Most of the satellite missions require orbital maneuvers to accomplish its goals. An orbital maneuver is an operation where the orbit of a satellite is changed, usually applying a type of propulsion. The maneuvers may have several purposes, such as the transfer of a satellite to its final orbit, the interception of another spacecraft, or the adjustment of the orbit to compensate the shifts caused by external forces. In this situation it is essential to minimize the fuel consumption to allow a greater number of maneuvers to be performed, and thus the lifetime of the satellite can be extended. There are several papers and studies which aim at the fuel minimization in maneuvers performed by space vehicles. In this context, this paper has two goals: (i) to develop an algorithm capable of finding optimal trajectories with continuous thrust that can fit different types of missions and constraints at the same time and (ii) to study the performance of two propulsion devices for orbital maneuvers under development at the Universidade de Brasilia, including a study of the effects of the errors in magnitude of these new devices.
The present paper is concerned with the optimization of spacecraft trajectories with minimum fuel consumption that uses a low thrust as a propulsion system. This type of propulsion system is the most economical type available in aerospace technology. It assumes that a force with a low magnitude is applied during a certain time to change the trajectory of the spacecraft.
This type of maneuver has been studied in the literature by several researches with different goals. Lawden [
After that, several researches were performed on this topic. A series of analytical methods considering the power-limited (PL) low-thrust transfer optimization using the two-body dynamics started with Beletsky and Egorov [
Another important application that uses the low-thrust propulsion system is the problem of station keeping of satellites. Any satellite moves away from its position due to perturbations or to the launch errors. So, station-keeping maneuvers are required to keep a satellite in its desired position. Geostationary satellites and constellations of satellites are good examples where this type of maneuver is required. Papers studying this problem are found in Boucher [
Applications of transfers from the Earth to the Moon can also be studied under this model, like done by Pierson and Kluever [
The first contribution of the present paper is a formulation and description of an algorithm developed to solve this problem, which has many important features to increase the number of maneuvers that get convergence and to increase the velocity of convergence in those situations. This problem is very sensible to small variation of parameters, so several actions are required to increase the efficiency of the convergence. This algorithm is based in Biggs [
The conditions imposed on the final orbit must be at least in one of its Keplerian elements, but can be in more elements, depending on the goals of the mission. The algorithm also allows the consideration of restrictions, like forbidding thrusting in certain parts of the orbits, specified by the interval between the true longitudes. This restriction implies that both the start and the stop of the thrust have to occur inside this interval. This is an important characteristic of the algorithm, because it is possible to avoid turning on the propulsion system when the satellite is not under direct observation from the ground. In addition, restrictions in the direction of the thrust can also be considered. This is important in order to accommodate propulsion systems that have limits in the pointing of the thrust. The present algorithm can also perform maneuvers with several arcs of propulsion, so it is possible to accommodate constraints of forbidden regions of burn that repeat along the orbit, like the restriction of turning off the propulsion system when the satellite is not under tracking from the ground. This is one of the most important and desired characteristics of the algorithm, because the operation of turning on the engine when the satellite is not under tracking is very risk, in particular when a new propulsion system is used. These constraints and considerations make the algorithm more complex but capable of finding optimal trajectories with realistic constraints. Another important point is the use of nonsingular variables, so equatorial and circular orbits can be used for the satellite.
Regarding the propulsion system, it is assumed that the thrust has a constant magnitude of any level, from very low to very high; its direction can be fully controlled, and it can be turned on and off at any time. In this way, to find the optimal maneuver, it is necessary to find the time to start and to stop the engine and the direction of the thrust at every instant of time, specified by the values of the pitch and yaw angles. So, it is applicable to transfers of large amplitudes or station-keeping maneuvers. It is also possible to include coasting arcs, which gives even more flexibility to the method.
The second task of this work is to simulate the optimal maneuver found by the algorithm described previously in a realistic system that can consider errors in the propulsion device with embedded closed loop PID (proportional, integral, and derivative) controller. This task is very important because one of the goals of the present paper is to test the performance of a new propulsion system under development at the Universidade of Brasilia, and the effects of its errors have to be considered.
The propulsion system developed at the Plasma Laboratory of the Universidade de Brasilia (Ferreira et al. [
The electric propulsion is already recognized as a successful technology for long-duration space missions (Ferreira et al. [
In this way, the present paper has the intention of showing an algorithm that is very flexible and realistic regarding constraints and other specific conditions and then using this algorithm to verify the performance of a new propulsion system that is under development, taking into account errors in magnitude of the thrust.
First of all it is necessary to introduce the notation for the orbital elements used in this paper and other important definitions. The notation is
The range angle “
Definition of the range angle and the arbitrary reference line.
There are also two important angles, pitch and yaw, which are used to define the direction of the thrust. The angle of pitch is designated by
The advantage of using these state equations is to avoid singularities if the inclination or the eccentricity is zero. The equations of motion and the control variables used here are shown in [
Some of the initial variables that should be guessed are the initial adjoint multipliers
The final complete algorithm has the following steps [ It starts from an initial guess for the set of variables ( The “adjoint-control” transformation is used to obtain the initial values of the Lagrange multipliers required for the numerical integrations. Now, with all the initial values needed to solve the problem, the equations of motion and the adjoint equations are simultaneously numerically integrated during the propulsion arc. The values of the pitch and yaw angles are obtained, at every step of the integration, by the principle of maximum of Pontryagin. At the end, the value of the final state The Keplerian elements of the orbit achieved are then obtained from the final state At this point, the satisfaction of the constraints is verified. If the magnitude of the vector that represents the constraints satisfaction is smaller than a specified tolerance provided by the user (the numerical zero), the algorithm proceeds to step (vii). On the other hand, if it is bigger, the algorithm proceeds to step (vi). At this step, the gradient of the constraints equations ( Once the constraints satisfaction is achieved, the next step is to search for the minimum of the fuel consumed. The direction of the search At this step, the magnitude of the vector In order to get better results, the initial data “ratio of contraction” At this step, the possibility of the control The current step is verified and, if it is not the last one, the algorithm goes to a new search for the minimum fuel consumption with a smaller direction search tolerance for the objective function. If this step is the last one, the algorithm is finalized.
There are several extensions of the algorithm that can be considered, in order to make it more flexible. The first one is the possibility to perform maneuvers with more than one arc of propulsion. Increasing the number of propulsion arcs means that the number of optimization variables is increased. For example, for one thrust arc there are
Another extension is the possibility to consider forbidden regions to apply the propulsion as a constraint. These regions are defined by the true longitude, as shown in (
In some situations, the initial range angles
The last extension is the possibility to consider constraints restricting the pitch and yaw angles. In this case the bounds considered on
As defined by Jahn [
The hall thruster, used in this work as the propulsion system, was fundamentally envisaged in Russia in 1960, and the first successful space mission with this thrust propulsion with 60 mN was used in the Meteor satellite series in 1972 (Zhurin et al. [
The working principle of the hall thrusters is the use of an electromagnet to produce the main magnetic field responsible for the plasma acceleration and generation (Ferreira et al. [
In the Plasma Laboratory of the Universidade de Brasília, it has been developed two kinds of hall thruster, called Phall I and Phall II. The first one to be constructed was the Phall I. It has a stainless steel chamber with 30 centimeters of diameter. Its magnetic field is produced by two concentric cylindrical arrays of ferrite permanent magnets: 32 bars in the outer shell and 10 bars in the inner shell. The magnetic field in the middle line of the plasma source’s channel is 200 Gauss (Ferreira et al. [
Comparative performance parameters measured in PHALL I and PHALL II.
PHALL1 | PHALL2 | |
---|---|---|
Maximum expected thrust (mN) | 126 | 126 |
Average measured thrust (mN) | 84,9 | 120,0 |
Thrust density (N/m²) | 4,68 | <6,0 |
Maximum specific impulse (s) | 1607 | 1607 |
Measured specific impulse (s) | 1083 | 1600 |
Ionized mass ratio (%) | 3,3 | ~30 |
Propellant consumption (Kg/s) |
|
|
Energy consumption (W) | 350 | 250–350 |
Electrical efficiency (%) | 33,9 | 60 |
Total efficiency (%) | 10,12 | 50,0 |
In this paper, it was found first the optimal maneuvers with the thrust parameters shown in Table
The maneuver simulator used to simulate the optimal maneuvers in a more realistic way is the spacecraft trajectory simulator (STRS). The STRS can consider constructional features and operation, such as nonlinearities, failures, errors, and external and internal disturbances, in order to determine the deviation in the reproduction of the optimal solution previously found.
Furthermore, maneuvers with continuous thrust are used during a long period of time. This kind of maneuvers requires a PID control system to guarantee the achievement of the final conditions of the maneuver. The operating structure of the STRS can be described as follows (Rocco [ The simulation occurs in a discrete way, that is, at every simulation step, the state of the vehicle (position and velocity) must be computed. For all purposes, the disturbances, nonlinearities, errors, and failures must be considered. Since a closed loop system was used to control the trajectory, the reference state is determined by applying the optimal maneuver. Then, the current state follows the reference but considers the effects of disturbances and the constructive characteristics of the vehicle. The difference between the reference state and the current state creates an error signal that is inserted into a PID control. The controller produces a control signal according to the PID control law and the gains that have been set. The control signal is inserted into the actuator model (propellant thrusters), where the nonlinearities inherent to the construction of the actuator can be considered. Thus, the behavior of the propellant thrusters can be reproduced by the appropriate adjustment of the model parameters, which are supplied by the manufacturer of the thrust actuator. Finally, an actuation signal With the actuation signal added with the possible disturbances, the model of the orbital dynamics provides the state (position and velocity) after the application of the propulsion. Through the use of sensors that were modeled considering its constructive aspects, the current state is determined and compared with the reference state in order to close the control loop.
Before performing tests with the propulsion system Phalls I and II, that is one of the goals of the present paper, it is important to test some of the particular characteristics of the algorithm developed here. Those tests are performed in maneuvers that have similar results available in the literature, so a comparison to validate the algorithm and the software developed can be made. After that, the results of the optimal maneuvers using the new propulsion devices, as well as the simulations of them in the STRS, are presented.
This first maneuver described has a constraint at the pitch angle, in order to test this capability of the proposed algorithm.
Initial orbital elements: Vehicle mass = 300 kg; thrust = 1.0 Newton, initial position Condition on the final orbit: Other constraints: 5°
Final orbital elements obtained after the maneuver:
Fuel consumption: 2.4235 kg; fuel consumption in Biggs [
Figure
Optimal pitch angles of the maneuver 1.
Now, with the constraint in the direction of propulsion, the fuel consumption was increased when compared with a maneuver without this constraint [
This maneuver has the constraint of a region prohibited for thrusting. This is one of the most important capabilities of the algorithm developed here.
Initial orbital elements: Vehicle mass = 300 kg; thrust = 1.0 Newton, initial position Condition on the final orbit: Other constraints: no thrust between the true longitudes 120° and 180°.
Final orbital elements:
Fuel consumption: 2.7852 kg; fuel consumption in Biggs [
Figure
Optimal pitch angles of the maneuver 2.
In Figure
This maneuver has the requirements of changing three orbital elements of the orbit at the same time.
Initial orbital elements: Vehicle mass = 300 kg; thrust = 1.0 Newton, initial position Condition on the final orbit:
Final orbital elements:
Fuel consumption: 5.565 kg; fuel consumption in Biggs [
Figures
Optimal pitch angles of the maneuver 3 for the first arc.
Optimal yaw angles of the maneuver 3 for the first arc.
Optimal pitch angles of the maneuver 3 for the second arc.
Optimal yaw angles of the maneuver 3 for the second arc.
In this maneuver there are three Keplerian elements fixed on the final orbit: semimajor axis, eccentricity, and inclination. Since the maneuver includes a change in the orbital plane, the yaw angle is not zero. The final conditions were satisfied and the fuel consumption was reduced by 0.014 kg, when compared to Prado [
In this section we present the optimal maneuvers found by the algorithm presented in the present paper when using Phalls I and II. The optimal thrust angles are presented, and the specific impulse considered for each maneuver is given in Table
Optimal pitch angles for maneuver 4.
Optimal pitch angles for the first arc of the maneuver 5.
Optimal pitch angles for the second arc of maneuver 5.
Optimal pitch angle for maneuver 6.
Optimal pitch angles for the first arc of maneuver 7.
Optimal pitch angles for the second arc of maneuver 7.
Semimajor axis deviation in maneuver 4.
Eccentricity deviation in maneuver 4.
Position deviation in maneuver 4.
Velocity deviation in maneuver 4.
Thrust applied in maneuver 4.
Semimajor axis deviation in maneuver 6.
Eccentricity deviation in maneuver 6.
Position deviation in maneuver 6.
Velocity deviation in maneuver 6.
Thrust applied in maneuver 6.
All the maneuvers considered here have the same initial orbital elements and final condition imposed for the final orbit. Only the number of arcs and the propulsion system is different at each maneuver. The parameters of the maneuver proposed are as follows. Initial orbital elements: Vehicle mass = 300 kg. Condition imposed on the final orbit:
For this maneuver it was considered the existence of three Phall I thrusters. The propulsion force of each one is assumed to be the average measured value. The parameters considered in this maneuver are as follows. Total thrust = 0.252 Newton; specific impulse: 1083 s.
Solution: Final orbital elements: Fuel consumption: 1.046346 kg. Duration of the maneuver: 44066.96 s.
This maneuver has the same conditions of maneuver 4, but now two thrust arcs are used, instead of one. The parameters considered in this maneuver are as follows. Total thrust = 0.252 Newton; specific impulse: 1083 s.
Solution: Final orbital elements after the first arc: Final orbital elements after the second arc: Fuel consumption: 1.013402 kg. Duration of the maneuver: 42679.54 s.
Note that the use of two arcs generates savings in the fuel consumption, as expected.
For this maneuver it was considered the existence of three Phall II thrusters. The propulsion force of each one is assumed to be the average measured value. The parameters considered in this maneuver are as follows. Total thrust = 0.360 Newton; specific impulse: 1600 s.
Solution: Final orbital elements: Fuel consumption: 0.697930 kg. Duration of the maneuver: 30398.73 s.
Note that the use of the Phall II generates savings in the fuel consumption since it has better parameters when compared to the Phall I device.
This maneuver has the same conditions of maneuver 6, but now with two thrust arcs instead of one. The parameters considered in this maneuver are as follows. Total thrust = 0.360 Newton; specific impulse: 1600 s.
Solution: Final orbital elements after the first arc: Final orbital elements after the second arc: Fuel consumption: 0.686314 kg. Duration of the maneuver: 29892.85 s.
Note that the use of two arcs generates savings in the fuel consumption one more time, as expected.
Now, with the help of the optimal maneuvers found previously, we shall simulate those maneuvers in the STRS using the optimal angles. Some different aspects of the propulsion system were considered to simulate a more realistic environment, and also the PID control was considered.
In this section, for the propulsion system, it was considered that at each inertial direction
In this case it is considered maneuver 4, including the solution found there. Final orbital elements: Fuel consumption: 1.0470 kg. Propellant mass difference from the optimal maneuver: 6.66 × 10−4 kg.
The error inserted in the propulsion system resulted in a deviation of the orbital elements, as we can see in Figures
In this case maneuver 5 is considered, including the solution found. Final orbital elements after the first arc: Final orbital elements after the second arc: Fuel consumption: 1.013362 kg. Propellant mass difference from the optimal maneuver: 6.384 × 10−4 kg.
Now maneuver 6 is considered. Final orbital elements: Fuel consumption: 0.6979 kg. Propellant mass difference from the optimal maneuver: 2.19 × 10−4 kg.
The same analysis for the figures made in Section
Maneuver 7 is now studied. Final orbital elements after the first arc: Final orbital elements after the second arc: Fuel consumption: 0.686279 kg. Propellant mass difference from the optimal maneuver: 2.1469 × 10−4 kg.
This maneuver uses the optimal angles given in maneuver 4 but includes the errors proposed for the propulsion system. In addition, it was considered other nonlinearities for the actuator: two seconds delay to respond the control signal; a rate limiter for the variation of the satelite velocity, which the absolute limit of the rate is 0.00025 m/s; dead zone where the propulsion system is not turned on if the required velocity increment is lower than 0.0002 m/s. Considering these nonlinearities for the propulsion system, the PID control was able to achieve the maneuver although the fuel consumption was slightly increased. The results for this simulation are as follows. Final orbital elements: Fuel consumption: 1.1276322 kg. Propellant mass difference from the optimal maneuver: 0.081429 kg.
The maneuver simulated here uses the optimal angles given in maneuver 6, with the errors proposed for the propulsion system. In addition, it was proposed the same nonlinearities (rate limiter, dead zone, and time delay) of maneuver 8. Considering these nonlinearities for the propulsion system, the PID control was able to achieve the maneuver although the fuel consumption was increased. The results for this maneuver are as follows. Final orbital elements: Fuel consumption: 0.721921 kg. The propellant mass difference from the optimal maneuver: 0.02402 kg.
Primarily, this paper established an algorithm to solve the problem of orbital maneuvers using a low-thrust control for a spacecraft that is travelling around the Earth. Then, all the maneuvers studied in this paper were successfully solved with the final conditions and constraints proposed and with reasonable fuel consumptions.
It was possible to verify that maneuvers with more thrusting arcs consume less fuel, as expected. This occurs because the algorithm has more variables to optimize, so it is possible to reduce the fuel consumption.
The second part of this work was to simulate the optimal maneuvers found by the algorithm proposed here in a more realistic environment, which can consider errors on the actuator that was not considered in the search for the optimal maneuver.
The maneuvers simulated in the STRS were coherent with the optimal maneuvers, validating the integration of both softwares.
The error considered in the propulsion system opens the possibility to study how the system would react to those nonlinearities. Although errors were inserted in the propulsion system, the PID control was able to correct and accomplish the maneuver.
It is possible to analyze the shifts caused by the error as well as the deviations in semimajor axis and eccentricity. It is also possible to see that the propulsion thrust was not linear.
For the maneuvers shown in Sections
Evidently, the phall II was more efficient than the phall I, as noticed by comparing maneuver 4 with maneuver 6 and maneuver 5 with maneuver 7. This was expected, since the phall II has a higher specific impulse and a higher average propulsion thrust as well. Nevertheless, the PID control was able to control all the maneuvers and reaches the final orbital constraints.