adaptive finite-time control for spacecraft rendezvous ...research article adaptive finite-time...

Research ArticleAdaptive Finite-Time Control for Spacecraft Rendezvous underUnknown System Parameters

Yuan Liu,1 Guojian Tang,1 Yuhang Li,2 Hang Li,1 Jing Ren,1 and Sai Zhang 3

1School of Astronautics, National University of Defense Technology, Changsha 410022, China2School of Electronic and Information Engineering, Northeast Agricultural University, Harbin 150030, China3College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Sai Zhang; [email protected]

Received 22 February 2020; Revised 23 June 2020; Accepted 11 July 2020; Published 5 August 2020

Academic Editor: Maj D. Mirmirani

Copyright © 2020 Yuan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this study, we investigated the sliding mode control (SMC) technology for the spacecraft rendezvous maneuver under unknownsystem parameters and external disturbance. With no knowledge of the mass and inertial matrix of the pursuer spacecraft, anadaptive SMC approach was devised using the hyperbolic tangent function to realize the control objective of reducing thechattering problem. In addition, the finite-time stability of the relative dynamics and the boundedness of the signals in theclosed-loop system were derived under proposed method. The effectiveness and advantages of the proposed method wereverified through theoretical analysis and numerical simulations.

1. Introduction

With rapid developments in the aerospace industry, space-craft rendezvous technology has been extensively applied inspace missions, such as deep space exploration, the establish-ment of space stations, and the detection of various compo-nents on Mars. Due to the fact that control systems are oneof the most crucial technologies of the rendezvous maneuver,extensive attention of researchers has been attracted in thepast decades. To guarantee the success of space missions, var-ious methods are studied for spacecraft rendezvous maneu-ver. However, designing controllers for the spacecraftrendezvous maneuver is still a challenging task because ofthe strongly coupled nonlinear dynamics and unknownexternal environment. Current attitude controls of spacecraftinclude backstepping control [1, 2], adaptive control [3–5],and sliding mode control (SMC) [6–8].

However, the methods in [1–7] cannot be used in thespacecraft rendezvous maneuver, which severely limits theirapplication. Considerable efforts are required during the con-troller design process in rendezvous missions to address theeffects of complex external disturbance and coupled nonlin-

ear dynamics. To reduce the effect of external disturbances,backstepping-based controllers, in which globally asymptoticstability can be achieved for closed-loop systems, have beeninvestigated in detail in [9–11]. The common drawback in[9–11] is that the convergence rate of the control system isasymptotical, that is, the control objective can be achievedwhen time is infinite. To ensure finite-time convergencefor the entire system, Wang et al. investigated controlmethods to accomplish the rendezvous mission by usingthe backstepping design [12]. However, actuator faults inactual spacecraft activities have not been considered in[9–12]. Unexpected and complex failures frequently occurin the actuators in spacecraft rendezvous missions. Thesefailures result in the degradation of control performance.Furthermore, failures may occur during a mission in theevent of limited communication bandwidth and transmis-sion delays. To ensure mission success and avoid unexpectedfailures, adaptive SMC algorithms have been investigated toimprove reliability [13, 14]. Another aspect that deserves spe-cial attention is collision avoidance during the rendezvousmaneuver, which has been ignored in previous articles.Artificial-potential-function-based SMC and backstepping

HindawiInternational Journal of Aerospace EngineeringVolume 2020, Article ID 3648260, 13 pageshttps://doi.org/10.1155/2020/3648260

https://orcid.org/0000-0003-4249-6481https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/3648260

control are proposed in [15, 16], respectively, for avoidingcollisions during rendezvous maneuvers.

Results in [9–16] were obtained by regarding the inertialparameters as available valuables. However, inertial parame-ters are not always available to designers during real missionsmainly because of fuel consumption in the pursuer space-craft. Moreover, solar radiation pressure and disturbancesin the space environment influence the inertial parameters.Considering these aspects, an adaptive SMCmethod was pre-sented in [17] for controlling spacecraft maneuver underunknown inertia. The adaptive SMC of [17] was improvedto an adaptive control algorithm based on gain and neuralnetworks in [18] for addressing uncertainties and externaldisturbances. For spacecraft proximity operations with para-metric uncertainties, an integrated adaptive backsteppingcontrol method and an adaptive control algorithm using dualquaternions were designed in [19, 20], respectively. However,the results from [17–20] cannot be directly applied to thespacecraft rendezvous maneuver. On the basis of [18–20],the uncertainties of the relative dynamics were compensatedusing robust adaptive backstepping control in [21] by intro-ducing radial basis function neural networks. In contrast to[21], the problem of time-varying inertial parameters wasstudied in [22]. Similarly, the continuous adaptive controlalgorithm was combined with the projection algorithm in[23] for controlling the spacecraft rendezvous maneuverunder time-varying inertia parameters and actuator faults.

The effective controllers in [17–23] that address uncer-tainties are asymptotically stable, that is, the system statesconverge to equilibrium when time is infinite. Unlikeasymptotically stable controllers, finite-time controlschemes have been widely studied and applied in space-craft attitude control because of their high convergencerate and superior control performance [8, 24, 25]. Control-lers for the unwinding phenomenon, which was not con-sidered in [24, 25], are presented in [8]. Moreover, inputsaturation constraints, which influence the performanceof spacecraft, were not considered in [8, 24, 25]. Therefore,an adaptive finite-time control algorithm was investigatedin [26] to address the problem of unavoidable input satura-tions for spacecraft. As a continuation of [26], the collisionproblem between the pursuer spacecraft and the target space-craft was studied using an adaptive finite-time antisaturationcontroller in [27].

Excellent finite-time stability can be achieved for thesystem by using the controllers in [8, 24–27]. However, thechattering phenomenon, which is the main cause of actuatordamage, was not considered in these studies. A boundarylayer function was incorporated into the controller in [28]to alleviate the chattering phenomenon. Similarly, continu-ous and chatter-free controllers were introduced in [29] forspacecraft rendezvous and docking. In this study, the finite-time control problem for spacecraft rendezvous was studiedin terms of the existing chattering problem in the SMC andthe unknown time-varying inertial parameters in real mis-sions. The contributions of this paper are as follows:

(i) In contrast to the existing spacecraft controlschemes for known inertia [10–13], unknown

parameters are considered in this paper, which con-siderably extends the application of control methodsfor the spacecraft rendezvous maneuver

(ii) Unlike the controllers in [17–23], finite-time stabil-ity can be achieved for the system with a high con-vergence rate by using the proposed method evenwhen the inertial parameters are unavailable todesigners

(iii) The hyperbolic tangent function can be used in thecontrol law to avoid the chattering problem. Fur-thermore, the sliding mode method was adopted inthis study.

The remainder of this paper is arranged as follows. Thedynamics model of the spacecraft is established in Section2. The finite-time controller is described in Section 3. Theeffectiveness of the controller is proved through simulationsin Section 4. Finally, the conclusion of this paper is presentedin Section 5.

2. Spacecraft Model and Preliminaries

2.1. Relative Attitude Dynamic Model. The control equationsfor the attitude motion of a rigid spacecraft can be establishedby using the unit quaternion. According to [23], the rotation

matrix R ∈ SOð3Þ and the unit quaternion Q = ½q0, qT�T ∈ Ξwith Ξ = fQ ∈ℝ ×ℝ3×3jq20 + qTq = 1g are applied in attitudeformulation. Furthermore, Qp and Qt represent the attitudeof the pursuer and the target, respectively. Consequently,the relative attitude between the pursuer and the target canbe expressed as follows:

~Q = ~q0, ~qT� �T =Q−1t ⊙Qp: ð1Þ

The relative attitude kinematics model [25] can beexpressed as follows:

_~q0 = −12~qv

T ~ω,

_~qv =12

~qv× + ~q0I3ð Þ~ω,

ð2Þ

where ~ω =ωp − ~Rωt is the relative angular velocity, with ωpand ωt denoting the angular velocity of the pursuer andtarget, respectively. For a vector a = ½a1, a2, a3�T , ax canbe defined using Eq. (3). The rotation matrix is definedin Eq. (4).

ax =

0 −a3 a2a3 0 −a1−a2 a1 0

26643775, ð3Þ

~R ≜ R ~qð Þ = ~q20 − ~qTv ~qv� �

I3 + 2~qv~qTv − 2~q0~qxv : ð4Þ

2 International Journal of Aerospace Engineering

According to the Euler-Newton formulas of the pur-suer and target [27], the corresponding attitude dynamicscan be expressed as follows:

Jt _ωt + ωt×Jtωt = 0, ð5Þ

J _ωp + ωp×Jωp = τ + τd , ð6Þwhere Jt ∈ R3×3 and J ∈ R3×3 denote the inertial matricesand τ ∈ R3 and τd ∈ R3 represent the control and externaldisturbance torques, respectively.

The derivative of ~ω satisfies the following expression:

_~ω = _ωp − _~Rωt − ~R _ωt: ð7Þ

Combining Eqs. (5)–(7) and considering that _~R = ~R~ω×,the following equation is obtained:

J _~ω = −Cr~ω − nr + τ + τd , ð8Þ

where Cr = Jð~RωtÞ× + ð~RωtÞ×J − ðJð~ω + ~RωtÞÞ× and nr =ð~RωtÞ×J~Rωt + J~R _ωt .2.2. Relative Orbit Dynamics Model. According to the theoryof relative motion, rp and νp are used to express the positionand velocity of the pursuer, as presented in Eqs. (9) and (10),respectively.

rp =~r + ~R rt + σtð Þ, ð9Þ

νp = ~ν + ~R νt + ω×t σtð Þ, ð10Þwhere rt and νt are the position and velocity of the target,respectively; ~r and ~ν are the relative position and relativevelocity, respectively; and the constant vector σt ∈ R3 denotesthe desired rendezvous position. The derivative of Eq. (9) canbe expressed as follows:

_rp = _~r + _~R rt + σtð Þ + ~R_rt: ð11Þ

According to the description of kinematic principles in[30], the following equations are obtained:

_rt = νt − ω×t rt , ð12Þ

_rp = νp − ω×prp: ð13ÞBy combining Eqs. (11) and (13), the following expres-

sion is obtained:

_~r + _~R rt + σtð Þ + ~R_rt = νp − ω×p rp: ð14Þ

From the aforementioned equation, the following expres-sion is obtained:

_~r = ~ν − Ct~r, ð15Þ

where Ct = ð~ω + ~RωtÞ×. Consequently, the derivative of νp isexpressed as follows:

_νp = _~ν + _~R νt + ω×t σtð Þ + ~R _νt + _ω×t σtð Þ: ð16Þ

According to the theory in [30], Eqs. (17) and (18) definethe position dynamics of spacecraft.

mt _νt +mtω×t νt = 0, ð17Þ

mp _νp +mpω×pνp = f + fd , ð18Þwhere mt and mp are constants that define the masses of thetarget and purser, respectively, and f ∈ RN and fd ∈ R3 denotethe control and external disturbance forces, respectively.From Eqs. (16) and (18), the following equation can beobtained [23]:

mp _~v + _~R vt + ω×t σtð Þ + ~R _vt + _ω×t σtð Þh i

+mpω×pνp = f + fd:

ð19Þ

Thus, the following equation is obtained:

mp _~v = −mpCt~v −mpnt + f + fd , ð20Þ

where nt = ð~RωtÞ×~Rvt + ~R _vt + ~ω×~Rσ×t ωt − ~Rσ×t _ωt .For convenience, the control torques can be expressed as

follows:

τ = εr , f = εt , ð21Þ

where εr ∈ RN and εt ∈ RN . By combining Eqs. (8), (20), and(21), the relative dynamics can be established as follows:

J _~ω = −Cr~ω − nr + εr + dr , ð22Þ

m _~ν = −mCt~ν −mnt + εt + dt , ð23Þwhere dr = τd and dt = fd .

During the controller design process, the exact motioninformation of the target spacecraft is assumed to be availableto the tracker spacecraft. In this paper, we focus on designingcontrollers for the dynamics expressed in Eqs. (21) and (22)to ensure the finite-time stability of the closed-loop systemeven in the presence of unknown inertial parameters andexternal disturbance.

To facilitate the controller design, the following assump-tions were made:

Assumption 1. The dynamics of the target are stable, whichimplies that ωt , _ωt , vt , and _vt are bounded and satisfy kωtk≤ a1, k _ωtk ≤ a2, and k _vtk ≤ a4, where a1, a2, a3, and a4 arepositive constants.

Assumption 2. The inertial matrix J and the mass of thepursuer m are unknown and satisfy λ1I3×3 ≤ J ≤ λ2I3×3 andm ≤ λ3, where λi ði = 1, 2, 3Þ are unknown positive constants.

3International Journal of Aerospace Engineering

Assumption 3. The external disturbance dr and dt areunknown bounded vectors, which satisfy kdrk ≤Dr < c1 andkdtk ≤Dt < c2, where c1 and c2 are positive constants.2.3. Preliminaries

Lemma 4 (see [24]). For an arbitrary real number x ∈ R,μ > 0, and κ = 0:2785, the following relation exists:

0 < xj j − x tanh μxð Þ ≤ κμ: ð24Þ

Lemma 5 (see [26]). For the system expressed in Eqs.(21) and (22), if the Lyapunov function V1 exists, itsatisfies the following expression:

_V1 ≤ −αVp1 + σ, ð25Þ

where α > 0, 0 < p < 1, and 0 < σ 0, i = 5, 6, 7, 8 and μi > 0, i = 1, 2, 3, 4. In theaforementioned equation, D̂r , D̂t , bα1, and bα2 are theestimations of Dr , Dt , α1, and α2, respectively. The

terms _̂Dr ,_̂Dt , bα1, and bα2 are defined as follows:

_̂Dr = c1 s1k k tanhs1k k3μ1

− c5D̂r

, ð31Þ

_̂Dt = c2 s2k k tanhs2k k3μ2

− c6D̂t

, ð32Þ

_bα1 = c3 ς1 s1k k tanh ς1 s1k k3μ3

− c7bα1 , ð33Þ_bα2 = c4 ς2 s2k k tanh ς2 s2k k3μ4

− c8bα2 , ð34Þ

where ci > 0, i = 1, 2,⋯, 8. The estimation errors aredefined as follows:

~Dr =Dr − D̂r ,_~Dt =Dt − D̂t ,

~α1 = α1 − bα1,~α2 = α2 − bα2:

ð35Þ

Theorem 6. For the dynamics expressed in Eqs. (22) and(23) with unknown system parameters J and m, thefinite-time stability of the system can be achieved usingthe controller proposed in Eqs. (29)–(34).

Proof. To prove the stability of the system, the Lyapunovfunction is designed as follows:

V1 =12sT1 Js1 +

12sT2ms2 +

12c1

~D2r +

12c2

~D2t +

12c3

α21 +12c4

α22:

ð36Þ

According to the relative system dynamics and controllaws, the following expressions are obtained:


_V1 = sT1 J_s1 + sT2m_s2 +1c1

~Dr_~Dr +

1c2

~Dt_~Dt +

1c3~α1 _~α1 +

1c4~α2 _~α2

= sT1 J_s1 + sT2m_s2 −1c1

~Dr_̂Dr −

1c2

~Dt_̂Dt −

1c3~α1

_bα1 − 1c4 ~α2 _bα2= sT1 −Cr~ω − nr + εr + dr

h+12J k1 + k2 1 + tanh

T ~qvð Þ tanh ~qvð Þ� �h i

~qv× + ~q0I3ð Þ~ωi

+ sT2 −mCt~ν −mnt + εt + dth

+m k3 + k4 1 + tanhT ~rð Þ tanh ~rð Þ

� �h i~ν − Ct~rð Þ

i−

1c1

~Dr_̂Dr −

1c2

~Dt_̂Dt −

1c3~α1

_bα1 − 1c4 ~α2 _bα2:ð37Þ

Then, the following equations are derived:

sT1 −Cr~ω − nr +12J k1 + k2 1 + tanh


~qv× + ~q0I3ð Þ~ω� �≤ α1ς1 s1k k,

ð38Þ

sT2 −mCt~ν −mnt +m k3 + k4 1 + tanhT ~rð Þ tanh ~rð Þ� �h i

~ν − Ct~rð Þh i

≤ α2ς2 s2k k:ð39Þ

The following equations are obtained from Eqs.(37)–(39):

_V1 ≤ α1ς1 s1k k + α2ς2 s2k k + sT1 εr + drð Þ + sT2 εt + dtð Þ−

1c1

~Dr_̂Dr −

1c2

~Dt_̂Dt −

1c3~α1

_bα1 − 1c4 ~α2 _bα2: ð40ÞSubstituting the equations of the proposed controllers

into Eq. (40), we obtain the following expression:

_V1 ≤ α1ς1 s1k k + α2ς2 s2k k + sT1dr + sT2dt+ sT1 −k5 tanh s1ð Þ − k6s1 − D̂r tanh

s1μ1

− bα1ς1 tanh ς1s1μ3

+ sT2 −k7 tanh s2ð Þð

− k8s2 − D̂t tanhs2μ2


−

1c1

~Dr_̂Dr −

1c2

~Dt_̂Dt −

1c3~α1

_bα1 − 1c4 ~α2 _bα2≤ α1ς1 s1k k + α2ς2 s2k k + s1k kDr + s2k kDt − k5 s1k k

− k6 s1k k2 − sT1 D̂r tanhs1μ1

− sT1 bα1ς1 tanh ς1s1μ3

− k7 s2k k − k8 s2k k2 − sT2 D̂t tanh

s2μ2

− sT2 bα2ς2 tanh ς2s2μ4

−

1c1

~Dr_̂Dr −

1c2

~Dt_̂Dt

−1c3~α1

_bα1 − 1c4 ~α2 _bα2 + 3κ k5 + k7ð Þ:

ð41Þ

From Lemma 4, we obtain the following condition:

−D̂rsT1 tanhs1μ1

≤ − s1k kD̂r + 3D̂rμ1κ,

−D̂tsT2 tanhs2μ2

≤ − s2k kD̂t + 3D̂tμ2κ,

−ς1bα1sT1 tanh ς1s1μ3

≤ −ς1bα1 s1k k + 3bα1μ3κ,−ς2bα2sT2 tanh ς2s2μ4

≤ −ς2bα2 s2k k + 3bα2μ4κ:

ð42Þ

Consequently, Eq. (41) can be rewritten as follows:

_V1 ≤ α1ς1 s1k k + α2ς2 s2k k + s1k kDr + s2k kDt − k5 s1k k− k6 s1k k2 − s1k kD̂r + 3D̂rμ1κ − bα1ς1 s1k k + 3bα1ς1μ3κ− k7 s2k k − k8 s2k k2 − s2k kD̂t + 3D̂tμ2κ − bα2ς2 s2k k+ 3bα2ς2μ4κ − 1c1 ~Dr _̂Dr − 1c2 ~Dt _̂Dt − 1c3 ~α1 _bα1 − 1c4 ~α2 _bα2+ 3κ k5 + k7ð Þ = ~α1ς1 s1k k + ~α2ς2 s2k k + s1k k~Dr+ s2k k~Dt − k5 s1k k − k6 s1k k2 − k7 s2k k − k8 s2k k2

−1c1

~Dr_̂Dr −

1c2

~Dt_̂Dt −

1c3~α1

_bα1 − 1c4 ~α2 _bα2+ 3κ k5 + k7 + D̂rμ1 + D̂tμ2 + bα1ς1μ3 + bα2ς2μ4� �:

ð43Þ

Combining the control laws defined in Eqs. (31)–(34),the following expression can be obtained:

_V1 ≤ ~α1ς1 s1k k + ~α2ς2 s2k k + s1k k~Dr + s2k k~Dt − k6 s1k k2

− k8 s2k k2 − ~Dr s1k k tanhs1k k3μ1

− c5D̂r

− ~Dt s2k k tanh

s2k k3μ2

− c6D̂t

− ~α1 ς1 s1k k tanh

ς1 s1k k3μ3

− c7bα1

− ~α2 ς2 s2k k tanhς2 s2k k3μ4

− c8bα2

+ 3κ k5 + k7 + D̂rμ1 + D̂tμ2 + bα1ς1μ3 + bα2ς2μ4� �= ~α1ς1 s1k k + ~α2ς2 s2k k + s1k k~Dr + s2k k~Dt − k6 s1k k2

− k8 s2k k2 − ~Dr s1k k tanhs1k k3μ1

− ~Dt s2k k tanh

s2k k3μ2

+ c5~DrD̂r + c6~DtD̂t

− ~α1ς1 s1k k tanhς1 s1k k3μ3

− ~α2ς2 s2k k tanh

ς2 s2k k3μ4

+ c7~α1bα1 + c8~α2bα2 + 3κ k5 + k7ð Þ + κD̂rμ1 + κD̂tμ2+ 3κbα1ς1μ3 + 3κbα2ς2μ4 ≤ ~α1ς1 s1k k + ~α2ς2 s2k k+ s1k k~Dr + s2k k~Dt − k6 s1k k2 − k8 s2k k2 − ~Dr s1k k+ 3~Drμ1κ − ~Dt s2k k + 3~Dtμ2κ − ~α1ς1 s1k k + 3~α1ς1μ3κ− ~α2ς2 s2k k + 3~α2ς2μ4κ + c5~DrD̂r + c6~DtD̂t + c7~α1bα1+ c8~α2bα2 + 3κ k5 + k7 + D̂rμ1 + D̂tμ2 + bα1ς1μ3 + bα2ς2μ4� �

≤ −k6 s1k k2 − k8 s2k k2 + c5~DrD̂r + c6~DtD̂t + c7~α1bα1+ c8~α2bα2 + 3κ k5 + k7 +Drμ1 +Dtμ2 + α1ς1μ3 + α2ς2μ4ð Þ

≤ −k6 s1k k2 − k8 s2k k2 − Dr − D̂r� �2 − Dt − D̂t� �2

− α1 − bα1ð Þ2 − α2 − bα2ð Þ2 + Δ1,

ð44Þ


where Δ1 = ðc5D2r /4ðc5 − 1ÞÞ + ðc6D2t /4ðc6 − 1ÞÞ + ðc7α21/4ðc7− 1ÞÞ + ðc8α22/4ðc8 − 1ÞÞ + 3κðk5 + k7 +Drμ1 +Dtμ2 + α1ς1μ3+ α2ς2μ4Þ:.

Equation (44) can be further rewritten as follows:

_V1 ≤ −2k6

λmax Jð Þ12sT1 Js1

−2k8m

12sT2ms2

− 2c1

12c1

~D2r

− 2c2

12c2

~D2t

− 2c3

12c3

α21

− 2c4

12c4

α22

+ Δ1

≤ −ρ1V1 + Δ1,ð45Þ

where ρ1 = min fð2k6/λmaxðJÞÞ, ð2k8/mÞ, 2c1, 2c2, 2c3, 2c4g.Consequently, according to the aforementioned equa-

tion, we conclude that s1, s2, ~Dr , ~Dt , ~α1, and ~α2 exponentiallyconverge to a bounded region with respect to Δ1. Then, thepositive constants �Dr , �Dt , �α1, and �α2 that satisfy the �Dr ≥Dr ,�Dr ≥ D̂r , �Dt ≥Dt , �Dt ≥ D̂t and �α1 ≥ α1, �α1 ≥ bα1, �α2 ≥ α2, �α2 ≥bα2 conditions must exist.

To illustrate the finite-time stability, the Lyapunov func-tion is defined as follows:

V2 =12sT1 Js1 +

12sT2ms2 +

1c1

�Dr − D̂r� �2 + 1

c2�Dt − D̂t

� �2+

1c3

�α1 − bα1ð Þ2 + 1c4 �α2 − bα2ð Þ2:ð46Þ

Combining the aforementioned schemes, the derivativeof V2 satisfies the following expression:

_V2 = sT1 J_s1 + sT2m_s2 −2c1

�Dr − D̂r� � _̂Dr

−2c2

�Dt − D̂t� � _̂Dt − 2c3 �α1 − bα1ð Þ _bα1

−2c4

�α2 − bα2ð Þ _bα2= sT1 −Cr~ω − nr + εr + dr½

+12J k1 + k2 1 + tanh


� ~qv× + ~q0I3ð Þ~ωi+ sT2 −mCt~ν −mnt + εt + dt

h+m k3 + k4 1 + tanh

T ~rð Þ tanh ~rð Þ� �h i

~ν − Ct~rð Þi

−2c1

�Dr − D̂r� � _̂Dr − 2c2 �Dt − D̂t� � _̂Dt

−2c3

�α1 − bα1ð Þ _bα1 − 2c4 �α2 − bα2ð Þ _bα2≤ α1ς1 s1k k + α2ς2 s2k k + sT1 εr + drð Þ

+ sT2 εt + dtð Þ −2c1

�Dr − D̂r� � _̂Dr

−2c2

�Dt − D̂t� � _̂Dt − 2c3 �α1 − bα1ð Þ _bα1

−2c4

�α2 − bα2ð Þ _bα2:

ð47Þ

By using the proposed control scheme, the followingexpression can be obtained:

_V2 ≤ α1ς1 s1k k + α2ς2 s2k k + s1k kDr + s2k kDt+ sT1 −k5 tanh s1ð Þ − k6s1 − D̂r tanh

s1μ1


+ sT2 −k7 tanh s2ð Þ − k8s2ð

− D̂t tanhs2μ2


−

2c1

�Dr − D̂r� � _̂Dr − 2c2 �Dt − D̂t� � _̂Dt

−2c3

�α1 − bα1ð Þ _bα1 − 2c4 �α2 − bα2ð Þ _bα2≤ α1ς1 s1k k + α2ς2 s2k k + s1k kDr + s2k kDt

− k5 s1k k − k6 s1k k2 − D̂rsT1 tanhs1μ1

− bα1ς1sT1 tanh ς1s1μ3

− k7 s2k k − k8 s2k k2

− D̂tsT2 tanhs2μ2

− bα2ς2sT2 tanh ς2s2μ4

−

2c1

�Dr − D̂r� � _̂Dr − 2c2 �Dt − D̂t� � _̂Dt

−2c3

�α1 − bα1ð Þ _bα1 − 2c4 �α2 − bα2ð Þ _bα2 + 3κ k5 + k7ð Þ≤ α1ς1 s1k k + α2ς2 s2k k + s1k kDr + s2k kDt − k5 s1k k

− k6 s1k k2 − D̂r s1k k + 3D̂rμ1κ − bα1ς1 s1k k+ 3bα1ς1μ3κ − k7 s2k k − k8 s2k k2 − D̂t s2k k+ 3D̂tμ2κ − bα2ς2 s2k k + 3bα2ς2μ4κ−

2c1

�Dr − D̂r� � _̂Dr − 2c2 �Dt − D̂t� � _̂Dt

−2c3

�α1 − bα1ð Þ _bα1 − 2c4 �α2 − bα2ð Þ _bα2 + 3κ k5 + k7ð Þ≤ −k5 s1k k − k7 s2k k + ς1 s1k k �α1 − bα1ð Þ

+ ς2 s2k k �α2 − bα2ð Þ + s1k k �Dr − D̂r� �+ s2k k �Dt − D̂t

� �−

2c1

�Dr − D̂r� � _̂Dr

−2c2

�Dt − D̂t� � _̂Dt − 2c3 �α1 − bα1ð Þ _bα1

−2c4

�α2 − bα2ð Þ _bα2+ 3κ k5 + k7 + D̂rμ1 + D̂tμ2 + bα1ς1μ3 + bα2ς2μ4� �:

ð48Þ

According to the designed adaptive laws, the followingexpression can be obtained:


_V2 ≤ −k5 s1k k − k7 s2k k + ς1 s1k k �α1 − bα1ð Þ + ς2 s2k k �α2 − bα2ð Þ+ s1k k �Dr − D̂r

� �+ s2k k �Dt − D̂t

� �− 2 �Dr − D̂r

� �s1k k tanh

s1k k3μ1

− c5D̂r

− 2 �Dt − D̂t

� �s2k k tanh

s2k k3μ2

− c6D̂t

− 2 �α1 − bα1ð Þ ς1 s1k k tanh ς1 s1k k3μ3

− c7bα1

− 2 �α2 − bα2ð Þ ς2 s2k k tanh ς2 s2k k3μ4

− c8bα2

+ 3κ k5 + k7 + D̂rμ1 + D̂tμ2 + bα1ς1μ3 + bα2ς2μ4� �

= −k5 s1k k − k7 s2k k + ς1 s1k k �α1 − bα1ð Þ + ς2 s2k k �α2 − bα2ð Þ+ s1k k �Dr − D̂r

� �+ s2k k �Dt − D̂t

� �− 2 �Dr − D̂r

� �s1k k tanh

s1k k3μ1

− 2 �Dt − D̂t

� �s2k k tanh

s2k k3μ2

− 2 �α1 − bα1ð Þς1 s1k k tanh ς1 s1k k3μ3

− 2 �α2 − bα2ð Þς2 s2k k tanh ς2 s2k k3μ4

+ 2c5D̂r �Dr − D̂r

� �+ 2c6D̂t �Dt − D̂t

� �+ 2c7bα1 �α1 − bα1ð Þ + 2c8bα2 �α2 − bα2ð Þ+ 3κ k5 + k7 + D̂rμ1 + D̂tμ2 + bα1ς1μ3 + bα2ς2μ4� �:

ð49Þ

According to Lemma 4, the following expression can beobtained:

_V2 ≤ −k5 s1k k − k7 s2k k + ς1 s1k k �α1 − bα1ð Þ+ ς2 s2k k �α2 − bα2ð Þ + s1k k �Dr − D̂r� � + s2k k �Dt − D̂t� �− 2 �Dr − D̂r

� �s1k k + 6μ1κ �Dr − D̂r

� �− 2 �Dt − D̂t

� �s2k k

+ 6μ2κ �Dt − D̂t� �

− 2 �α1 − bα1ð Þς1 s1k k + 6μ3ς1κ �α1 − bα1ð Þ− 2 �α2 − bα2ð Þς2 s2k k + 6μ4ς2κ �α2 − bα2ð Þ + 2c5D̂r �Dr − D̂r� �+ 2c6D̂t �Dt − D̂t

� �+ 2c7bα1 �α1 − bα1ð Þ + 2c8bα2 �α2 − bα2ð Þ

+ 3κ k5 + k7 + D̂rμ1 + D̂tμ2 + bα1ς1μ3 + bα2ς2μ4� �= −k5 s1k k − k7 s2k k − s1k k − 3μ1κð Þ �Dr − D̂r

� �− s2k k − 3μ2κð Þ �Dt − D̂t

� �− ς1 s1k k − 3μ3ς1κð Þ �α1 − bα1ð Þ

− ς2 s2k k − 3μ4ς2κð Þ �α2 − bα2ð Þ + 2c5D̂r �Dr − D̂r� �+ 2c6D̂t �Dt − D̂t

� �+ 2c7bα1 �α1 − bα1ð Þ + 2c8bα2 �α2 − bα2ð Þ

+ 3κ k5 + k7 + �Drμ1 + �Dtμ2 + �α1ς1μ3 + �α2ς2μ4� �

= −k5 s1k k − k7 s2k k − s1k k − 3μ1κð Þ �Dr − D̂r� �

− s2k k − 3μ2κð Þ �Dt − D̂t� �

− ς1 s1k k − 3μ3ς1κð Þ �α1 − bα1ð Þ− ς2 s2k k − 3μ4ς2κð Þ �α2 − bα2ð Þ + Δ2,

ð50Þ

w h e r e Δ2 = ðc5/2Þ�D2r + ðc6/2Þ�D2t + ðc7/2Þ�α21 + ðc8/2Þ�α22 + 3κðk5 + k7 + �Drμ1 + �Dtμ2 + �α1ς1μ3 + �α2ς2μ4Þ .

Equation (50) can be rewritten as follows:

_V2 ≤ −k5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

λmax Jð Þ

s12sT1 Js1

1/2− k7

ffiffiffiffi2m

r12sT2ms2

1/2−

ffiffiffiffic1

p s1k k − 3μ1κð Þ1c1

�Dr − D̂r� �2 1/2

−ffiffiffiffic2

p s2k k − 3μ2κð Þ1c2

�Dt − D̂t� �2 1/2

−ffiffiffiffic3

pς1 s1k k − 3μ3ς1κð Þ

1c3

�α1 − bα1ð Þ2 1/2−

ffiffiffiffic4

pς2 s2k k − 3μ4ς2κð Þ

1c4

�α2 − bα2ð Þ2 1/2 + Δ2≤ −ρ2V1/22 + Δ2,

ð51Þ

where ρ2 = min fk5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2/λmaxðJÞ

p, k7

ffiffiffiffiffiffiffiffi2/m

p, ffiffiffiffic1p ðks1k − 3μ1κÞ,ffiffiffiffi

c2p ðks2k − 3μ2κÞ,

ffiffiffiffic3

p ðς1ks1k − 3μ3ς1κÞ,ffiffiffiffic4

p ðς2ks2k − 3μ4ς2κÞg.

Consequently, s1 and s2 converge to two small regionsΘ1and Θ2 in finite time, where Θ1 and Θ2 are positive con-stants. The following equations can be obtained from the def-inition of s1 and s2:

~ωi + k1~qvi + k2 tanh ~qvið Þ = ζi, ζij j ≤Θ1 i = 1, 2, 3,~vi + k3~ri + k4 tanh ~rið Þ = ϑi, ϑij j ≤Θ2 i = 1, 2, 3:

ð52Þ

Thus, the following expressions can be obtained:

~ωi + k1 −ζi2~qvi

~qvi + k2 −

ζi2 tanh ~qvið Þ

tanh ~qvið Þ = 0

i = 1, 2, 3,

~vi + k3 −ϑi2~ri

~ri + k4 −

ϑi2 tanh ~rið Þ

tanh ~rið Þ = 0

i = 1, 2, 3:ð53Þ

Consequently, the relative dynamic system is stabilized infinite time under the conditions k1 − ζi/2~qvi > 0, k2 − ζi/2tanh ð~qviÞ > 0, and k3 − ϑi/2~ri > 0, k4 − ϑi/2 tanh ð~riÞ > 0,which implies that the errors ~qvi and ~r converge to the follow-ing region in finite time:

Δ~qv =maxζi2k1

,12ln

k2 − ζik2

−12

,

Δ~r =maxϑi2k3

,12ln

k4 − ϑik4

−12

:

ð54Þ

Furthermore, the relative angular velocity ~ωi and relativevelocity ~vi converge to the following regions in finite time:


~ωij j ≤ ζij j + k1i ~qvij j + k2 tanh ~qvið Þj j≤Θ1 + k1Δ~qv + k2 tanh Δ~qv

� �=Θqv ,

~vij j ≤ ϑij j + k3i ~rij j + k4i tanh ~rið Þj j≤Θ2 + �k3Δ~r + �k4 tanh Δ~rð Þ =Θr:

ð55Þ

Therefore, Theorem 6 is proved.

4. Simulation Results

In this section, the effectiveness and advantage of the devel-oped control strategy are presented. Detailed informationregarding the orbit and these two spacecrafts is presented asfollows [23]. In the simulation scenarios, a pursuer spacecraftis forced to rendezvous with the target spacecraft in anelliptical orbit with a perigee altitude rpa = 400km and

15

10

5

0

–5

–10

Rela

tive p

ositi

on/(

m)

0 20 40 60t/s

80 100 120 140 150

0.150.1

0.050

–0.05–0.1

100 110 120 130 140 150

i = 1i = 2i = 3

Figure 1: Relative position.

0 20 40 60t/s

80 100 120 140 150–3.5

–3

–2.5

–2

–1.5

–0.5

–1

Rela

tive v

eloc

ity/(

m/s

) 0

0.5

1

1.5

100 110 120 130 140 150

–3–4

–2

–1

0

1

23 × 10

–3

i = 1i = 2i = 3

Figure 2: Relative velocity.


eccentricity e = 0:3. The gravitational constant is μ =3:986 × 1014Nm2/kg and the radius of Earth is RE =6371km. The mass and initial matrix of the purserspacecraft and the target spacecraft are chosen as m =90kg, J = diag ð20, 20, 15Þkg ⋅m2 and mt = 150kg, Jt = diagð26, 16, 21Þkg ⋅m2, respectively. The initial true anomalyis vð0Þ = 10°.

The target spacecraft is supposed to service with the fol-lowing position:

rt = rt , 0, 0½ �T, rt =a 1 − e2� �

1 + e cos v, ð56Þ

0–15

–10

–5Eule

r ang

le er

ror/

(deg

ree)

0

5

10

15

20

20 40 60t/s

80 100 120 140 150

0.03

0.02

0.01

0100 110 120 130 140 150

i = 1i = 2i = 3

Figure 3: Relative Euler angle.

0 20 40 60 80 100 120 140 150t/s

–0.2

–0.15

–0.1

–0.05

0

0.05

0.1

0.15

Relat

ive a

ngul

er v

eloc

ity/(

rad/

s)

100 110 120 130 140 150

–14–12–10

–8–6–4

10–6

i = 1i = 2i = 3

Figure 4: Relative angular velocity.


where a = RE + ðrpa/1 − eÞ denotes the semimajor axis. Theparameter v is expressed as follows:

_v =n 1 + e cos vð Þ2

1 − e2ð Þ3/2, €v =

2n2e 1 + e cos vð Þ3 sin v1 − e2ð Þ3

, ð57Þ

where n =ffiffiffiffiffiffiffiffiffiu/a3

p. For the pursuer spacecraft, its rendezvous

position is expressed as δt = ½0, 5, 0�T in the body coordinateframe of the target. The angular velocity of the target andthe external disturbances are expressed as follows:

τd = 0:02 × 1 + cosπ

150t

� �+ sin

π

150t

� �� 1 ; 1 ; 1½ �TN ⋅m,

fd = 0:05 × 1 + cosπ

150t

� �+ sin

π

150t

� �� 1 ; 1 ; 1½ �TN ⋅m:

ð58Þ

The initial states are set as follows: the initial Euler angleerror is Θð0Þ = ½19:9984 − 9:9987 15:0050�T deg, ~ω = ½0 0 0�T,~rð0Þ = ½10, 10,−10�T, and ~vð0Þ = ½0 0 0�T.

The design parameters are set as follows: k1 = 2, k2 = 2,k3 = 0:5, k4 = 0:1, k5 = 0:05, k6 = 4, k7 = 10, k8 = 10, c1 = 0:02,c2 = 0:001, c3 = 0:05, c4 = 0:001, c5 = 1:1, c6 = 5, c7 = 1:1, and

0 50 100 150 200t/s

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Estim

ated

par

amet

ers

Figure 5: Estimation of βr .

0 50 100 150 200t/s

0

0.5

1

1.5

2

2.5

Esti

mat

ed p

aram

eter

s

Figure 6: Estimation of βt .


0 50 100 150t/s

0

0.005

0.01

0.015

0.02

0.025

0.03

Esti

mat

ed p

aram

eter

Figure 7: Estimation of Dr .

0 50 100 150t/s

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Esti

mat

ed p

aram

eter

Figure 8: Estimation of Dt .

0 50 100 150t/s

–3

–2.5

–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

Con

trol t

orqu

e/(N

⁎m

)

110100 120 130 140 150–4

–3

–2

–1

0

1 10–3

i = 1i = 2i = 3

Figure 9: Control torque τr .


c8 = 5. The simulation results are illustrated in Figures 1–10.Figures 1–4 indicate that the relative attitude dynamicscan be stabilized within 40 s and that the finite-time stabil-ity for the relative orbit dynamics can be ensured in 50 s.Figures 5–8 present the estimated valuables, which havean upper bound under the proposed adaptive laws.Figures 9 and 10 display the control torques of the devel-oped controllers. These simulation results indicate that thecontrol objective can be achieved with satisfactory perfor-mance and that the chattering phenomenon does not occurin the controllers.

5. Conclusion

In this study, we focus on the finite-time model-free trackingcontrol problem for the spacecraft rendezvous maneuverunder external disturbances. By resorting to a well-definedsliding mode surface, finite-time convergence of the trackingerrors is ensured, and all closed-loop signals are upperbounded. Adaptive laws are embedded into the controlschemes such that external disturbances and system uncer-tainties could be compensated. Moreover, the chatteringissue inherent in sliding mode technologies could be tackledbased on the hyperbolic tangent function. Upon utilizing theproposed method, rendezvous maneuver could be accom-plished with satisfactory performance even when systemparameters remain inaccessible to designers. Theoreticalanalysis and simulation results have been presented to illus-trate the effectiveness of the proposed method.

Data Availability

No data were used to support this study.

Conflicts of Interest

Authors declare that there are no conflicts of interest regard-ing publication of this article.

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–60

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Adaptive Finite-Time Control for Spacecraft Rendezvous under Unknown System Parameters1. Introduction2. Spacecraft Model and Preliminaries2.1. Relative Attitude Dynamic Model2.2. Relative Orbit Dynamics Model2.3. Preliminaries

3. Attitude Controller Design3.1. Basic Controller Design

4. Simulation Results5. ConclusionData AvailabilityConflicts of Interest

adaptive finite-time control for spacecraft rendezvous ...research article adaptive finite-time...

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