Robust adaptive relative position and attitude control for spacecraft autonomous proximity

ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research Article

Robust adaptive relative position and attitude control for spacecraft autonomous proximity Liang Sun a, Wei Huo a,n, Zongxia Jiao b a The Seventh Research Division, Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, PR China b Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing, 100191, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 23 April 2015 Received in revised form 17 December 2015 Accepted 27 February 2016 This paper was recommended for publication by Q.-G. Wang

This paper provides new results of the dynamical modeling and controller designing for autonomous close proximity phase during rendezvous and docking in the presence of kinematic couplings and model uncertainties. A globally defined relative motion mechanical model for close proximity operations is introduced firstly. Then, in spite of the kinematic couplings and thrust misalignment between relative rotation and relative translation, robust adaptive relative position and relative attitude controllers are designed successively. Finally, stability of the overall system is proved that the relative position and relative attitude are uniformly ultimately bounded, and the size of the ultimate bound can be regulated small enough by control system parameters. Performance of the controlled overall system is demonstrated via a representative numerical example. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Spacecraft control Close proximity operations Kinematic couplings Thrust misalignment Adaptive control Robust control

1. Introduction Spacecraft proximity control is an essential and important problem and has been widely studied in many space missions, such as spacecraft rendezvous and docking, on-orbit servicing, fuel supplying, formation flying, and space station construction. Generally, two rigid spacecraft, namely a chaser and a target, are involved in close proximity missions. The relative position tracking and attitude synchronization between two spacecraft are required simultaneously in close proximity scenario. Compared with the rigid body position control in three-dimension space, rigid body attitude control is a more difficult problem that plays a central role in many mechanical system applications and has therefore received considerable attention over the years. An adaptive faulttolerant attitude controller was proposed in [1], and finite-time attitude controllers were redesigned in [2] and [3] based on pulse modulation synthesis and the adaptive fuzzy control method, respectively. A time-varying terminal sliding mode technique was used to design the finite-time attitude tracking controller in [4]. A fixed-time attitude tracking controller was developed in [5] based n

Corresponding author. Tel./fax: þ 86 10 82339382. E-mail address: [email protected] (W. Huo).

on planning the desired quaternion attitude trajectory. Except for the attitude control for single spacecraft, there are many attitude synchronization controllers for spacecraft formation, such as neural network-based terminal sliding mode distributed controller [6], adaptive nonsingular fast terminal sliding mode-based obstacle avoidance controller [7], extended Kalman filter-based fault detection and isolation controller [8], and distributed controller without angular velocities measurement [9]. The classical method for the stabilization and tracking of rigid bodies in attitude relies on a local paramerization of the rotation matrix [10], such as the Euler angles, quaternion or Rodrigues parameters. Compared with position dynamics, the distinct features of the attitude dynamics are that its state space is not linear [11]. This yields important and unique properties that cannot be observed from dynamic systems evolving on a linear space. Even though the state space for attitude control problem is nonlinear, this problem has been analyzed by using coordinate representations resulting in different attitude kinematics [12]. Recently, a coordinate-free approach has been proposed to deal with the attitude dynamics directly based on the rotation matrix [13]. One of the main advantages of this approach is that it allows a global analysis of the designed feedback control law. However, it has been proved that global asymptotic stability is not achievable through continuous feedback, and the notion of

http://dx.doi.org/10.1016/j.isatra.2016.02.022 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i

2

L. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

almost global asymptotic stability is the best achievement in continuous attitude control [14]. Many models and controllers for spacecraft relative motion have been developed from the control community in theory and practice. To the best of our knowledge, the existing literatures on relative attitude control in formation or proximity operations are all based on the coordinate-dependent approach. Quaternion is employed to establish the relative attitude dynamics in [15–20]. Modified Rodrigues parameters are used to model the attitude dynamics in [21–24]. Moreover, dual quaternion was proposed for relative attitude and relative position motions in [25–27]. Recently, based on the quaternion attitude dynamics, an inverse optimal sliding mode controller was designed for spacecraft relative pose motion in [28]. Except for singularities of quaternion and modified Rodrigues parameters, they also have ambiguities in representing an attitude. For instance, in a quaternion-based attitude control system, convergence to a single attitude implies convergence to either of the two disconnected and antipodal points [29]. Otherwise, it may also exhibit unwinding behavior, where the controller rotates a rigid body through unnecessarily large angles [14]. An adaptive sliding mode relative pose controller was designed directly on the Lie group in [30], but the relative pose model is assumed well known without uncertainties. Above schemes are available to establish the relative motion model and realize the spacecraft close proximity operations, but most of them have following two drawbacks: (i) the mechanical model of the relative motion between two spacecraft is not globally defined; and (ii) the inherently kinematic couplings between relative position and relative attitude motions, mass and inertia parametric uncertainties of two spacecraft, unknown thrust misalignment, and bounded external disturbances in the mechanical model are not considered simultaneously in the controller design. In order to dealing with these two challenges, this study focuses on the new development of the modeling and control for spacecraft autonomous proximity operations. The unique features of this study lie in the following aspects.

 Compared with the coordinate-dependent relative attitude





controller in [15–24], relative attitude controller in this paper is developed directly based on the rotation matrix to avoid the singularities associated with coordinate-dependent representations, a globally defined nonlinear relative motion model is presented for close proximity operations by using the coordinate-free approach, and compared with the model in [31–34], the gravity of two spacecraft has been considered in relative position dynamics. With treating the kinematic couplings and external disturbances as bounded lumped disturbances for relative rotational and relative translational systems, the relative position and relative attitude controllers are designed one after another based on robust adaptive control method, where the elementwise adaptive laws are used to estimate unknown mass and inertia of the chaser, the norm-wise adaptive laws are used to estimate thrust misalignment vector and the upper bounds of lumped disturbances. Specially, the amount of online estimate parameters in the proposed adaptive controllers is only 12, thus the computational burden of the controller is largely decreased for the spacecraft proximity operations compared with the larger amount 392 in [35] and 324 in [36]. A rigorous Lyapunov analysis, that explicitly considers the kinematic couplings and model uncertainties in relative dynamics, is presented to establish stability properties, then uniformly ultimately boundedness of relative position and relative attitude is derived, and the ultimate bounds can be regulated sufficiently small with appropriate controller parameters. Moreover, the discontinuous signum function in [31,32]

is replaced by hyperbolic tangent function to avoid the highfrequency chattering phenomenon in controllers. The rest of this paper is organized as follows. Section 2 gives some notations and properties used in this paper. Section 3 derives a global mechanical model for proximity operations and summarizes an objective of the controller design. Section 4 proposes the controller design and stability analysis of two closed-loop subsystems. Section 5 verifies the application of the controller to a scenario of autonomous proximity operations. Section 6 concludes the work.

2. Preliminaries Throughout this paper, the skew symmetric matrix SðaÞ A R33 derived from a vector a ¼ ½a1 ; a2 ; a3 T A R3 is defined as 2 3 a2 0  a3 6 0  a1 7 SðaÞ ¼ 4 a3 5:  a2 a1 0 Actually, SðÞ is a map R3 -soð3Þ, this identifies the Lie algebra soð3Þ with R3 using the vector cross product in R3 . The inverse of the map SðÞ is referred to as the vee map, that is V : soð3Þ-R3 . Some properties of the map SðÞ are listed as follows [11]: SðaÞb ¼ a  b ¼ b  a ¼  SðbÞa; J SðaÞ J ¼ J a J ;

aT SðaÞ ¼ 0;

aT SðbÞa ¼ 0;

SðaÞ ¼ ST ðaÞ;

1  trðSðaÞSðbÞÞ ¼ aT b; 2

trðSðaÞAÞ ¼ trðASðaÞÞ ¼  aT ðA AT ÞV ; SðaÞA þ AT SðaÞ ¼ Sð½trðAÞI 3  AaÞ;

RSðaÞRT ¼ SðRaÞ;

for any a; b A R3 ; A A R33 , and R A SOð3Þ. Moreover, trðAÞ denotes the matrix trace; J a J and J a J 1 denote vector 2-norm and vector 1-norm, respectively; J A J and J A J F are induced matrix 2-norm and Frobenius-norm, respectively; SOð3Þ 9 fR A R33 : RT R ¼ I 3 ; detðRÞ ¼ 1g denotes the special orthogonal group; In and On are n  n unit and zero matrices, respectively. Define tanhðaÞ ¼ ½tanhða1 Þ; tanhða2 Þ; tanhða3 ÞT , where the a e  ai hyperbolic tangent function tanhðai Þ ¼ eeaii  , i ¼ 1; 2; 3, and it þ e  ai satisfies j tanhðai Þj o 1, tanhðai Þ ¼ 0 if and only if ai ¼ 0. Moreover, the following inequality holds for any constant c 4 0 and ai A R [37]:

ϵ

0 r j ai j  ai tanhðcai Þ r ; c where

ϵ is a constant that satisfies ϵ ¼ e  ðϵ þ 1Þ , i.e., ϵ ¼0.2785.

3. Problem statement 3.1. Chaser and target dynamics The control problem that a chaser subject to uncertain inertia and unknown disturbance tracks a tumbling non-cooperative space target is considered in this study. Fig. 1 depicts the related frames and vectors, where F i 9 fOxi yi zi g represents an Earthcentered inertia frame, F c 9 fCxyzg and F t 9 fTxt yt zt g are the chaser and target body-fixed frames, respectively; O, C, and T are the centers of mass of the Earth, chaser, and target, respectively; P is the chaser desired approaching position along the direction of target docking port; fr; r e g and fr t ; pt ; r pt g are the related position vectors represented in frame F c and frame F t , respectively. The aim of this study is to design controller such that origin C tracks point P and frame F c tracks frame F t .

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i

L. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

zt y

yt

xt

Thus, the tumbling target's position r t , velocity vt and angular velocity ωt are always bounded.

z

x

3.2. Relative position dynamics

C

T target

pt

rt

re

P

In light of Fig. 1, the position and velocity of point P denoted in frame F t are ( r pt ¼ r t þpt ; ð9Þ vpt ¼ vt þ Sðωt Þpt ;

chaser

rp

r

t

zi

where the pt A R3 is a constant vector denoted in frame F t . The relative position, relative velocity and relative angular velocity are also denoted in frame F c by 8 r ¼ r  RTe r pt ; > > < e ve ¼ v  RTe vpt ; ð10Þ > > : ω ¼ ω RT ω ;

yi

xi

3

O

Fig. 1. Description of the related frames and vectors.

e

The position motion of center of mass C and the attitude motion 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 [38]:

e

t

where Re ¼ RTt R A SOð3Þ is the rotation matrix from F c to F t . Substituting (10) into (2) and (4), and using r_ pt ¼ vpt  Sðωt Þr pt yield the relative position motion model as r_ e ¼ ve  SðωÞr e ;

ð11Þ

R_ ¼ RSðωÞ;

ð1Þ

r_ ¼ v  SðωÞr;

ð2Þ

_ þ SðωÞJ ω ¼ τ þ SðρÞf þ dτ ; Jω

can be derived from (8)–(10) and where any a A R as

ð3Þ

_ t Þpt  ¼ RTe ½  Sðωt Þvt  μt r t  Sðpt Þω _ t RTe v_ pt ¼ RTe ½v_ t þ Sðω

mv_ þ mSðωÞv þ μmr ¼ f þ df ;

ð4Þ

where r A R3 is chaser's position and R A SOð3Þ denotes chaser's attitude; v and ω A R3 are velocity and angular velocity of the chaser, respectively; μ ¼ μg = J r J 3 , μg is the gravitational constant of the Earth; τ and f A R3 are control force and torque, respectively; ρ A R3 is the thrust misalignment vector from the center of mass C to the applied point of force f ; df and dτ A R3 are disturbance force and torque, respectively; m A R and J A R33 are the chaser mass and inertia matrix, respectively. Similarly, consider that target moves in the circular orbit. With only considering the gravity and ignoring the other external forces and torques, kinematics and dynamics of the tumbling target can be modeled in frame F t as follows [39]: R_ t ¼ Rt Sðωt Þ;

ð5Þ

r_ t ¼ vt  Sðωt Þr t ;

ð6Þ

_ t þ Sðωt ÞJ t ωt ¼ 0; Jt ω

ð7Þ

mt v_ t þ mt Sðωt Þvt þ μt mt r t ¼ 0;

ð8Þ

where r t A R3 is target's position and Rt A SOð3Þ denotes target's attitude; vt and ωt A R3 are velocity and angular velocity of the target, respectively; μt ¼ μg =r 3o , ro is the radius of the circular orbit; mt A R and J t A R33 are target's mass and positive definite symmetric inertia matrix, respectively. Remark 1. Since the external forces and torques are ignored except for the gravity in target dynamics, the radius of the target's circular orbit ro is a positive constant, and scalar μt is also a positive constant. By defining kinetic energy function En ðtÞ ¼ 12ðμt mt r Tt r t þ mt vTt vt þ ωTt J t ωt Þ Z 0 and calculating its time derivative E_ n ðtÞ ¼  μt mt r Tt Sðωt Þr t mt vTt Sðωt Þvt  ωTt Sðωt ÞJ t ωt  0, we know En ðtÞ  En ð0Þ ¼ 12½μt mt r Tt ð0Þr t ð0Þ þ mt vTt ð0Þvt ð0Þ þ ωTt ð0ÞJ t ωt ð0Þ o 1.

mv_ e ¼  m½SðωÞv þ μr þ RTe v_ pt  Sðωe Þðv  ve Þ þ f þ df ; RTe v_ pt 3

RTe SðaÞ ¼

ð12Þ SðRTe aÞRTe

for

_t ¼  SðRTe ωt Þ½RTe vpt  RTe Sðωt Þpt   μt RTe r t  RTe Sðpt Þω ¼  Sðω  ωe Þ½v  ve  Sðω  ωe ÞRTe pt  þ μt RTe pt  μt ðr  r e Þ _ t:  RTe Sðpt Þω Thus relative position dynamics (12) can be rewritten as mv_ e ¼  mg  nP þ f þdf ; where g ¼ SðωÞve þS ðω  ω nP ¼ mRTe Sðpt ÞJ t 1 Sðωt ÞJ t ωt . 2

ð13Þ T e ÞRe pt þ

μr þ μ

T t Re pt 

μt ðr  re Þ and

3.3. Relative attitude dynamics Define a relative attitude trace function Ψ A R and a relative attitude vector eR A R3 as ( Ψ ðRe Þ ¼ 12 trðK  KRe Þ; ð14Þ eR ðRe Þ ¼ 12 ðKRe  RTe KÞV ; where K ¼ diagfk1 ; k2 ; k3 g is a diagonal matrix with distinct and positive constants ki ði ¼ 1; 2; 3Þ. It has been proved in [40] that Ψ ðRe Þ is locally positive definite and satisfies b1 J eR ðRe Þ J 2 r J Ψ ðRe Þ J r b2 J eR ðRe Þ J 2 b1 ¼ h2 hþ1 h3

ð15Þ b2 ¼ h5 ðhh11 h4 b0 Þ,

and where b0 is a with the constants positive constant that strictly satisfies Ψ ðRe Þ o b0 o h1 , and hi ði ¼ 1; …; 5Þ are defined by h1 ¼ minfk1 þk2 ; k2 þ k3 ; k3 þ k1 g; h2 ¼ maxfðk1  k2 Þ2 ; ðk2  k3 Þ2 ; ðk3  k1 Þ2 g; h3 ¼ maxfðk1 þ k2 Þ2 ; ðk2 þ k3 Þ2 ; ðk3 þ k1 Þ2 g; h4 ¼ maxfk1 þ k2 ; k2 þ k3 ; k3 þ k1 g; h5 ¼ minfðk1 þ k2 Þ2 ; ðk2 þk3 Þ2 ; ðk3 þ k1 Þ2 g: Differentiating Ψ ðRe Þ and eR ðRe Þ in (14) with respect to time and using R_ e ¼ Re Sðωe Þ lead to

Ψ_ ðRe Þ ¼  12 trðKRe Sðωe ÞÞ ¼ 12 ωTe ðKRe  RTe KÞV ¼ eTR ωe ;

ð16Þ

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i

L. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

 V  V e_ R ðRe Þ ¼ 12 KRe Sðωe Þ  ST ðωe ÞRTe K ¼ 12 KRe Sðωe Þ þ Sðωe ÞRTe K

4. Robust adaptive controller design

 V h i ¼ 12 Sð½trðRTe KÞI3  RTe Kωe Þ ¼ 12 trðRTe KÞI3  RTe K ωe 9 EðRe Þωe ;

ð17Þ where EðRe Þ is bounded by J EðRe Þ J r p1ffiffi2trðKÞ. Substituting ωe in (10) into (1) and (3) yields the relative attitude dynamics as _ t þ SðωÞωe  þ τ þ SðρÞf þdτ ; _ e ¼ SðωÞJ ω  J½RTe ω Jω

ð18Þ

_ t can be derived from (7) and (14) as where ω

Robust adaptive control deals with redesigning or modifying adaptive control schemes to make them robust with respect to bounded disturbances. In this section, robust adaptive controllers are developed for the relative position and relative attitude in spacecraft proximity maneuvers. The adaptive control laws are used to compensate the parametric uncertainties. The robustness is achieved at the expense of replacing the asymptotic tracking property of the dynamics model without kinematic couplings and model uncertainties.

ω_ t ¼  J t 1 Sðωt ÞJ t ωt

4.1. Relative position controller

Thus relative attitude dynamics (18) can be rewritten as

Before developing the relative position controller, we can derive J df J r ρf from Assumption 1 and J ωt J 2 r 2EλnJ ð0Þ from

_ e ¼ SðωÞJ ω  JSðωÞωe þ nA þ τ þ SðρÞf þ dτ ; Jω where nA ¼

ð19Þ

ωt ÞJ t ωt .

JRTe J t 1 Sð

Remark 2. In fact, eR ðRe Þ asymptotically converges to zero does not imply that R-Rt as t-1, since there exist three additional non-degenerate critical points of Ψ ðRe Þ when eR ðRe Þ ¼ 0 (all of four critical points of the function Ψ ðRe Þ are I 3 ; diagf1;  1;  1g; diagf 1; 1;  1g; diagf 1;  1; 1g, but the unique minimum point of Ψ ðRe Þ is I3 [41]), this is due to the nonlinear structures of SOð3Þ, and these cannot be avoided for any continuous attitude control systems [42]. Remark 3. The globally defined mechanical model of relative attitude and relative position for autonomous proximity operations is established by (11), (13), (17) and (19). From the terms SðωÞr e in (11) and nP in (13), we know the information of relative attitude motion is involved in the relative position motion. Thus the relative position is significantly affected by the relative attitude between two spacecraft. This reflects that the spacecraft proximity systems have inherently kinematic couplings. Assumption 1. J is an unknown symmetric positive definite constant matrix and bounded by λm r J J J r λM with known constants λm and λM; m is an unknown positive constant; ρ is an unknown constant vector and bounded by J ρ J r ρ with an unknown constant ρ ; Jt is an unknown symmetric positive definite constant matrix, and it is bounded by J J t 1 J J J t J r λt with an unknown constant λt; df and dτ are unknown vectors and bounded by J df J r ρf and J dτ J r ρτ with unknown constants ρf and ρτ , respectively. Assumption 2. The circular-orbit radius ro of the target is known in advance for the chaser. The chaser's motion information f R; ω; r; vg and relative motion information fRe ; ωe ; r e ; ve g can be directly obtained by the measurement devices mounted on the chaser body [43,44]. However, the target's motion information f Rt ; ωt ; r t ; vt g are assumed to be unmeasurable directly by the chaser. 3.4. Control objective The objective in this study is to design controllers based on the proposed model to drive the chaser spacecraft at a desired position pt along the docking port direction of the space target and reorient the chaser's attitude synchronized with the tumbling target attitude. In view of (10) and (14), the objective is equivalent to design effective control inputs f and τ under Assumptions 1 and 2, such that r e and eR converge to sufficiently small neighborhood of zero.

t

Remark 1, where λJ t is the minimum eigenvalue of Jt, thus J df nP J r J df J þm J RTe J J Sðpt Þ J J ωt J 2 J J t 1 J J J t J ¼ J df J þ m J pt J J ωt J 2 J J t 1 J J J t J r

2En ð0Þmλt

λJ t

þ ρf 9 δ P : ð20Þ

where J Re J ¼ J RTe J ¼ 1 and J Sðpt Þ J ¼ J pt J . Relative position control system requires the knowledge of mass m and the bound δP, but it is difficult to measure the values of mass m and the bound δP exactly. In general, there are estimation errors:

δ~ P ¼ δP  δ^ P ;

~ ¼ m  m; ^ m

ð21Þ

^ where the exact mass and its estimate are denoted by m and m, respectively, δ^ P is the estimate of δP. Then, a robust adaptive relative position controller is designed in Theorem 1. Theorem 1. Consider the relative position model (11) and (13) in close proximity operations under Assumptions 1 and 2, if the relative position controller is designed by ^ þ νP ; f ¼  kr r e  kv eP þ mg

ð22Þ

νP ¼  δ^ P tanhðγ eP Þ;

ð23Þ

and the corresponding parameter adaptive updating law is assigned as ^ ^_ ¼  km ðeTP g þ σ 1 mÞ; m

ð24Þ

_

ð25Þ

δ^ P ¼  kP ½eTP tanhðγ eP Þ þ σ 1 δ^ P ; 3

where eP A R is an augmented error vector given by eP ¼ v e þ c1 r e :

ð26Þ

and kr 4 0; kv 4 0; km 4 0; kP 4 0; c1 4 0; σ 1 40; γ 4 0. Then estima~ δ~ P are bounded and relative position r e converges to tion errors m; the sufficiently small neighborhood of zero. Proof. Consider a Lyapunov candidate 1 1 1 1 ~2 ~ 2þ V 1 ¼ kr r Te r e þ meTP eP þ δ : m 2 2 2km 2kP P Then, we obtain V 1 ¼

μ1 J z1 J

2

r zT1 U 1 z1 r

ð27Þ

2 ~ 2 þ 2k1 ~ P zT1 U 1 z1 þ 2k1m m P

δ and

μ2 J z1 J ; 2

ð28Þ

where z1 ¼ ½ J r e J ; J eP J  A R ; μ1 and μ2 are the minimum eigenvalue and the maximum eigenvalue of matrix U1, respectively. Substituting (22) into (13) leads to the closed-loop position system T

2

U 1 ¼ 12diagfkr ; mg A R22 ;

~ þ νP  nP þ df : mv_ e ¼  kr r e  kv eP  mg

ð29Þ

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i

L. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

_ ~_ ¼  m ^_ , δ~_ P ¼  δ^ P , r Te SðωÞr e ¼ 0, (11), (26), and (29), the Using m time-derivative of V1 is given by 1 1 ~m ~_ þ δ~ P δ~_ P ~ TP g þ m V_ 1 ¼  kr c2 J r e J 2  kv J eP J 2  me km kP ~m ^ þ eTP ðdf  nP þ νP Þ ¼  kr c1 J r e J 2  kv J eP J 2 þ σ 1 m T T þ σ δ~ δ^  δ~ e tanhðγ e Þ þ e ðd  n þ ν Þ: 1

P

P

P P

P

P

f

P

ð30Þ

P

5

Theorem 2. Consider the relative attitude model (17) and (19) for close proximity operations under Assumptions 1 and 2, if the adaptive relative attitude controller is designed by

τ ¼ kR eR  kw ωe þ SðωÞJ^ω þ J^SðωÞωe  Sðρ^ Þf þ νA ;

ð40Þ

νA ¼  δ^ A tanhðγ eA Þ;

ð41Þ

and the corresponding parameter adaptive updating law is assigned as

Since eTP ðdf  nP þ νP Þ r δP J eP J 1  δ^ P eTP tanhðγ eP Þ " # 3 X ¼ δP j ePi j  eP i tanhðγ ePi Þ þ δ~ P eTP tanhðγ eP Þ

_ kJ J^ ¼ ½ SðωÞωe eTA  eA ðSðωÞωe ÞT þ ωωT SðeA Þ  SðeA ÞωωT  2σ 2 J^ ; 2 ð42Þ

i¼1

ϵδP ~ T þ δ P eP tanhðγ eP Þ; r 3γ

ð31Þ

ρ^_ ¼  kρ ½ST ðf ÞeA þ σ 2 ρ^ ;

ð43Þ

~ 2 þ 12 m2 ; ~m ^ ¼ mðm ~ ~ r  12 m m  mÞ

ð32Þ

δ^ A ¼ kA ½eTA tanhðγ eA Þ þ σ 2 δ^ A ;

_

ð44Þ

2 δ~ P δ^ P ¼ δ~ P ðδP  δ~ P Þ r  12 δ~ P þ 12 δ2P ;

ð33Þ

substituting (31)–(33) into (30) gives rise to V_ 1 r  zT1 U 2 z1 þ κ 1 ;

ð34Þ

where U 2 ¼ diagfkr c1 ; kv g, κ 1 ¼ 12σ 1 ðm2 þ δP Þ þ 3ϵγ δP . Since kr ; kv ; km ; c1 ; σ 1 are positive constants, U 1 ; U 2 are positive definite. Now, we have 2

μ V_ 1 r  3 V 1 þ κ 2 ;

ð35Þ

μ2

where μ3 is the minimum eigenvalue of matrix U2 and 2 κ 2 ¼ κ 1 þ μμ32 ð2k1m m~ 2 þ 2k1P δ~ P Þ. This implies that   μ μ κ 2  μ32 t μ2 κ 2 V 1 ðtÞ r V 1 ð0Þ  2 þ : ð36Þ e

μ3

μ3

~ δ~ P are bounded. Then from (28), we have This means that r e ; eP ; m; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 μ κ 2  μμ32 t μ2 κ 2 J r e ðtÞ J r J z1 ðtÞ J r : ð37Þ V 1 ð0Þ  2 þ e

μ1

μ3

μ1 μ3

qffiffiffiffiffiffiffiffi μ κ Thus, limt-1 J r e ðtÞ J r μ2 μ2 , and the ultimately uniformly 1 3 boundedness of the relative position is proved. Moreover, smaller σ1 and larger c1 ; kv ; γ ; km ; kP result in larger μ3 and smaller κ2, thus the sufficiently small ultimate bound of J r e ð1Þ J can be regulated.□ 4.2. Relative attitude controller Before developing the relative attitude controller, we also can derive J dτ J r ρτ from Assumption 1 and J ωt J 2 r 2EλnJ ð0Þ from t

Remark 1, thus J nA þ dτ J r J J J J RTe J J ωt J 2 J J t 1 J J J t J þ J dτ J 2En ð0ÞλM λt ¼ J J J J ωt J 2 J J t 1 J J J t J þ J dτ J r þ ρτ 9 δA ;

λJ t

ð38Þ

Relative attitude control system requires the knowledge of an inertia matrix J, thrust misalignment vector ρ and the bound δA, but it is difficult to measure them exactly. In general, there are estimation errors: J~ ¼ J  J^ ;

ρ~ ¼ ρ  ρ^ ;

δ~ A ¼ δA  δ^ A ;

ð39Þ

where the exact inertia matrix and its estimate are denoted by J and J^ , respectively, all of matrices J; J^ ; J~ are symmetric; the exact thrust misalignment vector and its estimate are denoted by ρ and ρ^ , respectively; δ^ A is the estimate of δA. Then, inspired by the single rigid body attitude tracking controller in [40], a robust adaptive controller can be designed for relative attitude in Theorem 2.

where eA A R3 is an augmented error vector given by eA ¼ ωe þc2 eR

ð45Þ

for a constant c2 satisfying 8sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 9 < 2b k λ = 2kw 4kR kw 1 R m 0 o c2 o min : ; ; 2 2 1 : λ trðKÞ M kw þ pffiffi2kR λM trðKÞ; λM

ð46Þ

and kR 4 0; kw 40; kJ 4 0; kA 40; kρ 40; σ 2 4 0; γ 4 0. Then estimation errors J~ ; ρ~ ; δ~ A are bounded and relative attitude eR converges to the sufficiently small neighborhood of zero. Proof. Consider a Lyapunov candidate 1 1 ~ 2 1 1 ~2 V 2 ¼ ωTe J ωe þ kR Ψ ðRe Þ þ c1 ωTe JeR þ JJ JF þ J ρ~ J 2 þ δ : 2 2kJ 2kρ 2kA A ð47Þ For a positive constant ψ o h1 , define D  SOð3Þ as D9 fRe A SOð3Þ : Ψ ðRe Þ o ψ oh1 g: From Assumption 1, V2 is bounded in D by zT2 W 1 z2 r V 2 r zT2 W 2 z2 þ

1 ~ 2 1 1 ~2 JJ JF þ J ρ~ J 2 þ δ ; 2kJ 2kρ 2kA A

ð48Þ

where z2 ¼ ½ J eR J ; J ωe J T A R2 , and the matrices W 1 ; W 2 A R22 are given by " # b1 kR 12c2 λM W1 ¼ 1 ; 1 2c2 λM 2λm " # b2 kR 12c2 λM W2 ¼ 1 : 1 2c2 λM 2λM Substituting (40) into (19) leads to the closed-loop attitude system _ e ¼  kR eR kw ωe SðωÞJ~ ω  J~ SðωÞωe þ Sðρ~ Þf þ νA þ nA þ dτ : Jω ð49Þ Using (16), (17) and (49), the time-derivative of V2 is given by V_ 2 ¼  kw J ωe J 2  c2 kR J eR J 2 þ c2 ωTe JEωe  c2 kw ωTe eR  eTA ½SðωÞJ~ ω 1 1 1 _ _ þ J~ SðωÞωe  þ eTA Sðρ~ Þf þ trðJ~ J~ Þ þ ρ~ T ρ~_ þ δ~ A δ~ A þ eTA ðνA þ nA þ dτ Þ: kJ kρ kJ

ð50Þ Using the following facts that x y ¼ trðxy Þ ¼ trðyx Þ and trðJ~ AÞ ¼ trðJ~ AT Þ for any x; y A R3 and A A R33 ; ðSðeA ÞωÞT ¼  ωT SðeA Þ; _ _ _ _ Sðρ~ Þf ¼ Sðf Þρ~ ; J~ ¼  J^ , ρ~_ ¼  ρ^_ and δ~ A ¼  δ^ A , then we get T

T

T

V_ 2 ¼  kw J ωe J 2  c2 kR J eR J 2 þ c2 ωTe JEωe  c2 kw ωTe eR

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i

L. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

     1_ 1 T þ tr J~  SðωÞωe eTA  ωðSðeA ÞωÞT þ J~   ρ~ ST ðf ÞeA  ρ~_ kJ kρ  δ~ eT tanhðγ e Þ þ σ δ~ δ^ þ eT ðν þ n þ d Þ ¼  k J ω J 2 A A

2

A

A

A

A

A

τ

A

w

T  c2 kR J eR J 2 þ c2 ωTe JEωe  c2 kw ωTe eR þ σ 2 trðJ~ J^ Þ þ σ 2 ρ~ ρ^  δ~ A eTA tanhðγ eA Þ þ σ 2 δ~ A δ^ A þ eTA ðνA þ nA þ dτ Þ: pffiffiffi pffiffiffi Since J J J F r 3 J J J r 3λM ,

larger c2 ; kw ; γ ; kJ ; kρ ; kA result in larger λ3 and smaller κ4, thus the sufficiently small ultimate bound of J r e ð1Þ J can be regulated.□

e

ð51Þ

eTA ðnA þdτ þ νA Þ r δA J eA J 1  δ^ A eTA tanhðγ eA Þ " # 3 X ¼ δA j eAi j  eAi tanhðγ eAi Þ þ δ~ A eTA tanhðγ eA Þ i¼1

ϵδ r A þ δ~ A eTA tanhðγ eA Þ; 3γ trðJ~ J^ Þ ¼ trðJ~ ðJ  J~ ÞÞ ¼

3 X

ð52Þ

2 ðJ ii J~ ii  J~ ii Þ r

i¼1

3  X i¼1

1 2 1  J~ ii þ J 2ii 2 2



1 1 1 3 2 ¼  J J~ J 2F þ J J J 2F r  J J~ J 2F þ λM ; 2 2 2 2

ð53Þ

ρ~ T ρ^ ¼ ρ~ T ðρ  ρ~ Þ r  12 J ρ~ J 2 þ 12 J ρ J 2 r  12 J ρ~ J 2 þ 12 ρ 2 ;

ð54Þ

2 δ~ A δ^ A ¼ δ~ A ðδA  δ~ A Þ r  12 δ~ A þ 12 δ2A ;

ð55Þ

then substituting (52)–(55) into (51) gives rise to V_ 2 r  zT2 W 3 z2 þ κ 3 ;

ð56Þ

where κ 3 ¼ 12σ 2 ð3λ ρ 2 þ δA Þ þ 3ϵγ δA , 2 3  12c2 kw c 2 kR 4 5: W3 ¼  12c2 kw kw  p1ffiffi2c2 λM trðKÞ 2 Mþ

2

The inequality (46) for the constant c2 guarantees that the matrices W 1 ; W 2 ; W 3 become positive definite. Then, we have

λ3 V_ 2 r  V 2 þ κ 4 ;

ð57Þ

λ2

where λ3 and λ2 represent the minimum eigenvalue and the maximum eigenvalue of matrices W3 and W2, respectively; 2 κ 4 ¼ κ 3 þ λλ32 ð2k1 J J J~ J 2F þ 2k1ρ J ρ~ J 2 þ 2k1A δ~ A Þ. Thus V_ 2 ðtÞ o 0 when λ2 κ 1 V 2 ðtÞ 4 λ , and 3   λ2 κ 4  λλ32 t λ2 κ 4 þ : ð58Þ e V 2 ðtÞ r V 2 ð0Þ 

λ3

λ3

This means that eR ðtÞ; ωe ; J~ ; ρ~ ; δ~ A are bounded. Furthermore, from (48), we have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 λ2 κ 4  λλ32 t λ2 κ 4 e V 2 ð0Þ  þ J eR ðtÞ J r J z2 ðtÞ J r :

λ1

λ3

λ1 λ3

where λ1 is the minimum eigenvalue of matrix W1. Let a sublevel set of V2 be Va 9f ðRe ; ωe ; J^ ; ρ^ ; δ^ A Þ A SOð3Þ  R3  R33  R3  R : V 2 o ψ a g for a constant ψ a 4 0. If the following inequality for ψa is satisfied ψ a o bψ2 λ1 9 κ a , then we can guarantee that Va  D  R3  R33  R3  R, since it implies that J z2 J 2 o bψ2 , which leads Ψ ðRe Þ r b2 J eR J 2 r b2 J z2 J 2 o ψ . Thus, from (57), a sublevel set of Va is a positively invariant set, when κ 4 o ψ a o κ a , and it becomes smaller until ψ a ¼ κ 4 . In order to guarantee the existence of such Va , the following inequality should be satisfied

ψ

κ 4 o λ1 ¼ κ a ; b2

ð59Þ

which can be achieved by choosing sufficiently small σ2 and large γ ; kJ ; kρ ; kA . Thus, for any initial condition satisfying V 2 ð0Þ o κ a , the relative attitude ultimately converges to the set qffiffiffiffiffiffiffiffi limt-1 J eR ðtÞ J r λλ2 κλ1 . Thus the ultimately uniformly bounded1 3 ness of the relative attitude is proved. Moreover, smaller σ2 and

Remark 4. Outside the set C9 fz2 : λ3 J z2 J 2 4 κ 4 g, we have V_ 2 r 0 n from (57). Inside of the set B 9 z2 : J z2 J 2 r bψ2  D  R3  R33 R3  Rg, Eqs. (48) and (57) hold. The inequality (59) guarantees that the smallest sublevel set Vb of V2, covering the set C, lies inside of the largest sublevel set Vc of V2 in B, that is to say Vb  Vc . Therefore, along any solution starting in Vc , V2 decreases until the solution enters Vb , thereby yielding uniform boundedness. Note that the above aforementioned sets satisfy C  Vb  Vc  B  D  R3  R33  R3  R. Remark 5. The proposed relative attitude controller (40) is developed based on the rotation matrix rather than the conventional coordinate-dependent design approaches. Since the nonsingular range of the euler-angle-based and quaternion-based relative attitude description are ð 2π ; π2 Þ and ð  π ; π Þ, respectively. Moreover, the modified Rodrigues parameter-based approach should employ the projection and switching mechanisms to achieve the global relative attitude description [45]. The rotation matrix that is used to directly design the relative attitude controller can avoid the singularity of the conventional attitude parameters and achieve the almost global relative attitude control. Remark 6. It is worthwhile to mention that the proposed robust adaptive controllers (22) and (40) are derived based on the standard σ-modification scheme [46] to ensure the stability of the estimations, but the convergence of the estimated parameters to their true values cannot be guaranteed, since the persistent excitation conditions are not satisfied [46]. Recently, a new adaptive method was developed in [47] to ensure that the estimated parameters exponentially converge to their true values. This adaptive method was also extended to estimate the unknown neural network weights and sprung mass for the active suspension systems [48] and was used to exactly estimate parameters of robotic systems in finitetime [49]. In spite of the exact estimation of this adaptive method, the regressor matrices should be adapted online such that the computational burden of the controller is also increased as well as the composite adaptive control method in [50]. 4.3. Stability analysis for complete system Theorem 3. Under Assumptions 1 and 2, for the globally defined close proximity operations model, including (11), (13), (17) and (19), if the controllers are defined by (22) and (40) and adaptive laws are assigned by (24), (25), (42) and (43), then closed-loop system states r e and eR converge to sufficiently small neighborhood of zero. Proof. Let V ¼ V 1 þ V 2 be the Lyapunov candidate of complete system. Using (28) and (48), the bound of V can be given as zT M 1 z rV r zT M 2 z þ ϑ J θ~ J 2 ; z 9 ½zT1 ; zT2 T A R4 ,

where

ϑ ¼ max

n

1 ; 1; 1; 1; 1 2km 2kP 2kJ 2kρ 2kA

ð60Þ

M 1 ¼ diagfU 1 ; W 1 g, M 2 ¼ diagfU 1 ; W 2 g, o ~ δ~ P ; J J~ J F ; J ρ~ J ; δ~ A T . , and θ~ ¼ ½m;

Using (35) and (51), the time derivative of V is given by V_ r zT M 3 z þ κ 0 ;

ð61Þ

where M 3 ¼ diagfU 2 ; W 3 g, κ 0 ¼ κ 1 þ κ 3 . Under the given positive designing parameters, all of the matrices M 1 ; M 2 ; M 3 are positive definite. Then, we have

η V_ r  3 V þ κ ; η2

ð62Þ

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i

L. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

7

where η3 and η2 represent the minimum eigenvalue and the maximum eigenvalue of matrices M3 and M2, respectively; κ ¼ κ 0 þ ηη32 ϑ J θ~ J 2 . Thus we know   η κ  η3 t η κ VðtÞ r Vð0Þ  2 e η2 þ 2 : ð63Þ

η3

η3

This means that z and θ~ are bounded. Furthermore, from (60), we conclude sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " #   r e ðtÞ 1 η κ  η3 t η κ Vð0Þ  2 e η2 þ 2 ; r J zðtÞ J r eR ðtÞ η3 η1 η3 η1 where

η1 is the minimum eigenvalue of matrix M1. Thus,

lim J ½r Te ðtÞ; eTR ðtÞT J r

t-1

rffiffiffiffiffiffiffiffiffiffi η2 κ ;

η1 η3

and smaller σ 1 ; σ 2 and larger c1 ; c2 ; kv ; kw ; γ ; km ; kP ; kJ ; kρ ; kA result in larger η3 and smaller κ, thus the sufficiently small ultimate bounds of relative position and relative attitude can be regulated.□ Remark 7. Note that the feedback gains fkr ; kv ; c1 g in the relative position controller (22) and fkR ; kw ; c2 g in the relative attitude controller (40) can be chosen independently to satisfy their own bandwidths.

5. Numerical example An example of close proximity mission in orbit for rendezvous and docking is simulated in this section. Assume that the chaser's initial position, velocity, attitude, and angular velocity are r ¼ ½1; 1; 1T  7:078  108 m, v ¼ ½2; 3;  2T m=s, Rð0Þ ¼ I 3 , and ω ¼ ½0; 0; 0T rad=s, respectively. The initial relative position, relative velocity, pffiffiffi relativepffiffiffiattitude, and relative angular velocity are r e ¼ ½50= 2; 0;  50= 2T m, ve ¼ ½0:5;  0:5; 0:5T m=s, Re ð0Þ ¼ I 3 , and ωe ¼ ½0:02;  0:02; 0:02T rad=s, respectively. The target's orbital radius is r o ¼ 7:078  108 m and the gravitational constant of the Earth is μg ¼ 3:986  1014 m3 =s2 . Desired position of the chaser denoted in frame F t is given as pt ¼ ½0; 5; 0T m. The prior known bounds of the chaser's inertia matrix are λm ¼ 250 and λM ¼ 610; the bound of thrust misalignment vector is ρ ¼ 0:05. We use the proposed controllers (22) and (40) to achieve the objective for the chaser approaching a tumbling non-cooperative target. The controller parameters are chosen as K ¼ 0:6  diagf1; 2; 3g, kR ¼5, kw ¼60, kr ¼ 0.5, kv ¼6, c1 ¼0.1, c2 ¼ 0.2, kJ ¼ km ¼ 0:5, kA ¼ kP ¼ 0:2,

Fig. 3. Relative attitude motion under controller (40).

^ kρ ¼ 0:2, σ 1 ¼ σ 2 ¼ 0:01, γ ¼50, mð0Þ ¼ δ^ A ð0Þ ¼ δ^ P ð0Þ ¼ 0; ρ^ ð0Þ ^ ¼ 0; J ð0Þ ¼ O3 . In the simulation, parameters of the chaser, the target and disturbance are listed as:

ρ ¼ ½0:02; 0:03; 0:025T m;

m ¼ 58:2 kg; 3  51:5 6  22:5 424:4 2  27 7 J¼4 5 kg m ;  51:5  27 263:6 2 3 3336:3  135:4  154:2 6  135:4 3184:5  148:5 7 2 Jt ¼ 4 5 kg m ;  154:2  148:5 2423:7 2 3 1 þ 7 sin ðωo tÞ 2 cos ðωo tÞ 6 7 dτ ¼ 4 8  5 sin ðωo tÞ þ3 cos ðωo tÞ 5  10  5 N m; 1 þ 5 sin ðωo tÞ 2 cos ðωo tÞ 2 3 1  8 sin ðωo tÞ þ 2 cos ðωo tÞ 6 7 df ¼ 4 8  6 sin ðωo tÞ  3 cos ðωo tÞ 5  10  4 N: 1 þ 5 sin ðωo tÞ þ 2 cos ðωo tÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ωo ¼ μg = J r J 3 is the chaser's orbital angular velocity. Fig. 2 shows the time histories of relative position and relative velocity expressed in frame F c . The task of the position control in this simulation is to drive the chaser to a desired position pt . The relative position and relative velocity tending to steady state in 150 s imply that the desired position tracking is achieved. The time history of relative attitude is shown in Fig. 3. The relative attitude and relative angular velocities tending to steady state in 150 s indicate that the chaser attitude is synchronized with the attitude of the target. The control forces and control torques presented in Fig. 4 show that the large control efforts of chaser spacecraft are used initially in order to track the desired position and coincide with the tumbling target attitude quickly. Figs. 5 and 6 show that the estimation errors of unknown parameters are bounded by using the proposed controllers. The simulation results demonstrate that the proposed control strategy can drive the chaser to track the position and attitude of the target precisely. In order to show the advantage of the proposed relative attitude controller (40), the modified Rodrigues parameter-based relative attitude controller is designed by 2

598:3

 22:5

τ ¼ kR σ e  kw ωe þ SðωÞJ^ω þ J^SðωÞωe  Sðρ^ Þf  δ^ A tanhðγ eA Þ; Fig. 2. Relative position motion under controller (22).

ð64Þ

where eA ¼ ωe þ c2 σ e ; σ e is the relative attitude described by modified Rodrigues parameters; the estimations J^ ; ρ^ ; δ^ A are still

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i

8

L. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Fig. 7. Relative attitude motion under controller (64) with small gains. Fig. 4. Control forces and torques under controllers (22) and (40).

Fig. 8. Relative attitude motion under controller (64) with large gains.

^ J^ , and ρ^ under controllers (22) and (40). Fig. 5. Parameters estimation m,

adjusted by adaptive laws (42) and (43); attitude controller parameter is chosen as kR ¼60, while kw ; c2 ; kJ ; kA ; kρ ; σ 2 ; γ and initial values of estimations are still chosen as controller (40). Moreover, the relative position controller (22) is still employed. By using controller (64), the time history of relative attitude motion is shown in Fig. 7. Furthermore, when the feedback gains of controller (64) are enlarged as kR ¼ kw ¼ 160; c2 ¼ 2, relative attitude motion simulation result is shown in Fig. 8. Comparing Fig. 3 with Figs. 7 and 8, it is clearly seen that the transient response and steady-state performance of the closed-loop relative attitude system with proposed controller (40) are obviously better than ones with the modified Rodrigues parameter-based relative attitude controller (64).

6. Conclusion

Fig. 6. Parameters estimation δ^ P and δ^ A under controllers (22) and (40).

The six degrees-of-freedom relative motion control problem of a chaser spacecraft approaching a tumbling space target was investigated in this paper. With proposing a globally defined relative motion model, robust adaptive controllers were developed for relative position and relative attitude dynamics, in spite of the presence of the kinematic couplings, thrust misalignment, parametric uncertainties and bounded external disturbances. Despite

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i

L. Sun et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

the independently designing of relative position and attitude controllers, uniformly ultimately boundedness of the six degreesof-freedom relative motion was derived based on the Lyapunov analysis. A numerical example verified that the chaser can track both the desired relative position and target's attitude precisely by using the proposed controllers. However, the thruster output of the chaser and the approaching time of proximity missions are limited in practice, thus the performance of the closed-loop system may be deteriorated under the control input constraint and finite-time convergence. In future works, an extension of the proposed controller in this study used to handle control input constraint and finite-time convergence problems should be carried out.

Acknowledgments This work was supported by National Science Foundation of China (Grant nos. 61134005, 61327807) and National Key Basic Research Program of China (Grant no. 2012CB821204).

References [1] Han Y, Biggs JD, Cui N. Adaptive fault-tolerant control of spacecraft attitude dynamics with actuator failures. J Guid Control Dyn 2015;38(10):2033–42. [2] Mazinan AH, Pasand M, Soltani B. Full quaternion based finite-time cascade attitude control approach via pulse modulation synthesis for a spacecraft. ISA Trans 2015;58:567–85. [3] Huo B, Xia Y, Lu K, Fu M. Adaptive fuzzy finite-time fault-tolerant attitude control of rigid spacecraft. J Frankl Inst 2015;352(10):4225–46. [4] Zhao L, Jia Y. Finite-time attitude tracking control for a rigid spacecraft using time-varying terminal sliding mode techniques. Int J Control 2015;88(6): 1150–62. [5] Gao J, Cai Y. Fixed-time control for spacecraft attitude tracking based on quaternion. Acta Astronaut 2015;115:303–13. [6] Zhao L, Jia Y. Neural network-based distributed adaptive attitude synchronization control of spacecraft formation under modified fast terminal sliding mode. Neurocomputing 2016;171:230–41. [7] Zhou N, Xia Y. Coordination control design for formation reconfiguration of multiple spacecraft. IET Control Theory Appl 2015;9(15):2222–31. [8] Ghasemi S, Khorasani K. Fault detection and isolation of the attitude control subsystem of spacecraft formation flying using extended Kalman filters. Int J Control 2015;88(10):2154–79. [9] Mehrabian A, Khorasani K. Distributed and cooperative quaternion-based attitude synchronization and tracking control for a network of heterogeneous spacecraft formation flying mission. J Frankl Inst 2015;352(9): 3885–913. [10] Cabecinhas D, Cunha R, Silvestre C. Almost global stabilization of fullyactuated rigid bodies. Syst Control Lett 2009;58(9):639–45. [11] Lee T. Exponential stability of an attitude tracking control system on SOð3Þ for large-angle rotational maneuvers. Syst Control Lett 2012;61(1):231–7. [12] Younes AB, Mortari D, Turner JD. Attitude error kinematics. J Guid Control Dyn 2014;37(1):330–6. [13] Chaturvedi NA, Sanyal AK, McClamroch NH. Rigid body attitude control. IEEE Control Syst Mag 2011;31(3):30–51. [14] Bhat SP, Bernstein DS. A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Syst Control Lett 2000;39(1):63–70. [15] Lv Y, Hu Q, Ma G, Zhou J. 6DOF synchronized control for spacecraft formation flying with input constrint and parameter uncertainties. ISA Trans 2011;50 (4):573–80. [16] Bustan D, Pariz N, Hosseini Sani SK. Robust fault-tolerant tracking control design for spacecraft under control input saturation. ISA Trans 2014;53 (4):1073–80. [17] Yuichi I, Takashi K, Tomoyuki N. Nonlinear tracking control of rigid spacecraft under disturbance using PD and PID type H1 state feedback. In: Proceedings of the 50th IEEE conference on decision and control & European control conference. Piscataway, NJ: IEEE Publication; 2011. p. 6184–91. [18] Xin M, Pan H. Integrated nonlinear optimal control of spacecraft in proximity operations. Int J Control 2010;83(2):347–63.

9

[19] Hu Q, Li B, Zhang Y. Robust attitude control design for spacecraft under assigned velocity and control constraints. ISA Trans 2013;52(4):480–93. [20] Zhang F, Duan G. Robust adaptive integrated translation and rotation finitetime control of a rigid spacecraft with actuator misalignment and unknown mass property. Int J Syst Sci 2014;45(5):1007–34. [21] Zou A. Finite-time output feedback attitude tracking control for rigid spacecraft. IEEE Trans Control Syst Technol 2014;22(1):338–45. [22] Singla P, Subbarao K, Junkins JL. Adaptive output feedback control for spacecraft rendezvous and docking under measurement uncertainty. J Guid Control Dyn 2006;29(4):892–902. [23] Subbarao K, Welsh S. Nonlinear control of motion synchronization for satellite proximity operations. J Guid Control Dyn 2008;31(5):1284–94. [24] Zhang F, Duan G. Integrated translational and rotational finite-time maneuver of a rigid spacecraft with actuator misalignment. IET Control Theory Appl 2012;6(9):1192–204. [25] Wang X, Yu C. Unit dual quaternion-based feedback linearization tracking problem for attitude and position dynamics. Syst Control Lett 2013;62(3):225– 33. [26] Wang J, Liang H, Sun Z, Wu S, Zhang S. Relative motion coupled control based on dual quaternion. Aerosp Sci Technol 2013;25(1):102–13. [27] Wu J, Liu K, Han D. Adaptive sliding mode control for six-DOF relative motion of spacecraft with input constraint. Acta Astronaut 2013;87:64–76. [28] Pukdeboon C. Inverse optimal sliding mode control of spacecraft with coupled translation and attitude dynamics. Int J Syst Sci 2015;46(13):2421–38. [29] Mayhew CG, Sanfelice RG, Teel AR. Quaternion-based hybrid control for robust global attitude tracking. IEEE Trans Autom Control 2011;56(11):2555–66. [30] Lee D, Vukovich G. Adaptive sliding mode control for spacecraft body-fixed hovering in the proximity of an asteroid. Aerosp Sci Technol 2015;46:471–83. [31] Sun L, Huo W. Robust adaptive control of spacecraft proximity maneuvers under dynamic coupling and uncertainty. Adv Space Res 2015;56(10):2206–17. [32] Sun L, Huo W. Robust adaptive relative position tracking and attitude synchronization for spacecraft rendezvous. Aerosp Sci Technol 2015;41:28–35. [33] Sun L, Huo W. Nonlinear robust adaptive trajectory tracking control for spacecraft proximity operations. J Aerosp Eng 2015;229(13):2429–40. [34] Sun L, Huo W. 6-DOF intergrated adaptive backstepping control for spacecraft proximity operations. IEEE Trans Aerosp Electron Syst 2015;51(3):2433–43. [35] Pan H, Kapila V. Adaptive nonlinear control for spacecraft formation flying with coupled translational and attitude dynamics. In: Proceedings of the 40th IEEE conference on decision and control. Piscataway, NJ: IEEE Publication; 2001. p. 2057–62. [36] Zhang F, Duan G. Integrated relative position and attitude control of spacecraft in proximity operation missions. Int J Autom Comput 2012;9(4):342–51. [37] Ploycarpou MM. A robust adaptive nonlinear control design. Automatica 1996;32(3):423–7. [38] Cunha R, Silvestre C, Hespanha J. Output-feedback control for stabilization on SEð3Þ. Syst Control Lett 2008;57(12):1013–22. [39] Xin M, Pan H. Nonlinear optimal control of spacecraft approaching a tumbling target. Aerosp Sci Technol 2011;15(2):79–89. [40] Lee T. Robust adaptive attitude tracking on SOð3Þ with an application to a quadrotor UAV. IEEE Trans Control Syst Technol 2013;21(5):1924–30. [41] Sanyal AK, Chaturvedi NA. Almost global robust attitude tracking control of spacecraft in gravity. AIAA guidiance, navigation and control conference exhibit. AIAA Paper 2008-6979; 2008. [42] Koditschek D. Application of a new Lyapunov function to global adaptive tracking. In: Proceedings of the 27th IEEE conference on decision and control. Piscataway, NJ: IEEE Publication; 1988. p. 63–8. [43] Kim SG, Crassidis JL, Cheng Y, Fosbury AM, Junkins JL. Kalman filtering for relative spacecraft attitude and position estimation. J Guid Control Dyn 2007;30(1):133–43. [44] Segal S, Carmi A, Gurfil P. Stereovision-based estimation of relative dynamics between noncooperative satellite: theory and experiments. IEEE Trans Control Syst Technol 2014;22(2):568–84. [45] Shuster MD. A survey of attitude representations. J Astronaut Sci 1993;41 (4):439–517. [46] Krstic M, Kanellakopoulos I, Kokotovic P. Nonlinear and adaptive control design. New York: John Wiley & Sons; 1995. [47] Adetola V, Guay M. Performance improvement in adaptive control of linearly parameterized nonlinear systems. IEEE Trans Autom Control 2010;55 (9):2182–6. [48] Huang Y, Na J, Wu X, Liu X, Guo Y. Adaptive control of nonlinear uncertain active suspension systems with prescribed performance. ISA Trans 2015;54:145–55. [49] Na J, Mahyuddin MN, Herrmann G, Ren X, Barber P. Robust adaptive finitetime parameter estimation and control for robotic systems. Int J Robust Nonlinear Control 25(16):3045-71. [50] Slotine JJE, Li W. Applied nonlinear control. Englewood Cliffs, NJ: Pretice Hall; 1991.

Please cite this article as: Sun L, et al. Robust adaptive relative position and attitude control for spacecraft autonomous proximity. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.02.022i