An active disturbance rejection stationkeeping control scheme is derived and analyzed for stationkeeping missions of spacecraft along a class of unstable periodic orbits near collinear libration points of the SunEarth system. It is an error driven, rather than modelbased control law, essentially accounting for the independence of model accuracy and linearization. An extended state observer is designed to estimate the states in real time by setting an extended state, that is, the sum of unmodeled dynamic and external disturbance. This total disturbance is compensated by a nonlinear state error feedback controller based on the extended state observer. A nonlinear tracking differentiator is designed to obtain the velocity of the spacecraft since only position signals are available. In addition, the system contradiction between rapid response and overshoot can be effectively solved via arranging the transient process in tracking differentiator. Simulation results illustrate that the proposed method is adequate for stationkeeping of unstable Halo orbits in the presence of system uncertainties, initial injection errors, solar radiation pressure, and perturbations of the eccentric nature of the Earth's orbit. It is also shown that the closedloop control system performance is improved significantly using our method comparing with the general LQR method.
In recent years there has been an increasing interest in libration points missions. The libration points, which are normally called equilibrium points or Lagrangian points, correspond to regions in space where the centrifugal forces and the gravitational forces from the Sun and the Earth cancel each other. The existences of periodic orbits and quasiperiodic orbits in the vicinity of collinear libration points have been proved and analyzed rigorously in celestial mechanics [
However, these libration point orbits are inherently unstable but controllable. Thus, additional control force is needed for a spacecraft to remain close to the nominal orbit. The challenges of stationkeeping control emerge from highly accuracy, low computation burden, and minimal fuel cost control requirements under the condition of spacecraft dynamic uncertainties, unmodeled perturbations, and initial orbit injection errors [
The study of stationkeeping control on libration point orbits has become a popular research topic ever since the problem was firstly proposed. A vast majority of the stationkeeping control methods are designed based on LTI model via local linearization at the libration points due to the high nonlinearity of the dynamic equation of libration point orbit. The nonlinear dynamic equation is obtained utilizing a ClohessyWiltshire (CW) like reference frame, which is widely used for Earthcentered spacecraft dynamic analysis [
To improve the modeling accuracy, LTV method is employed instead of the LTI approach. Gurfil and Kasdin [
Besides the LTI and LTV modelbased method, existing works also involve methodologies directly developed from original nonlinear dynamical equation of motion. Rahmani et al. [
Consequently, this paper derives an active disturbance rejection stationkeeping control method considering mainly the following issues.
Proposing an error driven, rather than modelbased control law which takes into account system uncertainties, unmodeled disturbance, and orbit injection errors to achieve better robustness.
Considering output feedback from practical viewpoint rather than full information, since only position signals of spacecraft can be measured.
Better stationkeeping performance as well as simpler computation burden.
A new nonlinear stationkeeping control law based on active disturbance rejection control (ADRC) method, which refers to the socalled active disturbance rejection stationkeeping control (ADRSC) method in this paper, is proposed and analyzed. ADRC was firstly proposed by Han [
The remainder of this paper is organized as follows. Section
In this section, the dynamic models of spacecraft based on the circular restricted threebody problem (CR3BP) are established. Additionally, the relative Halo reference orbits derived from LP map method are presented.
The CR3BP is one of the most common nonlinear models which investigate the relative motion around the libration points. As is shown in Figure
Restricted threebody problem.
We can find libration points of the SunEarth system, denoted by
Periodic solutions of the nonlinear equations of motion can be constructed using the method of successive approximations in conjunction with a technique similar to the LindstedtPoincare method [
The equations of the Halo orbit to the third order are given by
The values of the various constants in (
Parameters  Value  Parameters  Value  Parameters  Value 


2.08645 

3.22927 

2.09270 



















−2.84508 

−2.30206 

−1.87037 






Due to complexity of modern systems, more attention has been paid on datadriven control scheme recently [
Han’s ADRC consists of three parts, a nonlinear tracking differentiator (TD) [
Consider the following system:
The TD has the ability to track the given input reference signal with quick response and no overshoot by providing transition process for expected input
One feasible secondorder TD can be designed as [
ESO is used to estimate
NLSEF generates control voltage
A nonlinear combination of errors signal can be constructed as [
The controller is designed as
As is mentioned in Section
The structure of ADRSC algorithm.
Assuming that only position signals of spacecraft can be detected and used for feedback, there are two different kinds of discrete TDs designed for stationkeeping control.
It is designed for reference Halo orbit tracking
It is designed for flight position tracking and estimating of the velocity of the spacecraft
The equations of stationkeeping of Halo orbits using ADRSC can be generally defined as
It is necessary to point out that a distinct improvement can be obtained using ADRSC for stationkeeping since the dynamic model does not need to be expressively known which is different with the aforementioned studies. Define
One can rewrite (
Since the states, unmodeled dynamics, and disturbances have been estimated by ESO presented in Section
The parameter
In this paper, a parameter selfturning approach is firstly proposed for
In any actual mission, the perturbation factors and the injection errors, coupled with the inherent unstable nature of the Halo orbits around the collinear libration points, will cause a spacecraft to drift from the periodic reference orbit [
The most important perturbative effects in the CR3BP are the eccentric nature of the Earth’s orbit and the gravitational force of the moon [
In the deepspace mission, SRP is another disturbance to account for. Here, we adopt a widely used model [
To compare the performance of ADRSC with linearization method, the simulation of an LQR controller [
The control forces for stationkeeping are provided by an ionic engine with the maximum thrust 60 mN which is currently commercially available [
The disturbing forces described in 4.4 are considered in the simulation. The eccentricity of Earth orbit
In the simulation, considering the system is not full state feedback, the TD is designed to track the spacecraft flight position and obtain the velocity as well. Figure
The TD tracking result.
In dimensionless coordinates the initial orbit parameters without injection errors of ISEE3 are given in Table
Initial orbits parameters of Halo missions without injection errors.
Mission orbits 







ISEE3 (110,000 km)  0.988872932669558  0 

0  0.008853797264729  0 
Halo (800,000 km)  0.989390221855232  0  0.006248517078177  0  0.012547011257083  0 
Figures
ADRSC of ISEE3 (
The relative position errors, velocity errors, and control input of ADRSC control of ISEE3.
The relative position errors, velocity errors, and control input of LQR control of ISEE3.
In order to describe the results of ADRSC more accurately when the spacecraft moves steady on the Halo orbit, here we introduce the popular evaluation parameters for stationkeeping [
The velocity increment and the mean absolute value of the position errors of ISEE3 stationkeeping with orbit injection errors and disturbance are given in Table
The velocity increments and the mean absolute value of the position errors of ADRSC and LQR control.
Mission orbits 









ADRSC: ISEE3 (110,000 km)  24.872  25.948  6.721  36.567  0.135  0.343  0.119  0.389 
LQR: ISEE3 (110,000 km)  25.030  25.991  6.727  36.705  1.991  2.126  0.555  2.965 
ADRSC: Halo (800,000 km)  160.770  81.501  75.049  195.248  0.380  0.607  0.895  1.146 
LQR: Halo (800,000 km)  160.815  81.703  75.115  195.394  14.325  6.516  6.647  17.084 
The parameters of ADRSC of Halo orbit (
Figures
ADRSC control of Halo orbit (
The relative position errors, velocity errors, and control input of ADRSC control of Halo orbit.
The relative position errors, velocity errors, and control input of LQR control of Halo orbit.
The velocity increment and the mean absolute value of the position errors of both ADRSC and LQR are presented in Table
From the above analyses and control results between ADRSC and LQR, we can find the following.
The ADRSC, which is an error driven control method, is adequate for the stationkeeping control of unstable orbits without any knowledge about the spacecraft dynamic model.
The ADRSC presents praiseworthy stationkeeping performance with orbit injection errors as well as unmodeled disturbances such as the SRP and the perturbative forces due to the eccentric nature of the Earth’s orbit using an ionic engine with maximum thrust 60 mN.
The ADRSC approach has better stationkeeping control ability and higher orbit maintenance accuracy compared with LQR controller despite of the same level of both the control methods thrust usage.
The ADRSC has better robust performance compared with the general stationkeeping method depending on spacecraft dynamic model.
There has been a great deal of commendable research on unstable libration point orbits stationkeeping as mentioned before. A major point of departure between ADRSC method and earlier approaches to stationkeeping is that ADRSC is an error driven rather than modelbased control law, which can inherently get across the dependency on model accuracy as well as the drawbacks of linearization. With the combination of TD, ESO, and NLSEF, the unmodeled disturbances as well as the unmodeled system dynamic can be compensated in real time. Thus, fast responsetime requirement and high accuracy of orbit maintenance requirement can be satisfied by ADRSC.
It is important to keep in mind not only the tracking accuracy but also the robustness of the stationkeeping controller, as the space environment cannot be accurately modeled as well as the internal and external disturbance. ADRSC extends the unmodeled spacecraft dynamic and the disturbance as a state, which can be estimated from ESO and then “rejected” from nonlinear feedback control. Thus, no matter the spacecraft system is model known or unknown, linear or nonlinear, time invariant or time variant, with disturbance or without, ADRSC is able to show desired performance.
A successful ADRSC approach for stationkeeping in the SunEarth
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by National Nature Science Foundation of China under Grant no. 61004017 and by the PolishNorwegian Research Programme in the frame of Project Contract no. PolNor/200957/47/2013. The authors would like to thank the editors and the anonymous reviewers for their keen and insightful comments which greatly improve the paper.