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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Integrating the advantage of magnetic bearings with a double gimble control moment gyroscope (DGCMG), a magnetically suspended DGCMG (MSDGCMG) is an ideal actuator in high-precision, long life, and rapid maneuver attitude control systems. The work presented here mainly focuses on performance testing of a MSDGCMG independently developed by Beihang University, based on the single axis air bearing table. In this paper, taking into sufficient consideration to the moving-gimbal effects and the response bandwidth limit of the gimbal, a special MSDGCMG steering law is proposed subject to the limits of gimbal angle rate and angle acceleration. Finally, multiple experiments are carried out, with different MSDGCMG angular momenta as well as different desired attitude angles. The experimental results indicate that the MSDGCMG has a good gimbal angle rate and output torque tracking capabilities, and that the attitude stability with MSDGCMG as actuator is superior to 10^{−3}°/s. The MSDGCMG performance testing in this paper, carried out under moving-base condition, will offer a technique base for the future research and application of MSDGCMGs.

Because a control moment gyroscope (CMG) is capable of generating large control torques and storing large angular momentum over long periods of time, it is often favored for precision pointing and tracking control of agile spacecraft in low Earth orbit and momentum management of large space vehicles [

The manner of support of the high-rate rotor is a deciding factor in the comprehensive performance of a CMG [

In the 1960s, the US devoted itself to the research and development of DGCMGs. In 1973, three orthogonally mounted DGCMGs were firstly employed on NASA's Skylab as the main actuator [

In this paper, we focus our attention on the performance test of MSDGCMGs under the moving-base condition, such as gimbal angle rate and MSDGCMG output torque tracking capacity as well as attitude stability with an MSDGCMG as an attitude actuator. The remainder of this paper is organized as follows: Section 2 briefly introduces the mathematical model of the single axis air bearing table. In Section 3, a parameter segment control law is used for the rapid maneuver and quick stability subjected to the control torque saturation. In view of the moving-gimbal effect on the stability of the magnetic bearing and the response bandwidth limit of the MSDGCMG gimbal, Section 4 presents a special steering law subject to the constraints of MSDGCMG gimbal angle rate and acceleration. Section 5 gives a detailed description of the semi-physical simulation platform of the MSDGCMG performance testing experiments. In Section 6, experiments and the corresponding results analysis are given in detail. Finally, some conclusions are drawn in Section 7.

The objective of this section is to give a mathematical model of the single axis air bearing table for developing the attitude control law in the next section. The models are introduced as follows, including the dynamic model and the error quaternion kinematics differential equation.

In the experiment, the single axis air bearing table is utilized for simulating the rotational motion of a certain physical axis of the satellite, and according to the feature of large angle and rapid maneuver, the pitch axis (_{d}

Assuming that the “3-1-2” Euler angle rotation is adopted, the relationship between the attitude quaternion and Euler angle is described as follows:
_{0} denotes the scalar part of the quaternion, _{1} _{2} _{3}]^{T}

If the Euler angle _{c}

Then the error quaternion, which represents the error between the current attitude quaternion and the command attitude quaternion, is defined as:

Substituting (

The attitude control system is nonlinear. Simultaneously, there exists strong coupling among the air bearing table, high-rate magnetically suspended rotor and the gimbal of the MSDGCMG. Moreover, the single axis air bearing table is subject to disturbance torques and the saturation limit of the control torque, and the moment of inertia of the air bearing table is uncertain for the reason of the rotation of the inner and outer gimbal. Considering the characteristics of the attitude control system mentioned above, a parameter segment control law is used in this paper and the parameters of attitude control law are chosen according to the error-angle to realize rapid maneuvers and quick stability.

In the light of the dynamics and kinematics

For the purpose of rapid maneuvering and quick stability, the parameters _{e}, θ_{e}_{c}_{e}_{e}

Where _{e1}_{e2}

Taking account of the torque output capacity of the MSDGCMG, the control torque _{max}, then the command control torque is:

Due to the coupling between the air bearing table, MSDGCMG gimbal and high-rate rotor, the movement of the disturbed rotor becomes more complex, which aggravates the runout of the rotor and even endangers the system stability. What's more, the quicker the angle velocity of the air bearing table is, the more serious the influence on the high-rate magnetically suspended rotor becomes. The maximum angle velocity of the air bearing table is closely related to _{max}. Therefore, a smaller value than the maximum output torque of MSDGCMG is used as the saturation limit _{max} to ensure MSDGCMG steadiness in our experiments.

The command control torque is derived by means of the above mentioned attitude control algorithm, and then a steering law is presented in this section to calculate the command gimbal angle rate on the basis of the command control torque and the current gimbal angle. The chief points of this section include: the introduction of the MSDGCMG and the corresponding coordinate system, MSDGCMG torque equation and the detailed development of the steering law.

The MSDGCMG consists of inner and outer gimbal servo systems as well as a high-rate rotor system. The high-rate rotor is mounted in the stator housing and provides a constant angular momentum. With a specific rotation of inner and outer gimbal, the vector direction of the rotor angular momentum is changed to produce the desired gyroscope torques to meet the requirements of the attitude control. The main parameters of the MSDGCMG used in this paper are:

Angular momentum: ≥15 Nms.

Maximum output torque: ≥10 Nm.

Nominal rotor speed: 30,000 r/min.

Gimbal angle range: 0°∼360°.

Furthermore, the circuit box of MSDGCMG is used to control the inner and outer gimbal servo systems as well as the high-rate rotor system. Assume the inner gimbal axis, outer gimbal axis and rotor axis are perpendicular to each other at the initial time, then the MSDGCMG coordinate system is defined as in

In this paper, MSDGCMG is put on the single axis air bearing table. At the initial time, the _{0} is the angular momentum magnitude of the individual MSDGCMG,

Therefore, the MSDGCMG output torque vector _{g}^{T}

It is clearly seen from the above equation that, the Jacobian matrix is a nonlinear function of the gimbal angles

For the attitude control of the single axis air bearing table, particular attention should be paid to the MSDGCMG output torque along the pitch axis. Therefore

The command gimbal rate

For the reason that the high-rate magnetically suspended rotor of the MSDGCMG is actually active-control elastic supporting with certain stiffness and damping, the gimbal movement will disturb the magnetically suspended rotor. For simplicity, the phenomena caused by the gimbal movement are called the moving-gimbal effects. On the other hand, the response bandwidth of the CMG gimbal is constrained by the output torque ability of gimbal motor, thus the gimbal servo systems are unable to respond to the large gimbal angle acceleration. Consequently, the limit of the gimbal angle velocity and the acceleration should be given full consideration, to make sure that the high-speed rotor would work steadily around the operating point.

Furthermore, the singularity problem is inherent in the CMG. When the MSDGCMG is trapped in the singular state, the gimbal of the MSDGCMG will vibrate seriously so as to influence the stability of the MSDGCMG. Therefore, the maximum gimbal angle acceleration _{max} is determined on the basis of the singularity measurement:
^{T}

In the experiment, the maximum limit of the gimbal angle acceleration is firstly taken into consideration. The inner and outer gimbal angle acceleration _{k}_{k}_{k}_{−1} are the command gimbal angle rates at moment _{s} denotes the control cycle. Then the gimbal angle acceleration is constrained as follows:
_{∞} is the infinite norm of the vector, and then the command gimbal rate at the moment

Finally, the gimbal angle velocity is calculated as follows:
_{max} denotes the allowed maximum gimbal angle rate. Then the command gimbal angle rate
_{k}

Note that the maximum limit of the gimbal angle rate and the acceleration defined as above has the same direction as that of themselves before maximum limit. That is, the MSDGCMG output torque is kept the same direction as the command control torque, but the magnitude of the actual torque is reduced.

The design of semi-physical simulation platform for MSDGCMG performance testing is introduced in this section. The single axis air bearing table is used to simulate the pitch axis of the satellite. The MSDGCMG is mounted on the air bearing table and is used as an attitude actuator to realize rapid maneuvers and quick stability of the air bearing table.

The hardware used in the semi-physical simulation platform is mainly composed of the MSDGCMG, the corresponding circuit box, single axis air bearing table, control cabinet of the air bearing table, attitude control real-time simulation computer, simulation management computer, power module, and so on. The hardware layout frame of the semi-physical simulation system is depicted in

The signal flow of the hardware is described as follows: the rotation angle of the air bearing table is measured by the photo-electric encoder fixed to the single axis air bearing table, and then is delivered to the control cabinet through CAN bus. The control cabinet collects the rotation angle information, derives the angle velocity through differential filtering of the rotation angle, and then delivers the rotation angle and angle velocity to the real-time simulation computer through the RS232 serial port. According to the error-angle between the current rotation angle and the command angle, the command torque is generated through the parameter segment control algorithm. In light of the command torque and the current MSDGCMG gimbal angle, the command gimbal rate is computed and is sent to the MSDGCMG circuit box through the CAN bus. The circuit box receives the command and then controls the gimbal servo system to rotate the gimbals. Because of the change of the direction of CMG angle momentum, the desired control torque of the MSDGCMG is finally exerted on the air bearing table.

In this experiment, the single axis air bearing table is the supporting platform of the attitude control semi-physical simulation system and the main parameters are described as:

Loading capacity: >200 Kg.

Friction torque: <5 × 10^{−4} Nm.

Rotation angle range: ±360°.

Rotation angle measurement accuracy: ±10^{−3}°.

Rotation angle minimum resolution: 2.5 × 10^{−4}°.

The attitude control real-time simulation system is a hardware-in-loop semi-physical simulation, which supports the Matlab/Simulink graphical module design, seamless connection with C coder, and automatic production of code. The hardware of this system is mainly composed of the simulation management computer (host computer), real-time simulation computer (target computer) and signal processing box. The main function of host computer here is to manage the real-time target computer. The host computer is based on the Windows platform, and Matlab/Simulink is applied for building the simulation model. The target computer adopts the Vxworks real-time operating system to guarantee the reliability of the semi-physical simulation. The communication between the host computer and the target computer is by means of Ethernet networks. What's more, the major role of the signal process box is to provide the signal processing and connection to the I/O interface, and to ensure the data communication between the semi-physical simulation platform and the hardware equipment.

Having designed the attitude control law, the MSDGCMG steering law as well as the semi-physical simulation platform, experiments are conducted in this section to validate the performance of the MSDGCMG. The following mainly includes the measurement of the moment of inertia, the measurement of the disturbance torques, experimental results and analysis. The experimental scenario is given in

The measurement of the air bearing table moment of inertia is important in the initial stage of the semi-physical simulation. Here the twisting vibration approach [

During the measurement of the moment of inertia, it should be ensured that the vibration amplitude of the air bearing table is small to satisfy the condition of micro-amplitude. However, the equivalent torsional stiffness is difficult to measure, and the change of torsional stiffness resulting from the reinstallation brings about large error in the moment of inertia. In order to obtain the moment of inertia _{1} denotes the system vibration frequency without the balance mass, _{2} denotes the system vibration frequency when the air bearing table is installed with the balance masses, Δ_{0} and

Here the vibration period is calculated by counting the zero-crossing number of the air bearing table angle velocity within a specified time. Multiple measurements are carried out, and an average measurement value is used from ten attempts. The experimental data is shown in

In the experiment, the air bearing table is subjected to some external disturbance torques, such as gravity torque resulted from incomplete balancing, friction torque produced from air bearing, the resistance moment caused by communication cable and power supply cable, and so on. Taking account of the influence of the disturbance torques on the control accuracy and attitude stability, the disturbance torque _{d}

Through multiple measurements, the average value of the disturbance torque is measured as 1.09 × 10^{−4} Nm.

In the experiment, the parameter segment attitude control law developed in this paper is now integrated to the MSDGCMG steering law. The parameters shown in the preceding sections are properly selected as in

Experiments of the following three cases are carried out with different MSDGCMG rotor speed and different command rotation angles of the air bearing table. Case 1, as shown in

From the experimental results of the three cases, it is indicated that the larger the rotor speed (angular momentum of the MSDGCMG) is, the shorter the time required to realize the same angle maneuver is, meaning that the greater the maneuver ability is. It's easily known from the above figures that, the inner and outer gimbal angle rates follow the command perfectly, and that MSDGCMG output torque almost agrees with the command torque under the moving-base condition. Moreover, the attitude stability with the MSDGCMG as an actuator is superior to 10^{−3}°/s, and there exists no transient overshoot during the process of the rapid maneuver and quick stability.

It is clearly seen from the

In the future, advanced control algorithms [

In order to fully validate the performance of the MSDGCMG developed by Beihang University, experiments under the moving-base condition are carried out, with different MSDGCMG rotor speeds and different command rotation angles. Here a parameter segment control law is used for the air bearing table rapid maneuver and quick stability subject to the output torque saturation limit. Meanwhile, in view of the moving-gimbal effects on the stability of the magnetic bearing and the gimbal response bandwidth limit, a special steering law is presented subjected to the limit of MSDGCMG gimbal angle rate and acceleration. Experimental results illustrate that the attitude stability of attitude control system is superior to 10^{−3}°/s by using the MSDGCMG and that the MSDGCMG output torque follows the command torque well. Moreover, it is drawn from the experiments that the larger the angle momentum is, the greater the maneuver capacity is. However, when the MSDGCMG gimbal quickly revolves and the satellite rapidly maneuvers, the moving-gimbal effects become strikingly serious. Therefore, in the future research, not only the limit of gimbal angle rate and acceleration should be taken consideration, but also particular attention should be paid to the limit of attitude velocity when an attitude control law is designed. These are the difference between the application of a mechanically suspended DGCMG and that of the MSDGCMG. The MSDGCMG performance testing in this paper, carried out under moving-base conditions, will offer a technical base for the future research and application of MSDGCMGs.

This research has been supported by National Natural Science Foundation of China under grant 61121003, the National Basic Research Program (973 Program) of China under grant 2009CB72400101C, National Civil Aerospace Pre-research Project.

Schematic Diagram of the MSDGCMG Coordinate System.

Hardware Layout Frame of Semi-Physical Simulation System.

Experiment Scene of Semi-Physical Simulation.

MSDGCMG Rotor Speed 6,000 rpm (Case 1).

MSDGCMG Rotor Speed 15,000 rpm (Case 2).

MSDGCMG Rotor Speed 20,000 rpm (Case 3).

Experimental Data of the Moment of Inertia.

Vibration Amplitude (°) | 1.03 | 1.100 | 1.040 | 1.090 | 1.120 | —— |

Frequency (Hz) | 0.32 | 0.297 | 0.297 | 0.297 | 0.297 | —— |

Moment of Inertia (kgm^{2}) |
—— | 53.976 | 54.026 | 54.113 | 54.088 | 54.051 |

Parameters and Values of Experiment.

Parameter Segment Control Law | _{1} = 25, _{2} = 50, _{3} = 100 | |

_{1} = 1, _{2} = 2, _{3} = 3 | ||

0.5 | ||

0.001 | ||

Piecewise points of error-angle _{e} |
_{e}_{1} = 0.1°, _{e}_{2} = 0.01° | |

Saturation limit of control torque _{max} |
Rotor speed 6,000 rpm: 0.70 Nm | |

Rotor speed 15,000 rpm: 1.75 Nm | ||

Rotor speed 20,000 rpm: 2.30 Nm | ||

Control cycle _{5} |
0.05 s | |

MSDGCMG Steering Law | Maximum gimbal angle rate _{max} |
6°/s |

Maximum gimbal angle acceleration _{max} |
_{max1} = 5.73°/s^{2}, _{max2} = 30°/s^{2} | |

Piecewise points of singularity measurement |
0.05 |

Experimental Results of Three Cases.

Case 1 | 0.51 × 10^{−3} |
0.0035 | 18.35 | 1.63 | 4.16 |

Case 2 | 0.34 × 10^{−3} |
0.0011 | 18.85 | 3.18 | 8.88 |

Case 3 | 0.18 × 10^{−3} |
0.0012 | 16.80 | 3.57 | 9.49 |