Robust adaptive control of spacecraft proximity maneuvers under dynamic coupling and uncertainty

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ScienceDirect Advances in Space Research xxx (2015) xxx–xxx www.elsevier.com/locate/asr

Robust adaptive control of spacecraft proximity maneuvers under dynamic coupling and uncertainty Liang Sun ⇑, Wei Huo Seventh Research Division, Science and Technology on Aircraft Control Laboratory, Beihang University, Beijing 100191, PR China Received 13 May 2015; received in revised form 10 August 2015; accepted 27 August 2015

Abstract This paper provides a solution for the position tracking and attitude synchronization problem of the close proximity phase in spacecraft rendezvous and docking. The chaser spacecraft must be driven to a certain fixed position along the docking port direction of the target spacecraft, while the attitude of the two spacecraft must be synchronized for subsequent docking operations. The kinematics and dynamics for relative position and relative attitude are modeled considering dynamic coupling, parametric uncertainties and external disturbances. The relative motion model has a new form with a novel definition of the unknown parameters. An original robust adaptive control method is developed for the concerned problem, and a proof of the asymptotic stability is given for the six degrees of freedom closed-loop system. A numerical example is displayed in simulation to verify the theoretical results. Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Spacecraft control; Proximity maneuvers; Adaptive control; Dynamic coupling effect; Model uncertainty

1. Introduction In recent years, relative motion control in spacecraft proximity maneuvers has received considerable attention for the potential applications, such as rendezvous and docking, repairing and refueling, and debris removing. Traditionally, the relative position and relative attitude controllers are designed separately. For instance, many relative position controllers for spacecraft relative translational motion were designed. Considering constraint of the obstacles in autonomous rendezvous, the terminal approaching guidance law was proposed by using artificial potential field in Lopez and Mclnnes (1995). Based on feedback linearization method, the nonlinear guidance laws for spacecraft rendezvous and hovering were designed in Kluever (1999) and Dang et al. (2014), respectively. In order to deal with the disturbances in rendezvous and ⇑

Corresponding author. E-mail address: [email protected] (L. Sun).

docking, a relative position controller was derived based on two-step sliding mode technique in Park et al. (1999). By using multi-pulse technology, the spacecraft relative navigation and guidance were solved in Hablani et al. (2002) to achieve rendezvous and docking in circular orbit. Consider the parametric uncertainties in relative position model, a robust controller based on linear matrix inequality was designed by Yang et al. (2010) so that the orbit transfer process can be achieved with small thrust. Based on state-space dynamics of relative position motion under a central gravitational field, a nonlinear feedback low-thrust controller was proposed in Gurfil (2007) to finish the orbital transfer missions. Multi-objective H 1 state feedback controller was proposed for orbital rendezvous in Gao et al. (2009) to achieve good robustness of system. Since the thruster failures of the spacecraft, a constrained reliable controller was developed by Lyapunov method in Yang and Gao (2013). A model predictive control system was designed and implemented in Hartley et al. (2012) to guide and control a chasing spacecraft during rendezvous

http://dx.doi.org/10.1016/j.asr.2015.08.029 0273-1177/Ó 2015 COSPAR. Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: Sun, L., Huo, W. Robust adaptive control of spacecraft proximity maneuvers under dynamic coupling and uncertainty. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.08.029

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L. Sun, W. Huo / Advances in Space Research xxx (2015) xxx–xxx

with a passive target spacecraft in an elliptical or circular orbit. In Guglieri et al. (2014), the design of the guidance navigation and control system of a ground test-bed was also presented for spacecraft rendezvous and docking experiments to validate the proportional–integrative–deri vative control law and pulse width modulators. An optimal control method based on h-D technique was used in Gao et al. (2013) to save control energy in spacecraft rendezvous missions. Consider the mass uncertainty and thrust uncertainty in the relative position dynamics, adaptive relative translational controllers were developed in Wu et al. (2013) and Yoon et al. (2014) for chasing spacecraft. Meanwhile, attitude synchronization controllers are mainly studied in the spacecraft formation problem, and many representative attitude synchronization controllers based on different communication protocols between multiple spacecraft were proposed in recent years, such as consensus-based approach (Okoloko and Kim, 2014), behavior-based method (Lawton and Beard, 2002), virtual structure approach (Ren and Beard, 2004), neighborhoodbased approach (Dimarogonas et al., 2009), passivitybased method (Bai et al., 2008), input-to-state stable proportional–derivative control (Hu et al., 2013). However, due to modeling simplification, the dynamic coupling effect between relative translation and relation rotation was neglected in above-mentioned works. Because of the dynamic coupling effect between two docking ports in close proximity maneuvers, the separate designing of translational and rotational controllers is not applicable. Thus, coupled control problem of relative translation and relative rotation between two spacecraft has received attention in recent years. In view of the robustness property of the state dependent Riccati equation technology, a unified robust controller for relative position and relative attitude was designed in Stansbery and Cloutier (2000). An asymptotically stable partial state feedback adaptive controller was proposed in Singla et al. (2006) to solve the autonomous spacecraft rendezvous and docking problem under measurement noises. A robust adaptive controller was presented in Subbarao and Welsh (2008) for autonomous rendezvous and docking, and ultimate boundedness of the errors was also derived. A kind of nonlinear model for relative translational and rotational motion between two spacecraft was established with considering kinematic coupling effect in Kristiansen et al. (2008). For the problem of the spacecraft relative motion dynamics in the absence of parametric uncertainties and external disturbances, a suboptimal solution to the Hamilton–Jacobi–Bell man (HJB) equation with adding perturbations to the cost function was firstly addressed in Xin and Pan (2010) by using a novel h-D technique. Then the problem of the autonomous spacecraft rendezvous and docking subject to model uncertainties was studied and the suboptimal controller based on h-D technique was extended in Xin and Pan (2012). With considering thrust magnitude constraints and spacecraft approach velocity, the relative position control problem in autonomous rendezvous and docking was

converted to a model predictive optimization problem in Di Cairano et al. (2010). Without considering the model uncertainty, a unit dual quaternion-based feedback linearization controller was proposed for the rigid body six degrees of freedom tracking in Seo (2015), Wang and Yu (2013) and Filipe and Tsiotras (2014), where the dualquaternion-based error dynamics was firstly deduced for the position and attitude tracking. In order to improve the computational efficiency of model predictive control when applied to the autonomous proximity maneuvers, an explicit control law combined with multi-parametric programming techniques was derived in Leomanni et al. (2014) for real-time implementation on simple hardware. In Di Mauro et al. (2015), a nonlinear control algorithm based on differential algebra was proposed to obtain a higher-order Taylor expansion of the state-dependent Ricatti equation solution in nonlinear optimal control problem, so that the computational effort of the spacecraft proximity controller can be reduced. By knowing the target motion information in real time, a relative motion controller was proposed in Zhang and Duan (2014) for the chaser spacecraft. A trajectory planning approach based on differential flatness and stochastic optimization was studied in Kobilarov and Pellegrino (2014) for shorttime-scale proximity operations of an under-actuated spacecraft. Many aforementioned controllers reveal some drawbacks, such as exactly knowing target inertial parameters, neglecting parametric uncertainties and external disturbances of chaser spacecraft. Motivated by aforementioned observations, the mechanical model of spacecraft proximity maneuvers is presented, and the novel adaptive feedback controllers for the relative translation and rotation are developed with direct estimation of unknown mass and inertia matrix of the chaser, the symmetric property of the estimated inertial matrix is always maintained based on a new proof of overall system stability. Compared with the adaptive controllers in Pan and Kapila (2001) and Zhang and Duan (2012), the amount of online estimated parameters in the proposed adaptive method is reduced largely from 392 and 324 to 11 by combining the element-wise and norm-wise adaption laws, so that the computational burden of the proximity controller is decreased. Compared with the uniformly bounded stability in Singla et al. (2006), Zhang and Duan (2014, 2012), asymptotical convergence of the closed-loop system states is guaranteed in this paper. Moreover, with sacrificing the asymptotical stability, undesired control chattering is avoided by employing a saturation function to replace the signum function in proposed controllers. The proposed adaptive control scheme is a unified design for the rigid body motion and can be easily extended to other engineering applications. The rest of this paper is arranged as follows. Section 2 describes the model of the spacecraft proximity maneuvers, and illustrates the objective of controller design. The designing procedure of a robust adaptive controller is given in detail in Section 3 proves the stability of the closed-loop

Please cite this article in press as: Sun, L., Huo, W. Robust adaptive control of spacecraft proximity maneuvers under dynamic coupling and uncertainty. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.08.029

L. Sun, W. Huo / Advances in Space Research xxx (2015) xxx–xxx

system. Simulation example is then shown in Section 4. Section 5 concludes the paper. Throughout this paper, the skew-symmetric matrix SðaÞ 2 R33 derived from a vector a ¼ ½a1 ; a2 ; a3 T 2 R3 is defined as 2 3 0 a3 a2 6 7 0 a1 5 SðaÞ ¼ 4 a3 a2 a1 0 and it satisfies aT SðaÞ ¼ 0; kSðaÞk ¼ kak, and SðaÞb ¼ SðbÞa; bT SðaÞb ¼ 0 for any b 2 R3 . kak and kak1 denote vector 2-norm and vector 1-norm of a. kAk and kAkF represent the induced matrix 2-norm and Frobenius-norm of 2 A 2 Rnn . trfAT Ag ¼ kAkF , where trfg is the trace operation for matrix. I n and On are n  n unit and zero matrices, T respectively. sgnðaÞ , ½sgnða1 Þ; sgnða2 Þ; sgnða3 Þ , where 8 > < 1; ai < 0 sgnðai Þ ¼ 0; ai ¼ 0 ; i ¼ 1; 2; 3: > : 1; ai > 0 2. Problem formulation 2.1. Chaser and target dynamics The control problem that a uncertain chaser spacecraft tracks a tumbling space target is investigated in this paper. Related frames and vectors are defined in Fig. 1, where F i , fOxi y i zi g is the Earth-centered inertial frame, F c , fCxyzg and F t , fTxt y t zt g are the chaser and target spacecraft body-fixed frames, respectively; origins C and T are mass centers of the chaser and target spacecraft, respectively; point P is the desired proximity position along the docking port direction of space target; solid arrows fr; re g and dashed arrows frt ; rpt ; pt g are related position vectors expressed in frame F c and F t , respectively. The objective is to control the chaser such that its mass center C tracks point P and frame F c tracks frame F t .

3

The position of mass center C and the attitude of frame F c with respect to frame F i can be described by following chaser kinematics and dynamics equations expressed in frame F c , if the modified Rodrigues parameters (MRP) are used for attitude parametrization (Xin and Pan, 2011). 8 r_ ¼ v  SðxÞr > > > < r_ ¼ GðrÞx ð1Þ > m_v þ mSðxÞv ¼ f þ d f > > : J x_ þ SðxÞJ x ¼ s þ d s where GðrÞ ¼ 14 ½ð1  rT rÞI 3 þ 2SðrÞ þ 2rrT ; r 2 R3 is the position and r is the MRP attitude; v; x 2 R3 are linear and angular velocities; f ; s 2 R3 are the control force and torque; d f ; d s 2 R3 are the disturbance force and torque; m 2 R and J 2 R33 are the chaser mass and the positive definite symmetric inertia matrix, respectively. With ignoring the external forces and torques, kinematics and dynamics of the tumbling target are described in frame F t by Xin and Pan (2011) 8 r_ t ¼ vt  Sðxt Þrt > > > < r_ ¼ Gðr Þx t t t ð2Þ > mt v_ t þ mt Sðxt Þvt ¼ 0 > > : J t x_ t þ Sðxt ÞJ t xt ¼ 0 where Gðrt Þ ¼ 14 ½ð1  rTt rt ÞI 3 þ 2Sðrt Þ þ 2rt rTt ; rt ; rt are target position and attitude; vt ; xt 2 R3 are target linear and angular velocities. mt 2 R and J t 2 R33 are the target mass and positive definite symmetric inertia matrix, respectively. Remark 1. For the tumbling target dynamics in (2), by defining kinetic energy function EðtÞ ¼ 12 ðmt vTt vt þ _ xTt J t xt Þ P 0 and calculating its time derivative EðtÞ ¼ mt vTt Sðxt Þvt  xTt Sðxt ÞJ t xt  0, we know EðtÞ  Eð0Þ < 1. Thus, the tumbling target linear velocity vt and angular velocity xt are always bounded.

2.2. Relative motion dynamics The MRP of relative attitude is defined by Shuster (1993) re ¼

rt ðrT r  1Þ þ rð1  rTt rt Þ  2Sðrt Þr 1 þ rTt rt rT r þ 2rTt r

ð3Þ

and the corresponding rotation matrix from F t to F c is R ¼ I3 

4ð1  rTe re Þ ð1 þ

rTe re Þ2

Sðre Þ þ

8 ð1 þ rTe re Þ2

S 2 ðre Þ

ð4Þ

According to Fig. 1, the position and velocity of point P with respect to O represented in frame F t can be obtained by Fig. 1. Definitions of frames and vectors.

r p t ¼ r t þ pt ;

vpt ¼ vt þ Sðxt Þpt

ð5Þ

Please cite this article in press as: Sun, L., Huo, W. Robust adaptive control of spacecraft proximity maneuvers under dynamic coupling and uncertainty. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.08.029

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where the pt 2 R3 is a constant vector in frame F t . The relative position, relative velocity, and relative angular velocity are described in frame F c by re ¼ r  Rrpt ;

ve ¼ v  Rvpt ;

xe ¼ x  Rxt

ð6Þ

Substituting (6) into (1) and using the identities R_ ¼ Sðxe ÞR; r_ pt ¼ vpt  Sðxt Þrpt and R1 ¼ RT yield the relative motion equations expressed in frame F c as 8 r_ e ¼ ve  SðxÞre > > > < r_ ¼ Gðr Þx e e e ð7Þ > m_ve ¼ m½SðxÞv þ R_vpt  Sðxe Þðv  ve Þ þ f þ d f > > : J x_ e ¼ SðxÞJ x  J ½Rx_ t þ SðxÞxe  þ s þ d s where Gðre Þ ¼ 14 ½ð1  rTe re ÞI 3 þ 2Sðre Þ þ 2re rTe ; R_vpt can be calculated from (5), (2), (6) and RSðaÞ ¼ SðRaÞR for any a 2 R3 as R_vpt ¼ R½_vt þ Sðx_ t Þpt  ¼ RSðxt Þvt  RSðpt Þx_ t ¼ SðRxt Þ½Rvpt  RSðxt Þpt   RSðpt Þx_ t ¼ Sðx  xe Þ½v  ve  Sðx  xe ÞRpt   RSðpt Þx_ t

ð8Þ

and x_ t can be calculated from (2) and (6) as x_ t ¼ J 1 t Sðxt ÞJ t xt T T ¼ J 1 t SðR ðx  xe ÞÞJ t R ðx  xe Þ

ð9Þ

To facilitate the controller design later, we rewrite (7) as 8 r_ e ¼ ve  SðxÞre > > > < r_ e ¼ Gðre Þxe > v_ e ¼ SðxÞve  S 2 ðx  xe ÞRpt þ n1 þ m1 f þ m1 d f > > : x_ e ¼ J 1 SðxÞJ x  SðxÞxe þ n2 þ J 1 s þ J 1 d s

Assumption 2. The chaser can directly measure its motion variables fr; v; r; xg and relative motion variables fre ; ve ; re ; xe g with the measurement devices mounted on the chaser body (Segal et al., 2014; Kim et al., 2007). However, the tumbling target motion variables frt ; vt ; rt ; xt g are assumed to be unavailable directly for the chaser. 2.3. Control objective Based on the model (10) under Assumptions 1 and 2, the problem of positioning the chaser at a safely desired position pt with respect to the target and rotating the chaser to coincide with the target attitude can be formulated as a regulation problem. The control objective is to design robust control inputs f and s in model (10), such that the controlled spacecraft proximity maneuvers system with above-mentioned uncertainties is capable to guarantee limt!1 re ¼ limt!1 ve ¼ limt!1 re ¼ limt!1 xe ¼ 0. Remark 3. It is worth mentioning that point P in Fig. 1 can be viewed as the position of target’s docking port or a certain fixed point along the direction of target’s docking port axis. Thus, if the control objective in this paper is finished, then the missions of spacecraft docking maneuvers or a proximity maneuver for acquisition of docking port axis can be achieved. Spacecraft proximity and docking maneuvers perform two significant tasks, including synchronizing the attitude of two spacecraft and driving the chaser to the point P, such that the docking ports of two spacecraft are aligned and docked safely.

3. Controller design and stability analysis

ð10Þ where T T n1 ¼ RSðpt ÞJ 1 t SðR ðx  xe ÞÞJ t R ðx  xe Þ;

n2 ¼

T RJ 1 t SðR ðx

 xe ÞÞJ t R ðx  xe Þ: T

Remark 2. SðxÞre and n1 in (10) reflect that the relative position motion between the two spacecraft is affected significantly by the relative attitude motion between them. This indicates the dynamic coupling effect for spacecraft proximity maneuvers. Besides, both n1 and n2 contain the target unknown inertia matrix J t , this is also the dynamic coupling effect. Assumption 1. m is an unknown positive scalar and J is an unknown symmetric positive definite matrix; J t is an unknown symmetric positive definite matrix; d f and d s are unknown disturbances, and they are bounded by unknown constant scalars d 1 and d 2 such that kd f k 6 d 1 and kd s k 6 d 2 .

Define two variables 

s 1 ¼ v e þ k1 r e s2 ¼ xe þ k2 re

ð11Þ

where ki > 0ði ¼ 1; 2Þ. Then (10) can be rewritten as 8 r_ e ¼ ve  SðxÞre > > > < r_ e ¼ Gðre Þxe > s_ 1 ¼ g 1 þ n1 þ m1 f þ m1 d f > > : s_ 2 ¼ g 2 þ n2  J 1 SðxÞJ x þ J 1 s þ J 1 d s

ð12Þ

where g1 ¼ k1 ½ve  SðxÞre   SðxÞve  S 2 ðx  xe ÞRpt ; g2 ¼ k2 Gðre Þxe  SðxÞxe . Defining unknown parameters a ¼ kJ 1 t kkJ t k; b ¼ 1 kJ kkJ k; df ¼ dm1 , and ds ¼ d 2 kJ 1 k, we can design novel adaptive controllers for spacecraft proximity maneuvers in following theorem. Theorem 1. Consider the system model (12) under Assumptions 1 and 2, if design the control force and torque inputs

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(

^ 1 f ¼ mh 2 f þ ^ h1 ¼ k 1 re þ k 2 s1 þ g1 þ ðd^ akpt kkx  xe k Þsgnðs1 Þ

ð13Þ 8 ^ > < s ¼  J h2 s þ ^ h2 ¼ 14 k 3 ð1 þ rTe re Þre þ k 4 s2 þ g 2 þ ðd^ akx  xe k2 > : 2 þ^ bkxk Þsgnðs2 Þ ð14Þ and the corresponding adaptive laws 8 > ^_ ¼ d1 sT1 h1 m > > > > > > J^_ ¼ 12 d2 ðs2 hT2 þ h2 sT2 Þ > > > > 2 > <^ a_ ¼ c1 kx  xe k ðkpt kks1 k1 þ ks2 k1 Þ _ ^ > b ¼ c2 kxk2 ks2 k1 > > > > _ > > > d^f ¼ c3 ks1 k1 > > > > : ^_ d s ¼ c4 ks2 k1

ð15Þ

where ^ aða ¼ m; J ; a; b; df ; ds Þ is the estimation of a and T J^ ¼ J^; k i > 0 and ci > 0ði ¼ 1; 2; 3; 4Þ; di > 0ði ¼ 1; 2Þ.  ; d^  are bounded, and ^ J^; ^ Then the estimations m; a; ^ b; d^ f

s

limt!1 re ¼ limt!1 ve ¼ limt!1 re ¼ limt!1 xe ¼ 0.

Proof. Substituting control inputs (13) and (14) into (12) gives the closed-loop system 8 r_ e ¼ ve  SðxÞre > > > < r_ e ¼ Gðre Þxe > s_ 1 ¼ g1 þ n1  mm^ h1 þ m1 d f > > : s_ 2 ¼ g2 þ n2  J 1 SðxÞJ x  J 1 J^h2 þ J 1 d s

Since J T ¼ J ; J T ¼ J 1 ; J^T ¼ J^, and J~T ¼ J~, then trfJ~_ J 1 J~g ¼ trfJ~J 1 J~_ g; sT2 J 1 J~h2 ¼ hT2 J~J 1 s2 ;     sT J 1 J~h2 ¼ tr h2 sT J 1 J~ ¼ tr J~J 1 s2 hT ; hT2 J~J 1 s2

¼

trfs2 hT2 J~J 1 g

2

¼ trfJ

1

1

1

þ d~f ks1 k1 þ d~s ks2 k1

ð18Þ

Then substituting h1 in (13) and h2 in (14) into (18) and using the facts rTe SðxÞre ¼ 0 and rTe Gðre Þ ¼ 14 ð1 þ rTe re ÞrTe lead to 1 V_ ðtÞ ¼ k 1 k1 rTe re  k 2 sT1 s1 þ sT1 n1 þ sT1 d f  m 2  sT d^f þ ^akpt kkx  xe k sgnðs1 Þ   1  k 3 k2 1 þ rTe re rTe re  k 4 sT2 s2 þ sT2 n2 4 2  sT2 J 1 SðxÞJ x þ sT2 J 1 d s  sT2 ðd^s þ ^akx  xe k 2 ^ þ bkxk Þsgnðs2 Þ þ ~akx  xe k2 ðkp kks1 k

ð16Þ

 1 1 ~2 V ðtÞ ¼ k 1 rTe re þ k 3 rTe re þ sT1 s1 þ sT2 s2 þ m 2 2d1 m 1 1 2 1 ~2 1 ~ 1 ~2 ~ d2f þ d þ trfJ~J 1 J~g þ a þ b þ 2d2 2c1 2c2 2c3 2c4 s ð17Þ

2

1 ~m ~_ m V_ ðtÞ ¼ k 1 rTe r_ e þ k 3 rTe r_ e þ sT1 s_ 1 þ sT2 s_ 2 þ d1 m 1 1 1 1 _ þ trfJ~J 1 J~_ g þ ~a~a_ þ ~b~b_ þ d~f d~f d2 c1 c2 c3 1 _ þ d~s d~s ¼ k 1 rTe ½s1  k1 re  SðxÞre  c4 þ k 3 rTe Gðre Þðs2  k2 re Þ   ~ þm 1 m h1 þ d f þ sT1 g 1 þ n1  m m

T 1 þ s2 g2 þ n2  J SðxÞJ x  J 1 ðJ~ þ J Þh2 þ J 1 d s   ~ 1  m þ sT1 h1 þ tr J~J 1 s2 hT2 þ h2 sT2 2 m 2 2 þ ~akx  xe k ðkpt kks1 k1 þ ks2 k1 Þ þ ~bkxk ks2 k1 þ d~f ks1 k1 þ d~s ks2 k1 ¼ k 1 rTe ½s1  k1 re  SðxÞre    1 þ k 3 rTe Gðre Þðs2  k2 re Þ þ sT1 g1 þ n1  h1 þ d f m

T 1 1 þ s2 g2 þ n2  J SðxÞJ x  h2 þ J d s 2 2 þ ~akx  xe k ðkpt kks1 k þ ks2 k Þ þ ~bkxk ks2 k

1

Define the parameter estimation errors ~ a ¼ ^a  aða ¼ m; J ; a; b; df ; ds Þ. The stability of the closed-loop system is proved by taking a function

2

5

1 ~

J h2 sT2 g:

Taking the time derivative of (17) along the trajectory of closed-loop system (16) gives rise to

t

1

2 þ ks2 k1 Þ þ ~bkxk ks2 k1 þ d~f ks1 k1 þ d~s ks2 k1

ð19Þ

Since sT1 n1 þ sT2 n2 6 akx  xe k2 ðkpt kks1 k1 þ ks2 k1 Þ; 1 T s d f þ sT2 J 1 d s 6 df ks1 k1 þ ds ks2 k1 ; m 1 sT2 J 1 SðxÞJ x 6 bkxk2 ks2 k1 ; then from ~a ¼ ^a  aða ¼ m; J ; a; b; df ; ds Þ, we have 1 V_ ðtÞ 6 k 1 k1 rTe re  k 3 k2 ð1 þ rTe re ÞrTe re  k 2 sT1 s1 4 T  k 4 s2 s2 þ ða  ^aÞkx  xe k2 ðkpt kks1 k1 þ ks2 k1 Þ 2 s Þks2 k þ ðb  ^bÞkxk ks2 k1 þ ðdf  d^f Þks1 k1 þ ðds  d^ 1 2 ~ ~   þ d ks k þ d ks k þ ~akx  x k ðkp kks k þ ks k Þ f

1 1

s

2 1

e

t

1 1

2 1

2 þ ~bkxk ks2 k1 1 2 6 k 1 k1 rTe re  k 3 k2 rTe re  k 2 sT1 s1  k 4 sT2 s2 6 l0 kfk 4 ð20Þ

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where l0 , minfk 1 k1 ; 14 k 3 k2 ; k 2 ; k 4 g; f , ½rTe ; rTe ; sT1 ; sT2  . Since V ðtÞ P 0 and V_ ðtÞ  0; V ðtÞ is monotonically decreasing along the closed-loop system trajectory and is bounded by zero. Hence V ðtÞ has a finite limit V ð1Þ as t ! 1 and 0 6 V ð1Þ 6 V ðtÞ 6 V ð0Þ < 1;

8t P 0

Meanwhile, Eq. (20) implies Z 1 Z 1 1 V ð0Þ  V ð1Þ V ð0Þ 2 V_ ðtÞdt 6 kfk dt 6  6 <1 l l0 l0 0 0 0 This means f is square integrable, thus re ; re , s1 , and s2 are square integrable. Since J 1 is a real symmetric and positive definite matrix, then there exists an orthogonal matrix U such pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi that U T C T CU ¼ J 1 , where C , diagf c1 ; c2 ; c3 g; ci ði ¼ 1; 2; 3Þ are three positive eigenvalues of J 1 ; U is derived by three corresponding eigenvectors. Thus, we know trfJ~J 1 J~g ¼ trfJ~T U T C T CU J~g ¼ kCU J~k2F . Recalling the definition of V ðtÞ yields 1 1 2 2 l1 kfk þ ckbk 6 V ðtÞ < 1 2 2

ð21Þ

l1 , minf1; k 1 ; k 3 g; c , minf1=ðd1 mÞ; 1=d2 ; 1=c1 ; T f ; d~ s  . ~ kCU J~kF ; ~ 1=c2 ; 1=c3 ; 1=c4 g, b , ½m; a; ~ b; d~ Thus, kbk < 1 and kfk < 1. From (11), kbk < 1, and the constant matrices C and U, we also have j^ aj < 1ða ¼   ^ m; a; b; d f ; d s Þ; kJ kF < 1; kre k < 1; kre k < 1; kve k < 1, and kxe k < 1. Furthermore, from (5), (6), kRk ¼ 1 and Remark 1, we can obtain kvk ¼ kve þ R½vt þ Sðxt Þpt k < 1; kxk ¼ kxe þ Rxt k < 1. From kfk < 1; j^ aj < 1ða ¼ m; a; b; df ; ds Þ and kJ^kF < 1, we have kgi k < 1ði ¼ 1; 2Þ and khi k < 1ði ¼ 1; 2Þ. Then from (13) and (14), we know kf k < 1 and ksk < 1. Moreover, from (16), we have k_re k < 1; kr_ e k < 1; k_s1 k < 1 and k_s2 k < 1. These mean re ; re ; s1 , and s2 are uniformly continuous. Based on the Barbalat Lemma (Sastry and Bodsom, 1989), we prove limt!1 re ¼ limt!1 re ¼ limt!1 s1 ¼ limt!1 s2 ¼ 0. Therefore, from (11), we can conclude limt!1 re ¼ limt!1 ve ¼ limt!1 re ¼ limt!1 xe ¼ 0. h where

Remark 4. In spite of the dynamic coupling between relative translation and relative rotation in the six degrees of freedom system, the robust adaptive controllers (13) and (14) are developed separately for relative position and relative attitude motions, such that the controller parameters fk 1 ; k 2 ; k1 g for relative translation and fk 3 ; k 4 ; k2 g for relative rotation can be regulated independently to satisfy their own bandwidths. Remark 5. In view of practical implementation, the data process limitation of the hardware leads to the high risk and heavy computational burden of six-degrees-offreedom controller on board computer, so the relative

translational controller and relative rotational controller are separated to implement in different hardware to decrease the computational burden and avoid the risk of data process limitation. Moreover, the amount of the online estimations in (15) is only 11, so that the computational resources are sharply reduced by comparing with the adaptive laws in Pan and Kapila (2001) and Zhang and Duan (2012). Remark 6. The adaptive terms in h1 and h2 are very important, since they are used to compensate the unknown parametric uncertainties, external disturbances, and dynamic coupling effect in system model. Moreover, the proposed controllers (13) and (14) become proportional-derivative controllers in the absence of adaptive terms, while the control accuracy and dynamic response performance cannot be guaranteed in spite of the simple structure of the proportional-derivative control. Considering designed controllers (13) and (14) with adaptive laws in (15), the actuator’s outputs of the chaser performs discontinuous since the signum function sgnðÞ is involved, which will lead to chattering in either numerical simulations or applications. In order to eliminate chattering, the terms sgnðs1 Þ and sgnðs2 Þ are replaced by a saturation function satðsi Þði ¼ 1; 2Þ yielding  sgnðsi Þ if ksi k > e satðsi Þ ¼ 1 ð22Þ s if ksi k 6 e e i where e > 0 is the bound to be designed. Subsequently, the stability analysis of the closed-loop system with the modified version of the controllers (13) and (14) is presented in the following theorem. Theorem 2. Consider the system model (12) under Assumptions 1 and 2, if the signum functions sgnðs1 Þ and sgnðs2 Þ in controllers (13) and (14) are replaced by saturation function (22), then it can be guaranteed that the closed-loop system states re ; re ; ve , and xe uniformly ultimately converge to the small neighborhood of zero. Proof. Substituting the saturation function (22) into controllers (13) and (14), we can obtain from (19) that 1 V_ ðtÞ ¼ k 1 k1 rTe re  k 2 sT1 s1 þ sT1 n1 þ sT1 d f m  2  sT1 d^f þ ^akpt kkx  xe k satðs1 Þ   1  k 3 k2 1 þ rTe re rTe re  k 4 sT2 s2 þ sT2 n2 4  sT2 J 1 SðxÞJ x þ sT2 J 1 d s  2 2  sT2 d^s þ ^akx  xe k þ ^bkxk satðs2 Þ 2

þ ~akx  xe k ðkpt kks1 k1 þ ks2 k1 Þ 2 þ ~bkxk ks2 k1 þ d~f ks1 k1 þ d~s ks2 k1

ð23Þ

From (22), if ks1 k > e and ks2 k > e, we know

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1 V_ ðtÞ 6 k 1 k1 rTe re  k 2 sT1 s1  k 3 k2 ð1 þ rTe re ÞrTe re  k 4 sT2 s2 4 ð24Þ Thus the boundedness of the s1 and s2 are ultimately guaranteed by ks1 ðtÞk 6 e and ks2 ðtÞk 6 e as t ! 1. Moreover, recalling (11), we get   

T  e e  T T T    ;  rTe ; xTe  6 ð25Þ  r e ; ve  6 kmin ðQ1 Þ kmin ðQ2 Þ where Q1 ¼ ½k1 I 3 ; I 3 ; Q2 ¼ ½k2 I 3 ; I 3 , and kmin ðQi Þði ¼ 1; 2Þ are minimum singular values of matrices Qi ði ¼ 1; 2Þ. Therefore, the closed-loop system states re ; re ; ve , and xe uniformly ultimately converge to the small neighborhood of zero. Remark 7. As can be seen from (25), smaller e and larger kmin ðQi Þði ¼ 1; 2Þ will improve steady-state precision, but easily cause the chattering. Thus, in practice, a tradeoff should be made in choosing proper ki ði ¼ 1; 2Þ and e in order to guarantee high control accuracy and eliminate of chattering simultaneously. The design procedures of the proposed controllers (13) and (14) can be summarized as follows. First, from the estif ; d^ s and system states in the current sammations ^ a; ^ b; d^ pling time, terms h1 and h2 in the current sampling time are derived. Second, the control inputs f and s in the current sampling time can be calculated based on h1 ; h2 and ^ J^. Third, the relative motion model equations in (10) m; and the adaptive laws in (15) can be used to calculate the variations of system states and online estimations in the current sampling time, then by using the Euler integral method, all system states and estimations in the next sampling time can be derived. Fourth, estimations f ; d^ s ; m; ^ ^ J^ and system states in the next sampling time a; ^ b; d^ are computed, then terms h1 ; h2 and control inputs f ; s in the next sampling time are also derived successively. Thus the iteration of closed-loop control system can be realized step by step. By using the proposed controllers (13) and (14), the asymptotical convergence of the closed-loop system states is guaranteed, while the online estimations are uniformly bounded. If the signum functions in the proposed controllers are replaced by saturation (22), then the states of closed-loop system converge to the small neighborhood of zero rather than asymptotical stable. 4. Simulation example In this section, simulation describes an example of the autonomous proximity mission in orbit, in which the target spacecraft has a lower dynamic operating condition so that the docking can be safely carried out. After the relative position and relative attitude have been precisely controlled, the docking ports of two spacecraft will be well aligned without relative motions. Simulation results are demonstrated the performance of the developed controller.

7

Table 1 Initial values in simulation. Variable

Value

Unit T

½1; 1; 1  7:078  10 ½2; 3; 2T ½0; 0; 0T ½0; 0;p0ffiffiffiT pffiffiffi T ½50= 2; 0; 50= 2 T ½0:5; 0:5; 0:5 ½0:5; 0:6; 0:7T ½0:02; 0:02; 0:02T

r v r x re ve re xe

6

m m=s  rad=s m m=s  rad=s

The initial simulation values are shown in Table 1. The T desired position for chaser in frame F t is pt ¼ ½0; 5; 0 ðmÞ. The initial values of the adaptive parameters are set as ^ d^f ð0Þ ¼ d^s ð0Þ ¼ 0; ^að0Þ ¼ ^bð0Þ ¼ 1:2; mð0Þ ¼ 50 ðkgÞ, and 2 3 600 20 50 6 7 J^ð0Þ ¼ 4 20 420 30 5 ðkgm2 Þ: 50 30

260

The controller parameters in (13)–(15) with saturation function (22) are selected as k 1 ¼ k 2 ¼ 0:01; k 3 ¼ k 4 ¼ 0:3; k1 ¼ 0:1; k2 ¼ 0:15; d1 ¼ 0:1; d2 ¼ 100; ci ¼ 2  105 ði ¼ 1; 2; 3; 4Þ; e ¼ 0:001. In the simulation, suppose the physical parameters of the chaser and the target are m ¼ 58:2 ðkgÞ, 2 3 598:3 22:5 51:5 6 7 J ¼ 4 22:5 424:4 27 5 ðkgm2 Þ; 51:5 27 263:6 2 3 3336:3 135:4 154:2 6 7 J t ¼ 4 135:4 3184:5 148:5 5 ðkgm2 Þ: 154:2

148:5

2423:7

The external disturbances on the chaser spacecraft are 2 3 3  2 sinðxo tÞ  4 cosðxo tÞ 6 7 d s ¼ 4 3  3 sinðxo tÞ þ 2 cosðxo tÞ 5  105 ðNmÞ; 2

2 þ 4 sinðxo tÞ þ 2 cosðxo tÞ

3 2  3 sinðxo tÞ þ 2 cosðxo tÞ 6 7 d f ¼ 4 1 þ 2 sinðxo tÞ  2 cosðxo tÞ 5  104 ðNÞ; 2  3 sinðxo tÞ þ 3 cosðxo tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 where xo ¼ lg =krk is the mean orbital rate of the chaser, lg ¼ 3:986  1014 ðm3 =s2 Þ is the gravitational constant. Fig. 2 gives the three-dimensional position motion trajectory of two spacecraft under the proposed controllers in this paper. The relative attitude and relative angular velocities are depicted in Fig. 3, which exhibits that the current attitude and angular velocity of the chaser and the ones of the target are coincident with each other after 80 ðsÞ, thus the attitude synchronization is achieved almost totally after 80 ðsÞ. The relative position control objective is

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8

L. Sun, W. Huo / Advances in Space Research xxx (2015) xxx–xxx

x 10

40

7

Chaser position

re (m)

2

re3

0 −20 −40 0

1

z

re2

20

3

20

40

60

80

100

t(s) 2

−1 4

0

2

2

7

ve (m/s)

0

x 10

re1

Target position

0

0

y

−2

−2 −4

x 10

7

ve1

ve3

−2 −4 −6 0

x

ve2

20

40

60

80

100

t(s)

Fig. 2. Closed-loop position trajectory of two spacecraft.

Fig. 4. Time response of relative position and velocity. 1

σe1

σe2

σe3 20

0

f (N)

σe

0.5

−0.5 −1 0

20

40

60

80 ωe2

0

50

−0.1 40

60

40

80

60

80

100

t(s) 100

20

20

ωe3

τ (Nm)

ωe (rad/s)

−20

0.1

−0.2 0

f3

0

−40 0

ωe1

f2

100

t(s) 0.2

f1

τ1

τ3

0 −50

100

−100 0

t(s)

τ2

20

40

60

80

100

t(s)

Fig. 3. Time response of relative attitude and angular velocity.

Fig. 5. Time response of control inputs of the chaser. T

m(kg) ˆ

54 52 50 0

20

40

60

80

100

60

80

100

60

80

100

t(s)

Jˆ

F

784 783 782 0

20

40

t(s)

−5

4

ˆ¯ d τ

to drive the chaser to a position of pt ¼ ½0; 5; 0 ðmÞ in frame F t . The relative position and relative velocity are depicted in Fig. 4, which shows that the desired relative position is reached in about 90 ðsÞ and the relative velocity goes to zero so that there is no relative translational motion. According to Fig. 5, it is shown that the modified controllers can effectively eliminated the chattering with almost the same response performances. Meanwhile, the control forces and torques presented in Fig. 5 show that the initial control inputs are large in order to drive the chaser to the desired position and attitude quickly. They decrease rapidly after the desired position and attitude are achieved. Figs. 6 and 7 show the adaptive estimations of the unknown parameters. As can be seen in Figs. 3–7, convergence of the closed-loop system errors also can be guaranteed and estimations always keep bounded. It is easily seen that the estimations do not converge to their true values, since the persistent excitation condition is not satisfied. Moreover, in order to illustrate the advantages

x 10

2 0 0

20

40

t(s) ^ kJ^kF , and d^s . Fig. 6. Time response of parameter estimations m;

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L. Sun, W. Huo / Advances in Space Research xxx (2015) xxx–xxx

9

−3

ˆ¯ d f

4

x 10

7

2 0 0

Chaser position

x 10

Target position

3 20

40

60

80

100

t(s)

2

a ˆ

z

1.2001

1 1.2 0

20

40

60

80

100

t(s)

1.2

0 4

1.2 0

2

2

ˆb

1.2

7

x 10 20

40

60

80

0

100

y

t(s)

where controller parameters are also set as ^ and k 1 ¼ k 2 ¼ 0:01; k 3 ¼ k 4 ¼ 0:3; k1 ¼ 0:1; k2 ¼ 0:15; m ^ J are set as their initial values. The corresponding simulation results of the relative motions are shown in Figs. 8–10. By further comparing Figs. 9 and 10 with Figs. 3 and 4, we know that attitude synchronization and position tracking are achieved in about 120 ðsÞ to reach the desired performance, this convergent time is clearly longer 30 ðsÞ than the one by using the proposed adaptive controllers. Moreover, defining the mean performance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi RT 2 error index MPE ¼ T1 kx k dt with tracking error e 0

−4

x 10

7

x

1

σe1

σe2

σe3

0.5

σe

0 −0.5 −1 0

20

40

60

80

100

120

t(s)

ωe (rad/s)

0.1

ωe1

ωe2

ωe3

0.05 0 −0.05 −0.1 0

20

40

60

80

100

120

t(s) Fig. 9. Time response of relative attitude and angular velocity without adaptation.

40

re1

re2

re3

20

re (m)

of the proposed controllers, by comparing proposed adaptive laws in (15) with ones in Pan and Kapila (2001) and Zhang and Duan (2012), we know that the proposed adaptive laws in this paper can decrease the amount of estimations online to 11 from 392 and 324, thus the computation burden of the adaptive controller is also decreased largely. Simulation results demonstrate that the proposed control strategies in this paper is able to track the position and attitude of the target precisely. Thus, the chaser spacecraft can acquire the docking axis at last and maintain the docking axis to align with the non-cooperative tumbling target. Since the tumbling motion of the target is bounded as stated in Remark 1, so the proposed adaptive controllers can easily finish the proximity missions under the small target’s angular velocity. But the proposed adaptive controllers may be failure when the target’s angular velocity is very large, since the largely target angular motion conditions leads to the fast position and attitude control commands for the chaser spacecraft with limited maneuvering ability. In order to investigate the effectiveness of the proposed adaptive control, the controllers (13) and (14) can be expressed as the simple proportional-derivative structure without adaptive control, that are  ^ f ¼ mð0Þðk 1 re þ k 2 s1 þ g 1 Þ 1 s ¼ J^ð0Þ½4 k 3 ð1 þ rTe re Þre þ k 4 s2 þ g2 

−2

−2

Fig. 8. Closed-loop position trajectory of two spacecraft without adaptation.

0 −20 −40 0

20

40

60

80

100

120

t(s) 2

ve (m/s)

Fig. 7. Time response of parameter estimations d^f ; ^a, and ^b.

0

ve1

ve2

ve3

0 −2 −4 0

20

40

60

80

100

120

t(s) Fig. 10. Time response of relative position and velocity without adaptation.

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10

L. Sun, W. Huo / Advances in Space Research xxx (2015) xxx–xxx T

σe1

σe2

re (m)

−50 0

20

40

60

80

100

t(s)

ve (m/s)

2

ve1

ve2

ve3

0 −2 −4 0

20

40

60

80

100

t(s) Fig. 12. Time response of relative position and velocity under small control gains.

1

σe1

σe2

σe3

0.5

σe

0 −0.5 −1 0

20

40

60

80

100

t(s) 0.2

ωe1

ωe2

ωe3

0.1 0 −0.1 −0.2 0

20

40

60

80

100

t(s) Fig. 13. Time response of relative attitude and angular velocity under large uncertainties.

re (m)

σe

re3

re1

re2

re3

20

0

0 −20

−0.5 20

40

60

80

100

−40 0

0.1

ωe1

ωe2

ve (m/s)

0 −0.05 40

60

80

40

60

2

ωe3

0.05

20

20

100

t(s) Fig. 11. Time response of relative attitude and angular velocity under small control gains.

80

100

t(s)

t(s)

ωe (rad/s)

re2

0

σe3

0.5

−0.1 0

re1

40

1

−1 0

50

ωe (rad/s)

xe ¼ ½rTe ; rTe ; vTe ; xTe  and simulation time T ¼ 100ðsÞ, the performance error index can be directly calculated by MPE1 ¼ 1:5903 and MPE2 ¼ 1:6464 for the cases of adaptive and non-adaptive control, respectively. Clearly, MPE2 is larger than MPE1 , thus we can conclude that the proposed adaptive estimation method is effective and the dynamic response performance of the closed-loop system is also improved by using adaptation. In order to show the response performance and robustness of the proposed adaptive controller, two cases of additional simulations are performed below to compared with the simulation results of the first case shown in Figs. 3 and 4. It is well known that the response performance and robustness of the nonlinear closed-loop systems are mainly determined by controller gains in the Lyapunov framework. Generally, in order to obtain better response performance and stronger robustness of the closed-loop system, suitable controller gains should be specified by trial-and-error. The simulation results of the second case is displayed in Figs. 11 and 12 by using small controller gains k i ¼ 0:002ði ¼ 1; 2; 3; 4Þ under given external disturbances and parametric uncertainties to test the response performance of the proposed adaptive controllers. The simulation results of the third case is shown in Figs. 13 and 14 by using the controller gains as the same in the first case under larger parametric uncertainties ^ ^ and mð0Þ ¼ 45 ðkgÞ; J ð0Þ ¼ diagf550; 400; 300g ðkgm2 Þ larger external disturbances 100  d s ðNmÞ and 100  d f ðNÞ to test the robustness of the proposed adaptive controllers. By comparing Figs. 11 and 12 with Figs. 3 and 4, we can see that the performance of convergent time and steady-state precision in Figs. 3 and 4 is clearly better than ones in Figs. 11 and 12 under the same initial states and modeling uncertainties. This implies that larger controller gains result in fast convergent time and high steady-state precision for the closed-loop system (16). The robustness of the closed-loop systems can be evaluated

ve1

ve2

ve3

0 −2 −4 −6 0

20

40

60

80

100

t(s) Fig. 14. Time response of relative position and velocity under large uncertainties.

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L. Sun, W. Huo / Advances in Space Research xxx (2015) xxx–xxx

by comparison the simulation results between Figs. 13,14 and 3, 4. As can be seen in Figs. 13 and 14, the proposed controllers ensure that attitude synchronization and relative position tracking are achieved in 90 ðsÞ under large modeling uncertainties. Furthermore, the transient response curves in Figs. 13,14 and 3, 4 are almost the same, this means that the proposed adaptive controllers have good robustness in spite of the large modeling uncertainties. 5. Conclusions A novel robust adaptive control method with standard measurable information was proposed for spacecraft close proximity operations with modeling uncertainties. The conventional adaptive control based on linear parametrization is replaced by combining element-wise and norm-wise adaptive methods to compensate the dynamic coupling effect and model uncertainties. Especially, the symmetric property of the estimated inertial matrix is always guaranteed in the proposed adaptive controller. The proposed controllers can guarantee asymptotic stability of the spacecraft relative motion system. In addition, a modified version of the controller was complemented to avoid the chattering phenomenon in proposed controller such that the closed-loop system sates converge to the tunable small neighborhood of the origin. The proposed controller can drive the chaser spacecraft to the desired position and attitude precisely in approach of space target, this was demonstrated by a simulation example. Except for the spacecraft rendezvous and docking missions, this controller design method can be also used in many other scenarios, such as spacecraft attitude tracking, space robot operation, space debris removing and space station construction. Future works will focus on the more complicated problem by extending the method in this work, such as trajectory tracking control, finite-time control and input constraint control. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No. 61327807, National Key Program of Natural Science Foundation of China under Grant No. 61134005 and National Key Development Program for Basic Research of China under Grant No. 2012CB821204. References Bai, H., Arcak, M., Wen, J.T., 2008. Rigid body attitude coordination without inertial frame information. Automatica 44 (12), 3107–3175. Dang, Z., Wang, Z., Zhang, Y., 2014. Modeling and analysis of relative hovering control for spacecraft. J. Guid. Control Dyn. 37 (4), 1091– 1102. Di Cairano, S., Park, H., Kolmanovsky, I., 2010. Model predictive control approach for guidance of spacecraft rendezvous and proximity maneuvering. Int. J. Robust Nonlinear Control 22 (12), 1398–1427.

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Dimarogonas, D.V., Tsiotras, P., Kyriakopoulos, K.J., 2009. Leaderfollower cooperative attitude control of multiple rigid bodies. Syst. Control Lett. 58 (6), 429–435. Di Mauro, G., Schlotterer, M., Theil, S., Lavagna, M., 2015. Nonlinear control for proximity operations based on differential algebra. J. Guid. Control Dyn. http://dx.doi.org/10.2514/1.G000842. Filipe, N., Tsiotras, P., 2014. Adaptive position and attitude tracking controller for satellite proximity operations using dual quaternions. J. Guid. Control Dyn. http://dx.doi.org/10.2514/1.G000054. Gao, H., Yang, X., Shi, P., 2009. Multi-objective robust H 1 control of spacecraft rendezvous. IEEE Trans. Control Syst. Technol. 17 (4), 794–802. Gao, X., Teo, K.L., Duan, G., 2013. An optimal control approach to robust control of nonlinear spacecraft rendezvous system with h-D technique. Int. J. Innovative Comput. Inf. Control 9 (5), 2099–2110. Guglieri, G., Maroglio, F., Pellegrino, P., Torre, L., 2014. Design and development of guidance navigation and control algorithms for spacecraft rendezvous and docking experimentation. Acta Astronaut. 94 (1), 395–408. Gurfil, P., 2007. Nonlinear feedback control of low-thrust orbital transfer in a central gravitational field. Acta Astronaut. 60 (8/9), 631–648. Hablani, H.B., Tapper, M.L., Dana-Bashian, D.J., 2002. Guidance and relative navigation for autonomous rendezvous in a circular orbit. J. Guid. Control Dyn. 25 (3), 553–562. Hartley, E.N., Trodden, P.A., Richards, A.G., Maciejowski, J.M., 2012. Model predictive control system design and implementation for spacecraft rendezvous. Control Eng. Pract. 20 (7), 695–713. Hu, Q., Lv, Y., Zhang, J., Ma, G., 2013. Input-to-state stable cooperative attitude regulation of spacecraft formation flying. J. Franklin Inst. 350 (1), 50–71. Kim, S.G., Crassidis, J.L., Cheng, Y., Fosbury, A.M., Junkins, J.L., 2007. Kalman filtering for relative spacecraft attitude and position estimation. J. Guid. Control Dyn. 30 (1), 133–143. Kluever, C.A., 1999. Feedback control for spacecraft rendezvous and docking. J. Guid. Control Dyn. 22 (4), 609–611. Kobilarov, M., Pellegrino, S., 2014. Trajectory planning for CubeSat short-time-scale proximity operations. J. Guid. Control Dyn. 37 (2), 566–579. Kristiansen, R., Nicklasson, P.J., Gravdahl, J.T., 2008. Spacecraft coordination control in 6DOF: integrated backstepping vs passivitybased control. Automatica 44 (11), 2896–2901. Lawton, J., Beard, R.W., 2002. Synchronized multiple spacecraft rotations. Automatica 38 (8), 1359–1364. Leomanni, M., Rogers, E., Gabriel, S.B., 2014. Explicit model predictive control approach for low-thrust spacecraft proximity operations. J. Guid. Control Dyn. 37 (6), 1780–1790. Lopez, I., Mclnnes, C.R., 1995. Autonomous rendezvous using artificial potential function guidance. J. Guid. Control Dyn. 18 (2), 237–241. Okoloko, I., Kim, Y., 2014. Attitude synchronization of multiple spacecraft with cone avoidance constraints. Syst. Control Lett. 69, 73–79. Pan, H., Kapila, V., 2001. Adaptive nonlinear control for spacecraft formation flying with coupled translational and attitude dynamics. In: Proc. 40th IEEE Conf. Decision Control, Orlando, USA. Dec. 4–7, 2001, pp. 2057–2062. Park, J.U., Choi, K.H., Lee, S., 1999. Orbital rendezvous using two-step sliding mode control. Aerosp. Sci. Technol. 3 (4), 239–245. Ren, W., Beard, R.W., 2004. Decentralized scheme for spacecraft formation flying via the virtual structure approach. J. Guid. Control Dyn. 27 (1), 73–82. Sastry, S., Bodsom, M., 1989. Adaptive Control: Stability, Convergence, and Robustness. Prentice Hall, pp. 18–19. Segal, S., Carmi, A., Gurfil, P., 2014. Stereovision-based estimation of relative dynamics between noncooperative satellites: theory and experiments. IEEE Trans. Control Syst. Technol. 22 (2), 568–584. Seo, D., 2015. Fast adaptive pose tracking control for satellites via dual quaternion upon non-certainty equivalence principle. Acta Astronaut. 115, 32–39.

Please cite this article in press as: Sun, L., Huo, W. Robust adaptive control of spacecraft proximity maneuvers under dynamic coupling and uncertainty. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.08.029

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L. Sun, W. Huo / Advances in Space Research xxx (2015) xxx–xxx

Shuster, M.D., 1993. A survey of attitude representations. J. Astronaut. Sci. 41 (4), 439–517. Singla, P., Subbarao, K., Junkins, J.L., 2006. Adaptive output feedback control for spacecraft rendezvous and docking under measurement uncertainty. J. Guid. Control Dyn. 29 (4), 892–902. Stansbery, D.T., Cloutier, J.R., 2000. Position and attitude control of a spacecraft using the state dependent Riccati equation technique. In: Proceedings of the American Control Conference, Chicago, USA. 28– 30 Jun. 2000, pp. 1867–1871. Subbarao, K., Welsh, S., 2008. Nonlinear control of motion synchronization for satellite proximity operations. J. Guid. Control Dyn. 31 (5), 1284–1294. Wang, X., Yu, C., 2013. Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics. Syst. Control Lett. 62, 225–233. Wu, S., Wu, Z., Radice, G., Wang, R., 2013. Adaptive control for spacecraft relative translation with parametric uncertainty. Aerosp. Sci. Technol. 31 (1), 53–58. Xin, M., Pan, H., 2010. Integrated nonlinear optimal control of spacecraft in proximity operations. Int. J. Control 83 (2), 347– 363.

Xin, M., Pan, H., 2011. Nonlinear optimal control of spacecraft approaching a tumbling target. Aerosp. Sci. Technol. 15 (2), 79–89. Xin, M., Pan, H., 2012. Indirect robust control of spacecraft via optimal control solution. IEEE Trans. Aerosp. Electron. Syst. 48 (2), 1798– 1809. Yang, X., Gao, H., 2013. Robust reliable control for autonomous spacecraft rendezvous with limited-thrust. Aerosp. Sci. Technol. 24 (1), 161–168. Yang, X., Gao, H., Shi, P., 2010. Robust orbital transfer for low earth orbit spacecraft with small-thrust. J. Franklin Inst. 347 (10), 1863– 1887. Yoon, H., Eun, Y., Park, C., 2014. Adaptive tracking control of spacecraft relative motion with mass and thruster uncertainties. Aerosp. Sci. Technol. 34, 75–83. Zhang, F., Duan, G., 2012. Integrated relative position and attitude control of spacecraft in proximity operation missions. Int. J. Autom. Comput. 9 (4), 342–351. Zhang, F., Duan, G., 2014. Robust adaptive integrated translation and rotation finite-time control of a rigid spacecraft with actuator misalignment and unknown mass property. Int. J. Syst. Sci. 45 (5), 1007–1034.

Please cite this article in press as: Sun, L., Huo, W. Robust adaptive control of spacecraft proximity maneuvers under dynamic coupling and uncertainty. Adv. Space Res. (2015), http://dx.doi.org/10.1016/j.asr.2015.08.029