Robust adaptive position and attitude-tracking controller for satellite proximity operations

Acta Astronautica 167 (2020) 135–145

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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro

Research paper

Robust adaptive position and attitude-tracking controller for satellite proximity operations

T

Bang-Zhao Zhou, Xiao-Feng Liu, Guo-Ping Cai∗ Department of Engineering Mechanics, State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Satellite proximity operations Pose tracking Adaptive sliding-mode control Unscented Kalman filter

This paper studies the pose tracking control problem for satellite proximity operations between a target and a chaser satellite, by which we mean that the chaser is required to track a desired time-varying trajectory given in advance with respect to the target. Firstly, by consulting an adaptive sliding-mode control method in literature developed for a class of nonlinear uncertain systems, an effective pose tracking controller is obtained. This controller requires no information about the mass and inertia matrix of the chaser, and takes into account the gravitational acceleration, the gravity-gradient torque, the J2 perturbing acceleration, and unknown bounded disturbance forces and torques. Then, an updated controller is proposed by combining the aforementioned controller and the unscented Kalman filter (UKF). This updated controller estimates the inertial parameters of the chaser through UKF, so it is of better adaptive ability to the initial estimation of the inertial parameters. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed controllers. The simulation results show that the updated controller is more accurate.

1. Introduction Several agencies and organizations around the world are investigating satellite proximity operations as an enabling technology for several space missions such as berthing, refueling, repairing, upgrading, transporting, rescuing, orbital debris removal, on-orbit satellite inspection, health monitoring, surveillance, and optical interferometry [1–4]. To the best of our knowledge, there are two kinds of solutions for the control problem of proximity operations between a target and a chaser satellite. (1) For the first kind, the chaser is controlled to approach the vicinity of the target, and several important constraints is considered, e.g., (a) the direction of the chaser's motion with respect to the target is constrained to avoid collision [5,6], and (b) the chaser's attitude is constrained for well observation of the target [7]. (2) As for the second kind, the current desired motion of the chaser is firstly planned according to the motion of the target, then a pose tracking controller is applied to make the chaser move as the planned motion. In this method, the planned desired motion makes sure that if the chaser moves along this motion, it will safely arrive at the docking port of the target with a suitable relative attitude, and its attitude will be always suitable to let its camera observe the target well. Therefore, pose tracking control is of great significance for the proximity operations, and it make the chaser track a desired time-varying trajectory given in

∗

advance with respect to the target. As a result, pose (position and attitude) tracking has been a hot area of research for several years. In the early studies about pose tracking, the researchers considered the circumstance where the target satellite moves in a circular orbit. In Ref. [8], a novel control algorithm based on hybrid linear quadratic regulator/artificial potential function was introduced for close proximity multiple spacecraft autonomous maneuvers. In Ref. [9], a model predictive control approach was applied to spacecraft rendezvous and proximity maneuvering problems in the orbital plane. However, since Clohessy-Wiltshire-Hill [10] linear equations are used in their dynamic models, the above two methods are not suitable for targets in elliptic orbit. Considering the circumstance where the target satellite may move in an elliptic orbit, the Tschauner-Hempel equations are applied in most latter researches. In Ref. [11], a Riccati-based tracking controller with collision avoidance capabilities was presented for proximity operations of spacecraft formation flying near elliptic reference orbits. Zhang and Duan [12] proposed an integrated relative position and attitude control strategy based on θ-D nonlinear optimal control technique for a chaser flying to a space target in proximity operation missions. Singla et al. [13] proposed a velocity-free adaptive control law to solve the spacecraft rendezvous and docking problem under measurement noises. In view of the robustness property of sliding mode technique, a super-twisting controller based on second order sliding mode

Corresponding author. E-mail addresses: [email protected] (B.-Z. Zhou), [email protected] (X.-F. Liu), [email protected] (G.-P. Cai).

https://doi.org/10.1016/j.actaastro.2019.10.035 Received 30 July 2019; Received in revised form 29 September 2019; Accepted 16 October 2019 Available online 05 November 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.

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algorithm was proposed in Ref. [14], and the finite time convergence of the closed-loop system was derived theoretically. Huang and Jia [15,16] investigated the fixed-time relative position tracking and attitude synchronization control problem of a spacecraft fly-around mission for a non-cooperative target in the presence of parameter uncertainties and external disturbances. However, the J2 perturbing acceleration due to Earth's oblateness, which is typically the largest perturbing acceleration on a satellite below Geosynchronous Earth Orbit (GEO), is not included in the dynamic models of the above six studies, so their methods are imperfect. Considering the common circumstance where the target satellite is uncontrolled, some researchers studied the pose tracking of the chaser and obtained much helpful results. In Ref. [17], a tracking control scheme that ensures accurate relative position tracking as well as attitude synchronization was proposed for proximity operations between a target and a chaser. Jiang et al. [18] investigated the fixed-time faulttolerant control problem of spacecraft rendezvous and docking with a freely tumbling target in the presence of external disturbance and thruster faults. Sun et al. [19] studied relative pose control for a rigid spacecraft with parametric uncertainties approaching to an unknown tumbling target. Considering the close proximity phase, Welsh et al. [20] developed a novel adaptive control strategy to achieve both attitude synchronization and relative position tracking. However, since the targets in the above four studies are all uncontrolled, their methods of pose tracking are not suitable if the target is controlled. There is another kind of limitations in the existing literature: in the pose tracking control problem of some studies, the chaser is asked to directly track the pose of the docking port on the target. Xia and Huo [21] presented a robust adaptive neural networks control strategy for spacecraft rendezvous and docking with the coupled position and attitude dynamics under input saturation. In Ref. [22], almost global asymptotic tracking control was proposed for autonomous body-fixed hovering of a rigid spacecraft over an asteroid. Lee and Vukovich [23] addressed the relative pose tracking control for autonomous rendezvous and docking of two spacecraft. Based on the terminal sliding mode, a robust adaptive terminal sliding mode control scheme was proposed to ensure the finite-time convergence of the relative motion tracking errors using limited control inputs despite the presence of unknown disturbances and moment of inertia uncertainty. In Ref. [24], a pose tracking control scheme was presented for spacecraft formation flying with a decentralized collision-avoidance scheme. Sun and Huo [25,26] studied the relative position tracking and attitude synchronization of non-cooperative spacecraft rendezvous with model uncertainty and external disturbance. Sun et al. [27] presented a six-degree-of-freedom relative motion control method for autonomous spacecraft rendezvous and proximity operations subject to input saturation, full-state constraint, kinematic coupling, parametric uncertainty, and matched and mismatched disturbances. However, among the above seven studies, since the tracked pose (the pose of the docking port on the target) is stationary with respect to the target, their methods are not suitable if the trajectory to be tracked is time-varying. As can be seen from the foregoing, the study of the pose tracking in which the chaser is asked to track a time-varying pose trajectory with respect to the target has a wider application. There are also several researches belong to this kind of study. In Ref. [28], the control problem of the chaser tracking a desired time-varying pose trajectory was solved based upon the sliding mode method. Wang and Sun [4] addressed the pose tracking control problem of the target-chaser spacecraft formation and a robust adaptive terminal sliding mode control law was proposed to ensure the finite time convergence of the relative motion tracking errors despite the presence of model uncertainties and external disturbances. In Ref. [29], a pose tracking controller is proposed to make sure that: (1) the chaser will safely arrive at the docking port of the target, and (2) the chaser will be always suitably oriented to observe the target well. Filipe and Tsiotras [30,31] proposed a nonlinear adaptive pose controller for satellite proximity operations

between a target and a chaser. Their controller requires no information about the mass and inertia matrix of the chaser satellite, and takes into account the gravitational acceleration, the gravity-gradient torque, the J2 perturbing acceleration, and constant but otherwise unknown disturbance forces and torques. However, the methods in the above five studies are imperfect, too. In the numerical calculation, the subtraction calculation of two similar numbers is a morbid problem. For example, 1000000001.234–1000000000 = 1.234. If it is calculated with nine significant digits of precision, the calculation process is: 1.00000000e9 – 1.00000000e9 = 0. Therefore, the subtraction of two similar numbers needs more significant digits, otherwise the relative error of the result will be rather large. Among the above five studies, the dynamic equations and control laws contain subtraction calculation of two similar numbers, so more significant digits are needed for the implementation of their methods. In this paper, the pose tracking control problem for satellite proximity operations between a target and a chaser satellite is addressed, by which we mean that the chaser is required to track a desired timevarying trajectory given in advance with respect to the target. The target is in an elliptic orbit, and can be controlled or uncontrolled. In Ref. [32], a boundary layer adaptive sliding-mode control method was developed for a class of nonlinear uncertain systems based on the nonlinear disturbance observer. This method is applied in this paper, and then an effective pose tracking controller is obtained. This controller requires no information about the mass and inertia matrix of the chaser, and takes into account the gravitational acceleration, the gravity-gradient torque, the J2 perturbing acceleration, and unknown bounded disturbance forces and torques. Furthermore, an updated controller is obtained by combining the aforementioned controller and the unscented Kalman filter (UKF). Comparing with the aforementioned controller, this updated controller estimates the inertial parameters of the chaser through UKF, so it is of better adaptive ability to the initial estimation of the inertial parameters. It should be noted that the dynamic model and control law in this paper do not contain the morbid problem in Refs. [4,28–31], so our method is much easier to be implemented. The contents of this paper are organized as follows. In Section 2, the relative translational and rotational dynamics of the chaser are presented. Then in Section 3, a pose tracking controller is designed. After that numerical simulation results are shown in Section 4. Finally, the conclusions are stated in Section 5. 2. Relative motion kinematics and dynamics In this paper, the target and chaser are assumed to be rigid bodies, and the disturbances including the Earth oblateness (J2) effect, the gravity gradient moment, and unknown bounded disturbances are considered in our dynamic model. The mass center of the target moves freely in an elliptical orbit, and the rotational motion is tumbling. The chaser need to track a desired pose motion to gradually dock with the target. The desired motion is obtained by specifying the relative pose motion with respect to the target, see Section 4. The frame ΣD is called “desired frame” in this paper, and its pose motion is the desired motion. The relative motion kinematics and dynamics of the chaser with respect to ΣD is introduced in this section. This is necessary in designing the pose tracking controller in Section 3. 2.1. Translational motion differential equation In order to describe the chaser's motion, the following frames are introduced. As shown in Fig. 1, the frame ΣN is an Earth-centered inertial frame. Its x axis points to the vernal equinox, its z axis is along the Earth's axis of rotation. The frame ΣH built at the target's mass center T is local-level local-horizon Euler-Hill frame. Its x axis is along the position vector of the target with respect to ΣN, and the z axis is aligned with the orbit angular momentum of the target. The body-fixed frame 136

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with respect to ΣD, which is rather small when comparing with the aforementioned velocity. In their equations, in order to obtain the small number (the acceleration), there must be another large number approximate to the aforementioned velocity in the right end. Therefore, the translational motion differential equations of these five references contain subtraction calculation of two similar numbers. As shown in the introduction, such subtraction calculation is a morbid problem, and more significant digits are needed to implement this calculation. Moreover, there is the same problem in the control law of these five papers. In this paper, this morbid problem is avoided (see Eqs. (3) and (4)), so our method is much easier to be implemented. 2.2. Rotational motion differential equation The rotation vector and angular velocity of ΣC with respect to ΣD are denoted by Θr = θr pr and ωr respectively. Here the unit vector pr is the axis of rotation and θr = ‖Θr ‖ ∈ [0, π ] the angle of rotation. The kinematic equation is

Θ˙ r = A (Θr ) ωCr

Fig. 1. Some frames in this paper.

where A (Θr ) = I3 + matrix of Θr and

ΣC of the chaser is built at its mass center. The position vector of ΣD with respect to ΣH is ρD, ρC is that of ΣC, and ρr represents the position vector of ΣC with respect to ΣD. If the target's mass center moves freely in an elliptical orbit, the relative translational motion [33] of the chaser with respect to ΣH is

a CH = f ρ (ρCH , v CH , t ) +

1

fθ (θr ) =

H

d (ρCH ) dt

=[x˙ C , y˙C , z˙ C ]T , a CH =

H 2

d dt 2

a rH = Gρ (ρrH , v rH , t ) +

mC

+ ΔaH ρ (t )

C

the gravity gradient moment. Meanwhile, ωr is the angular velocity of ΣC with respect to ΣD, so ωCr = ωCC − ACDωDD . Here ωD is the angular velocity of ΣD with respect to ΣN, and ACD is the direction cosine matrix from ΣD to ΣC. Therefore, the rotational motion differential equation of the chaser with respect to ΣD can be written as:

μ ⎤ rT2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(ρ H D (t ) ,

(7)

where JC is the chaser moment of inertia, is the applied control 3μ torque, dCω is the unknown disturbance torque, and MCg = r 5 rCC × JC rCC is

⎧ Θ˙ r = A (Θr ) ωr ⎨ ω˙ Cr = Gω (Θr , ωCr , t ) + J−C 1 [MCC (t ) + dCω (t )] ⎩ C

where

(2)

ω˙ Cr

=

Gω (Θr , ωCr ,

C

d (ωCr ) , dt

t) =

(8)

and

˜ CC JC ωCC J−C 1 [−ω

˜ Cr ACDωDD + MCg (t )] − ACDω˙ DD + ω

(9)

Here ωCC = ACDωDD + ωCr , and the parameters (ωDD (t ) , ω˙ DD (t ) ) represent the known desired rotational motion.

H 2

d

3. Controller design Sliding mode controller is well known for its robustness to system parameter variations and external disturbances [34]. In applying a sliding mode controller, firstly the controller drives and constrains the system states to lie within a neighborhood of the sliding manifold; then since when in the sliding manifold, the closed-loop response of the system becomes totally insensitive to both internal parameter uncertainties and external disturbances, all the system states will gradually converge to zero. In Ref. [32], a boundary layer adaptive slidingmode (ASM) controller is developed for a class of nonlinear uncertain systems based on the nonlinear disturbance observer (NDO). Considering the uncertainties of the system and the external disturbance's upper bound, they designed the boundary layer adaptive sliding-mode control scheme for eliminating the chattering phenomenon appeared in the traditional sliding-mode control and making the tracking error to

(3)

where H H H H H H ⎧ Gρ (ρr , v r , t ) = f ρ (ρC , v C , t ) + Δa J2 (t ) − aD (t ) H H H ⎨ ρCH = ρrH + ρ H v C = v r + v D (t ) D (t ), ⎩

(6)

MCC

H H H H ρr = ρC – ρD, so a rH = a CH − aH D , here a r = dt 2 (ρr ) and aD = dt 2 (ρ D ) . Therefore, the translational motion differential equation of the chaser with respect to ΣD can be written as

FCH (t )

(θr = 0)

˜ CC JC ωCC + MCC (t ) + MCg (t ) + dCω (t ) JC ω˙ CC = −ω

where rT is the distance between the target and the Earth, f is the true anomaly of the target, and μ = 3.986032 × 1014 m3/s2 is the gravitational constant. Assuming that the motion of the target is known, namely the variables (rT, f, f˙ and f¨ ) are given. As is shown in Fig. 1, d

(θr ≠ 0, π )

θr → 0

applied control force, mC is the mass of the chaser, Δa H J2 is the relative effect of the Earth oblateness due to J2 (see Appendix A), ΔaH ρ is the sum of all other unknown disturbance accelerations, and f ρ (ρCH , v CH , t ) is given by

H 2

1 + cos θr 2θr sin θr

is θr = π; namely Eq. (5) is almost completely applicable in this paper. Denote ωC as the angular velocity of ΣC with respect to ΣN. The Euler's dynamic equation of the chaser is

(1)

(ρCH ) =[¨x C , y¨C , z¨C ]T , FCH is the

μ (r T + x C ) 2 + ⎡ 2y˙C f˙ + yC f¨ + x C f˙ − 3/2 [(r T + x C )2 + yC2 + z C2 ] ⎢ μyC ⎢ 2 f ρ (ρCH , v CH , t ) = ⎢− 2x˙ C f˙ − x C f¨ + yC f˙ − 3/2 [(r T + x C )2 + yC2 + z C2 ] ⎢ μz C ⎢− 2 2 2 3/2 ⎢ ⎣ [(rT + xC ) + yC + zC ]

⎧ θ2 − r ⎨1/12 ⎩

˜ r is the skew-symmetric ˜ r )2 , here Θ + fθ (θr )(Θ

Therefore, lim+ fθ (θr ) = 1/12 , and the sole singular point of Eq. (5)

FCH (t ) H H + Δa H J2 (ρC , t ) + Δaρ (t ) mC

where ρCH = [x C , yC , z C ]T is the coordinate array of ρC in ΣH (in this paper, the superscript “T” represents transposition, a vector with a superscript “H” denotes the coordinate array of this vector in ΣH, note that vT does not represent the transposition “vT” but the coordinate matrix of v in ΣT), v CH =

(5) 1 ˜ Θ 2 r

(4)

H vH D (t ) , aD (t ) )

Here represents the known desired translational motion. In the translational motion differential equations of Refs. [4,28–31], the right end contains the velocity of the chaser with respect to ΣN, which is rather large; while the left end is the acceleration of the chaser 137

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and G in Eq. (10) respectively. It should be noted that, in order to make the chaser track the desired translational motion, the desired value x d, tran of x tran (=v rH ) should not be set as 0 simply. Because this only make sure that v rH converges to 0, while ρrH may not converges to 0. Instead, the derivative of x d, tran should be

approach zero. Since their control method is suitable for a class of nonlinear uncertain systems, it can be applied in pose tracking. In this section, their control method is firstly introduced; then a pose tracking controller is obtained by applying their method in our problem; at last, an updated controller is obtained by combining the aforementioned controller and the unscented Kalman filter (UKF).

x˙ d, tran = −2ωn v rH − ωn2 ρrH

3.1. Adaptive sliding-mode controller

where ωn is a positive constant, and x˙ d, tran denotes the desired value of a rH . This equation means that the desired value of ρrH will tend to 0 according to the motion law of the free critically damped spring oscillator. By substituting the above Ftran , Gtran , x d, tran , and x˙ d, tran into the ASM control law Eq. (13), then the control law for position tracking can be obtained as

The control method in Ref. [32] is introduced in this subsection to design a pose tracking controller. Assume the dynamic model of a nonlinear uncertain system is

x˙ = G (x, t ) + F (x) u + D (t , x, u)

(10)

where x∈ Rn is the state vector, u∈ Rn is the control input, G(x,t)∈ Rn , F (x)∈ Rn × n , and D (t , x, u) ∈ Rn is an unknown disturbance of the system. In order to make x(t) track a bounded reference signals xd(t), they designed the following switch function σ

σ = C (x − x d)

ˆ tran FCH = −mC C−1CFdt − mC D

(11)

x˙ d, rot = −2ωn ωCr − ωn2 Θr

(12)

where ϕ (λˆ , σ) = [ϕ (λˆ , σ1), …, ϕ (λˆ , σn )]T , k > 0, ϕ (λˆ , σi ) =

ˆ rot ˜ CC JC ωCC − MCg − JC C−1CFdr − JC D MCC = ω

( ) ( )

Here

φ (λˆ , σ) = [φ (λˆ , σ1), …, φ (λˆ , σn )]T ,

η1 > 0, and η2 > 0. In practice, if the sampling

(18)

where ~ C (ACDωD) − ACDω˙ D] − Cx˙ ˆ ˆ CFdr = C [ω d, rot + k σrot + ‖C‖βd ϕ (λ , σrot) , r D D ˆ rot is the estimation of the disturbance, see σrot = C (ωCr − x d, rot) , and D Eq. (12). 3.3. Modified ASM

(13)

Even if the inertial parameters of the chaser are unknown, the ASM can be also applied in pose tracking. However, if the error of estimation of the inertial parameters is too great, the pose tracking error will be bigger. Since the UKF can be used to estimate the inertial parameters of the chaser, the ASM controller is modified by combining ASM and UKF in this subsection, and the modified controller is called “ASM-up” in this paper. The system disturbance D (t , x, u) in Eq. (10) contains the following two parts: (1) the disturbance aroused by the uncertainty of chaser's inertial parameters, and (2) the unknown disturbance forces and torques in space. In this subsection, the UKF is used to estimate the inertial parameters of the chaser. With the estimation converging to the exact value, the disturbance aroused by the uncertainty of chaser's inertial parameters could be gradually eradicated from the system disturbance D (t , x, u) . At last, the precision of the pose tracking controller is refined. The UKF uses the pose measurement data, control input, and an accurate dynamic model to estimate the motion state and inertial parameters of the chaser. Since it is a mature method, it is not introduced here. The details of using UKF can be found in Appendix B. Assuming that before the estimations of the chaser's inertial parameters ˆ C , JˆC) denotes the estimations of inertial are updated by UKF, (m ˆ tran , D ˆ rot ) for the estimations of the system disparameters, and (D turbances. After the update, the above four estimations are denoted by ˆ ′tran , and D ˆ ′rot respectively. In order to make sure that control ˆ ′C , Jˆ ′C , D m input remain unchanged, the following equations can be established according to Eqs. (16) and (18).

ˆ i) 1 − exp(−λσ ˆ i) , 1 + exp(−λσ

βˆd and λˆ are adaptive parameters updated according to T T −1 T ∂x ˆ˙ ˆ T ⎧ ⎪ βd = η1 ϕ (λ , σ) ((CF (x)) ) ∂u C σ T ⎨ λˆ˙ = η βˆ φ (λˆ , σ)T ((CF (x))−1)T ∂x CTσ ⎪ 2 d ∂u ⎩

(17)

By substituting the above Frot , Grot , x d, rot , and x˙ d, rot into the ASM control law Eq. (13), then the control law for attitude motion tracking can be obtained as

ˆ (t , x, u) ∈ Rn is the output of NDO, z is the state vector of NDO, where D Q (x) = [Q1 (x), ⋯, Qn (x)]T∈ Rn is a nonlinear function vector to be ∂Q (x) designed, and L (x) = ∂x . In order to simplify the design, L(x) can be designed as a diagonal matrix, namely L(x) = diag(L1(x), L2(x), …, Ln(x)), Li(x) > 0, i = 1, 2, …, n. This NDO has a simple structure and a small amount of computation. It is not necessary to assume that the disturbance D (t , x, u) changes very slowly, and a highly accurate estimation value of D (t , x, u) can be obtained. At last, they proposed the following control law ˆ + k σ + ‖C‖βˆ ϕ (λˆ , σ)] u = −(CF (x))−1 [CG (x, t ) − Cx˙ d + CD d

(16)

σtran = CFdt = CGρ − Cx˙ d, tran + k σtran + ‖C‖βˆd ϕ (λˆ , σtran) , where H ˆ tran is the estimation of the disturbance, see Eq. C (v r − x d, tran) , and D (12). The attitude motion tracking controller is described below. Similar to the process of applying ASM controller to position tracking, the state vector x rot = ωCr , the control input urot = MCC , Frot = J−C 1, Grot = Gω (Θr , ωCr , t ) , and the derivative of x d, rot is

c c … c1n ⎡ 11 12 ⎤ c21 c22 … c2n ⎥ ⎢ , c > 0 , cin s n − 1 + ci, n − 1 s n − 2 + ⋯+ci1 = 0 where C = ⎢ ⋮ ⋮ ⋱ ⋮ ⎥ ij ⎢ ⎣ cn1 cn2 … cnn ⎥ ⎦ satisfies Hurwitz criterion, and (CF (x))−1 exists. Then the following nonlinear disturbance observer (NDO) is designed. ˆ (t , x, u) = z + Q (x) ⎧D ⎨ ⎩ z˙ = −L (x) z − L (x)[Q (x) + G (x, t) + Q (x) u]

(15)

(14) ˆ i) σ exp(−λσ φ (λˆ , σi ) = i ˆ i ))2 , (1 + exp(−λσ time is small enough, ∂x ∂u

can be approximately replaced by Δx . It should be noted that, it has Δu been proved in Ref. [32] that even the unknown disturbance D (t , x, u) is considered, the state x(t) in will converge to the desired value xd(t) given in advance if the control law (Eq. (13)) is applied. Therefore, this control law is robust. 3.2. Pose tracking controller Since the aforementioned method ASM is developed for a class of nonlinear uncertain systems, it can be applied in pose tracking, and then a pose tracking controller can be obtained. Firstly, the controller for position tracking is introduced. From Eq. (3) we know that the control input of the position tracking controller is utran = FCH ∈ R3 , the state vector is x tran = v rH ∈ R3 . Through comparing Eqs. (3) and (10), it is obvious that Ftran = 1/ mC and Gtran = Gρ (ρrH , v rH , t ) , here Ftran and Gtran correspond to the functions F 138

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ˆ ′tran = −m ˆ tran ˆ ′C C−1CFdt − m ˆ ′C D ˆ C C−1CFdt − m ˆ CD ⎧ −m ⎪ ~C ~ C Jˆ ωC − MC − ˆ ′rot = ω ωC Jˆ ′C ωCC − MCg − Jˆ ′C C−1CFdr − Jˆ ′C D C C C g ⎨ ⎪ ˆC C−1CFdr − JˆC D ˆ rot J ⎩

Table 1 Orbital elements of the target satellite [29].

(19)

Then the updated estimation of the disturbance can be calculated as

ˆ ′tran = [(m ˆ tran ]/ m ˆC − m ˆ ′C ) C−1CFdt + m ˆ CD ˆ ′C ⎧D ⎪ C C ~ − − 1 ˆ ′rot = (Jˆ ′C ) [ωC Jˆ ′C ωC − Jˆ ′C C 1CFdr − D ⎨ ~ C Jˆ ωC + Jˆ C−1C + Jˆ D ⎪ ˆ ω C Fdr C rot ] C C C ⎩

26553.937 0.729677 63.4 −90 0

T T T ⎧ ρC (t0) = [0, −y1 , 0] + 0.25[r1, r2, r3] ⎨ v TC (t0) = 0.05[r4, r5, r6]T (m / s ) ⎩

(m) (22)

where r1, r2, …, r5 and r6 are uniformly distributed pseudorandom numbers on [-1, 1]. It should be noted that here ρTC represent ρC in ΣT, while [r1, r2, r3]T represents the transposition. As for the initial conditions for attitude motion, the initial rotation angle between the chaser's attitude and that of ΣD is a uniformly distributed pseudorandom numbers on [5,15] deg, and the elements of ωCC (t0) are all uniformly distributed pseudorandom numbers on [-0.25, 0.25] deg/s. The unknown disturbances in Eqs. (3) and (8) are

In this section, numerical simulations are given and two cases (the target is uncontrolled or controlled) are considered. In each case, three pose controllers are applied to control the chaser to track a timevarying pose trajectory, and then the pose tracking errors of each controller are compared. These three controller are the ASM controller, the ASM-up controller, and the non-singular terminal sliding mode (NTSM) controller. The NTSM controller is a common controller, and it can be designed with standard process. As mentioned before, pose tracking means that the chaser is controlled to track a desired time-varying pose trajectory given in advance with respect to the target. In this paper, the given desired trajectory is shown in Fig. 2. As shown in Fig. 2, the desired pose motion (the pose motion of ΣD with respect to ΣT) can be divided into the following three phases: (1) ΣD moves along the straight line P1P2, and do not rotate with respect to ΣT, namely ΣD is parallel with ΣT in this phase; (2) ΣD moves along the circular arc P2 P2 ′, and rotates about the xD axis (parallel with the xT axis) to keep the yD axis pointing to the point O; and (3) ΣD moves along the straight line P2 ′ P3 , and does not rotate with respect to ΣT, namely the yD axis keeps pointing to O. During every phase of these three, the magnitude of the velocity is

−4 + 10−4 [sin (t ) + sin (t ), ΔaH ⎧ 1 2 ρ (t ) = 10 ⎪ ⎪ −sin (t1) − sin (t2), cos (t1) + cos (t2)]T (m/s2) ⎨ dCω (t ) = 10−5 + 10−5 [sin (t1) + sin (t2), ⎪ ⎪ −sin (t1) − sin (t2 ), cos (t1) + cos (t2)]T (N⋅m) ⎩

(23)

where t1 = πt/125, and t2 = πt/200. The parameters of ASM controller are: C = [4, 1, 2; 3, 4, 3; 2, 2, 4], η1 = η2= βˆd (t0)= 10, 2, 0.1, 1; k= λˆ (t0)= ωn= ztran (t0) = zrot (t0) = x d, tran (t0) = x d, rot (t0) = [0, 0, 0]T, and Q(x) and L (x) are given by T 3 3 3 ⎧ Q (x) = [x1 + x1 /3, x2 + x 2 /3, x3 + x 3 /3] ⎨ L (x) = diag(1 + x12 , 1 + x 22 , 1 + x 32) ⎩

(24)

]T .

where x = [x1, x2 , x3 The sampling period is chosen as 0.05 s. The control inputs are required to be bounded by FCH, i ≤ mC × 2 m/s2 and MCC, i ≤ 1 N ⋅m . When estimating the inertial parameters of the chaser by UKF, the pose measurement data of the chaser is required. Assume the standard deviations of the measurement noise are 2 cm and 2 deg respectively. In this paper, (mC , JC ) are the exact values of the chaser's inertial parameters, their initial estimations are denoted by (mC , es0 , JC , es0 ). Let mC, es0 = ker mC and JC, es0 = ker JC . In each case (the target is uncontrolled or controlled), the following six simulation conditions are considered: ker ∈ {0.5, 0.7, 1, 1.2, 1.5, 2} . Since the chaser is own spacecraft, the error of the initial estimations will not be too big. Therefore, the above six simulation examples are reasonable.

(0 ≤ τ ≤ τsf ) (τsf < τ < τf − τsf ) (τf − τsf ≤ τ ≤ τf )

Semimajor axis, km Eccentricity Inclination, deg Argument of perigee, deg Right ascension of the ascending node, deg

0.025 m/s2, y1= 30 m, y2 = 15 m, and y3 = 5 m. The mass of the chaser is mC = 15 kg, and its inertia matrix in the body-fixed frame ΣC is JC = diag(3.0514, 2.6628, 2.1879) kg⋅m2 . The target's inertia matrix in its body-fixed frame ΣT is diag(3.85, 4.36, 4.90) kg⋅m2 . Assuming the target's mass center moves freely in a Molniya orbit, with the orbital elements given in Table 1. At the beginning, the target is at the perigee. The initial conditions for translational motion of the chaser are given by

4. Numerical simulations

am τ ⎧ ⎪ vm ⎨ ⎪ am (τf − τ ) ⎩

Values

(20)

ˆ ′C , Jˆ ′C ) are closer to their exact values Since the estimations (m respectively, the disturbance aroused by the uncertainty of chaser's ˆ tran and D ˆ rot . As a inertial parameters is gradually eradicated from D ˆ ′tran , D ˆ ′rot ) will be closer to the exact result, the updated estimations (D values of the disturbance forces and torques in space respectively. As ˆ tran , D ˆ = z + Q (x) , so once (D ˆ rot ) are updated, ztran shown in Eq. (12), D ˆ ′tran − Q (x tran) , and zrot should be updated should be updated as z′tran = D ˆ ′rot − Q (x rot) . At the next moment, the estimations of the as z′rot = D chaser's inertial parameters can be also updated by UKF according to the newly obtained pose measurement and control input. Then the estimated inertial parameters are more accurate, so the disturbance aroused by the uncertainty of chaser's inertial parameters is further eradicated. In this way, with the estimated inertial parameters converging to the exact value, the disturbance aroused by the uncertainty of chaser's inertial parameters is gradually eradicated. At last, the precision of the pose tracking controller is refined.

vD / T =

Orbital elements

(21)

where τ = t – tsi, tsi is the initial moment of Phase i (i = 1, 2, 3), τsf = vm/am, and τf = (rs - rf)/vm + τsf. The yT-coordinates of the points P1, P2, P2 ′, and P3 are -y1, -y2, y2, and y3. Here vm= 0.1 m/s, am=

4.1. Case A: an uncontrolled tumbling target An uncontrolled tumbling target is common in on-orbit servicing (OOS) missions. This kind of target is considered in this subsection. In simulating the attitude motion of the target, the gravity gradient moment and small disturbance torque dTω, T = dCω on the target are considered, here dTω, T represent d ω, T in ΣT, and the initial conditions are given by

Fig. 2. Desired pose motion of the chaser with respect to ΣT. 139

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Fig. 5. Control force (ker = 1).

Fig. 3. Position tracking error (ker = 1). T ⎧ ΘT /N (t0) = [0,0,0] T ⎨ ωT /N (t0) = [0.025, 0.025, 0.05]T (rad/s) ⎩

(25)

where ωTT /N represents ωT /N in ΣT, and [0, 0, 0]T represents the transposition of [0, 0, 0]. In the condition of ker = 1, the initial estimations of the chaser's inertial parameters are completely exact, the pose tracking error is shown in Figs. 3 and 4. As shown in these two figures:

Fig. 6. Control torque (ker = 1).

(1) After the pose tracking error becomes stable, the position tracking error is within 1 mm, and 0.002 deg for the attitude tracking error. Namely these three controllers are all rather accurate. (2) According to the sequence of high to low control accuracy, the sequences of the three controllers are: ASM-up, ASM, and NTSM; and the accuracy of the first two methods is much higher than that of NTSM. (3) According to the sequence of high to low convergence velocity, the sequences of the three controllers are: ASM-up, NTSM, and ASM. (4) On the curve of position tracking error, there are tiny pinnacles at the moments when the acceleration of the desired motion changes suddenly (when t = 154 s, 629.2 s, and 733.2 s). (5) On the curve of attitude tracking error, there are tiny pinnacles at the moments when the angular acceleration of the desired motion changes suddenly (when t = 154 s and 629.2 s). Fig. 7. Translational motion of the chaser in ΣT (ker = 1).

In the above simulation, the control inputs corresponding to the ASM-up controller are shown in Figs. 5 and 6. As shown in these two figures, the control inputs change suddenly at the moments when the linear and angular acceleration of the desired motion change suddenly. If the chaser is controlled by the ASM-up controller, the trajectory of the chaser's mass center with respect to ΣT is shown in Fig. 7. In this figure, the center of the sphere is at [0 0 0]T, and the radius of the sphere is 10 m; the dotted line represents the desired trajectory, and the solid line for the real trajectory. As shown in this figure, when the chaser is controlled by the ASM-up controller, its real trajectory is very close to the desired one.

Fig. 8. Position tracking error (ker = 1.2).

In the conditions of ker = 1.2, 1.5, 0.7, 2, and 0.5, the pose tracking error increases gradually, this is shown in Figs. 8–17, and the picture in picture shows some details. As shown in these figures: (1) The pose tracking error increases as the initial estimation error of the chaser's inertial parameters increases. (2) For the NTSM controller, even in the conditions of ker = 2 and 0.5, the attitude tracking error is still within 0.2 deg; while the position

Fig. 4. Attitude tracking error (ker = 1). 140

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Fig. 9. Attitude tracking error (ker = 1.2).

Fig. 13. Attitude tracking error (ker = 0.7).

Fig. 10. Position tracking error (ker = 1.5).

Fig. 14. Position tracking error (ker = 2).

Fig. 11. Attitude tracking error (ker = 1.5).

Fig. 15. Attitude tracking error (ker = 2).

Fig. 12. Position tracking error (ker = 0.7).

Fig. 16. Position tracking error (ker = 0.5).

tracking error is up to 5 cm in the condition of ker = 1.2, and it is up to 50 cm if ker = 0.5. (3) For the ASM controller, in the above five conditions, the attitude tracking error is always rather small (within 0.01 deg). Although the position tracking error increases as the initial estimation error of the chaser's inertial parameters increases, the error is still within

2 mm even in the conditions of ker = 2 and 0.5. Therefore, these simulations indicate that the nonlinear disturbance observer (NDO) in ASM can indeed cope with the disturbance caused by the uncertainty of the inertial parameters to some extent. (4) For the ASM-up controller, since the inertial parameters are estimated by UKF, see Figs. 18 and 19, the initial estimation error of 141

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Fig. 17. Attitude tracking error (ker = 0.5).

Fig. 20. Position tracking error (ker = 1).

Fig. 21. Attitude tracking error (ker = 1).

Fig. 18. Estimation and exact value of the elements of JC (ker = 0.5).

Fig. 22. Position tracking error (ker = 1.2).

Fig. 19. Estimation and exact value of mC (ker = 0.5).

the chaser's inertial parameters has little effect on the accuracy of pose tracking. In the above five conditions, the pose tracking error is always rather small (the position tracking error is within 0.1 mm, and the attitude tracking error is within 0.001 deg). 4.2. Case B: a controlled tumbling target In this subsection, an attitude-controlled tumbling target is considered. Except for the rotational motion of the target, the remaining conditions are consistent with the previous subsection. The rotational motion is given by Ref. [29]. T T T ⎧ ΘT /N (t0) = [0,0,0] , ωT /N (t0) = [0,0,0] T ⎨ ω˙ T /N (t ) = 0.01 × [− 0.45 sin (0.1t ), cos (0.2t ), 0.5 cos (0.1t )]T ⎩

Fig. 23. Attitude tracking error (ker = 1.2).

picture in picture shows some details. As shown in these figures: (1) In the case of attitude-controlled tumbling target (Case B), the pose tracking error is bigger when comparing with the case where the target is uncontrolled tumbling (Case A). (2) The pose tracking error increases as the initial estimation error of the chaser's inertial parameters increases.

(26)

Note that here ωTT /N represents ωT /N in ΣT, while [0, 0, 0]T represents the transposition of [0, 0, 0]. In the conditions of ker = 1, 1.2, 1.5, 0.7, 2, and 0.5, the pose tracking error increases gradually, this is shown in Figs. 20–31, and the 142

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Fig. 24. Position tracking error (ker = 1.5).

Fig. 28. Position tracking error (ker = 2).

Fig. 25. Attitude tracking error (ker = 1.5).

Fig. 29. Attitude tracking error (ker = 2).

Fig. 26. Position tracking error (ker = 0.7).

Fig. 30. Position tracking error (ker = 0.5).

Fig. 27. Attitude tracking error (ker = 0.7).

Fig. 31. Attitude tracking error (ker = 0.5).

(3) For the NTSM controller, even in the conditions of ker = 2 and 0.5, the attitude tracking error is still within 2 deg; while the position tracking error is up to 15 cm in the condition of ker = 1.2, and it is up to 110 cm if ker = 0.5. (4) For the ASM controller, in the above six conditions, the attitude tracking error is always rather small (within 0.04 deg). Although the position tracking error increases as the initial estimation error

of the chaser's inertial parameters increases, the error is still within 10 mm even in the conditions of ker = 2 and 0.5. Therefore, these simulations indicate that the nonlinear disturbance observer (NDO) in ASM can indeed cope with the disturbance caused by the uncertainty of the inertial parameters to some extent. (5) For the ASM-up controller, since the inertial parameters are estimated by UKF, the initial estimation error of the chaser's inertial 143

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the initial estimation of the inertial parameters. Moreover, our methods (the ASM and ASM-up controller) are superior to the methods in the existing literatures, e.g., the tracked pose trajectory can be time-varying with respect to the target, the target can be controlled or uncontrolled, and the dynamic model and control law in this paper do not contain morbid problem like subtraction calculation of two similar numbers. Therefore, the research in this paper solves the problem of pose tracking control more perfectly.

parameters has little effect on the accuracy of pose tracking. In the above six conditions, the pose tracking error is always rather small (the position tracking error is within 0.1 mm, and the attitude tracking error is within 0.001 deg). 5. Conclusions In this paper, the pose tracking control problem for satellite proximity operations between a target and a chaser satellite is addressed, and an effective adaptive sliding-mode (ASM) controller is designed. This controller requires no information about the mass and inertia matrix of the chaser, and takes into account the gravitational acceleration, the gravity-gradient torque, the J2 perturbing acceleration, and unknown bounded disturbance forces and torques. Furthermore, an updated controller is obtained by combining ASM and UKF. Comparing with the ASM controller, this updated controller estimates the inertial parameters of the chaser through UKF, so it is of better adaptive ability to

Acknowledgments This work was supported by the Natural Science Foundation of China [grant numbers 11772187, 11802174], the China Postdoctoral Science Foundation [grant number 2018M632104], and the research project of the Key Laboratory of Infrared System Detection and Imaging Technology of Chinese Academy of Sciences [grant number CASIR201702]. The authors greatly appreciate their financial support.

Appendix A. Earth oblateness effect In this section, the relative effect of the Earth oblateness due to J2 is calculated. For an Earth satellite, the perturbing acceleration due to the J2 effect in the inertial frame ΣN is described as

a NJ2 (r N )

x (1 − 5(z / r )2) ⎤ 3μJ2 Re2 ⎡ ⎢ =− y (1 − 5(z / r )2) ⎥ ⎥ 2r 5 ⎢ 2 ⎣ z (3 − 5(z / r ) ) ⎦

(A1)

where r = [x, y, z] is the position vector of the satellite with respect to ΣN, J2 = 0.0010826267, Re = 6378.137 km, and μ = 3.986032 × 10 s2. Therefore, the Δa H J2 in Eq. (1) can be computed as N

T

HN a N (r N ) − a N (r N ) Δa H [ J2 C J2 = A J2 T ]

where

r CN

14

m3/

(A2)

is the position vector of the chaser with respect to ΣN, rTN is that of the target, and AHN represents the direction cosine matrix from ΣN to ΣH.

Appendix B. measurement and process model for the unscented Kalman filter The unscented Kalman filter (UKF) uses the pose measurement data, control input, and an accurate dynamic model to estimate the motion state and inertial parameters of the chaser. Here the motion state consists of pose, linear and angular velocities; while the inertial parameters include the inertial matrix JC and mC. In order to implement UKF, the process models are established in this section. Since the translational (Eq. (B2)) and rotational dynamic equations (Eq. (B4)) of the chaser are decoupled, the estimation of the motion state and inertial parameters can be performed using two separate UKF [35]. These two filters are called translational UKF and rotational UKF respectively in this paper. This separation helps to reduce the dimension of the covariance matrix in UKF. The state vector xtran and the measurement vector ytran for the translational UKF are denoted respectively as H ⎧ ⎡ ρC ⎤ ⎪ ⎪ x tran = ⎢ v H ⎥ ⎢ C⎥ ⎨ ⎣ mC ⎦ ⎪ ⎪ ytran = ρˆ CH = ρCH + wtran ⎩

where

ρˆ CH

(B1)

is the measurement of

ρCH ,

and wtran is the measurement error. According to Eq. (1), the process model of xtran is

H ⎡vC

⎢ x˙ tran = ⎢ f ρ (ρ H , v H , t ) + C C ⎢ 0 ⎣

H (t ) FC

mC

+

H Δa H J2 (ρC ,

t) +

⎤ ⎥

ΔaH ρ (t ) ⎥ ⎥ ⎦

(B2)

The state vector xrot and the measurement vector yrot for the rotational UKF are denoted respectively as

q ⎧ ⎡ C⎤ ⎪ x rot = ⎢ ωC ⎥ C ⎢p ⎥ ⎨ ⎣ lin ⎦ ⎪ ˆ ⎩ yrot = qC = qC ⊗ qw

(B3)

where qC is the attitude quaternion describing the orientation of ΣC with respect to ΣN, the definition of ⊗ can be found in Ref. [35], plin= [Jxx , Jyy , Jzz , Jxy , Jxz , Jyz ]T , (Jxx , ⋯, Jyz ) are the elements of JC, and the noise quaternion qw is a random variable, whose expected value represents a zero rotation. According to Eq. (7), the process model of xrot is

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x˙ rot

C ⎡ Ω (ωC ) qC ⎤ ⎢ = J−C 1 (−ω ˜ CC JC ωCC + MCC (t ) + MCg (t ) + dCω (t )) ⎥ ⎢ ⎥ ⎢ ⎥ ⎣0 ⎦

(B4)

⎡ 0 − ωx − ωy − ωz ⎤ ⎢ ωx ωz − ω y ⎥ 0 1 C Here Ω (ωCC ) = 2 ⎢ ⎥, and ωC = [ωx , ωy , ωz ]T . − ω ω ωx ⎥ 0 y z ⎢ ⎢ ωz ω y − ωx 0 ⎥ ⎣ ⎦ Then the motion state and inertial parameters of the chaser can be estimated by implementing UKF. The detailed process for implementing the rotational UKF can be found in Ref. [35]; as for that of the translational UKF, the standard UKF [36] is used in this paper.

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