1. Introduction
In performing space development in recent years, the on-orbit servicing system attracts attention. On-orbit services support spacecrafts by performing supply, maintenance, recover, repair, and assemble etc. on an orbit. By maintaining the spacecraft periodically, the spacecraft can be employed stably over a long time period. It can prevent the spacecraft from becoming debris by refueling, recovering and repairing the spacecraft whose control system become impossible by failure, fuel loss etc. By fixing and reusing the repaired spacecraft, the immense expense of the launch can be saved. In addition, by capturing the broken spacecraft and investigating, the cause of failure which was not found in the past may be found. It can increase the reliability of a following spacecraft. Many technologies are required for realization of the on-orbit servicing system, and rendezvous docking control is also one of the key technologies for realizing this. It is necessary to control long-distance transfer of spacecrafts especially in a geostationary orbit. In this research, the performance of orbit control method based on the C-W-Hill equations, which is often used in rendezvous docking control, is examined.
The simulation is performed by using the following three control systems, and the results are compared. (1) Orbit control method based on the C-W-Hill equations. (2) Orbit control method based on the C-W-Hill equations by the improved version. (3) The optimal control by the gradient method which assumed the bang-bang control. The features of these three control methods are explained.
(1) This control method is regarded as the optimal control in a short distance; however, if the distance becomes large, the error will increase. This control method is weak to disturbance, because it is a feed-forward control by using the thrust pattern obtained by off-line calculation beforehand.
(2) By this control method, the relative positions and the relative velocities are measured four times during the access. Errors are reduced by calculating the measured data on on-board.
(3) Errors can be farther reduced compared with two preceding methods; the calculation takes a larger time by performing orbital simulation iteratively.
Next, a graphic simulator is explained. In order to visualize a simulation, 3D graphic simulator shown in Fig.1 was developed in the last year. It was difficult to grasp various data simultaneously until then, since information was able to be acquired only by digital data. The situation of the simulation can be intuitively grasped by visualizing with this graphic simulator.
Fig.1 3D graphic simulator
2. Explanation of coordinate systems, symbols, and the derivation of equations of motion
Fig. 2 shows the coordinate systems used in this study. The origin of the inertial cartesian coordinates system is fixed to the center of the earth, axis is coincided with the earth axis, and axis are in the equatorial plane, where the direction of axis towards vernal equinox and axis is to make a right hand system.
Fig.2 Coordinate systems
The origin of the target coordinates system and the orbit coordinates system are both fixed to the mass center of the target, the direction of axis towards center of the earth, to the direction perpendicular to the orbital plane, and axis is to make a right hand system. In the case of a circular orbit, the moving direction of a satellite and the direction are coincided. The axis and axis are coincided, the axis and axis, the axis and axis are coincided.
Fig.3 Definition of the vectors
Fig. 3 shows some symbols used in this study. The vector from the center of earth to the chaser is , the vector from the center of earth to the target is , and the vector from the target to the chaser is . The velocity vector of the chaser is . Next, the orbital equations of motion are derived. Approximating the earth as an ellipsoid, the gravity potential is expressed as
(1)
where, : (=GM) gravity constant, : gravitational constant, : mass of the earth, : distance from a satellite to the center of the earth, : geocentric latitude, : associated Legendre function, : zonal coefficients. Note that the term of and are far smaller compared with the term of , and below the term of can be disregarded. Therefore, the equation (1) can be written as follows.
(2)
The equation (2) is expressed with inertial coordinates.
(3)
The equations of motion of a satellite with thrusts is as follows.
(4)
The above equation is expressed as
(5)
(6)
(7)
where, : the target inertial coordinates, : control force components, : disturbance force components, : mass of the satellite.
The thrusts , , and of thrusters in the orbital coordinate system are transformed into the inertial coordinate system by using three orbital elements shown in Fig.4.
Fig.4 Orbital element
The symbols show, : argument of latitude, : orbital inclination, : right ascension of the node, respectively. The thrust in the inertial coordinate system are expressed as
(8)
where expresses , and expresses .
Next, the equations of Hill are derived. The vector from the center of the earth to the chaser is expressed as follows
(9)
The velocity vector of the chaser is expressed as
(10)
The acceleration vector of the chaser is expressed as
(11)
Considering the force balance, from the equations (9), (10), (11), the following equation is obtained.
(12)
Transforming the equation (12) to the orbital coordinate system, then the equations of Hill is expressed as follows.
(13)
The orbital angular velocity is expressed as
(14)
where : gravity acceleration on the equator, : radius of the equator, : distance between the target and the center of the earth, respectively. Since the simulation is performed on the 400km orbit in this study, orbital angular velocity is
(15)
The data of the satellite used in this simulation is shown in table.1.
Table.1 Data of the satellite
Mass [kg] 5500
Fuel [kg] 4000
Inertia moment [kg・ ]
Ixx = 6786
Iyy = 4721
Izz = 5425
Ixy = 0
Iyz = 0
Izx = 0
Thrust force [N](all directions) 400
Specific thrust [s] 313
3. Guidance Control Method
First, the orbit control method based on the C-W-Hill equations is explained. When thrusts of the chaser are constant, the equations of Hill are expressed as follows,
(16)
The terms , ,and express the acceleration of each direction, and obtained by
(17)
The solution of (16) is,
(18)
(19)
(20)
By differentiating above equations,
(21)
(22)
(23)
The thrust patterns of the thrusters are determined by using equations (18)〜(23), which patterns become in general like shown in Fig.5.
Fig.5 Thrust patterns
In this figure, the horizontal axis is the time, and the vertical axis is the magnitude of the thrust. At first, the chaser produces thrust in the direction, in order to approach the target. As the velocity of the chaser is increased, the chaser is pushed out of a circular orbit by the centrifugal force. In order to cancel it, the thrust is simultaneously produced in the direction. When the chaser approaches to the target, in order to slow down, the thrust is produced in the direction. At this time, the thrust is produced in the direction for returning to the original orbit.
Next, the thrust pattern determination algorithm by the equations of C-W-Hill is explained. First, the initial values of the thrust patterns are assumed, and calculation is performed following to the equations of C-W-Hill. The position errors and the velocity errors in terminal time are calculated. The thrust patterns are corrected by the deviation from the desired value and applying the proper gain. The simulation is performed again by using the corrected thrust patterns. Finally the thrust pattern is obtained by repeating this calculation until the errors are reduced to within the values we required.
Next, the outline of the simulation is explained. The thrust pattern, which is obtained by the preceding algorithm, is inputted into the orbital simulator based on the orbital equations. The simulation is performed and position errors and velocity errors are calculated by comparing data of the chaser with data of final desired point in terminal time. The final desired point is 200m behind from the target in this study.
Fig.6 Outline of the simulation on the orbital coordinate
In the orbital coordinates, the outline of this simulation corresponds to the phase 1 of Fig.6. If the position error is small in the phase 1, the error can be cancelled in the remaining phase 2 and phase 3.
Fig.7 Outline of the simulation
The outline of the simulation is shown in Fig.7. In this study, the simulation is performed for 12 cases of initial relative distance 1[km], 2 [km], 3 [km], 5 [km], 10 [km], 15 [km], 20 [km], 25 [km], 30 [km], 35 [km], 40 [km], and 45[km]. Fig. 8 and Fig. 9 show the position errors and velocity errors in relation to the initial relative distance, respectively.
Fig.8 Result of position errors
Fig.9 Result of velocity errors
The errors in the direction are 0, because the simulation is performed in the orbital plane and the thrust is not produced in the direction. If it is necessary to control the direction, it is easily performed because the direction is controlled independently of the and . From Fig.8, we see that the position errors become larger, as initial relative distance becomes larger. The velocity error of the direction is decreased, when the initial relative distance beyond about 40km. The absolute velocity error become larger, as initial relative distance becomes larger. If the position error increases above 200m, the chaser and the target may collide, therefore, it is difficult to use this method.
Second, the orbit control method based on the C-W-Hill equations by the improved version is explained.
By this method, first, the data of the orbital simulation is stored in the time just before the thrust pattern changes. Next, the thrust pattern is re-calculated again by employing the stored data as initial values. By updating the data, even if the simulation is performed from a fairly large distance, the errors do not become large.
In this case, the simulation is performed for initial relative distance is 30km. Because the thrust pattern changes in 234.390[s], 344.076[s], 1052.352[s], and 1167.677[s], therefore the update timing are conducted at 200[s], 300[s], 1000[s], and 1150[s].
The result is as follows.
Table.2 Comparison of a result
Normal version Improved version
dx。[m] -101.522 -6.087
dz。[m] -248.377 -20.518
dvx。[m/s] 0.072 0.005
dvz。[m/s] 0.174 -0.238
It turns out that the position error is reduced considerably compared with the normal version. The velocity errors increase slightly, however, they are in our acceptable range. As mentioned above, it is possible to employ the orbit control method based on the equations of C-W-Hill from a large distance by using this improved method.
Finally, the gradient method is explained that assumed the control is bang-bang type. By the gradient method, first, the thruster employment timing is changed slightly and the orbital simulation is performed. Next, according to the quantity of change of the performance function, the timing is changed in the direction that the performance function becomes better in proportion to the gradient. The performance function is expressed as follows
(24)
where, expresses the position error of the direction, expresses that of the direction, expresses the velocity error of the direction, expresses that of the direction, , , and express the weight of each direction. Weights of the position errors are larger, because we weighted on to reduce the position error, rather than the velocity error.
Table.3 Comparison of a result
Normal version Gradient method
dx。[m] -101.522 0.000
dz。[m] -248.377 3.418
dvx。[m/s] 0.072 0.033
dvz。[m/s] 0.174 -0.280
The result is shown in Table.3. The position errors are farther reduced. Velocity errors are increased slightly, however, it is in an acceptable range. As mentioned above, when using the gradient method which assumed the bang-bang control, it turned out that a better result can be obtained.
4. Result and Conclusion
As shown in Fig.10, it turns out that the position errors is reduced by the method of the improved version and by the gradient method than by the method of the normal version. As shown in Fig.11, it turns out that the velocity errors are somewhat larger in the improved version and the gradient method than normal version. As mentioned above, the error can be cancelled in the remaining phase 2 and phase 3.
Fig.10 Conclusion of a position errors
Fig.11 Conclusion of a velocity errors
Table.4 Burnout fuel
Method Burnout fuel [kg]
Normal version 114.602
Improve version 114.301
Gradient method 115.042
Fuel consumption are shown in table.4. It turned out that the differences are very small.
As mentioned above, in a long-distance rendezvous docking indispensable to the on-orbit servicing system, if the method of the improve version or the gradient method is used, the rendezvous docking guidance control can be performed in good accuracy.
Reference
[1]Fumiaki Imado, “A study on the Spacecraft Nonlinear Optimal Orbit Control with Low Thrusts”, Nonlinear Problems in Aviation & Aerospace 1:287-296, 1999.
[2]Shinichi Kimura, “Orbital Maintenance System”, the Institute of Electronics, Information and Communication Engineers Vol.82 No.8, 1999.
[3]Kinshi Wada, “On board Control of HOPE which combined computer navigation and proportionality navigation” ,Shinshu University,1999.
[4]Keiji Fujii, “A study on rendezvous docking control”, Shinshu University, 2003.