Adaptive variable-structure finite-time mode control for spacecraft proximity operations with actuator saturation

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Adaptive variable-structure finite-time mode control for spacecraft proximity operations with actuator saturation Daero Lee ⇑, George Vukovich, Haichao Gui Department of Earth and Space Science and Engineering, York University, 4700 Keele Street, 150 Atkinson College, Toronto, Ontario M3J 1P3, Canada Received 5 June 2016; received in revised form 1 November 2016; accepted 17 February 2017

Abstract This paper presents an adaptive variable-structure finite-time control for spacecraft proximity maneuvers under parameter uncertainties, external disturbances and actuator saturation. The coupled six degrees-of-freedom dynamics are modeled for spacecraft relative motion, where the exponential coordinates on the Lie group SE(3) are employed to describe relative configuration. No prior knowledge of inertia matrix and mass of the spacecraft is required for the proposed control law, which implies that the proposed control scheme can be applied in spacecraft systems with large parametric uncertainties in inertia matrix and mass. Finite-time convergence of the feedback system with the proposed control law is established. Numerical simulation results are presented to illustrate the effectiveness of the proposed control law for spacecraft proximity operations with actuator saturation. Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Adaptive variable-structure finite-time sliding mode control; Coupled six degrees-of-freedom dynamics; Actuator saturation; Lie group SE(3); Spacecraft proximity operations

1. Introduction Spacecraft proximity operation is a core technology for many important space mission such as space debris management, supply to the International Space Station, on-orbit satellite maintenance, and large-scale structure assembly and satellite networking (Goodman, 2006; Singla et al., 2006). Spacecraft proximity operations require extremely delicate and precise translational and rotational maneuvers (Goodman, 2006; Singla et al., 2006; Souza et al., 2007). Generally up to now, the orbit and attitude motions of spacecraft have been separately controlled, ignoring their coupling so that satisfying control requirements has been difficult. However, coupled translational and rotational maneuvers of spacecraft need ⇑ Corresponding author.

E-mail addresses: [email protected] (D. Lee), [email protected] (G. Vukovich), [email protected] (H. Gui).

to be considered when large angular maneuvers are required (Xin and Pan, 2012), which necessitates a sixdegree of-freedom (6-DOF) control system. The modeling and control of six-degree-of-freedom (6DOF) relative motion between spacecraft is a challenging task, particularly for space missions requiring great maneuverability and high precision (Xin and Pan, 2012; Xin and Pan, 2011). Rigid body modeling of the 6-DOF relative motion between a number of spacecraft has been extensively studied in recent years based on various formulations (Xin and Pan, 2012; Xin and Pan, 2011; Zhang and Duan, 2012; Sun and Huo, 2015; Wang and Sun, 2012; Gui and Vukovich, 2016; Gui and Vukovich, 2016). Here the spacecraft relative motion dynamics are modeled as undergoing large range of motion maneuvers in three-dimensional Euclidean space. Their configuration space is the Lie group SEð3Þ, which is the set of positions and orientations of the spacecraft moving in three-dimensional Euclidean space (Sanyal et al., 2011; Marsden and Ratiu, 1999; Bloch

http://dx.doi.org/10.1016/j.asr.2017.02.029 0273-1177/Ó 2017 COSPAR. Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: Lee, D., et al. Adaptive variable-structure finite-time mode control for spacecraft proximity operations with actuator saturation. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.02.029

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D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx

Nomenclature SEð3Þ

Lie group or set of positions and orientations of the rigid spacecraft moving in three-dimensional Euclidean space SOð3Þ Lie group of orientations of the rigid body soð3Þ Lie algebra of SOð3Þ, which is represented as the linear space of 3  3 skew-symmetric matrices seð3Þ semi-direct product of R3 and soð3Þ and isomorphic to the vector space R6 6 R six dimensional real Euclidean vector space R3 three-dimensional real Euclidean space of positions of the center of mass of the body b inertial position vector m translational velocity in body frame X angular velocity in body frame Fg gravity force in body frame Mg gravity gradient moment in body frame l gravitational parameter of the Earth g configuration of the follower spacecraft on SEð3Þ n vector of body velocities of the follower spacecraft Ad adjoint actions of g 2 SEð3Þ on X 2 seð3Þ ad adjoint representation of seð3Þ adH co-adjoint representation of seð3Þ et al., 2003; Bullo and Lewis, 2005; Varadarajan, 1984). The relative configuration (position and attitude) of a follower spacecraft with respect to a leader spacecraft is represented by exponential coordinates in SEð3Þ (Varadarajan, 1984; Lee et al., 2015). Thus, translational and rotational maneuvers of a spacecraft can then be performed without explicit reference states by transforming a tracking problem into a stabilization problem using exponential coordinates. Note that the exponential coordinates for the relative attitude (orientation) are not uniquely defined in this case (Sanyal et al., 2011; Lee et al., 2015). A spacecraft is generally subject to various disturbance forces and torques, including gravitational, aerodynamic, solar radiation, magnetic and other environmental and non-environmental influences. In addition, the inertia properties of spacecraft which are inertia matrix and mass of the spacecraft may not be known accurately, so that control design must account for inertia uncertainties. The presence of the external disturbances and inertia uncertainties makes the control problem more complicated. Another important problem that occurs in practice is input saturation which is one of the severe actuator nonlinearities in the spacecraft control system design (Hu et al., 2008). This constraint represents the physical limitation on control capacity which occurs in most actuators. Input saturation can severely restrict system performance, resulting in system instability or otherwise undesirable control response (Hu et al., 2008; Hu, 2009; Zhu et al., 2011). The integrated

ug uc hf h ~g expm logm ~ H ~ b ~n

vector of known gravity inputs (moments and force) of the follower spacecraft vector of control inputs (torque and force) of the follower spacecraft fixed relative configuration of the follower spacecraft to the virtual leader relative configuration of the follower spacecraft to the virtual leader exponential coordinate vector for the configuration error of the follower spacecraft exponential map (inverse of the logarithm map) from seð3Þ to SEð3Þ logarithm map (inverse of the exponential map) from SEð3Þ to seð3Þ exponential coordinate vector for the attitude tracking error (principal rotation vector) of the follower spacecraft exponential coordinate vector for the position tracking error of the follower spacecraft relative velocities the follower spacecraft with respect to the virtual leader

Superscript 0 virtual leader states and parameters

relative position and attitude tracking problem due to these input constraints are further complicated by external disturbances and inertia uncertainties. There has not been a great number of studies done on integrated relative position and attitude tracking with control limits. Ahmed et al. (1998) developed an adaptive tracking control law that requires no knowledge of the spacecraft inertia matrix. The tracking algorithm is able to identify the spacecraft inertia matrix by using periodic command signals. Crassidis et al. (2000) showed an optimal control law for asymptotic tracking of spacecraft maneuvers using variable-structurecontrol. Younes et al. (2013) presented a nonlinear feedback control strategy where the feedback control is augmented with feedback gain sensitivity partial derivatives for handling model uncertainties. The optimal gain is then computed as a Taylor series expansion in the Riccati gain as a function of the system model parameters. Hu et al. (2008, 2011) addressed the attitude control problem for a spacecraft model with actuator saturation in the presence of inertia uncertainty and external disturbances where no prior knowledge of inertias is required and the upper bound of disturbance is not known. Boskovic et al. (2001) proposed two globally stable control algorithms based on variable-structure control in the presence of control input saturation, parametric uncertainty, and external disturbances. In this paper, inspiring from actuator saturation and attitude stabilization problems (Hu et al., 2008; Hu, 2009;

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Zhu et al., 2011; Gui and Vukovich, 2016), we extend attitude stabilization problem to coupled translational and rotational tracking control with actuator saturation by proposing an adaptive variable-structure finite-time mode control (AVFTC) law on SEð3Þ for autonomous spacecraft proximity operations. A new form of terminal sliding mode (TSM) on SEð3Þ is defined and a fast-TSM type reaching law is used to achieve relative position and velocity tracking in finite time. The proposed control law is important for autonomous spacecraft proximity operation problem where actuator saturation may occur and finite-time state stabilization is required to carry out coupled translational and rotational tracking maneuver in scheduled time. Furthermore, the proposed control law does not require prior knowledge of inertia matrices and upper bounds of external disturbances. It employs a saturation function to treat actuator saturation as the variable-structure control inputs. It is also designed to reduce the relative configuration (pose or position and orientation) and velocities without explicit reference states in finite time from any given initial state, except those that differ in orientation by a p radian rotation from the desired states at the initial time. It guarantees almost global finite-time convergence of the coupled translational and rotational tracking errors highly required for autonomous proximity operations. Thus, the spacecraft with actuator saturation can achieve the desired configuration or pose and velocities in finite time by performing coupled translational and rotational tracking maneuver in the presences of inertia uncertainty and external disturbances. The proposed control method is based on a global analysis of the motion using the framework of geometric mechanics, and is applicable to all types of orbits of the leader and follower spacecraft about a central body, including open (parabolic or hyperbolic) trajectories. The contribution of this paper is as follows: An AVFTC law is proposed for autonomous spacecraft proximity operations under actuator saturation so that the spacecraft can achieve the desired configuration and velocities in finite time in the presences of inertia uncertainty and external disturbances. The proposed controller can attain superior control performance than the conventional terminal sliding mode control (TSMC) law (Hui and Li, 2009). In addition, the relative motion dynamics and AVFTC law has no restriction in the orbit type for proximity operations. This paper is organized as follows. In Section 2, rigidbody dynamics models for the leader and follower spacecraft, and the relative coupled translational and rotational dynamics are presented. In Section 3, the design procedure of the terminal sliding mode control (TSMC) on SEð3Þ is presented. In Section 4, the design procedure of the adaptive variable-structure terminal sliding mode control (TSMC) on SEð3Þ is presented. In Section 5, simulation results are presented. In Section 6, this paper concludes a summary. 2. Rigid body dynamics models The dynamics of the leader space spacecraft (which is not controlled) and the follower spacecraft are now

3

described. For the relative motion dynamics of the spacecraft, three coordinates frames are defined. The Earth Centered Inertial (ECI) frame is denoted by fNg ¼ fN X ; N y ; N z g. Both the leader and follower spacecraft are regarded as rigid bodies, and their body-fixed frames are denoted by fB 0 g ¼ fB0x ; B0y ; B0z g and fBg ¼ fBx ; By ; Bz g, respectively. For spacecraft proximity operations, the follower is tasked to perform coupled translational and rotational maneuvers as illustrated in Fig. 1. The desired trajectories of the spacecraft are obtained by specifying their relative pose (position and orientation) with respect to the leader’s trajectory. The translational dynamics models used here for the spacecraft include the effect of the Earth oblateness on orbits to the level of (J 2 ), the second zonal harmonics. 2.1. Dynamics model and state trajectory of the leader The leader spacecraft is modeled as a rigid spacecraft orbiting the Earth with its central gravity field. The configuration space of the leader is the special Euclidean group SEð3Þ, which is the set of all translational and rotational motions of a rigid body. SEð3Þ is also a Lie group and can be expressed as the semi-direct product SEð3Þ ¼ R3 nSOð3Þ, where SOð3Þ is the Lie group of special orthogonal matrices representing orientations of the spacecraft body, and R3 is the three-dimensional real Euclidean space of positions of the center of mass of the body. The superscript ðÞ0 specifies the leader spacecraft states and parameters. The leader attitude is represented by the rotation matrix R0 2 SOð3Þ that transforms the leader body-fixed frame to the Earthcentered inertial (ECI) frame. The leader’s position is expressed by the inertial position vector b0 2 R3 from the origin of the ECI frame to the center of mass of the leader. The translational and angular velocities of the leader are denoted by the vectors m 0 2 R3 and X0 2 R3 , respectively, as represented in the body-fixed frame fB0 g ¼ fB0x ; B0y ; B0z g. The kinematics of the leader is expressed as: h i R_ 0 ¼ R0 X0 ; ð1Þ b_ 0 ¼ R0 m 0 ;

ð2Þ

where the operator ½  : R3 ! soð3Þ is the cross-product and soð3Þ is the Lie algebra of SOð3Þ, which is represented as the linear space of 3  3 skew-symmetric matrices. The Lie algebra of SEð3Þ, denoted by seð3Þ, is a sixdimensional vector space that is tangent to SEð3Þ at the identity element. The algebra seð3Þ is a semi-direct product of R3 and soð3Þ, and is isomorphic to the vector space R6 . Let the leader’s mass be m0 and its moment of inertia matrix be J 0 in its body frame. The dynamics of the leader are given by Lee et al. (2015) h i _ 0 ¼ J 0 X0  X0 þ M 0 ðb0 ; R0 Þ; J 0X ð3Þ g    T m0 m_ 0 ¼ m0 m0 X0 þ F 0g ðb0 ; R0 Þ þ m0 R0 a0J 2 ðb0 Þ; ð4Þ

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Fig. 1. Spacecraft proximity operations performing coupled translational and rotational maneuvers.

where M 0g 2 R3 ; F 0g denote the gravity gradient moment and gravity force on the leader, respectively, as given by Lee et al. (2015) and Schaub and Junkins (2009) !   0  l 0 ½p J p ; ð5Þ Mg ¼ 3 0 5 kb k ! ! ! 0 T 0 m l l 15 lðp J pÞ F 0g ¼  p3 J0 p þ p; ð6Þ 2 kb0 k5 kb0 k7 kb0 k3 T

where p ¼ ðR0 Þ b0 ; J0 ¼ 12 traceðJ 0 ÞI 3 þ J 0 ; l is the gravitational constant of the Earth (398,600.44 km3 s2), and I n denote a n  n identity matrix. The perturbation due to the Earth’s oblateness, J 2 in Earth-Centered-Inertial coordinate is (Vallado, 2004):  3 2 2 3l J 2 R2 b0I 5b0K  1  2kb0 k5 kb0 k2 7 6 7 6   7 6 2 5b0K 7 6 3l J 2 R2 b0J 0 ð7Þ aJ 2 ¼ 6  2kbk5 1  kb0 k2 7; 7 6 7 6 5 2 4 3l J R2 b0  5b0  2kb20 k5 K 3  kb0Kk2 where b0I ; b0J and b0K are the components of  T b0 ¼ b0I b0J b0k ; J 2 ¼ 1:08263  103 , and R ¼ 6378 km is the Earth’s equatorial radius. The state space of the leader is TSEð3Þ, which is described in Marsden and Ratiu (1999), Bloch et al. (2003) and Bullo and Lewis (2005) as SEð3Þ  seð3Þ ’ SEð3Þ  R6 and its motion states are denoted by ðb0 ; R0 ; m0 ; X0 Þ. Here, seð3Þ denotes the Lie algebra (tangent space at the identity) of the Lie group SEð3Þ, which is isomorphic to R6 as a vector space. The configura-

tion of the leader on SEð3Þ can also be represented by the following 4  4 matrix: " # R0 b0 0 2 SEð3Þ: ð8Þ g ¼ 013 1 where 0mn denotes a m  n null matrix. The vector of body velocities of the leader is denoted by " # X0 0 n ¼ ð9Þ 2 R6 : m0 From here on, the kinematics of the leader (1) will be expressed as follows: " h i # X0 m0 0 _ 0 0 0 _ g_ ¼ g ðn Þ ; where ðn Þ ¼ 2 seð3Þ; ð10Þ 013 0 where ðÞ_ denotes the isomorphism from R6 to seð3Þ. The mass and inertia properties of the leader assigned by a 6  6 matrix, and the vector of gravitational forces and moments assigned by a 6  1 vector, are expressed as follows: " # " # 0 M 0g 0 J 33 I0 ¼ 2 R66 ; and u0g ¼ 2 R6 : T 033 m0 I 3 F 0g þ m0 R0 a0J 2 ð11Þ The adjoint operator and co-adjoint representations where the adjoint action of SEð3Þ on seð3Þ is defined in the same manner as that between a Lie group and its corresponding Lie algebra. The adjoint action of g0 2 SEð3Þ on X 2 seð3Þ is given by

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" h

Adg0 ¼

R0 i 

0

b 

033 R

0

R

#

0

_ ¼ J ½X X þ M g ðb; RÞ þ sc ðb; R; m; XÞ þ sd ; JX

2 R66 ; such that

1 j Adg0 X j ¼ g0 X ðg0 Þ ;

where ðÞj : seð3Þ ! R6 is the inverse of the vector space _ isomorphism ðÞ : R6 ! seð3Þ. The representation of SEð3Þ given by (12) is termed the adjoint representation of SEð3Þ on its Lie algebra. The adjoint representation of seð3Þ is expressed in matrix form as 2h 3 i  X0 033 6 7 adn0 ¼ 4   h  i 5 2 R66 : ð13Þ 0 0 X m The co-adjoint operation is described on the dual of the Lie algebra, which can also be identified with R6 via the vector space isomorphism ðÞj . Therefore, the co-adjoint operator can be expressed in matrix form as 2 h i   3   X0  m0 6 7 T h  i 5: adn0 ¼ ðadn0 Þ ¼ 4 ð14Þ 0 033  X The use of the co-adjoint operator allows the dynamics of a spacecraft or other object modeled as a rigid body to be compactly expressed; the dynamics of the leader (3) and (4) in the compact form (Lee et al., 2015) using (14) are: ð15Þ

Note that without the gravity moment and force, the above equation is the Euler-Poincare´ equation on SEð3Þ (Marsden and Ratiu, 1999). The kinematics (10), the dynamics (15), along with known initial states ðg0 ðt0 Þ; n0 ðt0 ÞÞ at time t0 , can be used to generate the state trajectory of the leader spacecraft for time t P t0 . 2.2. Follower spacecraft dynamics The configuration of the follower spacecraft is given by the position vector from the origin of the geocentric inertial frame to the center of mass of the follower spacecraft (denoted by b 2 R3 ), and the attitude is given by the rotation matrix from the follower’s body-fixed frame !

!

!

fBg ¼ fb x ; b y ; b z g to the geocentric inertial frame (denoted R 2 SOð3Þ). The kinematics for the follower spacecraft takes the same form as the kinematics for the leader by removing the upper script 0, and is given by: _

g_ ¼ gðnÞ :

ð16Þ

Let /c : SEð3Þ  seð3Þ ! R denote the feedback control force acting on the spacecraft, and let 3 sc : SEð3Þ  seð3Þ ! R denote the feedback control torque acting on the spacecraft. The dynamics equations of motion for the spacecraft are then given as follows: 3

ð17Þ

mm_ ¼ m½m  X þ F g ðb; RÞ þ mRT aJ 2 ðbÞ þ /c ðb; R; m; XÞ þ /d ;

ð12Þ

0 0 0 I0 n_ 0 ¼ adH n0 I n þ ug :

5

ð18Þ

where F g ; M g 2 R3 denote the gravity force and gravity gradient moment, /c ; sc 2 R3 denote the control force and moment, and /d ; sd 2 R3 denote unknown forces and moments on the follower spacecraft, respectively due to atmospheric drag, radiation pressure, etc. and other bounded uncertainties. The gravity force and moment on the follower spacecraft have the same form as the gravity force and moment on the leader, which are given by (5)– (6). The dynamics (17)–(18) can be expressed in the compact form (Lee et al., 2015): In_ ¼ adH n In þ ug þ uc þ ud ;



 Mg sc ¼ ug ¼ ; u ; c T F g þ mR aJ 2 /c



 sd J 033 and I ¼ ; ud ¼ /d 033 mI 3

ð19aÞ

ð19bÞ

where ug 2 R6 is the vector of known gravity inputs (moments and force), uc 2 R6 is the vector of control inputs (torque and force), and ud 2 R6 is the vector of external disturbances on the follower spacecraft. 2.3. Relative coupled translational and rotational dynamics In this section, the relative coupled translational and rotational dynamics between the leader and follower spacecraft is described using the actual relative configuration and desired or fixed relative configuration between the leader and follower spacecraft, given by h 2 SEð3Þ and hf 2 SEð3Þ, respectively. The leader spacecraft trajectory is generated using the compact form of the dynamics of (19a) without control (Lee et al., 2015). The superscript ðÞ0 specifies the states and parameters of the leader spacecraft. The fixed relative configuration hf provides the appropriate inter-spacecraft separations and relative orientations. Let ðg; nÞ 2 SEð3Þ  R6 denote the states (configuration and velocities) of the follower spacecraft. The leader’s trajectory is generated using (10) and (15), and the desired states of the follower spacecraft are gd ¼ g0 hf and nd ¼ Adðhf Þ1 n0 for t P t0 ;

ð20Þ

and the configuration error between the follower spacecraft and leader spacecraft is 1

h ¼ ðg0 Þ g for t P t0 :

ð21Þ

The configuration tracking error of the follower spacecraft is expressed in exponential coordinates using the logarithm map: _

1

1

ð~gÞ ¼ logmððhf Þ hÞ ¼ logmððgd Þ gÞ;

ð22Þ

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D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx

where logm : SEð3Þ ! seð3Þ is the logarithm map (inverse of the exponential map). This makes ~ g the exponential coordinate vector giving the relative configuration between the desired configuration and the actual configuration of the follower satellite in the formation. This exponential coordinate vector for the configuration tracking error for the leader spacecraft is " # ~ H ~ g¼ 2 R6 ; ð23Þ ~ b

nential coordinates (25) and the time evolution of the relative accelerations (28). 2.4. Trajectory tracking errors The desired state trajectory of the spacecraft required to maintain the formation is given by the desired configuration gd and its time derivative, which gives the desired velocities, nd ¼ Adðh f Þ1 n0 as in (20). Denote the desired

The relative velocities of the follower spacecraft with respect to the leader are obtained by taking the time derivative of both sides of (21), and substituting (10) and (16), which yields

state trajectory for time t P t0 by the desired position vector in the inertial frame bd ðtÞ, the desired attitude Rd ðtÞ, and the desired translational and angular velocities in the bodyfixed frame, md ðtÞ and Xd ðtÞ respectively. The kinematics in SEð3Þ still holds for the desired states. The trajectory tracking errors of the follower spacecraft is defined as Lee et al. (2015): aðtÞ ¼ bðtÞ  bd ðtÞ: position tracking error in the inertial frame, X ðtÞ ¼ R0 ðtÞT aðtÞ: position tracking error T in the leader’s body-fixed frame, QðtÞ ¼ R0 ðtÞ RðtÞ: attitude tracking error in rotation matrix (orientation),     tðtÞ ¼ mðtÞ  QT ðtÞ md þ Xd ðtÞ X ðtÞ translational velocT ity tracking error, and xðtÞ ¼ XðtÞ  QðtÞ Xd ðtÞ: angular velocity tracking error.

~ n ¼ n  Adh1 n0 ¼ n  Adðhd Þ1 nd ;

3. Terminal sliding mode control design

~ 2 R3 are the exponential coordinate ~ 2 R3 and b where H vectors for the attitude tracking error (principal rotation vector) and the position tracking error, respectively. Remark 1. The logarithm map log: SEð3Þ ! seð3Þ is bijective when the principal angle of rotation corresponding to RðHÞ has a magnitude less than p radians, i:e:; kHk < p. It is not uniquely defined when kHk is exactly p radians.

ð24Þ

~ is the relative velocity of ~_ Þ and n where hd ¼ h1 f h ¼ expðg the follower spacecraft with respect to the leader in the follower’s body-fixed frame. The quantities h0 ¼ hðt0 Þ; ~g0 ¼ gðt0 Þ; n0 ¼ nðt0 Þ and ~ n0 ¼ ~ nðt0 Þ at initial time t0 , which are known; quantities if the states of the follower spacecraft are known. The kinematics in exponential coordinates is given in Bullo and Murray (1995) as follows: ~ g_ ¼ Gð~ gÞ ~ n:

ð25Þ

The properties of Gð~ gÞ are described in the Appendix. Note that this result also follows from the expansion of GðXÞ in terms of adX as given by (70), since adX X ¼ 0 for any vector X 2 R6 . The vector of relative velocities can be expressed as ~ n¼n n; where  n ¼ Adh1 n0 :

ð26Þ

The expression for the relative accelerations, i.e., the time derivative of ~ n, can be obtained from the time derivative of  n in (26) as derived in the Appendix and is given by ~_ ¼ n_ þ adn Ad 1 n0  Ad 1 n_ 0 : n h h

ð27Þ

Substituting the dynamics (19a) into (27) produces the following equation for the time evolution of the relative accelerations: _ I~ n ¼ adH n In þ ug þ uc þ ud   þ I adn Adh1 n0  Adh1 n_ 0 :

ð28Þ

Thus, the coupled relative translational and rotational dynamics is described by the relative kinematics of expo-

In this section, based on the existing results on the standard terminal sliding mode control (TSMC), the design procdure for spacecraft position and attitude stabilization is presented. Before giving the main results, lemmas and assumptions are needed. Assumption 1. The external disturbances sd ðtÞ and /d ðtÞ are assumed to be bounded by some known constants F i j sd i j6 F j ; i; j ¼ 1; 2; 3; j /d i j6 F j ; i ¼ 1; 2; 3; j ¼ 4; 5; 6:

ð29Þ ð30Þ

Assumption 2. The configurations (g0 and g) and velocities (n0 and n) of the leader and follower are assumed to be available for the control law design for the leader through onboard navigations. Assumption 3. The symmetric positive definite inertia matrix and the spacecraft mass are assumed to satisfy the following inequalities: kJ k 6 kJ ;

mðtÞ 6 mðt0 Þ:

ð31Þ

where kJ > 0 is an upper bound on the norm of the inertia matrix, which is unknown due to the uncertainty existing in the inertia matrix, mðtÞ and mðt0 Þ are the time varying mass and initial mass of the spacecraft, respectively. Assumption 4. All six components of the control torque and force uc are constrained by respective saturation values, expressed by

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j uci ðtÞ j6 scmax ; i ¼ 1; 2; 3; j uci ðtÞ j6 /cmax ; i ¼ 4; 5; 6;

ð32aÞ ð32bÞ

where scmax and /cmax are the available maximum torque and force components in the actuators, respectively. Assumption 5. The control signal strictly dominate the unknown disturbances, that is j ud i ðtÞ j6 sd max < scmax ; j ud i ðtÞ j6 /d max < /cmax ;

i ¼ 1; 2; 3; i ¼ 4; 5; 6:

ð33Þ ð34Þ

3.1. Terminal sliding mode control A sliding plane on the Lie group SEð3Þ or SEð3Þ which consists of the exponential coordinates and relative velocities is defined to achieve position and attitude stabilities in finite or scheduled time as s¼ ~ n þ C~ gr ;

ð35Þ

fying the inequality k H i > F i ; i ¼ 1;2;...;6. Proof. Consider candidate:

the

Lemma 1. Suppose there exist a smooth function, v. If there exist c > 0 and 0 < a < 1 such that v is positive definite as well as v_ þ cva is negative semi-definite, then v can converge to zero in finite-time. Meanwhile, the convergence time T c , which the time interval to reach v ¼ 0, satisfies

Tc 6

v01a cð1  aÞ

ð36Þ

where v0 denotes the initial value of v (Bhat and Bernstein, 2000). Lemma 2. Consider the spacecraft system (28). For terminal sliding surface (35) satisfying s ¼ 061 , then ~ g  061 and ~ n  061 can be reached in finite or scheduled time (Lee and Vukovich, 2016).

following

Lyapunov

function

1 V 1 ¼ sT Is: 2

ð38Þ

Taking time derivative of the Lyapunov function V 1 yields  _ V_ 1 ¼ sT I ~n þ Cr~gðr1Þ ~g_ : ð39Þ Substituting the time evolution of the relative accelerations (25) and the kinematics of exponential coordinate (28) into (39) results in h V_ 1 ¼ sT adH n In þ ug þ uc þ ud   ð40Þ þ I adn Adh1 n0  Adh1 n_ 0 þ KGð~gÞ~n : Substituting the control law (37) into (40) yields 6 X    H  V_ 1 ¼ sT ud  kH sgnðsÞ 6  k i j si j F i j si j

T

where s ¼ ½s1 ; . . . ; s6  2 R6 is the sliding plane,  T T ~ g ¼ ½ g1 ; . . . ; g6  ; ~ nr ¼ nr1 ; . . . ; nr6 ; C ¼ diagðc1 ; . . . ; c6 Þ is a positive definite matrix, and 0 < r < 1. Before giving the main results, the following lemmas, properties and assumptions are needed.

7

pffiffiffi 1 ¼  2ckmin ðIÞV 21 ;

i¼1

ð41Þ

where 0 < c < 1 and kmin ðIÞ is the minimum eigenvalue of I. Therefore, by Lemma 1, the state trajectory of the system (25) and (28) will reach the sliding surface sðtÞ ¼ 061 in finite or scheduled time. By Lemma 2, the control law (37) globally stabilizes the closed-loop feedback system given by (25) and (28), and ð~g; ~nÞ ¼ ð061 ; 061 Þ 2 R6  R6 in finite-time and therefore tracks the trajectory ðg; nÞ ¼ ðgd ; n0 Þ in finite-time. Remark 2. The control law (37) is discontinuous when crossing the sliding surface sðtÞ ¼ 061 , which may result in undesirable chattering. The chattering due to the implementation of the signum function in the control scheme (53) can be mitigated by replacing the signum function with the continuous saturation function defined as:

satðsi ; eÞ ¼

si =e; sgnðsi Þ;

j si j6 e; for i ¼ 1; 2; . . . ; 6: j si jP e;

ð42Þ

Theorem 1. For the relative coupled translational and rotational dynamics described by (25) and (28), if the sliding plane is designed as the follower (35), the system motion will converge to zero along the sliding plane sðtÞ ¼ 0 in finite or scheduled time and the control scheme is as follows:

where e denotes the width of the boundary layer. There may exist a possible singularity in the TSMC as ~ g converges to 061 . But the singularity problem can be avoided when K H is properly chosen (Hui and Li, 2009). The TSMC then switches on to drive the system controlling errors zero in finite or scheduled time.

 0 _0 uc ¼  ug þ adH gÞ~nÞ þ K H sgnðsÞ ; n In þ Iðadn Adh1 n  Adh1 n þ KGð~

4. Controller design under input saturation

ð37Þ

 ðr1Þ ðr1Þ where K ¼ diag c1 r~g1 ;...;c6 r~ g6 ;sgnðsÞ ¼ ½sgnðs1 Þ;...; T

sgnðs6 Þ is a column of signum functions and and H K H ¼ diagðk H 1 ;...;k 6 Þ is a positive definite gain matrix satis-

In this section, an adaptive variable-structure TSMC or AVFTC law is proposed to achieve coupled position and attitude tracking in finite or scheduled time in the presences of bounded external disturbances, parametric uncertainties and actuator saturation. Consider the relative motion

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8

D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx

dynamics for a rigid spacecraft system with actuator saturation: _ I~ n ¼ adH n In þ ug þ satðuc Þ  þ I adn AdðhÞ1 n0  AdðhÞ1 n_ 0 þ ud :

ð43Þ

 T where satðuc Þ ¼ satðuc1 Þ; . . . ; satðuc6 Þ is the actual control torque and force generated by the actuators and satðuci Þ is a nonlinear characteristic which takes the form 8 scmax sgnðuci Þ if j uci j> scmax > > > < uci if j uci j6 scmax for i ¼ 1; 2; 3; satðuci Þ ¼ > /c sgnðuci Þ if j uci j> /cmax > > : max uci if j uci j6 /cmax for i ¼ 4; 5; 6: ð44Þ To deal with the control constraints for convenience, the saturation function satðuc Þ can be expressed as satðuc Þ ¼ vðuc Þ  uc :

ð45Þ

where vðuc Þ ¼ diagðv1 ðuc Þ; ;v6 ðuc ÞÞ with 8 scmax if j uci j> scmax > uci > > > > <1 if j uci j6 scmax for i ¼ 1; 2; 3; vi ðuc ðtÞÞ ¼ /c max > > > uci if j uci j> /cmax > > : 1 if j uci j6 ucmax for i ¼ 4; 5; 6:

ð46Þ

The coefficient 0 < vi ðuc ðtÞÞ 6 1 is an indicator for the saturation degree of the ith entry of the control vector (Zhu et al., 2011). No saturation occurs in the control component uci ðtÞ when vi ðuc ðtÞÞ is 1 whereas there is almost no feedback from the control component uci ðtÞ when vi ðuc ðtÞÞ approaches 0. Based on the TSM manifold surface (35), the proposed control law can guarantee coupled position and attitude tracking in finite or scheduled time under actuator constraints, bounded external disturbances and uncertainty in the inertial matrix. Before giving the theorem, the following lemmas and Assumptions should be defined. Lemma 3. Suppose a1 ; a2 ; . . . ; an and 0 < m < 2 are positive numbers; the following inequality holds (Yu et al., 2005).  2 m  2 ð47Þ a1 þ a22 þ    þ a2n 6 am1 þ am2 þ    þ amn : Lemma 4 Chen et al., 2013. Assume that there exists the continuous positive definite function V ðtÞ which satisfies the following inequality: V_ ðtÞ þ k1 V ðtÞ þ k2 V a ðtÞ 6 0;

8t > t0 :

ð48Þ

then V ðtÞ converges to the equilibrium point in finite time ts 1 k1 V 1a ðt0 Þ þ k2 log ts 6 t0 þ ; k1 ð1 þ aÞ k2 where k1 > 0; k2 > 0 and 0 < a < 1.

ð49Þ

Assumption 6. The external disturbance  T T sd ðtÞ /Td ðtÞ satisfies the following condition.

ud ðtÞ ¼

ksd ðtÞk 6 k 01 kws k; ð50aÞ k/d ðtÞk 6 k 02 kw/ k; ð50bÞ h i T ~ T ~nT 2 R6 ;C 1 ¼ diagðc1 ;c2 ; c3 Þ; C 2 ¼ where ws ¼ ðC 1 HÞ s h iT ~ns and ~ ~ T ~nT 2 R6 , diagðc4 ; c5 ; c6 Þ; w/ ¼ ðC 2 bÞ n/ are / three upper and lower components of relative velocities ~ n h iT T ~T ~ ~ respectively, defined by n ¼ ns n/ , and k 01 and k 02 are the unknown bounds. To design an adaptive TSMC law, an adaptive fast-TSM-type reaching law is defined as Yu et al. (2005) r s_ ðtÞ ¼ P s  LsigðsÞ ; r

r

ð51Þ r

T

where sigðsÞ ¼ ½j s1 j sgnðs1 Þ; . . . ; j s6 j sgnðs6 Þ ; P and L are positive definite matrices. In Gao and Hung (1993) and Yu et al. (1999), it was shown that the reaching law control can guarantee the convergence of the trajectory of the closed-loop system since it is driven onto the sliding plane in finite or scheduled time while the chattering is reduced by properly tuning the design parameters P and L. Assumption 7. There exist positive scalars k 1 and k 2 such that the following conditions are satisfied: (Zhu et al., 2011) " f ¼

fs f/

#

 0 _r ; _ 0 þ C~ In þ u þ I ad Ad 1 n  Ad 1 g n ¼ adH n g h h n ð52aÞ

kf s k 6 ðk 1  k 01 Þkws k;

ð52bÞ

kf / k 6 ðk 2  k 02 Þkw/ k;

ð52cÞ

where f s and f / are the upper and lower three components in function f , respectively. Note that kJ k 6 kJ and k 1 > 0, mðtÞ 6 mðt0 Þ and k 2 > 0 hold for the above assumption. Theorem 2. The rigid body spacecraft system (25) and (43) with the sliding surface (35) and saturation input constraints (46), the trajectory of the closed-loop system can be driven to the sliding surface sðtÞ ¼ 061 with the adaptive controller (53) and update law (55), and it will converge to the origin in finite or scheduled time. uc ðtÞ ¼ P s  LsigðsÞr  us ðtÞ; where the adaptive control law us ðtÞ is defined as h iT us ðtÞ ¼ uTss ðtÞ uTs/ ðtÞ ; ss ðtÞ ; uss ðtÞ ¼ q^c1 ^k 1 kws k kss ðtÞk s/ ðtÞ us/ ðtÞ ¼ q^c2 ^k 2 kw/ k : ks/ ðtÞk

ð53Þ

ð54aÞ ð54bÞ ð54cÞ

Please cite this article in press as: Lee, D., et al. Adaptive variable-structure finite-time mode control for spacecraft proximity operations with actuator saturation. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.02.029

D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx T

T

where ss ðtÞ ¼ ½s1 ; s2 ; s3  and s/ ðtÞ ¼ ½s4 ; s5 ; s6  and the update law for the estimated parameters ^k 1 ; ^k 2 ; ^c1 , and ^c2 are chosen as ^k_ 1 ðtÞ ¼ q1 kss ðtÞkkns k with ^k 1 ð0Þ > 0; ^k_ 2 ðtÞ ¼ q2 ks/ ðtÞkkn/ k with ^k 2 ð0Þ > 0; ^c_ 1 ðtÞ ¼ q^c31 ^k 1 kss ðtÞkkws k with ^c1 ð0Þ > 0; ^c_ 2 ðtÞ ¼ q^c3 ^k 2 ks/ ðtÞkkw k with ^c2 ð0Þ > 0; /

2

ð55aÞ ð55bÞ ð55cÞ ð55dÞ

where q > 1; q1 and q2 are the design parameters. Proof. Consider the following Lyapunov candidate:   1 1 ~2 1 ~2 2 2 V ¼V1þ k þ k þ ~c1 þ ~c2 ; 2 q1 1 q2 2

function

ð56Þ



1

2

V_ 6 kss kðk 1  k 01 Þkws k þ ks/ kðk 2  k 02 Þkw/ k  d1 sTs ðP 1 ss þ L1 sigðss Þr Þ  d2 sT/ ðP 2 s/ ss r þ L2 sigðs/ Þ Þ  d1 sTs q^c1 ^k 1 kws k kss k s/  ~k 1 kss kkws k  d2 sT/ q^c2 ^k 2 kw/ k ks/ k  ~k 2 ks/ kkw k þ ~c1^c2^c_ 1 þ ~c2^c2^c_ 2 ; /

1

2

ð57Þ

ð60Þ

where P 1 ¼ diagðp1 ; p2 ; p3 Þ; P 2 ¼ diagðp4 ; p5 ; p6 Þ; L1 ¼ diagð‘1 ; ‘2 ; ‘3 Þ, and L2 ¼ diagð‘4 ; ‘5 ; ‘6 Þ. From the defini~c2 ¼ d2  ^c1 tion of ~c1 ¼ d1  ^c1 1 ðtÞ; 2 ðtÞ, the following expression hold: ~c_ 1 ðtÞ ¼ ^c2 c_ 1 ðtÞ; 1 ðtÞ~ ~c2 ðtÞ ¼ ^c2 ðtÞ~c_ 2 ðtÞ:

ð61aÞ ð61bÞ

2

~ ^ ~c1 ¼ d1  ^c1 c2 ¼ d2  ^c1 where 1 ; ~ 2 ; k 1 ¼ k 1  k 1 ðtÞ; ~k 2 ¼ k 2  ^k 2 ðtÞ, and the parameters d1 and d2 will be defined later. After taking the time derivative of (56) and making appropriate substitutions from (25), (43), (53), and (55), the time derivative becomes h V_ ¼ sT ðtÞ adH n In þ ug þ satðuc Þ  i þ I adn AdðhÞ1 n0  AdðhÞ1 n_ 0 þ d 1 ~ ^_ 1 _ k 1 k 1  ~k 2 ^k 2 þ ~c1^c2 c_ 1 þ ~c2^c2 c_ 2 ; 1 ^ 2 ^ q1 q2 1 _ 1 _ ¼ sT ðtÞ½f þ vðuc ðtÞÞuc ðtÞ þ d   ~k 1 ^k 1  ~k 2 ^k 2 q1 q2 2 _ 2 _ þ ~c1^c ^c1 þ ~c2^c ^c2 :

9

Substituting ¼ d1^c1  1; obtain

~k 1 ¼ k 1  ^k 1 ; ~k 2 ¼ k 2  ^k 2 ; ~c1^c1 ~c2^c2 ¼ d2^ c2  1, and (61) into (60), we

r r V_ 6 d1 sTs ðP 1 ss þ L1 sigðss Þ Þ  d2 sT/ ðP 2 s/ þ L2 sigðs/ Þ Þ þ ð1  qÞ^k 1 ks/ kkw k þ ð1  qÞ^k 2 ks/ kkw k; ð62aÞ /

/

r V_ 6 d1 kmin ðP ÞsT s  d2 kmin ðLÞsT sigðsÞ þ ð1  qÞ^k 1 ks/ kkw/ k þ ð1  qÞ^k 2 ks/ kkw/ k;

ð62bÞ

where kmin ðP Þ and kmin ðLÞ are the minimum eigenvalues of P and L matrices, respectively. And the parameters k 1 and k 2 are estimated using the update laws (55a) and (55b) and their initial values must be chosen as ^k 1 ð0Þ > 0 and ^k 2 ð0Þ > 0 to guarantee that ^k 1 ðtÞ > 0 and ^k 2 ðtÞ > 0 for all t 2 ½0; 1Þ. The design parameters q are chosen as q > 1 to guarantee the terms ð1  qÞ^k 1 ks/ kkw/ k and ð1  qÞ^k 2 ks/ kkw/ k are negative. From Lemma 3, V_ can be written as

Using Assumption 6 and 7 gives

ðrþ1Þ=2 V_ 6 2d1 kmin ðP 1 ÞV 1  2ðrþ1Þ=2 d2 kmin ðP 2 ÞV 1

V_ 6 kss ðtÞkkf s k þ ks/ ðtÞkkf / k þ sT ðtÞvðuc ðtÞÞuc ðtÞ

The estimation errors ð~k 1 ; ~k 2 ; c1 ; c2 Þ are bounded and will converge to a certain constant value respectively. As discussed in Lemma 4, the state of the system ~g and ~ n will converge to zero in finite or scheduled time (Zhu and Yan, 2014). This implies that the closed-loop dynamics described by (25) and (43) tracks the desired trajectory ðg0d ; n0d Þ given by (20) in finite or scheduled time using the control law (53) and the update law (55). Moreover, the domain of attraction of this trajectory is almost global over the state space SEð3Þ  R6 . Therefore, the closed-loop system is almost globally asymptotically convergent over the state space SEð3Þ  R6 (Lee et al., 2015).

þ k 01 kss ðtÞkkf s k þ k 02 ks/ ðtÞkkf / k 1~ 1 k 1 q1 kss ðtÞkkws k  ~k 2 q2 ks/ ðtÞkkw/ k q1 q2 þ ~c1^c2^c_ 1 þ ~c2^c2^c_ 2 

1

2

ð58Þ

Since 0 < vi ðuc ðtÞÞ 6 1, according to the density property of real number (Gao and Hung, 1993), there exist constants d1 and d2 that satisfy 0 < d1 6 minðvi ðuc ðtÞÞÞ 6 1; 0 < d2 6 minðvi ðuc ðtÞÞÞ 6 1;

i ¼ 1; 2; 3; i ¼ 4; 5; 6:

ð59aÞ ð59bÞ

Substituting the control law (53) and the adaptation laws (55) yields

ð63Þ

Remark 3. The adaptive and update laws us ðtÞ (53) take the forms of the adaptive and update laws for spacecraft attitude stabilization (Zhu et al., 2011). However, the proposed control scheme is extended for coupled position and attitude tracking of a rigid spacecraft. The control law

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D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx

(53) is independent of the inertia matrix J and mass m. Thus, the proposed control scheme can be applied to the spacecraft system with a large parametric uncertainty in the inertia matrix and mass. Furthermore, the control law does not include the upper bound of the disturbance. Instead, the bound is estimated by the adaptive law.

Remark 4. The control law (53) is discontinuous when crossing the sliding surface sðtÞ ¼ 061 , which may induce undesirable chattering leading to instability and could cause damage. To overcome this problem, us ðtÞ is modified by introducing so-called boundary layer around the sliding surface (Wheeler et al., 1998; Yoo and Chung, 1992) 8 > ; if kss ðtÞk > e; uss ðtÞ ¼ q^c1 ^k 1 kws k kssss ðtÞ > ðtÞk > > > > 2 < uss ðtÞ ¼ q2^c2 ^k 3 kws k ss ðtÞ ; if kss ðtÞk 6 e; 1 1 e ð64Þ us ðtÞ ¼ s/ ðtÞ ^ > ðtÞ ¼ q^ c kw k k u > s/ 2 2 / ks/ ðtÞk ; if ks/ ðtÞk > e; > > > > : u ðtÞ ¼ q^c ^k kw k2 s/ ðtÞ ; if ks ðtÞk 6 e; s/ 2 2 / / e

where e > 0 is a small positive value of bound layer. The signum function of the proposed control law (53) is also replaced by the function in (42). The proposed control law can then provide a continuous approximation to the discontinuous SMC law by introducing a boundary layer and guarantee the output tracking error within a neighborhood of the sliding surface (Zhu et al., 2011). However, the system can only guarantee the bounded motion around the sliding surface and not asymptotic stability. Remark 5. The differential Eqs. (25), (28) and (43) can be implemented with a standard integration scheme like Euler’s method or Runge-Kutta methods, since ~ gðtÞ and ~ nðtÞ are vectors in R6 . Thereafter, the relative configuration is obtained from  _ hðtÞ ¼ hf expm ~ ð65Þ gðtÞ : Finally, the configuration gðtÞ and velocities nðtÞ of the follower spacecraft are computed by gðtÞ ¼ g0 ðtÞhðtÞ ¼ g0 ðtÞhf expmð~ gðtÞ_ Þ; nðtÞ ¼ ~ nðtÞ þ Ad 1 n0 ðtÞ; hðtÞ

assumed to behave a tumbling body which has no control capability to perform any maneuver. The follower spacecraft should achieve satisfactory position and attitude tracking results in the scheduled flight time by performing coupled translational and rotational maneuvers. That is, from a given initial configuration the follower spacecraft is required to achieve a fixed relative configuration hf whose desired position and velocity are ½0 5 0T m and ½0 0 0T m/s in the leader’s body-fixed frame while synchronizing its attitude with that of the leader. The initial relative position, translational velocity and angular velocity in the leader’s body-fixed frame, and initial relative attitude in principal angle given in Table 2. In this simulation the properties of the leader are: the mass and inertia matrix are m0 ¼ 60:7 kg and inertia matrix J 0 ¼ diagð4:855:104:76Þ kgm2, respectively. The follower spacecraft mass, m ¼ 56:7 kg, and inertia matrix, J ¼ diagð4:854:765:10Þ kgm2 are similar to the leader. The follower is assumed to have available continuous actuation in/about all body axes where maximum force and torque components are j sci j6 scmax ¼ 0:01 Nm and j /ci j6 /cmax 2 ¼ 0:1 m=s  m ¼ 5:67 N, respectively when corresponding actuator saturation occurs. The control parameters are selected as r ¼ 19=20; q ¼ 40; q1 ¼ q2 ¼ 0:1; e ¼ 0:98; C ¼ P ¼ 0:016I 6 , and K ¼ L ¼ diagð1; 1; 1; 5; 5; 5Þ. The initial parameter estimates are chosen as ^k 1 ð0Þ ¼ 40; ^k 2 ð0Þ ¼ 50; ^c1 ð0Þ ¼ 1:5  103 , and ^c2 ð0Þ ¼ 1:5  105 . The numerical simulation is implemented with MATLAB software. The fourth-order Runge Kutta method is used for numerical integration with 0.1 s step size for scheduled flight time tf ¼ 600 s. To examine the robustness of the control laws against external disturbances, the following disturbance models are given (Sun and Huo, 2015; Sun and Huo, 2015)

Table 1 Initial orbital elements of the leader.

ð66Þ

Orbital elements

Values

ð67Þ

Semi-major axis (km) Eccentricity Inclination (°) Right ascension of ascending node (°) Argument of perigee (°) True anomaly (°)

26628 0.7417 63.4 0 90 150

where expm is exponential map from seð3Þ to SEð3Þ which is inverse of logarithm mapping. 5. Numerical simulation results In this section the spacecraft proximity operation problem is simulated to demonstrate the effectiveness of the AVFTC law (53). The TSMC law (37) is also simulated to compare the control performance of the TSMC law with that of the AVFTC law. The leader spacecraft follows a Molniya orbit which is a highly elliptical orbit while the follower performs approaching maneuver neighboring orbit. The initial orbital elements of the leader are given in Table 1. The leader is

Table 2 Initial relative configuration and velocities of the follower spacecraft. Initial state Relative Relative Relative Relative

Values

position (m) ½ 500 500 300 T attitude (°) 120 translational velocity (m/s) ½ 3:17 1:39 1:89 T angular velocity (rad/s) ½ 3:105 1:397 1:891 T  103

Please cite this article in press as: Lee, D., et al. Adaptive variable-structure finite-time mode control for spacecraft proximity operations with actuator saturation. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.02.029

D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx

2

3 þ sinðpt=125Þ þ 4 sinðpt=200Þ

3

6 7 sd ðtÞ ¼ 4 5  sinðpt=125Þ  5 sinðpt=200Þ 5  105 Nm; 7 þ cosðpt=125Þ þ 5 cosðpt=200Þ 2

3

ð68Þ

8 þ sinðpt=125Þ þ 2 sinðpt=200Þ 6 7 /d ðtÞ ¼ 4 5  sinðpt=125Þ  3 cosðpt=200Þ 5  104 N; 6 þ sinðpt=125Þ þ 5 cosðpt=200Þ ð69Þ The robustness of the proposed controller was also tested by assuming that the elements of the inertia matrix J are perturbed by 30% uncertainty and the mass of the follower spacecraft m is perturbed by 10% uncertainty, respectively. Figs. 2 and 3 show translational and rotational tracking errors in the presences of the external disturbances and parameter uncertainties, and actuator saturation. The norm of position tracking error kaðtÞk of the AVFTC in Fig. 2(a) converges to less than 0.1 m at 600 s while the ~ norm of attitude tracking error kHðtÞk of the AVFTC in Fig. 3(a) converges to less than 0.05° at 600 s and 0.01°.

11

The norm of translational velocity tracking error ktk of the AVFTC in Fig. 2(b) converges to less than 102 m/s at 600 s while the angular velocity tracking error kxk of the AVFTC in Fig. 3(b) converges to less than 105 rad/s at 600 s. The bounded attitude tracking errors in Fig. 3 are due to the bounded external disturbance torques. In addition, the translational and rotational tracking errors of the AVFTC reduce to zero faster those of the TSMC in the spite of the external disturbances and parameter uncertainties, and actuator saturation. Note that this configuration tracking is obtained without explicit reference states from the given initial state to the desired configuration. Fig. 4 shows the parameters estimations (55a), (55b), (55c), (55d). It is clear that the parameters ^c1 ; ^c2 ; ^k 1 and ^k 2 converge to constants, respectively. Fig. 5 shows time histories of control forces and torques. Three component control force and torque of the AVSTMC are presented, respectively in Fig. 5(a). Relatively large control forces and torques are required initially to drive the follower spacecraft to the desired position and attitude quickly. The control efforts then decrease rapidly

Fig. 2. Translational tracking errors. (a) kak, (b) ktk.

Fig. 3. Rotational tracking errors. (a) kHk, (b) kxk.

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12

D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx

Fig. 4. Estimated parameters ^c1 ; ^c2 ; ^k 1 and ^k 2 .

Fig. 6. Relative configuration of the follower in the leader’s body-fixed frame.

after some oscillations to reach the desired position and attitude. In Fig. 5(a), it is obvious to observe that actuator saturation occurs in both control force and torque responses. To avoid actuator saturation, the AVFTC law uses the bounded control torque and force components 0.1 Nm and 5.67 N, respectively while the spacecraft performs coupled translational and rotational maneuvers. The large chattering due to the use of the signum function (53) is attenuated by the saturation function (42). Table 3 shows the control performances of both controllers. The control performance of the AVFTC law shows more precise tracking results than the TSMC law at the terminal flight time by saving control efforts. The proposed controller can effectively reduce the chattering with almost the system performance as Figs. 2 and 3. To further investigate the control performance of the AVFTC law, two additional flight times of 300 s and 1800 s are also used to simulate the same proximity operation scenario. It is obvious to observe that larger control

Rt Rt effort of Dv ¼ 0f k/c k=mdt and 0f ksc kdt are used to achieve the desired relative configuration as the scheduled flight time decreases. There are small changes in the position and attitude tracking errors in three flight times. Thus, the spacecraft proximity operations with the proposed controller can be performed by compromising the control effort and scheduled flight time. Fig. 6 shows the relative configuration of the follower in the leader’s body-fixed frame. Note that the follower’s body-fixed frame is plotted along the relative approach trajectory at 0, 60, 150, 300, 1800 and 600 s, respectively. The follower almost achieve the desired configuration at 300 s and maintains the desired configuration by the terminal flight time. This figure demonstrates that the spacecraft with the AVFTC can achieve accurate 6-DOF tracking maneuver for proximity operations though the actuator saturation limits, external disturbances and uncetain parameters are explicitly considered (see Table 4).

Fig. 5. Time histories of control forces and torques. (a) Control response of the AVFTC, (b) norms of control forces and torques.

Please cite this article in press as: Lee, D., et al. Adaptive variable-structure finite-time mode control for spacecraft proximity operations with actuator saturation. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.02.029

D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx Table 3 Comparison of control performances ðtf ¼ 600Þ s. Performance index

AVFTC

TSMC

kHðtf Þk (°) kaðtf Þk (m) ktðtf Þk (m/s) kxðtf Þk (rad/s) RDvtf (m/s) 0 ksc kdt (Nm s)

0.069 0.030 0.002 7:13  106 18.841 0.5214

0.115 0.065 0.003 1:32  105 20.514 0.6724

13

angle. Note that g~ is not uniquely defined when the corresponding principal rotation angle is p rad. Given T g ¼ ½HT bT  2 R6 , the function GðgÞ 2 R66 can be expressed as a block-triangular matrix, as follows (Bra´s et al., 2013):

 AðHÞ 033 GðgÞ ¼ ; ð71Þ T ðH; bÞ AðHÞ where

  1 1 1 þ cos h 2  AðHÞ ¼ I 3 þ ðHÞ þ 2  ðH Þ ; 2 2h sin h h

Table 4 Control performances of the AVFTC law about three filght times. Performance index

tf ¼ 300 s

tf ¼ 600 s

tf ¼ 1800 s

kHðtf Þk (°) kaðtf Þk (m) ktðtf Þk (m/s) kxðtf Þk (rad/s) RDvtf (m/s) 0 ksc kdt (Nm s)

0.012 0.011 0.003 2:03  106 24.94 0.4538

0.069 0.030 0.002 7:13  106 18.841 0.5214

0.077 0.08 0.002 6:57  106 8.5607 0.1339

SðHÞ ¼ I 3 þ

1  T ðH; bÞ ¼ ðSðHÞbÞ AðHÞ 2    1 1 þ cos h  T þ 2 Hb þ ðHT bÞAðHÞ 2h sin h h ð1 þ cos hÞðh  sin hÞ SðHÞbHT 2h sin2 h   ð1 þ cos hÞðh þ sin hÞ 2 þ HT bHHT ;  h4 2h3 sin2 h 

6. Conclusion Spacecraft proximity operation problem for a spacecraft model actuator saturation in the presence of parameter uncertainties and external disturbances are studied with a unified control law design. Based on the derived relative coupled dynamics, an adaptive variable-structure terminal sliding mode control law was proposed to force the state of the closed-loop system to converge to the origin in finite time. The domain of attraction of the trajectory is almost global over the state space SEð3Þ  R6 . Numerical simulation result demonstrates that the proposed control scheme can drive the follower spacecraft to achieve both the desired configuration and velocities for proximity operations by performing coupled translational and rotational maneuvers under actuator saturation in finite time despite the presences of external disturbances and parameter uncertainties. Furthermore, it shows that the proposed control law can be superior to the traditional terminal sliding mode control in control performance. Appendix A. From Bullo and Murray (1995), the following expansion for GðXÞ 2 R66 , where X 2 R6 , in terms of the adjoint representation of seð3Þ is obtained: 1 GðX Þ ¼ I 66 þ adX þ aðhÞad2X þ bðhÞad4X ; 2 3 h2 where h2 aðhÞ ¼ 2  h cotðh=2Þ  csc2 ðh=2Þ; 4 8 2 1 h and h2 bðhÞ ¼ 1  h cotðh=2Þ  csc2 ðh=2Þ; 4 8

1  cos h h  sin h  2 ðHÞ þ ðH Þ ; h2 h3

ð70Þ

where h ¼ kX R k is the Euclidean norm of X R 2 R3 , which is the vector of the first three components of the exponential coordinate vector X 2 R6 , and I 66 is the 6  6 identity matrix. Therefore, h corresponds to the principal rotation

where h ¼ kHk is the norm of H, which is the vector of the first three components of the exponential coordinate vector g 2 R6 ; therefore, h corresponds to the principal rotation angle. The exponential coordinate vector H and its time derivatives are obtained from Rodrigues’ formula for the rotation matrix in terms of the exponential coordinates on SOð3Þ: RðHÞ ¼ expSOð3Þ ðHÞ ¼ I 3 þ

sin h 1  cos h  2 ðHÞ þ ðH Þ ; h h2 ð72Þ

where expSOð3Þ is exponential map from R3 to SOð3Þ. The adjoint operator on the Lie algebra seð3Þ can be obtained as a derivative of the adjoint action of SEð3Þ on seð3Þ via the exponential map below ((73)). Let g ¼ expðeY Þ where Y 2 seð3Þ and e 2 R and let X 2 R6 . From (12); Adg X ¼ ðexpmðeY ÞX expðeY ÞÞ ¼ gXg1 :

ð73Þ

The derivative of this adjoint evaluated at e ¼ 0 is defined to be the adjoint operation on the Lie algebra, and is given by adY X :¼

d ðexpmðeY ÞX expmðeY ÞÞje¼0 ¼ YX  XY : de

If X ¼ g_ 2 seð3Þ for some g 2 R6 , and if

  H b Y ¼ f_ ¼ 2 seð3Þ; 013 0 then this operator can be expressed by the matrix

  H 033 adf ¼ ; b H

ð74Þ

ð75Þ

ð76Þ

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D. Lee et al. / Advances in Space Research xxx (2017) xxx–xxx

such that j

adf g ¼ ðYX  XY Þ :

ð77Þ

The co-adjoint operator is described on the dual of the Lie algebra, which can be identified with R6 via the vector j space isomorphism ðÞ . Therefore, the co-adjoint operator can be expressed in matrix form as

 H b  T adf ¼ adf ¼ : ð78Þ 033 H This co-adjoint operation is seen to arise in the dynamics model of an autonomous vehicle with control inputs, when the vehicle is modeled as a rigid body. The time derivative of the adjoint action Adh1 n0 in (26) is derived from the adjoint operation on the Lie algebra. Taking the time derivative of Adh1 n0 d d  1 0_ j Adh1 n0 ¼ h n h dt dt  j ¼ h_ 1 n0_ h þ h1 n_ 0_ h þ h1 n0_ h_ : ð79Þ Since

d 1 h dt

_ 1 ¼ ~ ¼ h1 hh n_ h1 ,

 _ 1 0_ j d Adh1 n0 ¼ ~ n h n h þ h1 n_ 0_ h þ h1 n0_ h~ n_ dt  _ 1 0_ j  j ¼ ~ n ðh n hÞ  ðh1 n0_ hÞ~ n_ þ h1 n_ 0_ h ð80Þ : Thus, (80) is arranged by using (12) and (77): d Adh1 n0 ¼  ad~n Adh1 n0 þ Adh1 n_ 0 dt ¼  adn Adh1 n0 þ Adh1 n_ 0 :

ð81Þ

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Please cite this article in press as: Lee, D., et al. Adaptive variable-structure finite-time mode control for spacecraft proximity operations with actuator saturation. Adv. Space Res. (2017), http://dx.doi.org/10.1016/j.asr.2017.02.029