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1st IAA Conference on Space Situational Awareness (ICSSA)

Orlando, FL, USA

IAA-ICSSA-17-0X-XX NONLINEAR RELATIVE MOTION STATE ESTIMATION AND BACKSTEPPING

CONTROL OF A SPACECRAFT HOVERING AROUND AN ASTEROID

Hong Yao(1)*, Dan Simon(2) (1) Cleveland State University, Cleveland, Ohio, h.yao16@csuohio.edu

(2) Cleveland State University, Cleveland, Ohio, d.j.simon@csuohio.edu

Keywords: Asteroid Exploration, Hovering Control, Ensemble Kalman Filter, Backstepping Control ABSTRACT In this paper, the relative motion model for fixed hovering of a spacecraft over an asteroid in a highly uncertain dynamic environment is described. Then translational and rotational motion state estimation and control is designed under the consideration of the coupling effect of the translational motion and the rotational motion. For translational motion, the linear Kalman filter (KF) and proportional-derivative (PD) control are applied; for rotational motion, the ensemble Kalman filter (EnKF) and backstepping control are applied. Simulation results demonstrate the feasibility of the proposed relative motion state estimation and control approach. 1 INTRODUCTION Asteroids represent the next step for space exploration. Their detailed study will address open questions on the formation of the Solar System and will support future projects for in-space resource utilization. Small bodies are also an ideal intermediate step for extending the capabilities of human space flight beyond the Earth-Moon system [1]. So far there have been several successful missions to small bodies. Notable examples include NEAR-Shoemaker, which performed the first soft landing on an asteroid; Hayabusa, which returned a small sample from the Itokawa asteroid; and most recently, the ROSETTA mission, which made the first landing on the surface of a comet [2]. Future sample return missions are also currently planned. An asteroid is one kind of typical small body. In order to ensure the success of asteroid exploration, one of the keys is the ability to perform proximity operations, which is also a way to deal with impact threats from near Earth objects (NEO). Spacecraft proximity operation is an enabling technology for space missions such as asteroid exploration, on-orbit satellite inspection, health monitoring, surveillance, servicing, refueling, and formation flying. One challenge associated with this technology is the need to accurately and simultaneously track relative position and attitude in order to achieve mission objectives [3]. For the asteroid exploration, one important stage is spacecraft hovering around the asteroid, during which time the spacecraft must keep a constant position and attitude with respect to the asteroid.

* Corresponding author is from Beihang University, Beijing, China, and is currently a visiting scholar at the Electrical Engineering and Computer Science Department of Cleveland State University.

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Therefore, the relative pose, which includes relative position and relative attitude, must be estimated and then controlled by feedback control laws. Relative pose estimation or determination has been a subject of considerable research for many years in computer vision, photogrammetry, and robotics, and many solutions have been proposed [4]. The Kalman filter is usually chosen as the estimator candidate due to its feature of optimality for linear systems. However, the system model for the relative pose estimation problem is nonlinear in nature, so the extended Kalman filter (EKF) has been used to deal with nonlinearities [5]. The EKF accounts for nonlinearities by linearizing the system about its last-known estimate while neglecting higher-order terms, resulting in a suboptimal solution. The unscented Kalman filter (UKF), a member of the sigma point filter family, attempts to overcome some of the shortcomings of the EKF for the estimation of nonlinear systems. The UKF uses a set of sample points, or sigma points, that are determined from the prior mean and covariance of the state, and then the posteriori mean and covariance of the state can be estimated from the nonlinearly transformed sigma points. This approach usually gives the UKF better accuracy than the EKF for nonlinear systems [6, 7]. The Rao-Blackwellized particle filter was also utilized for relative pose estimation, despite its heavy computational burden [8]. For relative pose control, there are two approaches: near-inertial hovering and body-fixed hovering. In near-inertial hovering, the spacecraft is stationed at a fixed, desired position relative to the asteroid in the sun-asteroid frame, implying that the asteroid rotates beneath the spacecraft. In body-fixed hovering, the spacecraft stays at a fixed pose (position and orientation) relative to the rotating asteroid [9]. Body-fixed hovering is essential for the spacecraft to sample the surface of the asteroid. Many works have applied nonlinear methods to achieve the control of translational and rotational motion. These nonlinear control methods include sliding mode control [10], robust adaptive control [11], Lyapunov methods [12], time delayed feedback control [13], etc. For a rigid spacecraft, the coupling effect between the translational and rotational motions exist, which is ignored by the most research works, although some research works exist which take the coupling effect into consideration [14]. In this paper, we consider the coupling effect of rotational motion and translational motion. In general, the particle filter is computationally intractable and impractical for high dimension problems. Therefore, in practice, this method suffers from the curse of dimensionality and its successful application is limited to problems with small or moderate dimensions [15]. The EnKF is a powerful tool for estimation of high-dimensional state-space models. Despite its successes, the EnKF is relatively under-utilized in the statistics community [16] and has been widely overlooked by signal processing researchers [17]. Thus far, to the best of our knowledge, there are only a few papers about the EnKF for spacecraft relative pose estimation. In this paper, a nonlinear filtering technique based on the EnKF is used for the rotational motion state estimation of a spacecraft hovering around an asteroid. Further, a nonlinear control law for rotational motion based on the backstepping technique is employed to track the desired rotation commands. The paper is organized as follows. First, the research background and motivation are introduced. Then, the relative kinematic and dynamic equations of translational and

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rotational motion for a spacecraft hovering around an asteroid are derived. In the next section, the relative rotational motion control law is derived using the backstepping technique. Then, the EnKF algorithm is described and the relative motion state estimation and control strategy is given. After that, the proposed state estimation and control algorithms are validated through a numerical example. Finally, conclusions are drawn. 2 DYNAMIC MODEL OF A SPACECRAFT HOVERING AROUND AN ASTEROID In order to design the state estimation and control system, the dynamic model has to be derived. The translation and rotation of relative motion have historically been analyzed independently, with separate control algorithms being developed for translation and rotation [18]. The definition of relative pose, including relative translation and rotation, involves an object reference frame and a base reference frame. The relative pose of the object reference frame comprises the three position coordinates of the origin of the object reference frame in the base reference frame, and the three orientation angles of the object reference frame with respect to the base reference frame. 2.1 DYNAMIC MODEL OF TRANSLATIONAL MOTION For the dynamic modeling of translation, the first step is to define the object reference frame and the base reference frame, in which the position of the spacecraft can be represented. The next step is to model the force disturbances affecting the motion of the spacecraft in the vicinity of the asteroid. The last step, the dynamics of the relative translation of the spacecraft with respect to the asteroid are established for control system design and simulation. Here the object reference frame is the spacecraft body frame (SBF), which is fixed at the spacecraft center of mass, with the coordinate axes aligned along the spacecraft’s principal moments of inertia. For the base reference frame, most previous research used a unique frame that is fixed to the asteroid and whose origin is at the asteroid’s center of mass, and whose coordinate axes are aligned along the asteroid’s principal moments of inertia. In contrast, in this paper the base reference frame is determined by the initial time of the hovering operation mission. The base reference frame is fixed to the asteroid with its origin at the asteroid’s center of mass, and with coordinate axes that coincide with the SBF at the initial time of the hovering operation. We call this base reference frame the asteroid centred frame (ACF) as shown in Fig. 1.

Figure 1. Spacecraft position vector in the base reference frame ACF

ACF

SBF

r

X

X

Y

Y

Z

Z

4

The frame ACF is used to describe the motion of the spacecraft relative to the rotating asteroid. Dynamic perturbations include non-central body gravity forces acting on a spacecraft, solar radiation pressure, and third-body effects from the sun. Previous research [5] compares the effects of these accelerations when the spacecraft is in a 1 km circular terminator orbit around the Itokawa asteroid. The conclusion is that the accelerations due to the central body are the main contributing factor to the spacecraft motion, while the perturbations due to solar radiation pressure and spherical harmonics are of the same order of magnitude. Therefore, for simplicity, here we do not take solar radiation pressure and third-body effects into consideration.

Define Tzyx ω as the rotation rate of the asteroid with respect to the

inertial coordinates, which is described in the ACF frame. Asteroid motion is assumed to be pure rotation, which is a reasonable assumption because most asteroids are in a stable rotation about the axis of their largest moment of inertia with a period of a few hours. Fast rotators and tumbling asteroids exist, but they are not predominant [1]. So here we consider the small body to be in uniform rotation; that is,

0ω .

Define Tzyxr as the position vector from the asteroid center of mass to the

spacecraft center of mass. The position vector r represents the relative position of the spacecraft with respect to the asteroid, which is described in the ACF frame. Considering the control acceleration, the dynamic equation of position can be found as follows [19]:

cgU ar

rωωrωr

)(2 (1)

where gU is gravity potential, and Tczcycxc aaaa is control acceleration given

by some control law. For spacecraft hovering, the position r will be nearly constant, and the rotation rate ω is constant under the assumption of uniform rotation. So we can simplify the dynamic equation for the purpose of state estimation.

Suppose Tzyxg cccU

)( rωω

r are slowly varying parameters. Then the

scalar form of the dynamic equation is

cxxyz aczyx 2-2 (2a)

cyyxz aczxy 22- (2b)

czzxy acyxz 2-2 (2c)

Let the desired hovering position be constant Tzyx 000 , and denote the deviation

of position as Tzyx . We then have xxx 0 , yyy 0 ,

zzz 0 , xx , yy , zz , xx , yy , zz . Then the

dynamic equations can be written as

cxxyz aczyx 22 (3a)

cyyxz aczxy 22- (3b)

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czzxy acyxz 2-2 (3c)

Take x , y , z , x , y , z , xc , yc , zc as the state variable X . Then the state

space equation used for the linear Kalman filter is

cz

cy

cx

z

y

x

xy

xz

yz

a

a

a

c

c

c

z

y

x

z

y

x

000

000

000

100

010

001

000

000

000

000000000

000000000

000000000

100022000

010202000

001220000

000100000

000010000

000001000

X (4)

In the above equation, the angular velocity Tzyx ω is unknown, but can

be supplied by the rotational motion state estimation filter. 2.2 DYNAMIC MODEL OF ROTATIONAL MOTION Rotational motion describes the relationship of the reference frame SBF with respect to the base reference frame ACF, and includes the relative attitude and the relative angular velocity. The relative attitude is given by the classical Rodrigues parameters

Tzyx ggg g , and the relative angular velocity is denoted as

Tzyx ω . The relative angular velocity is equal to the absolute

angular velocity Tbzbybxb ω of the spacecraft minus the angular velocity

ω of the asteroid. We use SAM to denote the rotation matrix from ACF to SBF. In

terms of the classical Rodrigues parameters, the rotation matrix can be written as follows [20]:

13333

GIGIMSA (5)

where 33I is the unit matrix and

0

0

0

xy

xz

yz

gg

gg

gg

G . Then the relative

angular velocity described in the reference frame SBF is

ωMωω SAb (6)

The kinematics equation of the relative rotation of the spacecraft is

ωggGIg

T33

2

1 (7)

The dynamics equation of the relative rotation motion of the spacecraft is derived as

)(11

1

ωωωIωITI

ωMωMTωIωI

ωMωMωω

bbbbbcb

SA

SAcbbbb

SA

SAb

(8)

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where bI is the matrix of the moment of inertia of the spacecraft, cT is the control

torque given by some attitude control law, and ω is the angular acceleration of the

asteroid. For uniform rotation, 0ω , and

0

0

0

xy

xz

yz

.

In the above dynamics equation, the angular velocity bω is known due to the

absolute attitude determination system of the spacecraft. 3 RELATIVE MOTION CONTROL Relative motion control includes position control and relative attitude control, which are treated separately here. For position control, according to the linear differential equations of Eq. (3), we can design a PD control law to keep the states stabilized near the zero. The unknown external interference will be estimated along with the system states and can be compensated by feed forward control. Here we focus only on the design of relative attitude control by the backstepping technique, which will be described in two steps as in [21]. First, an angular velocity control law is assumed, and next, torque commands are generated to keep the entire system stable. 3.1 Step 1 We define the first and the second variables of the backstepping controller as

gz 1 (9)

αωz -2 (10)

where α is a virtual control law. The time derivative of 1z is expressed as

ωgGωggGIgz )(2

1

2

1331

T (11)

where TggGIgG 33)( . The first Lyapunov function is defined as

1111 )( zzzV T (12)

Its time derivative is expressed as

αgGzzgGzzzV )()(2 121111 TTT (13)

In order to make 1V negative, α is chosen as

11 )( zgGkα T (14)

where 1k is a positive gain matrix. The time derivative of 1V becomes

211111 )()()(- zgGzzgGkgGzV TTT (15)

The term 21 )( zgGz T will be considered in the next step.

3.2 Step 2

The time derivative of 2z is expressed as

αωωωIωITIαωz -)(- 112

bbbbbcb (16)

The second Lyapunov function is defined as

22112122

1)(),( zIzzVzzV b

T (17)

Its time derivative is expressed as

7

12111

212111

2212

)(-)()()(-

-)()()()(-

zgGαIωωIωIωTzzgGkgGz

αIωωIωIωTzzgGzzgGkgGz

zIzVV

Tbbbbbbc

TTT

bbbbbbcTTTTT

bT

(18)

To make 2V negative, the control law cT is chosen as

221)()(- zkzgGαIωωIωIωT Tbbbbbbc (19)

where 2k is a positive gain matrix. The time derivative of 2V becomes

2221112 )()( zkzzgGkgGzV TTT (20)

According to Lyapunov theory, 1z and 2z will be asymptotically convergent, which

indicates that the state g and ω will also be convergent.

4 RELATIVE MOTION STATE ESTIMATION ALGORITHMS A closed loop control system requires input from a state estimator. In correspondence to the separate control system design for position and relative attitude, two different state estimation filters will be presented for translational and rotational motion state estimation, respectively. For translational motion state estimation, there are nine states, including three position deviation states, three velocity deviation states and three external disturbance states, all of which will be estimated by the linear Kalman filter [22]. For rotational motion state estimation, there are six states, including three relative attitude states and three relative angular velocity states, all of which will be estimated by the EnKF algorithm [20, Table 4.4]. The EnKF is based on the Kalman filter. But in the EnKF, the covariance matrix P is stored and mathematically manipulated only implicitly, via an ensemble

][ 1 mXXX of system states, where m is the ensemble size [23]. Through

linear or nonlinear state equations and measurement equations, the forecast of the state and output are conducted by each member of the ensemble. Suppose at time

k the forecast of the state and output by the ensemble are ][ 1 kmkk XXX and

][ 1 kmkk YYY ; then the mean

kX and

kY are

m

ikik

m 1

1XX (21)

m

ikik

m 1

1YY (22)

The priori covariance matrices xxeekP , yxee

kP and yyee

kP are

m

i

T

kkikkiee

km

xx

11

1XXXXP (23)

m

i

T

kkikki

ee

km

yx

11

1YYXXP (24)

m

i

T

kkikki

ee

km

yy

11

1YYYYP (25)

If the system measurement equation is linear, the measurement matrix is denoted as

kH and the measurement noise covariance matrix is kR , in which case we have [16]

8

Tk

eek

ee

kxxyx HPP (26)

kTk

eekk

ee

kxxyy RHPHP (27)

Now the Kalman gain matrix can be calculated as

1 yyyx ee

k

ee

kk PPK (28)

Then each element of the ensemble can be updated as

),,1( mikikkikkkiki XHvyKXX (29)

where ky is the measurement, and kiv ~ ),( kN R0 .

In state estimation, one important issue is how to acquire the measurement information. There are two popular methods of relative pose measurement. One is a vision based method in which point features are detected and extracted from a camera’s sequential images, and then features are matched to a database of landmarks. The database has to be prepared on the ground and uploaded to the spacecraft in advance during the characterization operations phase near the small body. With feature matching, the relative pose of the spacecraft with respect to the small body can be computed based on feature pairs (2D points in the image matched with 3D locations of a landmark) [2]. The second method of relative pose measurement uses an optical camera coupled with light detection and ranging (LiDAR) to directly measure the range from the spacecraft to points on the asteroid surface, while the camera obtains the orientation of the points in the camera frame. Then the 3D coordinates of a set of points can be obtained in each view. For matching 3D points in two different views, the iterative closest point (ICP) algorithm is commonly used. However, it is well known that implementations of the ICP algorithm involve a large computational cost. If the matching problem of two point sets has been solved by some image processing algorithm, then a more computationally efficient algorithm based on least squares can be used for the calculation of relative motion parameters [20]. With the second measurement method, the position deviation and relative attitude measurement information (the classical Rodrigues parameters) for the relative navigation filter can be obtained. Here the assumption is that the position

measurement error is 2 m (1σ) and the angle measurement error is 0.1°(1σ). The

measurement matrices for translational and rotational motion state estimation are

][ 6633 0I and ][ 3333 0I respectively. The relative motion state estimation and

control flow chart of the spacecraft hovering operation (including the relative motion control laws) is shown as Fig. 2.

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Figure 2. The flow chart of relative motion state estimation and control

5 NUMERICAL EXAMPLE A spacecraft hovering mission scenario is considered to demonstrate the proposed state estimation and control scheme in Figure 2, except for the “measurement and process” block, which is not the research objective in this paper and thus is not considered. We choose the Itokawa asteroid as the target for the spacecraft to hover over. The spacecraft is required to achieve body-fixed hovering over the rotating asteroid. The mission scenario aims to make the spacecraft can stay at a given initial position and attitude with respect to the asteroid, which are determined at the beginning of the hovering operation command. The initial spacecraft position is assumed to be [600 600 600]T m, which is used only for simulation and is not used for state estimation and control. The initial absolute attitude quaternion of the spacecraft is [0.888 0.195 0.260 0.325]T, which is known due to the absolute attitude determination system of the spacecraft. We desire the spacecraft to maintain the same position and relative attitude as is given at the initial time. Therefore, we need to keep the position deviation and the relative attitude deviation (the classical Rodrigues parameters) stabilized near zero. The numerical value for the gravitational parameter of the Itokawa asteroid is 2.36

m3/s2. The uniform rotation rate of the Itokawa asteroid is [0 0 1.4386]×10-4 rad/s. The asteroid rotation rate in the ACF Frame is [-0.0482 0.0742 0.1135]×10-3 rad/s, and is used only for the translational dynamic simulation; its estimate is given by the

rotational motion state estimation filter. The moment of inertia bI of the spacecraft is

diag[30 40 50]T kg·m2. Table 1 summarizes the parameters used for the simulation of the state estimation and control algorithms.

Translational dynamics Rotational dynamics

Spacecraft and asteroid motion model

Measurement and process

Position deviation measurement information

Relative attitude measurement information

EnKF KF ω

PD control law Backstepping control law

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Table 1. The parameters used for the numerical simulation

Translational Motion

Initial Conditions Position Deviation (m) [0 0 0]T

Velocity Deviation (m/s) [0.0236 -0.0970 0.0734]T

KF

Initial State Estimate [0 0 0 0 0 0 0 0 0]T

Initial Covariance Matrix diag[12 I3×3, 0.12 I3×3, 10-10I3×3]

Process Noise Covariance 09×9

Measurement Noise Covariance

22 I3×3

PD Controller kp 10-3 I3×3

kd 0.0443 I3×3

Rotational Motion

Initial Conditions

Relative Attitude [0 0 0]T

Relative Angular Velocity (rad/s)

10-3 ×[0.0482 -0.0742 -0.1135]

EnKF

Initial State Estimate [0 0 0 0 0 0]T

Initial Covariance Matrix diag[(4.4×10-4)2 I3×3, (8.7×10-4)2 I3×3]

Process Noise Covariance 06×6

Measurement Noise Covariance

(4.4×10-4)2 I3×3

Ensemble Size 50

Backstepping Controller

k1 0.2 I3×3

k2 0.6 Ib (Ib : The moment of inertia of

the spacecraft)

The simulation time is 1500 seconds and the simulation time step is 1 second. The simulation results of translational motion in the asteroid body-fixed ACF frame are shown in Fig. 3 and Fig. 4, the velocity and its estimation errors are shown in Fig. 5 and Fig. 6, the external disturbance and its estimation errors are shown in Fig. 7 and Fig. 8, and the control acceleration is shown in Fig. 9. The simulation results of rotational motion with respect to the ACF frame, the relative attitude and its estimation errors are shown in Fig. 10 and Fig. 11, the relative angular velocity and its estimation errors are shown in Fig. 12 and Fig. 13, and the control torque is shown in Fig. 14. The translational motion results in Fig. 3 and Fig. 5 show that the position and velocity converge to the desired values. Fig. 4 shows that after convergence, the position estimation errors are less than 0.5 m. Fig. 6 shows that the velocity estimation errors tend to zero. Fig. 7 and Fig. 8 show that the external disturbance is almost constant and the external disturbance errors also tend to zero but with slow convergence speed. Fig. 9 shows that initially, large accelerations are required to control the translational motion. The rotational motion results in Fig. 10 and Fig. 12 show that relative attitude and relative angular velocity converge to their desired values. Fig. 11 and Fig. 13 show that the relative attitude and angular velocity estimation errors tend to zero. Fig. 14 shows that initially, large torques are required to control the rotational motion, but the control torques eventually decrease to almost zero. In summary, the simulation results verify that the spacecraft can maintain the desired relative position in the asteroid body-fixed frame while keeping the relative attitude stationary with respect to the asteroid. The spacecraft hovering operation around the asteroid has been successfully simulated.

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Fig. 3. Position of the spacecraft Fig. 4. Estimation errors of position

Fig. 5. Velocity of the spacecraft Fig. 6. Estimation errors of velocity

Fig. 7. External disturbance Fig. 8. Estimation errors of disturbance

Fig. 9. Control acceleration

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Fig. 10. Relative attitude Fig. 11. Estimation errors of relative attitude

Fig. 12. Relative angular velocity Fig. 13. Estimation errors of relative angular

velocity

Fig. 14. Control torque

6 CONCLUSION The main contribution of this paper is a method for relative motion state estimation and control of spacecraft body-fixed hovering in a highly uncertain dynamical environment around an asteroid. Although the method assumes that the asteroid is rotating uniformly, it can be expected to be applicable for the time-varying situation of slowly tumbling asteroids, based on the weak coupling between translational motion and rotational motion of the spacecraft. The paper applied the EnKF to the rotational motion state estimation problem, but future research could apply the EnKF to the joint translation and rotation motion state estimation problem, which would enhance the effectiveness of the EnKF method but at the expense of a corresponding increase in the state dimension.

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ACKNOWLEDGMENTS The first author thanks the China Scholarship Council for research support at Cleveland State University as a visiting scholar. REFERENCES [1] Turconi A, Palmer P, Roberts M. Efficient Modelling of Small Bodies Gravitational Potential for Autonomous Proximity Operations. One of Chapters of Astrodynamics Network AstroNet-II [M]. Springer International Publishing, 2016: 257-272. [2] Kicman P, Lisowski J, Bidaux-Sokolowski A. Vision-Based Navigation Around Small Bodies. One of Chapters of Astrodynamics Network AstroNet-II [M]. Springer International Publishing, 2016: 137-149. [3] Filipe N, Tsiotras P. Adaptive Position And Attitude Tracking Controller For Satellite Proximity Operations Using Dual Quaternions[J]. Journal of Guidance, Control, and Dynamics, 2014. [4] Goddard J S. Pose and Motion Estimation From Vision Using Dual Quaternion-Based Extended Kalman Filtering[D]. University of Tennessee, Knoxville, Ph. D. Thesis, 1997. [5] Dietrich A B. Supporting Autonomous Navigation with Flash Lidar Images in Proximity to Small Celestial Bodies[D]. Dissertation, University of Colorado, 2017. [6] VanDyke M C, Schwartz J L, Hall C D. Unscented Kalman Filtering For Spacecraft Attitude State And Parameter Estimation[J]. Advances in the Astronautical Sciences, 2004, 118(1): 217-228. [7] Yang H X, Yang X X, Zhang W H. Orbit Determination of Spacecraft Formation Flying With Slowly Rotating Asteroids[J]. Journal of Navigation, 2014, 67(4): 687-710. [8] Cocaud C, Kubota T. Autonomous Navigation Near Asteroids Based On Visual SLAM[C]. Proceedings of the 23rd International Symposium on Space Flight Dynamics, Pasadena, California. 2012. [9] Lee D, Sanyal A K, Butcher E A, et al. Finite-Time Control For Spacecraft Body-Fixed Hovering Over An Asteroid[J]. IEEE Transactions on Aerospace and Electronic Systems, 2015, 51(1): 506-520. [10] Zhang P, Ma T, Zhao B, et al. Robust Linear Quadratic Regulator Via Sliding Mode Guidance For Spacecraft Orbiting A Tumbling Asteroid[J]. Mathematical Problems in Engineering, 2015. [11] Sun L, Huo W. Robust Adaptive Control Of Spacecraft Proximity Maneuvers Under Dynamic Coupling And Uncertainty[J]. Advances in Space Research, 2015, 56(10): 2206-2217. [12] Gui H, de Ruiter A H J. Velocity-Free Control of Spacecraft Body-Fixed Hovering Around Asteroids[C]. 27th AAS/AIAA Space Flight Mechanics Meeting, San Antonio, Texas, Feb. 5-9, 2017. [13] Li H, Hexi B. Time Delayed Feedback Control of Hovering and Orbiting About Asteroids[C]. 27th AAS/AIAA Space Flight Mechanics Meeting, San Antonio, Texas, Feb. 5-9, 2017. [14] Bolatti D A, de Ruiter A H J. Modeling Spacecraft Orbit-Attitude Coupled Dynamics in Close Proximity to Asteroids[C]. AIAA/AAS Astrodynamics Specialist Conference, 2016. [15] Santitissadeekorn N, Jones C. Two-Stage Filtering For Joint State-Parameter Estimation[J]. Monthly Weather Review, 2015, 143(6): 2028-2042. [16] Katzfuss M, Stroud J R, Wikle C K. Understanding The Ensemble Kalman Filter[J]. The American Statistician, 2016, 70(4): 350-357.

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