Neural network-based adaptive second order sliding mode control of Lorentz-augmented spacecraft formation

Author’s Accepted Manuscript Neural network-based adaptive second order sliding mode control of Lorentz-augmented spacecraft formation Xu Huang, Ye Yan, Yang Zhou www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)31199-7 http://dx.doi.org/10.1016/j.neucom.2016.10.021 NEUCOM17633

To appear in: Neurocomputing Received date: 3 November 2015 Revised date: 1 September 2016 Accepted date: 20 October 2016 Cite this article as: Xu Huang, Ye Yan and Yang Zhou, Neural network-based adaptive second order sliding mode control of Lorentz-augmented spacecraft formation, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.10.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Neural network-based adaptive second order sliding mode control of Lorentz-augmented spacecraft formation Xu Huang*, Ye Yan, Yang Zhou College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China [email protected] [email protected] [email protected] * Corresponding author. Tel: +86 15574868148; Fax: +86 073184573187.

Abstract

The Lorentz force acting on an electrostatically charged spacecraft provides a new means of propulsion for orbital maneuvers such as Lorentz-augmented spacecraft formation flying. Modeling the Earth’s magnetic field as a tilted dipole corotating with Earth, a nonlinear dynamical model that describes the orbital motion of a Lorentz spacecraft about an arbitrary elliptic orbit is developed. Then, the optimal open-loop control trajectories of formation establishment are solved by the pseudospectral method. To guarantee trajectory tracking in the presence of external perturbations and system uncertainties, a closed-loop neural network-based adaptive second order fast terminal sliding mode controller is designed using state feedback, which simultaneously solves the singularity and chattering problems that generally exist in conventional terminal sliding mode. Neural networks are employed to approximate the unknown nonlinearities in the system dynamics. Meanwhile, to ensure closed-loop tracking control without velocity measurements, an output feedback controller is also proposed with an observer introduced to capture the velocity signals. The overall stabilities for both control schemes are proved by a Lyapunov-based method. Numerical simulations are presented to verify the performance of the proposed controllers. Furthermore, the controllers could be applied to other Lorentz-augmented

1

relative orbital control problems.

Keywords: Neural network; Adaptive control; Second order sliding mode; Lorentz spacecraft; Spacecraft formation flying

1.

Introduction

Lorentz spacecraft, a new kind of conceptual space vehicle, would be advantageous over traditional spacecraft due to the promising ability of saving propellant. Different from conventional orbital maneuvers propelled by thrusters on board, a Lorentz spacecraft could manage such maneuvers in a propellantless way. It actively generates electrostatic charge on its surface to induce Lorentz force as electromagnetic propulsion via interaction with the ambient magnetic field. Despite of the fact that the Lorentz force could only act in the direction that is perpendicular to the local magnetic field and the vehicle’s velocity relative to the local magnetic field, the applications of Lorentz spacecraft are still promising, such as propellantless rendezvous [1-5], spacecraft formation flying [6-14], spacecraft hovering [15-17], planetary capture and escape [18, 19], and orbital inclination control [20] and so on. A Lorentz spacecraft is more effective in low Earth orbit (LEO) where the magnetic field is intenser and the vehicle travels much faster than high Earth orbit. More importantly, the key to evaluate the efficiency and ability of a Lorentz spacecraft is its specific charge, namely, the charge-to-mass ratio. Higher orders of specific charge imply superior ability to accelerate. Natural spacecraft charging levels may reach to the order of 10-8 C/kg due to the surrounding plasma environment. However, effective orbital maneuvers in LEO necessitate a specific charge on the order of at least 10 -5 C/kg [20]. It is generally accepted that charging levels on the order of 10 -3 to 10-2 C/kg are near-term feasible [18]. Peck [1] presented that 0.03 C/kg seems to be the maximum in near future, which could afford relative

2

orbital control in LEO rather than absolute orbital maneuvers with severer requirements on the specific charge. Lorentz-augmented spacecraft formation flying (LASFF), a potential application of relative orbital control, has raised research interest. Peck et al. [6] proposed the conception of LASFF and designed a triangular formation reconfiguration in equatorial circular orbits. By incorporation of the Lorentz accelerations into Gauss variational equations as perturbations, Peng and Gao [7] proposed a J2-invariant formation perturbed by the Lorentz force, based on the assumption of a constant magnetic dipole orientation within single orbital period. This assumption inevitably introduces model errors because the Earth’s magnetic field is corotating with Earth. By modeling the Earth’s magnetic field as a nontilted dipole with its magnetic pole aligned with the Earth’s geographic pole, Tsujii et al. [8] investigated the dynamics and control of LASFF by using both linearized and nonlinear relative dynamical models. Unlike this nontilted dipole model, Huang et al. [9] developed a nonlinear dynamical model of LASFF that explicitly includes the dipole tilt angle, which is more representative of the Earth’s magnetic field. Based on this model, the optimal trajectories of formation establishment and reconfiguration are solved by Gauss pseudospectral method (GPM). Sobiesiak and Damaren [10-12] studied the controllability of LASFF system and derived the conclusion that the relative spacecraft states are not fully controllable by only using the Lorentz force. Based on this conclusion, they designed an optimal linear quadratic regulator for LASFF using hybrid propulsion. Aforementioned works mainly dealt with the open-loop control of LASFF, and few of them studied the closed-loop control problems. Huang et al. [9,13] designed closed-loop sliding mode controllers for LASFF using hybrid inputs consisting of the specific charge and the thruster-generated control

3

acceleration of Lorentz spacecraft. But it is the net charge but not the specific charge that is the real control input. Thus, an accurate knowledge of the spacecraft mass is necessary to determine the required amount of the net charge to guarantee precise control input. However, the spacecraft mass is hardly determinable on orbit due to the fuel expenditure and other variations. Moreover, the model errors arising from the assumption of a tilted magnetic dipole for the Earth’s magnetic field further increase the system uncertainties. Therefore, to deal with these system uncertainties, adaptive controllers that estimate the uncertainties online remain to be designed. Conventional adaptive control techniques for SFF require that the uncertain nonlinearities satisfy the assumption of linearity in the parameters [14,21]. To get rid of this requirement, neural networks (NNs) have been widely used to estimate the nonlinear uncertainties in system dynamics because NNs are capable of approximating any smooth functions over compact sets to arbitrary accuracy [22-24]. Thus, the radial basis function NN (RBFNN) is used in this paper to approximate the unknown functions, and the adaptive tuning law of the RBFNN is derived via a Lyapunov-based approach. Notably, despite of the capability of high-precision approximation, the approximation error still exists. Therefore, in most of the previous works [22, 25-28], it could only guarantee that the state errors are uniformly ultimately bounded or that the state errors will be arbitrarily small if the feedback gains are sufficiently large [29]. In order words, these NN-based controllers cannot render the state errors converge to zero asymptotically. If the asymptotic stability of the closed-loop system is desired, a prior knowledge on the bound of the approximation error becomes a necessary prerequisite. However, the upper bound of the approximation error is hardly obtainable in advance. To solve this problem, another adaptation law is proposed in this paper to estimate the bound of the approximation error. Also, this adaptation law is

4

derived via a Lyapunov-based approach to ensure the asymptotic stability. In this way, the requirement on the prior knowledge about the upper bound of the approximate error is avoided, and the asymptotic stability could also be ensured. Furthermore, to render the closed-loop LASFF system robust against external perturbations and system uncertainties, sliding mode control (SMC) methodology has also been used in the controllers design due to its eminent feature of insensitivity to the matched disturbances and uncertainties [30-33]. The fast terminal SMC (FTSMC), a variant of SMC, is superior to linear SMC due to the advantage of finite-time convergence [34]. Also, it is advantageous over terminal SMC due to its fast convergence rate [34]. But the initial FTSMC has two main disadvantages [27,35]. One is the singularity problem, and the other is the chattering problem. To deal with the singularity, Zou et al. [27] proposed a solution by switching from the fast terminal sliding mode to a general one when the singularity may occur. Besides, to solve the chattering problem, the boundary layer method is generally used [17]. However, within the boundary layer, the ideal characteristic of insensitivity to the matched disturbances is lost because of the high-gain feedback asymptotic control in the layer. Meanwhile, the finite-time stability is also lost within the boundary layer due to the same reason [27]. Another alternative to the chattering problem is the second order SMC (SOSMC) approach [36-41]. In regular first order SMC, a discontinuous switch control item with sign function is generally included in the control law to ensure the system robustness against matched disturbances and uncertainties [36]. Thus, it may lead to a high-frequency oscillation around the sliding surface, that is, the chattering phenomenon. Differently, in SOSMC, the discontinuous switch item acts in the time derivative of the control input [36]. After integration, the actual control input is continuous, and the chattering is thus eliminated. Current

5

SOSMC algorithms mainly included twisting [37], super twisting [38], and suboptimal algorithm [36] and so on [39]. Moreover, by using the SOSMC method, the singularity problem existing in the original FTSMC could also be effectively avoided. In other words, the SOSMC method solves the singularity and chattering problem simultaneously. Details about this fact will be elaborated in this paper. Hence, the resulting closed-loop state feedback controller is a NN-based adaptive second order FTSMC scheme. Considering the fact that velocity measurements may not always be available and that eliminating the velocity sensors could reduce the control cost and structure mass, a second order observer is introduced for velocity estimation. Based on this observer, an output feedback controller is then designed for LASFF to deal with the unavailability of velocity measurements. Thus, compared with existing control schemes for LASFF [6-14], the main enhancements of this work are summarized as follows: 1) This control scheme is based on the assumption of a tilted magnetic dipole, which is more representative of Earth’s magnetic field; 2) This adaptation law gets rid of the assumption of linearity in uncertain parameters, and is able to estimate the nonlinearities online; 3) No prior information about the upper bound of the uncertainties is required in advance; 4) This control scheme could also accommodate the case without velocity measurements. The organization of this paper proceeds as follows. A nonlinear dynamical model of LASFF is developed in Section 2. Section 3 briefly introduces the generation of optimal open-loop control trajectories by GPM. The detailed designs of closed-loop state and output feedback controllers are, respectively, elaborated in Sections 4 and 5. The numerical scenarios are simulated in Section 6 to

6

testify the performance of both open-loop and closed-loop controllers. The final section concludes the paper.

2.

Dynamical model

2.1. Dynamic equations of LASFF The spacecraft in formation flying are referred to as the chief and deputy spacecraft, respectively. The deputy is assumed to be a charged Lorentz spacecraft and the chief is uncharged. As depicted in Fig. 1, OE X I YI Z I is an Earth-centered inertial (ECI) frame with OE being the center of Earth.

OC xyz is the relative motion (RM) frame, where x axis is in the radial direction, z axis is aligned with the normal direction of the chief’s orbital plane, and y axis completes the right-handed Cartesian frame. OC and OD refer to the center of mass of the chief and the deputy, respectively.

RC and RD are, respectively, the orbital radius vector of the chief and the deputy.

Fig. 1. Definition of coordinate frames. Denote ρ  RD  RC  [ x

y z]T as the relative position vector between the chief and the deputy,

the dynamic equations of LASFF (i.e., the control system model of LASFF) can be expressed in RM 7

frame as [16]

ρ  [ x y z ]T  F ( ρ, ρ)  GU

(1)

 2uC y  uC2 x  uC y  nC2 RC  nD2 ( RC  x)    F ( ρ, ρ)   2uC x  uC2 y  uC x  nD2 y  2    n z D  

(2)

with

where G  mD1 , with mD being the mass of the deputy. U  UC  U L  mD (aC  aL ) refers to the total control forces composed of the thruster-generated control force U C and the Lorentz force U L acting on the deputy, where aC  [ar

as

aw ]T and aL  [ax

ay

az ]T . nC   / RC3

and

nD   / RD3 , where  is the gravitational parameter of Earth and RD  [( RC  x)2  y 2  z 2 ]1/ 2 is the orbital radius of the deputy. uC is the argument of latitude of the chief. Thus, uC  [0 0 uC ]T and uC  [0 0 uC ]T are, respectively, the orbital angular velocity and acceleration vector of the chief. For a Keplerian elliptic orbit, the orbital motion of the chief is governed by following dynamics: RC  RC uC2   / RC2 RC uC  2 RC uC

(3)

2.2. The Lorentz acceleration By assuming that the Earth’s magnetic field could be modeled as a tilted dipole located at the center of Earth that corotates with Earth, and that the Lorentz spacecraft could be regarded as a charged point mass, then the Lorentz acceleration acting on the Lorentz spacecraft can be expressed as

aL  Vr  B

(4)

where   qD / mD is the specific charge (i.e., charge-to-mass ratio) of the deputy, with qD being the net charge. Vr refers to the velocity of the Lorentz spacecraft relative to the local magnetic field B , given by [2] 8

B  ( B0 / RD3 )[3(n0  RD0 ) RD0  n0 ]

(5)

where B0  8.0 1015 T  m3 is the Earth’s magnetic dipole moment. The superscript 0 denotes a unit vector in that direction. For example, RD0 is the unit orbital radius vector of the deputy, given by

RD0  (1/ RD )[ RC  x

y z]T

(6)

n0 represents the unit magnetic dipole moment vector, and can be described in RM frame as

 nx   (cos  cos uC  sin  cos iC sin uC )sin   sin iC sin uC cos   n   ny    (cos  sin uC  sin  cos iC cos uC )sin   sin iC cos uC cos    nz    sin  sin iC sin   cos iC cos  0

(7)

where iC is the orbital inclination of the chief.  is the dipole tilt angle with respect to the rotation axis of Earth, as shown in Fig. 2.  is defined as   M  C , where C is the right ascension of the ascending node of the chief, and  M is the inertial rotation angle of the dipole, given by

M  E t  0

(8)

where E is the rotational rate of Earth and 0 is the initial phase angle of the dipole.

Fig. 2. Definition of angles. Substitution of Eqs. (6) and (7) into Eq. (5) yields the expressions of the local magnetic field in RM frame. Given that the magnetic field is corotating with Earth, the relative velocity between the Lorentz 9

spacecraft and the local magnetic field is thus given by Vr 

dRD  ωE  RD  RC  ρ  (uC  ωE )  ( RC  ρ) dt

(9)

or in RM coordinates:

Vx   RC  x  y (uC  E cos iC )  zE sin iC cos uC    Vr  Vy    y  ( RC  x)(uC  E cos iC )  zE sin iC sin uC  Vz   z  ( RC  x)E sin iC cos uC  yE sin iC sin uC 

(10)

Therefore, by substituting Eqs. (5) and (10) into Eq. (4), the expressions of the Lorentz acceleration in RM frame can be derived as

aL   l  [lx

ly

lz ]T

(11)

where

lx  Vy Bz  Vz By , l y  Vz Bx  Vx Bz , lz  Vx By  Vy Bx

3.

(12)

Open-loop controller

The Lorentz-augmented formation establishment or reconfiguration problem is firstly described as a general trajectory optimization problem (TOP). Then the optimal open-loop control trajectory is solved by GPM, a direct transcription approach that has been widely used in spacecraft TOPs. Pseudospectral method transcribes a TOP into a nonlinear programming (NLP) by parameterizing the state and control variables using global orthogonal polynomials and approximating the dynamics at the collocation points derived from a Gaussian quadrature. The resulting NLP is then solved by appropriate numerical optimization methods. Problem formulation of LASFF TOP and detailed description of GPM are referred to [9,42]. Denote χ  [qD UCT ]T as the hybrid control inputs composed of the charge and thruster-generated control forces of Lorentz spacecraft, then the following optimal criterion is utilized in this paper, given by 10

tf

J a   χ T Qχdt

(13)

t0

where t0 and t f refer to the initial and final time, respectively. Q  diag(Qq , QC , QC , QC ) is a diagonal weight matrix with the elements satisfying Qq  0 and QC  0 . Remark 1. Previous researches indicate that the dynamical system of Lorentz-augmented relative motion is not fully controllable by singly using the geomagnetic Lorentz force. Other kinds of control forces, such as the thruster-generated control forces, are therefore necessary to render the system fully controllable [9-11]. This explains why an optimal criterion that consists of hybrid control inputs is considered here. Remark 2. The ratio of Qq and QC determines the weights on the consumption of the charge and thruster-generated control forces. To make the best of the propellantless Lorentz force and save the fuel on board, the values of Qq and QC are generally chosen as Qq

QC such that the fuel expenditure

is given priority. In this way, a nearly propellantless LASFF would be achieved. That is, the magnitude of the thruster-generated control forces could be reduced to the same or even smaller order than that of the differential external perturbations between typical LEO spacecraft in proximity.

4.

Closed-loop state feedback controller

4.1. RBFNN Because RBFNN is capable of approximating any continuous nonlinear function over a compact set to arbitrary accuracy, it has been widely used to approximate the unknown dynamics in adaptive controller design. For a continuous nonlinear function f ( X ) 

n

, it can be approximated by RBFNN

as [29]

f ( X )  W T h( X )  ε

11

(14)

with hj ( X )  exp( || X  C j ||2 /2b2j ),

j  1, 2,

where X is the input vector and m denotes the number of the neurons. h  [h1 with C j and b j being the center and width of the jth neuron, respectively. W   weight and ε is the approximation error that satisfies

(15)

,m

h2 mn

hm ]T , is the ideal

ε   m and  m  0 .

For the estimation of f in the controller design, it can be represented by fˆ ( X )  Wˆ T h( X )

(16)

where Wˆ denotes the estimated weight, which would be tuned by the adaptation laws to guarantee the system stability via the Lyapunov approach.

4.2. Second order fast terminal sliding mode Considering the external perturbations and system uncertainties, the perturbed dynamical system of Lorentz-augmented spacecraft relative motion can be modified from Eq. (1) as ρ  F ( ρ, ρ)  (G  G)U  d  F ( ρ, ρ)  (G  G){qD [l ( ρ, ρ)  l ( ρ, ρ)]  U C }  d

(17)

 F ( ρ, ρ)  G[qD l ( ρ, ρ)  U C ]  D

where G  (mD  mD )1  mD1 is the bounded mass uncertainty, and d is the bounded external perturbations satisfying

d  d m and d m  0 . U  qD (l  l )  UC . l  Vr  B refers to the

bounded model errors resulting from the assumption of modeling the Earth’s magnetic field as a tilted dipole, with B being the bounded magnetic field errors. Hence, the overall bounded model errors and external perturbations can be summarized as D  GU  GqD l  d that satisfies

D  Dm and

Dm  0 . The desired trajectory, namely, the open-loop optimal one generated by the GPM, is governed by the following dynamics, given by 12

ρd  Fd ( ρd , ρd )  G[qDd ld ( ρd , ρd )  UCd ]

(18)

where qDd and U Cd are, respectively, the desired optimal net charge and thruster-generated control forces derived via GPM, and U d  qDd ld  UCd . Denote e  ρ  ρd  [ex

ey

ez ]T and ev  ρ  ρd  [ex

ey

ez ]T as the relative position and

velocity errors, respectively. Then, by evaluating Eqs. (17) and (18), the error dynamics is given by

e  ev  F  GU  D  ρd

(19)

where U  qD l  UC . A linear sliding surface is first chosen as s  [ sx

sy

sz ]T  ce  ev

(20)

where c  diag(cx , cy , cz ) is a diagonal positive-definite parameter matrix. The first and second time derivatives of Eq. (20) are, respectively, given by

s  cev  ev , s  cev  ev

(21)

To complete the design of second order sliding mode, a fast terminal sliding surface is then designed as [34]

η  s  ζs  ξs q / p

(22)

where ζ  diag( x ,  y ,  z ) and ξ  diag( x ,  y ,  z ) are diagonal positive-definite matrices. The positive odd integers p and q are chosen such that 0  q / p  1 . With the selected second order FTSM, the adaptive controller design is elaborated in the next subsection.

4.3. Adaptive controller design Taking the time derivative of the sliding surface η yields

η  s  ζs  ξs q / p1 s where s q / p 1  diag(sxq / p 1 , syq / p 1 , szq / p 1 ) .

13

(23)

Substitution of Eqs. (19) and (21) into Eq. (23) yields that

η  cev  ev  ζs  ξs q / p 1 s  cev  F ( ρ, ρ, ρ)  GU  D( ρ, ρ, ρ)  ρd ( ρd , ρd , ρd )  ζs  ξs q / p 1 s

(24)

Denote f ( X )  D  ρd  W T h( X )  ε , then the estimation of f is expressed as fˆ ( X )  Wˆ T h( X )

where X  [ ρT

ρT

ρT

ρdT

ρdT

(25)

ρdT ]T .

Due to the inclusion of l ( ρ, ρ, ρ) in D , the input vector X consists of the relative acceleration vector ρ , which is hardly measurable on orbit. Because the relative velocity between spacecraft ρ is negligibly small as compared to the velocity of single spacecraft (i.e.,

ρ

dRi / dt , i  C, D ), then

the velocity of the deputy spacecraft relative to the local magnetic field Vr can be approximated by that of the chief spacecraft VrC  RC  (uC  ωE )  RC . Notably, VrC is only dependent on time but not on ρ . That is, Vr ( ρ)  VrC and thus l ( ρ, ρ)  Vr ( ρ)  B( ρ)  VrC  B( ρ)  lC ( ρ) . Meanwhile, considering the fact that the relative velocity between spacecraft ρ contributes little to the velocity of the deputy with respect to the local magnetic field Vr , it is also reasonable to assume that

l ( ρ, ρ)  l ( ρ, ρd ) . Provided that either of the aforementioned two approximations of l holds, then the input vector X could be modified as Xˆ  [ ρT f

ρT

ρdT

ρdT

ρdT ]T , and the estimation of

is hence revised as fˆ ( Xˆ )  Wˆ T h( Xˆ )

(26)

Substitution of Eq. (26) into Eq. (24) yields that

η  cev  F  GU  W T h( X )  W T h( Xˆ )  W T h( Xˆ )  ε  ζs  ξs q / p 1 s  ce  F  GU  W T h( Xˆ )  ζs  ξs q / p 1 s  δ

(27)

v

where δ  W T h( X )  W T h( Xˆ )  ε  W T h  ε with h  h( X )  h( Xˆ ) . Define  m  0 as the upper

14

bound of W T h  ε . Notably,  m is unknown. Then, the time derivative of equivalent control can be derived as

Ueq  G 1[cev  F  Wˆ T h( Xˆ )  ζs  ξs q / p 1 s  Δm ]

(28)

where Δm  ˆm sgn(η) and ˆm is the estimate of the unknown bound  m . sgn(η) is the sign function vector, given by sgn(η)  [sgn(x ) sgn( y ) sgn(z )]T . The time derivative of switch control is chosen as

U s  G 1[k1η  k2 sgn(η)]

(29)

where ki  diag(kix , kiy , kiz ) ( i  1, 2 ) is a positive-definite parameter matrix. Hence, the time derivative of the total control input is derived as

U  Ueq  U s

(30)

Integrating Eqs. (28) and (29) yields the equivalent control and switch control, given by t U eq (t )  G 1 (cev  F  ζs  ξs q / p )   G 1[Wˆ T h( Xˆ )  Δm ]d t0

(31)

and t

U s (t )   G 1[k1η  k2 sgn(η)]d t0

(32)

Furthermore, the adaptive tuning law of the estimated weight matrix is designed as

Wˆ  Γh( Xˆ )ηT where Γ  diag(1 , 2 ,

(33)

, m ) is a diagonal positive-definite parameter matrix.

Also, the adaptation law of the estimated bound ˆm is designed as

ˆm    | i |, i  x, y, z i

where   0 is a design constant. Therefore, the total control input of the closed-loop system is

15

(34)

U  Ueq  U s

(35)

Note that the actual control inputs are composed of the charge qD and the thruster-generated control forces U C , it remains to distribute the control input in a fuel-optimal way because propellant is a limited resource on board. To minimize the fuel consumption, the following optimal criterion is considered as tf

tf

t0

t0

J b   Ldt  

UC dt  

tf

t0

U  qD l dt

(36)

Solving the Euler-Lagrange equation

d  L  L 0   dt  qD  qD

(37)

yields the optimal trajectory of the net charge, given by [13]

(U  l )/ || l ||2 , || l || 0 qD   || l || 0 0,

(38)

Then, the optimal trajectory of the thruster-generated control forces are therefore derived as

U  [(U  l )l ]/ || l ||2 , || l || 0 U C   || l || 0 U ,

(39)

If the required charge exceeds the near-term feasible maximum qDm  m mD , then the control laws are modified as qD  qDm sgn(qD ) U C  U  qD l

(40)

Notably, when calculating the switch control U s , s is required. It can be derived via the following differentiator [37]: w (t )  v0 (t ) v0 (t )  v1 (t )  κd 0 | w (t )  s(t ) |1/ 2 sgn[ w (t )  s(t )] v1 (t )  κd 1 sgn[v1 (t )  v0 (t )] sˆ  v (t )

(41)

1

where | w(t )  s(t ) |1/ 2  diag(| wx  sx |1/ 2 ,| wy  s y |1/ 2 ,| wz  sz |1/ 2 ) . κdi  diag( dix ,  diy ,  diz ) ( i  0,1 )

16

is a constant parameter matrix. Also, the structure of the proposed NN-based adaptive second order fast terminal sliding mode controller (ASOFTSMC) is illustrated in Fig. 3.

Fig. 3. Block diagram of the closed-loop state feedback controller. Remark 3. The time derivative of equivalent control U eq may become singular when s  0 and

s  0 , but the actual equivalent control U eq contains no negative fractional powers and is therefore nonsingular. Also, the time derivative of switch control U s includes discontinuous term which may lead to high frequency chattering. But, after integration, the derived switch control U s contains no such item [41]. Therefore, the proposed second order FTSMC simultaneously solves the singularity and chattering problems existing in the conventional FTSMC.

4.4. Stability analysis Theorem 1. Consider the dynamical system of LASFF in Eq. (17), if the second order fast terminal sliding surfaces are chosen as Eqs. (20) and (22), the control laws are designed as Eqs. (38) and (39), and the adaptive tuning laws of the RBFNN and the upper bound of the approximation error are designed as Eqs. (33) and (34), respectively, then the system errors will converge to zero

17

asymptotically and the overall closed-loop system is asymptotically stable. Proof. Consider the following Lyapunov function:

1 1 1 V1  ηT η  tr{W T Γ1W }   1 m2 2 2 2

(42)

where W  W   Wˆ and W  Wˆ .  m   m  ˆm is the estimation error of  m , and  m  ˆm . Taking the time derivative of V1 yields that V1  ηT η  tr{W T Γ 1W }   1 m m  ηT [cev  F  GU  W T h( Xˆ )  ζs  ξs q / p 1 s  δ]  tr{W T Γ1Wˆ }   1 mˆm

(43)

By substituting Eqs. (30), (33), and (34) into Eq. (43), it is obtained that V1  ηT [k1η  k2 sgn(η)  (W T  Wˆ T )h( Xˆ )  Δ m  δ]  tr{W T h( Xˆ )ηT }   1 mˆm  ηT k1η  ηT k2 sgn(η)  ηT Δ m  ηT δ   1 mˆm  ηT k1η   k2i | i |  ˆm  | i |  (ˆm   m ) | i |   1 mˆm i

 ηT k1η   k2i | i |

i

(44)

i

i

where i  x, y, z . If k1i and k2i are chosen such that k1i  0 and k2i  0 , then it holds that V1  0 , η  0 . Therefore, the overall closed-loop system is asymptotically stable. This completes the proof.

5.

Closed-loop output feedback controller

5.1. Second order sliding mode observer For the control of the closed-loop system in the absence of velocity measurements, a finite-time second order sliding mode observer is introduced as [38, 43]

xˆ1  xˆ 2  κo1 | x1  xˆ1 |1/ 2 sgn( x1  xˆ1 ) xˆ 2  g (t , x1 , xˆ 2 , U )  κo 2 sgn( x1  xˆ1 )

(45)

where x1  ρ and x2  ρ . xˆ1 and xˆ 2 denote the estimates of ρ and ρ , respectively. The function g is defined as g  F ( x1 , xˆ 2 )  GU . κoi  diag(oix ,  oiy , oiz ) ( i  1, 2 ) is a constant parameter matrix to be determined by following inequalities:

18

 o1 j  0,

j  x, y, z

 o 2 j  3gm  2( gm / o1 j )2 , where g m  0 is the upper bound of

(46)

j  x, y, z

(47)

g(t , x1 , x2 , U )  g(t , x1 , xˆ 2 , U )  D .

Detailed proof of the finite-time convergence of the observer is referred to [43], which indicates that the estimated states converge to the actual states in finite time T f , namely, x1  xˆ1 and x2  xˆ 2 hold after time T f .

5.2. Controller design and stability analysis Similar to the state feedback controller design, the output feedback control law is designed as

U  Ueq  U s

(48)

U eq  G 1 (ceˆv  Fˆ  ζsˆ  ξsˆq / p )   G 1[Wˆ T h( Xˆ )  Δˆ m ]d

(49)

with t

t0

t

U s   G 1[k1ηˆ  k2 sgn(ηˆ )]d t0

(50)

where eˆv  xˆ 2  ρd , Fˆ  F ( x1 , xˆ 2 ) , sˆ  ce  eˆv , ηˆ  sˆ  ζsˆ  ξsˆq / p , and Δˆ m  ˆm sgn(ηˆ ) . The input vector of the RBFNN is modified as Xˆ  [ x1T

xˆ 2T

ρdT

ρdT

ρdT ]T . Also, the control input is

distributed as the net charge and the thruster-generated control forces based on the optimal distribution laws as shown in Eqs. (38) and (39). Likewise, the adaptive tuning law of the RBFNN is designed as

Wˆ  Γh( Xˆ )ηˆ T

(51)

Also, the adaptation law of ˆm is revised as

ˆm    | ˆi |, i  x, y, z

(52)

i

Theorem 2. Consider the dynamical system of LASFF in Eq. (17), if the second order fast terminal

19

sliding mode surfaces are chosen as Eqs. (20) and (22), the observer is designed as Eq. (45), the control law is designed as Eq. (48), and the adaptive tuning laws of the RBFNN and the upper bound of the approximation error are designed as Eqs. (51) and (52), respectively, then the system errors will converge to zero asymptotically and the overall closed-loop system is asymptotically stable. Proof. Consider the Lyapunov function

1 1 1 V2  ηˆ T ηˆ  tr{W T Γ1W }   1 m2 2 2 2

(53)

Using similar method, it can be obtained that

V2  ηˆ T k1ηˆ   k2i | ˆi |  0, i  x, y, z

(54)

i

Note that Eq. (54) indicates the asymptotic convergence of the estimated state errors. Nevertheless, with the introduced observer, the estimated system states will converge to the actual system states in finite time before the estimated state errors converge to zero as t   . Therefore, the asymptotic convergence of the actual state errors can also be guaranteed. This completes the proof.

6.

Numerical simulations and analysis

6.1. Open-loop controller The chief is assumed to be flying in a near circular LEO, of which the initial orbital elements are shown in Table 1. The initial phase angle of the tilted dipole is chosen as 0  60 and the dipole tilt angle is   11.3 . The deputy departures from the origin of the RM frame with zero relative velocity (i.e., ρd (0)  0 , ρd (0)  0 ). The mass of the deputy is assumed to be mD  50 kg. Denote T as the chief’s orbital period. To formulate an approximate projected circular orbit at the final time of

t f  T , the required terminal relative states are calculated via the energy-matching conditions [44] as: xd (T )  0, xd (T )  0.544,

yd (T )  1000, zd (T )  0 4 yd (T )  1.75 10 , zd (T )  1.089 20

(55)

where the relative position and velocity are in units m and m/s, respectively. 30 Legendre-Gauss (LG) collocation points are utilized in GPM, and the weight matrix is chosen as Q  diag(1,106 ,106 ,106 ) such that the propellant expenditure is given priority. Table 1 Initial orbit elements of the chief spacecraft Orbit element

Value

Semi-major axis / km

7000

Eccentricity

0.005

Inclination / deg

40

Right ascension of ascending node / deg

50

Argument of perigee / deg

0

True anomaly / deg

0

Fig. 4 depicts the optimal trajectories of the control inputs composed of the net charge and the thruster control forces of the Lorentz spacecraft. The discrete points denote the results derived via GPM, and the solid lines denote the Lagrange interpolation between the LG points. As can be seen, the net charge is on the order of 10-1 C, and the thruster-generated control forces are on the order of 10-4 N, which corresponds to a specific charge on the order of 10-3 C/kg, and the thruster-generated control accelerations on the order of 10-6 m/s2. Note that J2 perturbation, arising from the oblateness and nonhomogeneity of Earth, is one of the most dominant disturbances in LEO [45,46]. Its expression in the ECI frame is given by [47]

 X i3  X iYi 2  4 X i Z i2  3 J 2 RE2   a J ( Ri )   X i2Yi  Yi 3  4Yi Z i2  , i  C , D 2 || Ri ||7  2 3 X i Zi  3Yi 2 Z i  2Z i3    where RC  [ X C

YC

ZC ]T and RD  [ X D YD

(56)

Z D ]T refer to the position vector of the chief and

the deputy spacecraft expressed in the ECI frame, respectively. J 2  1.0826 103 is the second zonal harmonic coefficient and RE is the radius of Earth. 21

The differential relative J2-perturbation acceleration experienced by the chaser and the Lorentz spacecraft, which perturbs the relative orbital motion, is thus given by aJ  aJ ( RD )  aJ ( RC ) . Note that Eq. (56) is expressed in the ECI frame. By a simple coordination transformation from the ECI frame to the RM frame, the expression of a J in the RM frame can be derived. Generally, for spacecraft flying in typical LEOs, if the relative distance between spacecraft is within several kilometers, then the resulting differential J2-perturbation acceleration is on the order of 10 -6 m/s2.

Fig. 4. Time histories of open-loop optimal control inputs. Thus, as aforementioned, the necessary thruster-generated control accelerations are on the order of 10-6 m/s2 in this example. Furthermore, the formation size in this example is within several kilometers. Therefore, the thruster-generated control accelerations are nearly on the same order as the relative J2-perturbation accelerations for spacecraft within several kilometers, verifying the statement in Remark 2.

22

Fig. 5. Trajectories of relative transfer orbit and formation. The trajectories of the relative transfer orbit and the final formation are shown in Fig. 5. Likewise, the discrete points denote the results of GPM. To verify the validity of the results of GPM, the interpolated control trajectories are substituted into the nonlinear dynamical model Eq. (1) to derive the actual trajectories of the relative states by integrating the nonlinear equations using 4th order Runge Kutta method. The numerical results are denoted by solid lines in Fig. 5, which are nearly coincided with the GPM’s results. For these two methods, the terminal relative position and velocity errors are, respectively, on the order of 10-2 m and 10-5 m/s, thus proving the feasibility of GPM. With the Lorentz force as primary propulsion, the reduced percent of velocity increment consumption can be calculated via tf

   1 

t0 tf

t0

U C dt (57)

U dt

which is about   98.6% in this scenario, proving the ability of Lorentz spacecraft in saving fuel on board.

23

6.2. Closed-loop controller To verify the performance of the closed-loop state and output feedback controllers in the presence of external perturbations and model uncertainties, typical perturbations in LEOs are incorporated into the dynamical model as external disturbances, that is, the J2 perturbation and atmospheric drag. The expression of the J2-perturbation acceleration experienced by each spacecraft has been given in Eq. (56). The atmospheric-drag acceleration acting on each spacecraft can be represented as [48]

1 ad (Vri )   Gi di Cdi Ai || Vri ||2 Vˆri , i  C , D 2

(58)

where the subscript C or D refers to the chief or the deputy spacecraft. Gi  mi1 is the inverse of the mass. di  d 0 exp[(h  h0 ) / H ] is the local atmospheric density, where h is the orbital altitude,

 d 0 is the atmospheric density at the reference altitude h0 , and H is the scale altitude [48]. Cdi is the drag coefficient. Vri is the velocity of the spacecraft relative to the local atmosphere. Since it is assumed here that the atmosphere is corotating with Earth, then the vehicle’s velocity with respect to the local atmosphere is the same as the one with respect to the local magnetic field, which is given in Eq. (9). Ai is the projected area of the spacecraft perpendicular to the relative velocity Vri . Then, the differential atmospheric drag between the chief and the Lorentz spacecraft is thus given by

ad  ad (VrD )  ad (VrC ) . The initial relative state errors are chosen as e (0)  [100 100 50]T m and ev (0)  0 m/s. The initial estimates of the observer are chosen as xˆ1 (0)  0 m and xˆ 2 (0)  [0.1 0.1 0]T m/s. Moreover, the error vector is set as l  0.05l sin(nC t ) . Other simulation parameters are listed in Table 2, where i  x, y, z and j  1, 2,

,5 .

Furthermore, C j is the jth column of the parameter matrix C  by 24

155

used in the RBFNN, given

C  [C1T

C1T

C2T ]T

(59)

where

 700  1500   1000 C1    1  1.5   1.2

350 750 500 0.5 0.75 0.6

0 0 0 0 0 0

350 750 500 0.5 0.75 0.6

700  1500  1000   1  1.5   1.2 

(60)

and

 1.6 0.8 0 0.8 1.6  C 2  10   2 1 0 1 2   1.5 0.75 0 0.75 1.5  3

(61)

Table 2 Simulation parameters. Case

Parameter

b j  103 , Wˆ (0)  053 ,  j  5 105 , ˆm (0)  0 ,   3 105 , Controller

ci  1.5 103 ,  i  102 , i  2 102 , k1i  8 103 , k2i  108 , p  5, q  3;

Differentiator

 d 0i  102 ,  d1i  105 ;

Observer

 o1i  10 ,  o 2i  6.5 102 ;

Atmosphere

d 0  1.454 1013 kg/m3, h0  600 km, H  79 km;

Spacecraft

mC  mD  50 kg, CdC  CdD  2 , AC  AD  1.5 m2, mD  2 kg, m  0.03 C/kg.

Time histories of relative position and velocity errors are, respectively, depicted in Fig. 6 and Fig. 7, from which it is clear that all errors converge to small values after about half an orbital period for both state and output feedback controllers. Details during the first 50 s are enlarged in the right sides of both figures to show the differences more clearly. Furthermore, for both controllers, the terminal accuracy of the relative position error is on the order of 10 -2 m, and that of the relative velocity error is on the order of 10-5 m/s.

25

Fig. 6. Time histories of relative position errors.

Fig. 7. Time histories of relative velocity errors. As can be seen, for these two controllers, the error trajectories are similar but not exactly the same because relative velocity information is unavailable for the output feedback one. The velocity signals used by the output feedback controller are generated by the observer. Time histories of the real and estimated relative velocity for the output feedback controller are shown in Fig. 8. Obviously, the observer captures the velocity signals within a finite time of about 10 s, ensuring the success of the

26

output feedback control scheme.

Fig. 8. Comparisons of the real and estimated relative velocity for the output feedback controller. The required control inputs composed of the net charge and thruster-generated control forces for both controllers are compared in Fig. 9, and details of the first 50 s are enlarged in the right side of Fig. 9. It is obvious that the control inputs are nonsingular and no significant chattering is observed, verifying the arguments in Remark 3. Since that the near-term feasible maximal specific charge is about 0.03 C/kg, the maximal available net charge is therefore given by qDm  m mD  1.5 C for a Lorentz spacecraft with a mass of 50 kg. Also, the maximal required thruster-generated control forces are on the order of 10-1 N, which corresponds to the control accelerations on the order of 10 -3 m/s2. As can be seen, for both controllers, the magnitude of the required charge exceeds the possible maximum of 1.5 C during the first several seconds, thus it is set as the maximum with corresponding polarity. The control inputs of both controllers begin to track the desired ones since about half an orbital period. Similarly, by using Eq. (57), the percent of reduced velocity increment consumption for the closed-loop formation establishment is 85.3% or 84.9% for the state or output feedback controller. This simulation example

27

testifies the validity of the proposed state and output feedback controllers for LASFF in the presence of external perturbations and system uncertainties, including J2 perturbations, atmospheric drag, and the uncertainties of mass and magnetic field.

Fig. 9. Time histories of control inputs. 6.3. Comparisons with other methods To demonstrate the advantages of the proposed controller over other control approaches, another two controllers have also been introduced to conduct comparative studies. One is a second order sliding mode controller without NN, the other is a modified PID controller. In view of Eqs. (31) and (32), the former one is designed as

U  U eq  U s

(62)

t

where the switch control is still U s   G 1[k1η  k2 sgn(η)]d . However, the equivalent control is t0

revised as Ueq  G 1 (cev  F  ζs  ξs q / p ) . Notably, without NN to estimate the unknown nonlinear dynamics, the closed-loop system driven by the control law Eq. (62) is not asymptotically stable any 28

more. Instead, it is uniformly ultimately bounded. That is, the error states will not converge exactly to zero asymptotically. Instead, they will converge to small neighborhoods of zero. Detailed proof on the stability of this closed-loop system is given in the Appendix A. Another modified PID controller is designed as U  G 1 (F  ρd  k p e  ki  e dt  kd ev )

(63)

where k p , ki , and k d are the proportional, integral, and derivative gains, respectively [49]. Likewise, the closed-loop system with the control law Eq. (63) is not asymptotically stable. Instead, it is uniformly ultimately bounded. Given that PID control methodology is a commonly used one, analysis on the closed-loop system stability is not elaborated here for brevity. Quantitative comparisons are then made between these controllers, and the performance indices are defined as follows. Define || e || as the error distance. The settling time t s is defined as the time required by the error distance || e (t ) || to reach and stay within 1% of the initial value || e (0) || , that is, t  ts , || e (t ) || 1% || e (0) || . Then, the mean steady error distance is defined as

e s  mean{|| e (t ) ||} ts  t  t f

(64)

Furthermore, the reduced percent of velocity increment consumption (i.e.,  ) has been defined in Eq. (57), and the control energy consumption is defined as

Jc 

1 tf T U Udt 2 t0

(65)

Control parameters for the controllers Eqs. (62) and (63) are summarized in Table 3. Notably, the controller parameters in Table 3 are chosen such that these two controllers consume similar or almost identical control energy to complete the formation establishment as the NN-based state feedback controller in the above subsection does. In this way, the comparisons between these three controllers

29

will be more reasonable and justified. Also, the controller parameters for the NN-based one and other simulation parameters are chosen the same as those given in Table 2. Table 3 Controller parameters. Controller

Parameter

Without NN [Eq. (62)]

ci  1.5 103 ,  i  102 , i  2.04 102 , 3 8 p  5 , q  3 , k1i  8 10 , k2i  10 ;

k p  1.11104 , ki  6 108 , kd  8 102 .

PID [Eq. (63)]

Table 4 Performance indices of the controllers. Performance index Controller || e (t f ) || (m)

t s (orbit)

e s (m)

 (%)

J c (N2·s)

With NN [Eq. (35)]

0.02

0.53

0.36

85.28

6.32

Without NN [Eq. (62)]

0.18

0.52

0.55

85.24

6.32

PID [Eq. (63)]

2.96

>1.00

—

83.38

6.32

Fig. 10. Comparisons of the norms of the relative distance errors and control inputs. For each of the controllers, time histories of the norms of relative distance errors and control inputs are compared in Fig. 10. Quantitative results are summarized in Table 4. As can be seen, at the expense nearly equal control energy, the NN-based one presents the best controller performances except for a 30

bit longer settling time than the one without NN. More importantly, by using the NNs to estimate the unknown nonlinearity in system dynamics, the terminal relative distance error (i.e., || e (t f ) || ) of the NN-based controller is at least one order of magnitude smaller than those of the other two controllers. Thus, it can be concluded that the NN-based controller is able to enhance the control accuracy while not degrading other performance indices. These comparative results coincide with the theoretical proof that the NN-based controller could ensure asymptotic stability but the other two controllers could only ensure uniform ultimate boundedness. In this way, the advantages of the NN-based controller proposed in this paper have been proved both theoretically and numerically. Furthermore, it is also notable that with the Lorentz force as auxiliary propulsion, the consumption of velocity increment consumption could be greatly saved, regardless of the control methods. As proved in Section 6.1, the ability of Lorentz spacecraft in saving fuels has been verified here again.

7.

Conclusions

Two neural network-based adaptive second order fast terminal sliding mode controllers, which use state and output feedback respectively, have been designed for LASFF to track the open-loop optimal trajectories obtained by GPM. RBFNNs are used to approximate the uncertain nonlinearities in the system dynamics, and SOSMC technique is used to guarantee that the closed-loop system is robust against external perturbations and system uncertainties. The adaptive tuning laws of the RBFNNs and the upper bound of the approximation error are derived by a Lyapunov-based method to ensure the overall asymptotic stability of the closed-loop system. The main advantages of the proposed control scheme have been substantiated both theoretically and numerically, which are summarized as follows. 1) By using the SOSMC method, the singularity and chattering problems existing in the initial 31

FTSMC have been effectively solved, and the resulting control law is thus nonsingular and continuous; 2) With the adaptation law to estimate the upper bound of the approximate error, no prior knowledge about this bound is required; 3) The closed-loop controller ensures asymptotic stability that all state errors converge to zero asymptotically, thus presenting enhanced control accuracy as compared to those with uniform ultimate boundedness only. Furthermore, the proposed controllers could be directly applied to other Lorentz-augmented relative orbital control problems, such as spacecraft hovering and rendezvous, and could be simply modified to deal with other similar second order system control problems.

Acknowledgments

The authors express their gratitude to the reviewers and editors for their valuable and constructive comments that helped greatly enhance the quality of this paper. This work was partly supported by the Fund of Innovation by Graduate School of National University of Defense Technology under Grant B140106, and the Hunan Provincial Innovation Foundation for Postgraduate under Grant CX2014B006.

Appendix A

The stability analysis of the closed-loop system driven by the controller Eq. (62) is summarized in the following theorem. Theorem 3. Consider the dynamical system of LASFF in Eq. (17), if the second order fast terminal

32

sliding mode surfaces are chosen as Eqs. (20) and (22), the control law is designed as Eq. (62), then the system errors will converge to the neighborhood of zero, given by

| ei | ei , | ei | ei , i  x, y, z

(A.1)

where the detailed expressions of  ei and  ei will be given in the following proof. Proof. Consider the Lyapunov candidate V3  (1/ 2)ηT η  0 , η  0 . Taking the time derivative of

V3 along the system trajectory yields that V3  ηT η  ηT [cev  F  GU  f  ζs  ξs q / p 1 s]

(A.2)

where f  D  ρd is an unknown bounded nonlinear function, satisfying || f || f m and f m  0 . Notably, f m is unknown. Substitution of the control law Eq. (62) into Eq. (A.2) yields that

V3  ηT [k1η  k2 sgn(η)  f ]

 k1min || η ||2 k2 min  | i | || η || f m

(A.3)

i

where k1min and k2 min refer to the minimum eigenvalue of k1 and k 2 , respectively. Note that

| 

i

| || η || , Eq. (A.3) further reduces to

i

V3  (k1min  || η ||1 f m ) || η ||2 k2 min || η ||

(A.4)

V3  k1min || η ||2 (k2 min  f m ) || η ||

(A.5)

or

Obviously, if k1min  || η ||1 f m  0 or k2 min  f m  0 holds, V3 will continue to converge. However, since f m is unknown, one cannot predetermine a control gain k2 min to ensure the validity of the second inequality. Thus, in view of the first inequality, the terminal convergent region of η will be 1 || η ||   k1min fm

33

(A.6)

Thereafter, the system dynamics is governed by

i  si   i si  i siq / p , | i ||| η || 

(A.7)

where i  x, y, z . Rewrite Eq. (A.7) as si  ( i i si1 )si  i siq / p  0

(A.8)

si   i si  (i i si q / p )siq / p  0

(A.9)

or

Likewise, if  i i si1  0 or i i si q / p  0 holds, si will continue to converge. Thus, the terminal convergent region of si is given by

| si | si  min{ i1 ,(i1 ) p / q }

(A.10)

Similarly, the system dynamics is thereafter governed by

si  ci ei  ei , | si | si

(A.11)

Then, rewrite Eq. (A.11) as (ci  si ei1 )ei  ei  0 or ci ei  (1  si ei1 )ei  0 . Obviously, ei will continue to converge if ci  si ei1  0 or 1  si ei1  0 holds. Therefore, the terminal convergent regions of ei and ei are derived as | ei | ei  ci1si , | ei | ei  si

(A.12)

As can be seen, different from the NN-based controller Eq. (35) where the sliding surface η converges to zero asymptotically, the controller without NN could only guarantee the boundedness of

η , as demonstrated in Eq. (A.6). The boundedness of η further results in the boundedness of the state errors ei and ei , indicating that these state errors will converge to the neighborhood of zero but not converge exactly to zero. Obviously, to render the state errors sufficiently close to zero, the sliding surface parameters (i.e., ci ,  i , and  i ) and the control gains (i.e., k1i ) should be sufficiently large. However, large control parameters will also result in large control inputs.

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Above analysis proves theoretically that the NN-based controller is superior to the one without NN in control precision, as demonstrated by the numerical simulations in Section 6.3. This completes the proof.

References [1] M.A. Peck, Prospects and challenges for Lorentz-augmented orbits, in: AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California, 15–18 August, 2005. [2] G.E. Pollock, J.W. Gangestad, J.M., Longuski, Analytical solutions for the relative motion of spacecraft subject to Lorentz-force perturbations, Acta Astronaut. 68 (2011) 204–217. [3] X. Huang, Y. Yan, Y. Zhou, T. Yi, Improved analytical solutions for relative motion of Lorentz spacecraft with application to relative navigation in low Earth orbit, Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 228(11) (2014) 2138–2154. [4] X. Huang, Y. Yan, Y. Zhou, J. Yu, Output feedback control of Lorentz-augmented spacecraft rendezvous, Aerosp. Sci. Technol. 42 (2015) 241–248. [5] X. Huang, Y. Yan, Y. Zhou, H. Zhang, Pseudospectral method for optimal propellantless rendezvous using geomagnetic Lorentz force, Appl. Math. Mech. –Engl. Ed. 36(5) (2015) 609– 618. [6] M.A. Peck, B. Streetman, C.M. Saaj, V. Lappas, Spacecraft formation flying using Lorentz forces, J. Br. Interplanet. Soc. 60(7) (2007), 263–267. [7] C. Peng, Y. Gao, Lorentz-force-perturbed orbits with application to J2-invariant formation, Acta Astronaut. 77 (2012) 12–28. [8] S. Tsujii, M. Bando, H. Yamakawa, Spacecraft formation flying dynamics and control using the geomagnetic Lorentz force, J. Guid. Control Dyn. 36(1) (2013) 136–148. [9] X. Huang, Y. Yan, Y. Zhou, Optimal spacecraft formation establishment and reconfiguration propelled by the geomagnetic Lorentz force, Adv. Space Res. 54(11) (2014) 2318–2335. [10] L.A. Sobiesiak, C.J. Damaren, Optimal continuous/impulsive control for Lorentz-augmented spacecraft formations, J. Guid. Control Dyn. 38(1) (2015) 151–156. [11] L.A. Sobiesiak, C.J. Damaren, Controllability of Lorentz-augmented spacecraft formations, J. Guid. Control Dyn. 38(11) (2015) 2188–2195. [12] L.A. Sobiesiak, C.J. Damaren, Lorentz-augmented spacecraft formation reconfiguration, IEEE Trans. Control Syst. Technol. 24(2) (2016) 514–524. [13] X. Huang, Y. Yan, Y. Zhou, Optimal Lorentz-augmented spacecraft formation flying in elliptic

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orbits, Acta Astronaut. 111 (2015) 37–47. [14] X. Huang, Y. Yan, Y. Zhou, Adaptive fast nonsingular terminal sliding mode control of Lorentz-augmented spacecraft relative motion, Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. (2015). in press, http://dx.doi.org/10.1177/0954410015621089 [15] J.W. Gangestad, G.E. Pollock, J.M. Longuski, Propellantless stationkeeping at Enceladus via the electromagnetic Lorentz force, J. Guid. Control Dyn. 32(5) (2009) 1446–1475. [16] X. Huang, Y. Yan, Y. Zhou, Dynamics and control of spacecraft hovering using the geomagnetic Lorentz force, Adv. Space Res. 53(3) (2014) 518–531. [17] X. Huang, Y. Yan, Y. Zhou, H. Zhang, Sliding mode control for Lorentz-augmented spacecraft hovering around elliptic orbits, Acta Astronaut. 103(2014) 257–268. [18] J.W. Gangestad, G.E. Pollock, J.M. Longuski, Lagrange’s planetary equations for the motion of electrostatically charged spacecraft, Celest. Mech. Dyn. Astron. 108(2) (2010) 125–145. [19] J.W. Gangestad, G.E. Pollock, J.M. Longuski, Analytical expressions that characterize propellantless capture with electrostatically charged spacecraft, J. Guid. Control Dyn. 34(1) (2011) 247–258. [20] G.E. Pollock, J.W. Gangestad, J.M. Longuski, Inclination change in low-Earth orbit via the geomagnetic Lorentz force, J. Guid. Control Dyn. 33(5) (2010) 1387–1395. [21] H. Wong, V. Kapila, A.G. Sparks, Adaptive output feedback tracking control of spacecraft formation, Int. J. Robust Nonlinear Control 12(2–3) (2002) 117–139. [22] A. Zou, K.D. Kumar, Adaptive output feedback control of spacecraft formation flying using Chebyshev neural networks, J. Aerosp. Eng. 24(3) (2011) 361–372. [23] S. Mefoued, A second order sliding mode control and a neural network to drive a knee joint actuated orthosis, Neurocomputing 155(2015) 71–79. [24] L. Zhao, Y. Jia, Neural network-based distributed adaptive attitude synchronization control of spacecraft formation under modified fast terminal sliding mode, Neurocomputing 171(2016) 230– 241. [25] A. Zou, K.D. Kumar, Adaptive attitude control of spacecraft without velocity measurements using Chebyshev neural network, Acta Astronaut. 66 (2010) 769–779. [26] A. Zou, K.D. Kumar., Z. Hou, Quaternion-based adaptive output feedback attitude control of spacecraft using Chebyshev neural networks, IEEE Trans. Neural Netw. 21(9) (2010) 1457–1471. [27] A. Zou, K.D. Kumar., Z. Hou, Finite-time attitude tracking control for spacecraft using terminal sliding mode and Chebyshev neural network, IEEE Trans. Syst. Man, Cybern. B, Cybern. 41(4) (2011) 950–963. [28] X. Huang, Y. Yan, Y. Zhou, H. Xu, Output feedback sliding mode control of Lorentz-augmented spacecraft hovering using neural networks, Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.

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229(10) (2015) 939–948. [29] T. Sun, H. Pei, Y. Pan, H. Zhou, C. Zhang, Neural network-based sliding mode adaptive control for robot manipulators, Neurocomputing 74(14) (2011) 2377–2384. [30] C. Hsu, C. Chang, Intelligent dynamic sliding-mode neural control using recurrent perturbation fuzzy neural networks, Neurocomputing 173(2016) 734–743. [31] H. Li, J. Wang, P. Shi, Output-feedback based sliding mode control for fuzzy systems with actuator saturation, IEEE Trans. Fuzzy Syst. (2015). in press, http://dx.doi.org/10.1109/ TFUZZ.2015.2513085 [32] H. Li, J. Wang, H. Lam, Q. Zhou, H. Du, Adaptive sliding mode control for interval type-2 fuzzy systems, IEEE Trans. Syst., Man, Cybern., Syst. (2016). in press, http://dx.doi.org/10.1109/ TSMC.2016.2531676 [33] H. Li, P. Shi, D. Yao, L. Wu, Observer-based adaptive sliding mode control for nonlinear Markovian jump systems, Automatica 64(2016) 132–142. [34] X. Yu, Z. Man, Fast terminal sliding-mode control design for nonlinear dynamical systems, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(2) (2002) 261–264. [35] L. Qi, H. Shi, Adaptive position tracking control of permanent magnet synchronous motor based on RBF fast terminal sliding mode control, Neurocomputing 115(2013) 23–30. [36] A. Ferrara, M. Rubagotti, A sub-optimal second order sliding mode controller for systems with suturing actuators, IEEE Trans. Autom. Control 54(5) (2009) 1082–1087. [37] A. Levant, Higher-order sliding modes, differentiation and output-feedback control, Int. J. Control 76(9–10) (2003) 924–941. [38] J. Davila, L. Fridman, A. Levant, Second-order sliding-mode observer for mechanical systems, IEEE Trans. Autom. Control 50(11) (2005) 1785–1789. [39] S. Mondal, C. Mahanta, A fast converging robust controller using adaptive second order sliding mode, ISA Trans. 51 (2012) 713–721. [40] S. Mondal, C. Mahanta, Adaptive second-order sliding mode controller for a twin rotor multi-input-multi-output system, IET Control Theory Appl. 6(14) (2012) 2157–2167. [41] S. Mondal, C. Mahanta, Adaptive second order terminal sliding mode controller for robotic manipulators, J. Frankl. Inst. 351(4) (2014) 2356–2377. [42] D.A. Benson, G.T. Huntington, T.P. Thorvaldsen, A.V. Rao, Direct trajectory optimization and costate estimation via an orthogonal collocation method, J. Guid. Control Dyn. 29(6) (2006) 1435–1440. [43] M. Van, H. Kang, Y. Suh, Second order sliding mode-based output feedback tracking control for uncertain robot manipulators, Int. J. Adv. Robot Syst. 10(16) (2013) 1–9. [44] P. Gurfil, Relative motion between elliptic orbits: generalized boundedness conditions and optimal 37

formationkeeping, J. Guid. Control. Dyn. 28(4) (2005) 761–767. [45] X. Huang, Y. Yan, Y. Zhou, H. Zhang, Nonlinear relative dynamics of Lorentz spacecraft about J2-perturbed orbit, Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 229(3) (2015) 467–478. [46] X. Huang, Y. Yan, Y. Zhou, Nonlinear control of underactuated spacecraft hovering, J. Guid. Control Dyn. 39(3) (2016) 684–693. [47] C.J. Damaren, Almost periodic relative orbits under J2 perturbations, Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 221(5) (2007) 767–774. [48] S. Varma, K.D. Kumar, Multiple satellite formation flying using differential aerodynamic drag, J. Spacecr. Rockets 49(2) (2012) 325–336. [49] K.H. Ang, G. Chong, Y. Li, PID control system analysis, design, and technology, IEEE Trans. Control Syst. Technol. 13(4) (2005) 559–576.

Vitae Xu Huang was born in Yangzhou, China, in 1990. He received the B.S. degree in aerospace engineering from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2011, and the M.S. degree in aerospace engineering from National University of Defense Technology, Changsha, China, in 2013, where he is currently working toward the Ph.D. degree in the College of Aerospace Science and Engineering. He was a Summer Intern with the Helicopter Research and Development Institute, Aviation Industry Corporation of China, in 2010. His current research interests include spacecraft dynamics and control, intelligent control theory and applications, and electrostatic astrodynamics.

Ye Yan was born in Chengde, China, in 1971. He received the B.S. degree in automation from National University of Defense Technology, Changsha, China, in 1994, and the M.S. and Ph. D. degrees in aerospace engineering from the same university in 1997 and 2000, respectively. He is currently a Professor with the College of Aerospace Science and Engineering, National University of Defense Technology. From August to October in 2015, he was a Visiting Scholar with the Department of Engineering, University of Cambridge, U.K. His current research interests include space systems engineering, and intelligent control theory with aerospace applications.

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Yang Zhou was born in Hangzhou, China, in 1989. He received the B.S. degree in materials processing engineering from Northeastern University, Shenyang, China, in 2011, and the M.S. degree in aerospace engineering from National University of Defense Technology, Changsha, China, in 2013, where he is currently working toward the Ph.D. degree in the College of Aerospace Science and Engineering. His current research interests include spacecraft dynamics and control, intelligent optimization theory with aerospace applications, and mission planning for on-orbit servicing.

Highlights

   

An almost propellantless formation establishment is achieved via Lorentz force. Optimal open-loop hybrid inputs are derived by pseudospectral method. A closed-loop adaptive sliding mode controller using neural networks is proposed. An output feedback controller is designed for cases without velocity measurements.

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