algorithm diminishes angular velocities of reaction wheels in their various configurations. The desaturation process has a significant impact on satellite attitude control and indicates that in the momentum unloading mode, a satellite should not be commanded to perform tasks which require high accuracy pointing.

An attitude control system was developed which fulfils two main tasks: provides a satellite with sufficient attitude control capabilities in the detumbling and normal modes of operation and ensures adequate performance of control actuators by the momentum unloading control process. The actuators of the control system are magnetorquers and reaction wheels. The software architecture allows scalability of the system to various numbers and configurations of actuators as well as changing PID regulator parameters, adjusting to a particular case. The model of the attitude control system was integrated with the satellite simulation software developed at the Warsaw University of Technology. The results of simulations of the control system operation are satisfying. The model may be a useful tool for development of an attitude control system to be operating in real time and embedded into nanosatellite hardware. Also, the simulation results presented in the paper may be useful for similar system development. In further research, the software is intended to be used for other missions and other control strategies.

Bold

Satellite inertia matrices

Viscous friction coefficient of the RW motor shaft

Magnetic field vector

The Earth magnetic field in a magnetorquer coordinate system

Transformation matrix from coordinate system

Nominal magnitude of magnetic dipole generated by one magnetorquer

Magnetic dipole generated by the magnetorquer coil,

Diagonal matrix composed of magnetorquers nominal dipole moments

Force generated by (

Satellite loads

Matrix representing actuator configuration

Angular momentum of the reaction wheel

Armature current of the reaction wheel motor

Satellite moments of inertia

Satellite products of inertia

Moment of inertia of the reaction wheel about its spin axis

Satellite moments of inertia matrix

Diagonal matrix containing the moments of inertia of reaction wheels

Torque constant of the RW motor

Gain matrices: proportional, derivative, and integral

Torque

Satellite mass

Total number of actuators

Number of devices

Quaternion of a satellite attitude in the inertial frame

Matrix of angular velocities in attitude equation

Satellite position in the inertial coordinate system (ECI),

Satellite yaw rate

Diagonal matrix containing reaction wheel motor constants,

Static mass moments of the satellite

Time

Time constant of the integral controller

Satellite orbital period

Actuator control signals

Satellite linear velocity in the satellite body frame,

Satellite state vector

Specified control dumping ratio

Attitude error vector

Specified linear control bandwidth

Satellite angular velocity in the satellite body frame,

Satellite angular velocity in the actuator coordinate system

Spin velocity of a single reaction wheel

Satellite linear and angular velocity matrix

Reaction wheel angular velocities.

Reflects the device or coordinate system

Aerodynamics

Control

Commanded

Difference between the commanded and current values

In the satellite body coordinate system

Gravity

Magnetic field (acting on the bus)

Magnetorquers

Reaction wheels

Solar radiation pressure

Propulsion (not used in this study).

Matrix pseudoinverse

Matrix from vectors of cross-product

Vector, matrix transpose

Differentiation in time.

The data described in the article is a result of numerical simulation. Although the data itself is not available, a description of how it was obtained is provided. The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there is no conflict of interest.

This research was supported by the European Commission’s Horizon 2020 Program [grant number 687490].