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Reliable spacecraft rendezvous without velocity measurement
Acta Astronautica 144 (2018) 52–60
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Reliable spacecraft rendezvous without velocity measurement Shaoming He *, Defu Lin School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, People's Republic of China
A R T I C L E I N F O
A B S T R A C T
Keywords: Spacecraft rendezvous Velocity-free Finite-time convergence External disturbances Actuator faults
This paper investigates the problem of finite-time velocity-free autonomous rendezvous for spacecraft in the presence of external disturbances during the terminal phase. First of all, to address the problem of lack of relative velocity measurement, a robust observer is proposed to estimate the unknown relative velocity information in a finite time. It is shown that the effect of external disturbances on the estimation precision can be suppressed to a relatively low level. With the reconstructed velocity information, a finite-time output feedback control law is then formulated to stabilize the rendezvous system. Theoretical analysis and rigorous proof show that the relative position and its rate can converge to a small compacted region in finite time. Numerical simulations are performed to evaluate the performance of the proposed approach in the presence of external disturbances and actuator faults.
1. Introduction Owing to the stringent performance requirement, guidance law design for spacecraft rendezvous is still an active research area and a benchmark problem for engineers [1–6]. Spacecraft rendezvous mission refers to the trajectory maneuver command that guides a chaser spacecraft to arrive at the target spacecraft with the same velocity. Such mission is typically divided into many phases, including far-range closing, close-range closing, and final mating [7]. This paper mainly focuses on guidance scheme design for the close-range phase, where the main concern of this phase is the rendezvous precision instead of the energy consumption. Since the chaser spacecraft is moving in the vicinity of the target vehicle during the close-range rendezvous, the relative motion between these two vehicles can be described by the well-known Clohessy-Wiltshire (C-W) differential equations [8] or Tschauner-Hempel (T-H) differential equations [9,10]. The difference between these two models lies in that the C-W equations are for circular target orbit while the T-H equations are for elliptical target orbit. This paper mainly focuses on the elliptical orbit rendezvous problem. With the development of modern control theory, many elegant guidance schemes have been proposed in recent years to achieve autonomous rendezvous. To cite a few, the authors in Ref. [11] proposed a robust guidance law for circular orbital rendezvous based on multi-objective H∞ control approach. This controller can be easily obtained by a convex optimization problem with linear matrix inequality constraints, but it only considers the state-feedback control scheme. To
* Corresponding author. E-mail addresses: [email protected] (S. He), [email protected] (D. Lin). https://doi.org/10.1016/j.actaastro.2017.12.016 Received 3 July 2017; Received in revised form 29 August 2017; Accepted 11 December 2017 Available online 14 December 2017 0094-5765/© 2017 IAA. Published by Elsevier Ltd. All rights reserved.
improve the overall guidance performance, the parametric Lyapunov differential equation approach was also a powerful tool that can be used to design rendezvous laws [12,13] for uncertainty-free rendezvous system. Compared to the rendezvous guidance laws based on the quadratic regulation theory, the parametric Lyapunov approach is easy to implement. To cope with uncertainties in the plant parameters and reduce the computational burden for onboard implementation, a passivity-based direct adaptive spacecraft proximity operational strategy was derived upon the simple adaptive control methodology in Ref. [14]. Under the framework of integral sliding mode control technique, the authors in Ref. [15] developed a robust control law for spacecraft formation flying, where the equivalent part was based on linear quadratic optimal control method. By combining backstepping methodology with sliding mode approach, the authors in Ref. [16] presented a new guidance law for uncertain spacecraft rendezvous system. Although the sliding mode rendezvous guidance law is robust to external perturbations, the inherent chattering problem limits its practical applications. To eliminate the reaching phase and guarantee global sliding manifold, an adaptive robust rendezvous guidance law was proposed in Ref. [17] based on the time-varying sliding mode control approach with a complicated sliding surface design. Integrating extended state observer with nonsingular terminal sliding mode, the authors in Ref. [18] proposed a finite-time convergence rendezvous law with full state feedback. The authors in Ref. [19] considered the application of State-Dependent Riccati Equation (SDRE) for coupled orbital and attitude relative motion of formation flying or rendezvous. The SDRE controller can provide optimal control performance in terms of a meaningful index for the spacecraft formation
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Acta Astronautica 144 (2018) 52–60
system, but it is sensitive to external disturbances. As model predictive control (MPC) is one of the promising nonlinear techniques for regulating nonlinear systems while considering multiple constraints, the authors in Refs. [20,21] considers the application of MPC methodology to rendezvous control law design. Despite its advantages, the MPC method usually requires high computational load and thus is difficult for the implementation in the onboard embed system. It should be noted that most of the above mentioned rendezvous laws are full state feedback control laws. However, the relative velocity measurement is always an issue for sensor-less low-cost space vehicles. Considering this, output feedback design using only position measurement is more desirable for spacecraft rendezvous from the viewpoint of real application. The typical solution to the output feedback for spacecraft applications includes the design of an ad hoc dynamic filter, i.e., Kalman filter for example, driven by the output only [22,23]. However, in these works, no external disturbances were considered for the filter design, although the perturbations are inevitable in real applications. To this end, the authors in Ref. [24] proposed an output feedback rendezvous law based on high-gain observer and input-to-state stability to suppress the effect of the external disturbances to a specified level. However, only asymptotical stability is considered in this reference, which means that the convergence of the relative range and its rate is exponential. Obviously, the control law with infinite settling time is not desirable for the critical high-value rendezvous missions, where the control precision, especially during the final close-in phase, is of great importance for many space operations. It is well known that finite-time convergence ensures better disturbance rejection performance and smaller steady-state tracking errors than asymptotically stable systems [25,26]. Therefore, it is more desirable to design finite-time velocity-free rendezvous control laws. Motivated by the above analysis, this paper investigates the problem of designing a robust finite-time convergence guidance law for elliptical orbital spacecraft rendezvous using only relative position measurements. Comparison results show that the proposed law has better disturbance rejection performance and smaller steady-state tracking errors than the asymptotical convergence law. The key features and contributions of the proposed approach are summarized as follows.
Fig. 1. Target-orbital rotating coordinate system.
target spacecraft. R and r represent the vector from the center planet to the target spacecraft and the vector from the target spacecraft to the chaser spacecraft, respectively. As this paper considers elliptical orbit rendezvous problem, the relative motion kinematics of the chaser with reference to the target spacecraft can be formulated as [27]. d2 r Rþr R ¼ μ d2 t jjR þ rjj3 jjRjj3
! þuþw
(1)
where μ represents the gravity constant, u the controlled acceleration input vector and w the lumped disturbance. Assumption 1. The unknown external disturbances, in space, including environmental disturbance, solar radiation and magnetic effect, are all bounded in reality. With this mind, the norm of the lumped disturbance satisfies jjwjj δ, where δ > 0. For simplicity, defining the notations as r ¼ ½x; y; zT , u ¼ ½ux ; uy ; uz T and w ¼ ½wx ; wy ; wz T , then, system (1) can be written as 8 μx > x€ ¼ ω2 x þ 2ωz_ þ ω_ z þ ux þ wx > > > þ rjj3 jjR > > > > > μy < y€ ¼ þ uy þ wy kR þ rk3 > > ! > > > > zR 1 > 2 > _ _ x z € ¼ ω z 2 ω ω x μ þ þ uz þ wz > 3 : R2 kR þ rk
(1) Using homogeneous and Lyapunov theories, a robust observer is proposed to precisely estimate the relative velocity between two neighboring spacecraft despite external disturbances in finite time. (2) With the reconstructed velocity information, a novel output feedback rendezvous law is then synthesized to regulate the relative distance and its rate to a small compact region in finite time. Moreover, analysis also proves that the proposed rendezvous law is robust against actuator faults and thus is a reliable control approach.
(2)
where ω denotes the angular rate of the target orbit, which is governed by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μð1 þ e cos θÞ ω¼ R3
The rest of the paper is organized as follows. In Sec. 2, some backgrounds and preliminaries are stated, followed by the proposed output feedback rendezvous law derived in Sec. 3. Finally, some simulation results and conclusion remarks are offered.
(3)
where e 2 ½0; 1Þ represents the eccentricity of the orbit, and θ stands for the true anomaly. During the close-range closing phase, the distance between the chaser and the target spacecraft r is much smaller than the distance between the planet center and the target R, i.e. r≪R. With this fact in mind, system (2) can be further simplified as [27].
2. Backgrounds and preliminaries
8 μx 2 > > > x€ ¼ ω x þ 2ωz_ þ ω_ z R3 þ ux þ wx > > > < μy y€ ¼ 3 þ uy þ wy R > > > > > > : z€ ¼ ω2 z 2ωx_ ω_ x þ 2μz þ uz þ wz R3
2.1. Problem formulation This work assumes the chaser spacecraft is equipped with a highperformance low-level attitude control system that provides attitude stabilization and maneuver tracking, i.e., there is no time delay in tracking the attitude command provided by the guidance loop. This study aims to design the guidance input to this low-level controller to achieve autonomous rendezvous. Consider the target-orbital rotating coordinate system x y z, as shown in Fig. 1, where the origin is fixed at the center of mass of the
(4)
In order to determine the target orbit, the following two augmented equations are required [28].
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Acta Astronautica 144 (2018) 52–60
μ R€ ¼ Rω2 2 R
(5)
R_ ω_ ¼ 2 ω R
(6)
2 6 6 6 f ðq; vÞ ¼ 6 6 6 4
The control interest here is to design a rendezvous law using the relative position information only to drive arbitrary initial states of system (4) to a small compact region in finite time.
ω2 x þ 2ωz_ þ ω_ z μy 3 R ω2 z 2ωx_ ω_ x þ
μx 3 R3 7 7 7 7 7 7 2μz 5 R3
b, b Denote q v as the estimated information of q, v, respectively. The proposed observer for relative velocity estimation is formulated as
2.2. Some fundamental facts
~ Þα1 b_ ¼ b q v þ ϑβ1 sigðq _b ~Þα2 v ¼ f ðq; b v Þ þ u þ ϑ2 β2 sigðq
Notations. Throughout the paper the following notations will be used. Let 0m and I m denote the m m zero matrix and m m identity matrix, respectively. For any positive definite matrix P, let λmax ðPÞ and λmin ðPÞ be the maximum and minimum eigenvalues of P, respectively.
~ ¼qq b , ϑ, β1 , β2 , α1 2 ð0:5; 1Þ and α2 ¼ 2α1 1 are positive where q design parameters. ~ and v~ ¼ v b Let the observer estimation errors be q v . Then, it follows from (12) and (13) that
For a given vector x ¼ ½x1 ; x2 ; …; xn T 2 ℝn and a given constant a 2 ℝ,
x_ ¼ f ðxÞ;
xa ¼ ½x1a ; x2a ; …; xna T ,
α1 α2 e e v_ e ¼ f ðq; vÞ f q; b v ϑ2 β2 sig qe þw q_ e ¼ v ϑβ1 sig q
sigðxÞa ¼ ½sigðx1 Þa ; sigðx2 Þa ; … pffiffiffiffiffiffiffiffiffi a T ; sigðxn Þ , sgnðxÞ ¼ ½sgnðx1 Þ; sgnðx2 Þ; …; sgnðxn ÞT and jjxjj ¼ xT x. Lemma 1. [29] Consider the following system define
f ð0Þ ¼ 0;
x⊂ℝn ;
and
xð0Þ ¼ x0
(14) Consider the following coordinate transformation
(7)
ε_ 2 ¼ ϑα2 β2 sigðε1 Þα2 þ
!c n X c xi j i¼1
v~ ϑ2
(15)
f ðq; vÞ f ðq; b vÞ þ w ϑ2
(16)
Define A as
(8)
A¼
β1 I 3 β2 I 3
I3 03
(17)
It is easy to verify that the matrix A is Hurwitz. Therefore, there exists a unique matrix P ¼ PT > 0 such that AT P þ PA ¼ Q for any given matrix Q ¼ QT > 0. Theorem 1. For the estimation error system (14) under Assumptions 1, ~; v~Þ can if the observer is designed as (13), the estimation errors ðq converge to a nonzero bounded residual around zero in finite time.
(9)
Lemma 3. [31] For any xi 2 ℝ, i ¼ 1; 2; …; n and a real number c 2 ð0; 1, the following inequality holds !c n n X X c xi xi j n1c i¼1 i¼1
ε2 ¼
ε_ 1 ¼ ϑε2 ϑα1 β1 sigðq~ Þα1
where Vðx0 Þ is the initial value of VðxÞ. Lemma 2. [30] For any x 2 ℝ, y 2 ℝ and c > 0, d > 0, the following inequality holds c d xj yj c xjcþd þ d yjcþd cþd c þ d
~ q ϑ
ε1 ¼ ;
Then, the error system (14) can be rewritten as
domain of attraction of the origin and the settling time Treach satisfies Treach
T
ε ¼ εT1 ; εT2 ;
where f : U→ℝn is continuous on an open neighborhood U of the origin x ¼ 0. Suppose there exists a Lyapunov function VðxÞ, defined on a neigh_ borhood N⊂ℝn of the origin, and VðxÞ l½VðxÞa þ kVðxÞ; 8x 2 N\f0g, þ where a 2 ð0; 1Þ, l; k 2 ℝ . Then, the origin of system (7) is locally finite time stable. The set Ω ¼ fx½VðxÞ1a < l=kg \ N is contained in the 1 k ln 1 ½Vðx0 Þ1a ka k l
(13)
Proof. For system (16), if the term ðf ðq; vÞ f ðq; b v Þ þ wÞ=ϑ2 is omitted, then system (16) reduces to
(10)
ε_ 1 ¼ ϑε2 ϑα1 β1 sigðq~ Þα1 ε_ 2 ¼ ϑα2 β2 sigðε1 Þα2
Lemma 4. [31] For any x 2 ℝ, y 2 ℝ and 0 < ρ ¼ ρ1 =ρ2 1, where ρ1 , ρ2 are two positive odd integers, the following inequality holds
(18)
(11)
For α1 2 ð0:5; 1Þ and α2 ¼ 2α1 1 2 ð0; 1Þ, it is clear that system (18) is homogeneous of degree α1 1 < 0 with respect to the weights ð1; α1 Þ. ~; v~Þ ¼ ζ T Pζ as a Lyapunov function candidate for system Consider Vα ðq
In this section, we first propose a novel finite-time convergence observer to estimate the relative velocity. Then, a new robust output feedback guidance law for spacecraft rendezvous is synthesized.
(18), where ζ ¼ ½ζ T1 ; ζ T2 , ζ 1 ¼ sigðε1 Þ1=r , ζ 2 ¼ sigðε2 Þ1=ðrα1 Þ , r ¼ α1 α2 . Furthermore, denote fα as the vector field of system (18), and let ~ ; v~Þ be the Lie bracket of the vector fields q ~ and v~. Thus, it can be Lfα Vðq ~ ; v~Þ and Lfα Vðq ~; v~Þ are homogeneous of degree 2=r easily verified that Vðq and 2=r þ α1 1, respectively. According to Lemma 4.2 in [32], there exist
jxρ yρ j 21ρ x yjρ
T
3. Robust output feedback rendezvous scheme
c1 ðα1 ; ϑÞ ¼ min Lfα VðzÞ fz:VðzÞ¼1g
3.1. Finite-time observer design for relative velocity estimation
c2 ðα1 ; ϑÞ ¼ max Lfα VðzÞ
(19)
fz:VðzÞ¼1g
_ T , then, (4) can then be rewritten in _ y; _ z Denoting q ¼ ½x; y; zT , v ¼ ½x; a compact form as
such that
q_ ¼ v;
~ ; v~Þβ Lfα Vðq ~ ; v~Þ c2 ðα1 ; ϑÞ½Vðq ~ ; v~Þβ c1 ðα1 ; ϑÞ½Vðq
v_ ¼ f ðq; vÞ þ u þ w
(12)
where β ¼ rð2=r þ α1 1Þ=2 < 1.
where 54
(20)
S. He, D. Lin
Acta Astronautica 144 (2018) 52–60
Substituting (28), (29) to (25) yields
Using the same approach as shown in [32,33], one can obtain that lim c2 ðα1 ; ϑÞ
α1 →1
ϑ λmax ðPÞ
pffiffiffi 6 3λmax ðPÞL 2 2 3ðrα1 þ1Þ=2 δλmax ðPÞ 2rα1 _ q ~; v~Þ c2 Vðq ~ ; v~Þβ þ Vð kζk þ kζk r α1 rα1 ϑ2 c ~; v~Þβ þ c3 Vðq ~ ; v~Þ þ 42 Vðq ~ ; v~Þ1rα1 =2 c2 Vðq ϑ (30)
(21)
In view of the estimation error system (16), consider the following Lyapunov function candidate ~; v~Þ ¼ ζ T Pζ Vðq
pffiffi ðr α1 þ1Þ=2 ðPÞL ρ2 þδÞλmax ðPÞ with c3 ¼ 6r α31λλmax , c4 ¼ 23 r α1 λðmin . ðPÞ min ðPÞ
(22)
~ ; v~Þ along system (16) Then, evaluating the time derivative of Vðq gives
For further analysis, the following two cases are considered. ~ ; v~Þ > 1. Then, it follows from (30) that Case 1: Vðq
2 3 0 d 7 _ q ~; v~Þ þ 2zetaP6 ~ ; v~Þ ¼ Vα ðq f ðq;vÞ f ðq; b Vð 4 vÞ þ w5 dt diag ε2 j1=ðrα1 Þ1 2 r α1 ϑ 3 2 0 7 ~ ; v~Þβ þ 2ζP6 f ðq;vÞ f ðq; b c2 Vðq 4 vÞ5 diag ε2 j1=ðrα1 Þ1 rα1 ϑ2 3 2 0 6 w 7 þ2ζP4 5 diag ε2 j1=ðrα1 Þ1 rα1 ϑ2 X3 1=ðrα1 Þ1 2λmax ðPÞjjζjjjjf ðq;vÞ f ðq; b v Þjj ε2i j i¼1 β ~ ; v~Þ þ c2 Vðq r α1 ϑ2 X3 1=ðrα1 Þ1 2δλmax ðPÞkζk ε 2i j i¼1 þ rα1 ϑ2 (23)
c _ q ~ ; v~Þβ þ c3 þ 42 Vðq ~ ; v~Þ c2 Vðq ~; v~Þ Vð ϑ
~; v~Þ 1 can be According to Lemma 1, one can conclude that Vðq reached in finite time. ~ ; v~Þ 1. Inequality (30) can be further rewritten as Case 2: Vðq c _ q ~; v~Þ c2 Vðq ~ ; v~Þβ þ c3 Vðq ~; v~Þβ þ 42 Vðq ~; v~Þ1r=2 Vð ϑ
c ~ ; v~Þβ1þr=2 c3 Vðq ~ ; v~Þ1r=2 ~; v~Þβ1þr=2 þ 42 Vðq c2 ð1 ϑ0 ÞVðq ϑ
where ϑ0 2 ℝþ . If the inequality ~ ; v~Þβ1þr=2 c3 Vðq ~ ; v~Þβ1þr=2 þ c2 ð1 ϑ0 ÞVðq
_ q ~ ; v~Þ Vð
~; v~Þ þ c2 Vðq
2λmax ðPÞjjζjjL
(24)
~ ; v~Þ Vðq
X3 X3 1=ðrα1 Þ1 ε2i ε2i j i¼1 i¼1
r α1 X3 1=ðrα1 Þ1 2δλmax ðPÞjjζjj ε2i j i¼1 þ rα1 ϑ2
c4 ϑ2
2
2β2þr
c2 ð1 ϑ0 Þ c3
¼
c4 c2 ϑ2 ð1 ϑ0 Þ c3 ϑ2
2
2β2þr
1
2β2þr 1 c5 kζk μ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 λmin ðPÞ c2 ϑ ð1 ϑ0 Þ c3 ϑ
Using Lemma 2 gives ! 3 X 1=ðrα1 Þ þ rα1 ε2i 3ð1 rα1 Þε2j j i¼1
(33)
(34)
To guarantee the feasibility of inequality (34), one can choose ϑ such that c2 ð1 ϑ0 Þ > c3 holds. From the preceding analysis, the residual set of the estimation error ζ can be obtained from inequality (34) as
(25) 1=ðrα1 Þ1 ε2j j
c4 0 ϑ2
holds, then, from inequality (33), the system tends to be finite-time ~; v~Þ can be calculated as attractive and the residual set of Vðq
Substituting inequality (24) into (23) yields β
(32)
~ ; v~Þβ c2 ϑ0 Vðq
Because of the Lipschitz property of function f , there exists a positive constant L such that v Þjj Lkv~k ¼ Lϑ2 jjε2 jj jjf ðq; vÞ f ðq; b
(31)
(35)
Furthermore, using Lemma 3 one can imply that ! 3 X 1=ðrα1 Þ ε2i j i¼1
(26)
~ ; v~Þk kðq
jε1i j þ ϑ2 !
3 X 2=r r=2 ¼ϑ ε1i j i¼1
2
þϑ
3 X
jε2i j
i¼1
3 X 2=ðrα1 Þ rα1 =2 ε2i j i¼1
!r=2 3 X 2=r þ 31rα1 =2 ϑ2 31r=2 ϑ ε1i j i¼1
(27)
31r=2 ϑjjζjjr þ 31rα1 =2 ϑ2 jjζjjrα1
!
!rα1 =2 2=ðrα1 Þ ε2i j i¼1
(36)
3 X
Substituting (35) into (36) one can finally derive the residual set of ~; v~Þ as the estimation error ðq
Applying the inequality ða1 þ a2 Þ2 2a21 þ 2a22 to (27) yields ! ! 3 3 X X pffiffiffi pffiffiffi 1=ðrα1 Þ1 ε2j j ε2i 3 3jjζ 2 jj 3 3kζk j¼1 i¼1
3 X i¼1
where j ¼ 1; 2; 3. Then, one can imply that ! ! ! 3 3 3 X X X 1=ðrα1 Þ1 1=ðrα1 Þ ε2j j ε2i 3 ε2j j j¼1 i¼1 j¼1
ϑ
~ ; v~Þjj 31r=2 ϑμr þ 31rα1 =2 ϑ2 μrα1 jjðq
(28)
(37)
This completes the proof. QED. By Lemma 3, one has 3 X 1=ðrα1 Þ1 3rα1 ε2i j i¼1
!1rα1 3 X 1=ðrα1 Þ 3ðrα1 þ1Þ=2 kζk1rα1 ε1i j i¼1
Remark 1. If one sets α1 ¼ 1, then observer (13) reduces to the linear state observer and therefore only guarantees asymptotical stability, which means that the settling time is infinite. If α1 ¼ 0:5, the proposed observer (13) becomes the well-known super-twisting algorithm for
(29)
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Acta Astronautica 144 (2018) 52–60
system state identification. Obviously, the velocity estimation with α1 ¼ 0:5 involves the undesired discontinuous sign function and thus suffers from the chattering problem. Consequently, α1 2 ð0:5; 1Þ can be viewed as the link between the continuous linear control and the discontinuous nonlinear control.
Substituting (46) into (45) gives V_ i
v þvb v Þ ¼ qT v~ þ qT ηd þ qT ðη ηd Þ V_ 0 ¼ qT q_ ¼ qT ð b
ηd ¼ k1 qυ
Proof. For the closed-loop system, consider Ω ¼ V þ W as a Lyapunov function candidate. Computing the time derivative of Ω gives
(39)
_ Ω
þ
where k1 > 0, 1=2 < υ ¼ υ1 =υ2 < 1, where υ1 and υ2 are two positive odd integers. Using (40), one has V_ 0 ¼ k1 qT qυ þ qT v~ þ qT ðη ηd Þ
i¼1
i¼1
1þυ 2 qi j þ υσ i j1þυ 1þυ
3 X 21υ 1þυ qi j þ υσ i j1þυ 1 þ υ i¼1
3 X
Vi ;
Vi ¼
i¼1
1
2υ η υ ∫ i s1=υ η1= ds di 1þ1=υ η
ð2 υÞ21υ k1
di
V_ i
_ c2 Vðq ~; v~Þβ k3 Ω
_ c2 Vðq ~; v~Þβ k3 Ω
(44)
(50)
3 3 X X 1þυ 1þυ qi j k3 σ i j þ M1 i¼1 i¼1
(51) P3 vi σ i =k1 . i¼1 ~
3 3 X X 1þυ 1þυ qi j k3 σ i j þ cm i¼1 i¼1
(52)
where cm is a positive constant. Inequality (52) shows that the closed-loop system is UUB by 8 < Θ ¼ ζ; q; σ k ζ k :
(45)
cm
!1=ð2βÞ
c2 λβmin ðPÞ
9 3 3 X cm X cm = 1þυ 1þυ ; qi j k ; σ i j k ; 3 i¼1 3 i¼1 (53)
Next, it follows from Lemma 4 that
21υ jηi jjσ i jυ jσ i j1υ
ðjσ i jjηi ηdi j þ jσ i jjηdi jÞ 2 21υ σ i σ i jυ þ k1 σ i qi jυ ! ½k1 þ 21υ ð1 þ υÞ σ i j1þυ k1 υσ i j1þυ 21υ þ 1þυ 1þυ 1υ
jσ i j1þυ
i¼1
~; v~; q; σ ÞjV þ W Vmax g. Then, M1 Consider the compact set Θ1 ¼ fðq is bounded in the compact set Θ1 . With this in mind, it follows from (51) that
1 2υ 2 Vi ηi ηdi σ i j σ i j 1þ1= υ 1þ1= υ ð2 υÞ21υ k1 ð2 υÞk1 1
1υ
2
3 X
k1 þ 21υ ð1 þ υÞ þ k3 ð1 þ υÞk1
~ ; v~Þ þ c4 Vðq ~; v~Þ1r α1 =2 =ϑ2 þ qT v~ þ where M1 ¼ c3 Vðq
(43)
It follows from Lemmas 3 and 4 that jηi jjηi ηdi jjσ i j1υ
!
with k3 > 0. Then, one can imply that
(42)
Differentiating Vi with respect to time yields 1υ ~vi þ η η 1 1=υ υ ∂ηi ¼ 1υ i ∫ ηi s1=υ ηdi ds þ σ 2 i di ∂t 2 k1 ð2 υÞ21υ k11þ1=υ 1 1 ∂η 1υ ðj~vi j þ jηi jÞηi ηdi σ i j1υ þ σ 2υ i 1þ1=υ i ∂t 2 k1 ð2 υÞ21υ k1
21υ þ υ þ k3 1þυ 1þ1=υ
To determine the real control law, consider the following Lyapunov function W ¼ V0 þ
(49)
k1 þ 21υ ð1 þ υÞ k2 1þ1=υ ð1 þ υÞk1 ð2 υÞ21υ k1
k2 ð2 υÞ21υ k1
Substituting (42) into (41) gives V_ 0 k1 qT qυ þ qT v~ þ
3 X jv~i jjσ i j þ k1 i¼1
k1
υ Denoting σ ¼ η1=υ η1= d , then, it follows from Lemmas 3 and 4 that
1υ
c4 ~ ; v~Þ1rα1 =2 Vðq ϑ2
Choose k1 , k2 such that
(41)
3 X 3 P υ 1υ qi η η 2 qi σ i j di i
~; v~Þβ þ c3 Vðq ~; v~Þ þ c2 Vðq
3 21υ þ υ X k1 jqi j1þυ þ qT v~ 1þυ i¼1
(40)
(48)
Theorem 2. For system (12) under control law (48) with observer (13), the closed-loop system is uniformly ultimately bounded (UUB). Moreover, the relative distance and its rate between two neighboring spacecraft can converge to a small compact region around zero in finite time.
where η ¼ b v and ηd is the virtual control law to be determined. For system (39), the virtual control law ηd is designed as
3 P
σ
(47)
∂ηi ∂t
where k2 > 0 is a design parameter.
In order to achieve autonomous rendezvous, consider V0 ¼ 0:5qT q as a Lyapunov function. Differentiating V0 with respect to time yields
i¼1
2υ 1þ1=υ i Þ21υ k1
~Þα2 v Þ ϑ2 β2 sigðq u ¼ k2 σ 2υ1 f ðq; b
(38)
3.2. Output feedback rendezvous scheme design
1
Then, the real control law is designed as
to accelerate the convergence rate.
qT ðη ηd Þ
1þυ υσ i j1þυ 1 ½k1 þ 21υ ð1 þ υÞ σ i j þ ~vi σ i þ k1 ð1 þ υÞk1 1þυ þ ð2 υ
Remark 2. Moreover, for large initial estimation errors, that is, ~ð0Þk≫1, one can add linear terms as kq ~ b_ ¼ b ~Þα1 þ ϑβ1 q q v þ ϑβ1 sigðq _v ¼ f ðq; b ~ ~Þα2 þ ϑ2 β2 q b v Þ þ u þ ϑβ2 sigðq
(46)
(54)
P P Then, one can imply that W c5 ð 3i¼1 q2i þ 3i¼1 σ 2i Þ, where c5 is a 1þ1=υ
constant and is defined as c5 ¼ maxf1=2; 1=½ð2 υÞk1 follows from Lemma 3 that
56
g. Finally, it
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3 3 3 X X X j~vi jjσ i j 1þυ 1þυ k3 qi j k3 σ i j þ qT v~ þ k1 i¼1 i¼1 i¼1 ! 3 3 3 X X j~vi jjσ i j 1þυ X 1þυ k3 þ qT v~ þ qi j þ σ i j k1 i¼1 i¼1 i¼1
Table 1 Summary of required model parameters for simulation.
(55)
3 X k3 W ð1þυÞ=2 j~vi jjσ i j ð1þυÞ=2 þ qT v~ þ k1 c5 i¼1
Parameter
Value
Target orbital parameters
the semi-major axis a ¼ 2:4616 107 m, the eccentricity e ¼ 0:73074, and the gravity constant μ ¼ 3:986 1014 m3 =s2
Initial relative kinematics
initial relative position: ½ 50m 70m 60m T , initial relative velocity: ½ 0
0
0 T , initial true anomaly: θð0Þ ¼ 0
ϑ ¼ 5, β1 ¼ 1, β2 ¼ 0:2, α1 ¼ 0:8, k1 ¼ 0:05, k2 ¼ 0:15, υ ¼ 17=23. Considering the control input of real spacecraft is limited in practice, the maximum controlled acceleration of the chaser is assumed to be bounded by umax ¼ 0:5m=s2 . The required model parameters for simulation are summarized in Table 1. Since the external perturbations from other objects in space may cause periodic variations [34], these effects can be formulated as periodic disturbances as
Since the closed-loop system is UUB, (55) becomes P ð1þυÞ=2 W_ c6 W ð1þυÞ=2 þ cn , where c6 ¼ k3 =c5 > 0, cn ¼ qT v~ þ 3i¼1 ~vi σ i =k1 is bounded. As 0 < ð1 þ υÞ=2 < 1, Lemma 1 reveals that the relative distance and its rate can converge to a small region around zero in finite time. This completes the proof. QED. Remark 3. From the guidance command (48), it is clear that the control input only requires the position measurement in practical implementation, whereas most previous rendezvous guidance laws require full state feedback information [11,14–16]. This means that the cost of the proposed approach related to the onboard sensors can be reduced and, therefore, the low-cost autonomous rendezvous is achievable.
wx ¼ 0:005sinð0:2tÞ; wy ¼ 0:005cosð0:2tÞ wz ¼ 0:005½0:6sinð0:2tÞ þ 0:6cosð0:2tÞ To show the superiority of finite-time stable control law, the asymptotical convergence high-gain observer-based output feedback rendezvous control law [24] is also performed in simulations for the purpose of comparison. This rendezvous control law is defined as
Remark 4. In Ref. [24], the authors proposed an output feedback law for spacecraft rendezvous by utilizing a high-gain observer to reconstruct the relative velocity information. Although the effect of the external disturbances can be rejected to a specified level by the high-gain feedback, only asymptotical convergence can be guaranteed by this approach. As a comparison, the proposed rendenzvous law leverages the geometric homogeneity concept to provide finite-time convergence, which gives better performance in terms of the control precision, as shown in the simulations.
8 μx a b > ux ¼ ω2 x þ 2ωbz_ þ ω_ z 3 2 x bx_ > > > ε R ε 0 > 0 > > > < μy a bb uy ¼ 3 2 y y_ ε0 ε0 R > > > > > > 2μz a b > > : uz ¼ ω2 z 2ωbx_ ω_ x þ 3 2 z bz_ ε0 R ε0
3.3. Design procedure of the proposed rendezvous law Although the proposed output feedback rendezvous guidance law seems to be quite complicated, it only requires simple design procedures and low computational burdens. This feature is of great importance for the online implementation of the proposed rendezvous law when the computation power is limited. The detailed design procedure of the proposed output-feedback guidance law is summarized as follows.
(56)
_ respectively, and are governed _ y, _ z, _ b _ b z_ are the estimations of x, where b x, y, by the following high-gain observer 8 l1 > > bx_ ¼ bx_ þ ðx bx Þ > > > ε > > > > > a b b l2 _ > b > > > x_ ¼ ε2 x ε0 x_ þ ε2 ðx bx Þ > 0 > > > > > > _ b l1 > > < by ¼ y_ þ ε ðy by Þ
(1) Choose a suitable homogeneity power α1 to achieve desired convergence time of the velocity estimator. Obviously, smaller α1 provides faster convergence rate of the estimation error, since it is more close to the discontinuous controller. This, however, also requires larger control effort and thus trade-off design is required under certain conditions. Once α1 is fixed, calculating α2 as α2 ¼ 2α1 1. (2) Select a suitable observer gain ϑ and the scaling factors β1 ; β2 . As shown in (35), these three parameter mainly determine the estimation accuracy of the proposed state observer since ϑ directly affects the convergence region and β1 ; β2 determine P. (3) Design a suitable controller power term υ, which mainly affects the convergence time of the proposed output-feedback rendezvous law. Once υ is fixed, the guidance gains k1 and k2 can be properly designed according to condition (50).
> > by__ ¼ a y b by_ þ l2 ðy by Þ > > > ε0 ε20 ε2 > > > > > > > _ b l1 > bz ¼ z_ þ ðz bz Þ > > > ε > > > > a b l2 > b_ > > : z_ ¼ 2 z ε bz_ þ 2 ðz bz Þ
ε0
0
(57)
ε
The design parameters for control law (56) and observer (57) are all chosen from Ref. [24] without any further modifications. It follows from the results in Ref. [24] that control law (56) ensures that the relative distance and its rate converge to a small region around zero asymptotically.
4. Simulation results 4.2. Comparison results without actuator faults In this section, the effectiveness and robustness of the proposed rendezvous law is validated through numerical simulations by an example of close-range approaching phase in an elliptical spacecraft rendezvous mission.
The response curves of the relative distance and its rate between two spacecraft under both guidance laws are depicted in Figs. 2 and 3, respectively. The control input power of these two control laws are presented in Fig. 4. From these three figures, one can observe that the relative position and its rate between these two neighboring flying vehicles converge to a small region around zero with desired transient performance and the control input efforts under these two laws are
4.1. Simulation setup The design parameters of the proposed rendezvous law are selected as 57
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Acta Astronautica 144 (2018) 52–60
Fig. 2. Relative distance. Fig. 5. Relative distance estimations.
Fig. 3. Relative velocity.
Fig. 6. Relative velocity estimations.
the real system states in a finite time. Since the proposed output feedback rendezvous guidance law only utilizes the position measurement and is given in a closed-form, it requires low computational loads, with the recorded computational time of each step being less than 0:5ms in our computer simulations. This feature is of great importance for the online implementation of the proposed rendezvous law when the computation power is limited. As we consider the periodic external disturbances, the maximum steady-state position error eq and velocity error ev are leveraged as the performance index for evaluations. The mean position error eq and velocity error ev of 200 Monte-Carlo runs are summarized in Table 2. From this table, one can note that the proposed guidance law exhibits smaller steady-state tracking errors and thus is much more suitable for high-value spacecraft rendezvous missions. Fig. 4. Control input power.
4.3. Comparison results with actuator faults comparable. However, the zoomed-in graphs in Figs. 2 and 3 show that the proposed law exhibit more accurate response, demonstrating the superiority of disturbance rejection performance of finite-time convergence compared with asymptotical convergence. The estimation performance of the proposed observer in the considered scenario is presented in Figs. 5 and 6. It can be seen that the observed system states converge to
In practice, actuator fault is an important issue encountered in real spacecraft. In this subsection, it will be shown that, due to the inherent robustness property of the proposed rendezvous law, it can actually achieve the same objective even for time-varying actuator faults. As shown in Ref. [35], there usually exist two types of actuator faults in 58
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Table 2 Mean position error eq and velocity error ev of 200 Monte-Carlo runs. Parameter
eq
ev
Rendezvous law (56) [24] Proposed law (48)
0:12m 0:01m
0:024m=s 0:001m=s
practical systems. One is the additive fault, which means that the bounded faults enter the system control channel in an additive way, and the other one is the losing of control effectiveness. Consider the uncertain rendezvous dynamics with actuator fault as q_ ¼ v;
v_ ¼ f ðq; vÞ þ ðlu þ FÞ þ w
(58)
where F ¼ ½F1 ; F2 ; F3 T with kFk lF represents the additive fault, l ¼ ½l1 ; l2 ; l3 T with 0 li 1 stand for losing of control effectiveness. Note that li ¼ 1, li ¼ 0 and 0 < li < 1 mean that the ith actuator works normally, has completely failed and has partially lost its effectiveness, respectively. Define a new vector as Δ ¼ ðlu þ FÞ þ w u, then system (58) can be rewritten as q_ ¼ v;
v_ ¼ f ðq; vÞ þ u þ Δ
Fig. 8. Relative velocity.
(59)
Since the actual control input of spacecraft is always bounded due to physical limits, it follows from Assumption 1 that Δ is also bounded. Note that (59) has similar structure as compared with (12) and therefore the proposed rendezvous law is also feasible to faulty system (59). The time-varying actuator faults in simulations are selected as
8 <
Fi ¼ 0; Fi ¼ 0:01 þ 0:05sinð0:2tÞ;
li ¼ 1; li ¼ 0; : li ¼ 0:8 þ 0:1sinð0:2tÞ;
0 < t < 100s otherwise
0 < t < 50s 20s t < 70s otherwise
The response curves of the relative distance and its rate between two spacecraft under both guidance laws without any parameter tunings are depicted in Figs. 7 and 8, respectively. The control input power of these two control laws are presented in Fig. 9. It follows from Figs. 7 and 8 that the proposed law also achieves high-precision rendezvous while control law (56) has relatively larger steady-state error in the presence of timevarying actuator faults. As the actuators completely failed during 20s t < 70s, the convergence time is relatively longer than that of the fault-free case. These results clearly show that the proposed rendezvous law has strong robustness against actuator faults and therefore is more reliable in real applications.
Fig. 9. Control input power.
The mean position error eq and velocity error ev of 200 Monte-Carlo runs are summarized in Table 3. From this table, one can note that the proposed guidance law shows similar performance as the case without actuator failure, whereas the rendezvous law (56) generates much larger steady-state tracking errors. 5. Conclusions The problem of autonomous spacecraft rendezvous during the closerange closing phase in elliptical orbit is investigated based on finite-time control approach. The proposed formulation is essentially an output feedback control approach, which requires no relative velocity and any priori information of the lumped disturbance. Stability analysis shows that the relative distance and its rate between two neighboring spacecraft can converge to a small region around zero in finite time. Numerical simulations reveal that the proposed rendezvous law has better disturbance rejection performance, strong robustness against actuator faults and smaller steady-state tracking errors than the asymptotical rendezvous law. Future work includes considering control input constraint for the future onboard implementation of the proposed control law. Table 3 Mean position error eq and velocity error ev of 200 Monte-Carlo runs.
Fig. 7. Relative distance. 59
Parameter
eq
ev
Rendezvous law (56) [24] Proposed law (48)
4:1m 0:19m
0:26m=s 0:02m=s
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Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Reliable spacecraft rendezvous without velocity measurement Shaoming He *, Defu Lin School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, People's Republic of China
A R T I C L E I N F O
A B S T R A C T
Keywords: Spacecraft rendezvous Velocity-free Finite-time convergence External disturbances Actuator faults
This paper investigates the problem of finite-time velocity-free autonomous rendezvous for spacecraft in the presence of external disturbances during the terminal phase. First of all, to address the problem of lack of relative velocity measurement, a robust observer is proposed to estimate the unknown relative velocity information in a finite time. It is shown that the effect of external disturbances on the estimation precision can be suppressed to a relatively low level. With the reconstructed velocity information, a finite-time output feedback control law is then formulated to stabilize the rendezvous system. Theoretical analysis and rigorous proof show that the relative position and its rate can converge to a small compacted region in finite time. Numerical simulations are performed to evaluate the performance of the proposed approach in the presence of external disturbances and actuator faults.
1. Introduction Owing to the stringent performance requirement, guidance law design for spacecraft rendezvous is still an active research area and a benchmark problem for engineers [1–6]. Spacecraft rendezvous mission refers to the trajectory maneuver command that guides a chaser spacecraft to arrive at the target spacecraft with the same velocity. Such mission is typically divided into many phases, including far-range closing, close-range closing, and final mating [7]. This paper mainly focuses on guidance scheme design for the close-range phase, where the main concern of this phase is the rendezvous precision instead of the energy consumption. Since the chaser spacecraft is moving in the vicinity of the target vehicle during the close-range rendezvous, the relative motion between these two vehicles can be described by the well-known Clohessy-Wiltshire (C-W) differential equations [8] or Tschauner-Hempel (T-H) differential equations [9,10]. The difference between these two models lies in that the C-W equations are for circular target orbit while the T-H equations are for elliptical target orbit. This paper mainly focuses on the elliptical orbit rendezvous problem. With the development of modern control theory, many elegant guidance schemes have been proposed in recent years to achieve autonomous rendezvous. To cite a few, the authors in Ref. [11] proposed a robust guidance law for circular orbital rendezvous based on multi-objective H∞ control approach. This controller can be easily obtained by a convex optimization problem with linear matrix inequality constraints, but it only considers the state-feedback control scheme. To
* Corresponding author. E-mail addresses: [email protected] (S. He), [email protected] (D. Lin). https://doi.org/10.1016/j.actaastro.2017.12.016 Received 3 July 2017; Received in revised form 29 August 2017; Accepted 11 December 2017 Available online 14 December 2017 0094-5765/© 2017 IAA. Published by Elsevier Ltd. All rights reserved.
improve the overall guidance performance, the parametric Lyapunov differential equation approach was also a powerful tool that can be used to design rendezvous laws [12,13] for uncertainty-free rendezvous system. Compared to the rendezvous guidance laws based on the quadratic regulation theory, the parametric Lyapunov approach is easy to implement. To cope with uncertainties in the plant parameters and reduce the computational burden for onboard implementation, a passivity-based direct adaptive spacecraft proximity operational strategy was derived upon the simple adaptive control methodology in Ref. [14]. Under the framework of integral sliding mode control technique, the authors in Ref. [15] developed a robust control law for spacecraft formation flying, where the equivalent part was based on linear quadratic optimal control method. By combining backstepping methodology with sliding mode approach, the authors in Ref. [16] presented a new guidance law for uncertain spacecraft rendezvous system. Although the sliding mode rendezvous guidance law is robust to external perturbations, the inherent chattering problem limits its practical applications. To eliminate the reaching phase and guarantee global sliding manifold, an adaptive robust rendezvous guidance law was proposed in Ref. [17] based on the time-varying sliding mode control approach with a complicated sliding surface design. Integrating extended state observer with nonsingular terminal sliding mode, the authors in Ref. [18] proposed a finite-time convergence rendezvous law with full state feedback. The authors in Ref. [19] considered the application of State-Dependent Riccati Equation (SDRE) for coupled orbital and attitude relative motion of formation flying or rendezvous. The SDRE controller can provide optimal control performance in terms of a meaningful index for the spacecraft formation
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system, but it is sensitive to external disturbances. As model predictive control (MPC) is one of the promising nonlinear techniques for regulating nonlinear systems while considering multiple constraints, the authors in Refs. [20,21] considers the application of MPC methodology to rendezvous control law design. Despite its advantages, the MPC method usually requires high computational load and thus is difficult for the implementation in the onboard embed system. It should be noted that most of the above mentioned rendezvous laws are full state feedback control laws. However, the relative velocity measurement is always an issue for sensor-less low-cost space vehicles. Considering this, output feedback design using only position measurement is more desirable for spacecraft rendezvous from the viewpoint of real application. The typical solution to the output feedback for spacecraft applications includes the design of an ad hoc dynamic filter, i.e., Kalman filter for example, driven by the output only [22,23]. However, in these works, no external disturbances were considered for the filter design, although the perturbations are inevitable in real applications. To this end, the authors in Ref. [24] proposed an output feedback rendezvous law based on high-gain observer and input-to-state stability to suppress the effect of the external disturbances to a specified level. However, only asymptotical stability is considered in this reference, which means that the convergence of the relative range and its rate is exponential. Obviously, the control law with infinite settling time is not desirable for the critical high-value rendezvous missions, where the control precision, especially during the final close-in phase, is of great importance for many space operations. It is well known that finite-time convergence ensures better disturbance rejection performance and smaller steady-state tracking errors than asymptotically stable systems [25,26]. Therefore, it is more desirable to design finite-time velocity-free rendezvous control laws. Motivated by the above analysis, this paper investigates the problem of designing a robust finite-time convergence guidance law for elliptical orbital spacecraft rendezvous using only relative position measurements. Comparison results show that the proposed law has better disturbance rejection performance and smaller steady-state tracking errors than the asymptotical convergence law. The key features and contributions of the proposed approach are summarized as follows.
Fig. 1. Target-orbital rotating coordinate system.
target spacecraft. R and r represent the vector from the center planet to the target spacecraft and the vector from the target spacecraft to the chaser spacecraft, respectively. As this paper considers elliptical orbit rendezvous problem, the relative motion kinematics of the chaser with reference to the target spacecraft can be formulated as [27]. d2 r Rþr R ¼ μ d2 t jjR þ rjj3 jjRjj3
! þuþw
(1)
where μ represents the gravity constant, u the controlled acceleration input vector and w the lumped disturbance. Assumption 1. The unknown external disturbances, in space, including environmental disturbance, solar radiation and magnetic effect, are all bounded in reality. With this mind, the norm of the lumped disturbance satisfies jjwjj δ, where δ > 0. For simplicity, defining the notations as r ¼ ½x; y; zT , u ¼ ½ux ; uy ; uz T and w ¼ ½wx ; wy ; wz T , then, system (1) can be written as 8 μx > x€ ¼ ω2 x þ 2ωz_ þ ω_ z þ ux þ wx > > > þ rjj3 jjR > > > > > μy < y€ ¼ þ uy þ wy kR þ rk3 > > ! > > > > zR 1 > 2 > _ _ x z € ¼ ω z 2 ω ω x μ þ þ uz þ wz > 3 : R2 kR þ rk
(1) Using homogeneous and Lyapunov theories, a robust observer is proposed to precisely estimate the relative velocity between two neighboring spacecraft despite external disturbances in finite time. (2) With the reconstructed velocity information, a novel output feedback rendezvous law is then synthesized to regulate the relative distance and its rate to a small compact region in finite time. Moreover, analysis also proves that the proposed rendezvous law is robust against actuator faults and thus is a reliable control approach.
(2)
where ω denotes the angular rate of the target orbit, which is governed by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μð1 þ e cos θÞ ω¼ R3
The rest of the paper is organized as follows. In Sec. 2, some backgrounds and preliminaries are stated, followed by the proposed output feedback rendezvous law derived in Sec. 3. Finally, some simulation results and conclusion remarks are offered.
(3)
where e 2 ½0; 1Þ represents the eccentricity of the orbit, and θ stands for the true anomaly. During the close-range closing phase, the distance between the chaser and the target spacecraft r is much smaller than the distance between the planet center and the target R, i.e. r≪R. With this fact in mind, system (2) can be further simplified as [27].
2. Backgrounds and preliminaries
8 μx 2 > > > x€ ¼ ω x þ 2ωz_ þ ω_ z R3 þ ux þ wx > > > < μy y€ ¼ 3 þ uy þ wy R > > > > > > : z€ ¼ ω2 z 2ωx_ ω_ x þ 2μz þ uz þ wz R3
2.1. Problem formulation This work assumes the chaser spacecraft is equipped with a highperformance low-level attitude control system that provides attitude stabilization and maneuver tracking, i.e., there is no time delay in tracking the attitude command provided by the guidance loop. This study aims to design the guidance input to this low-level controller to achieve autonomous rendezvous. Consider the target-orbital rotating coordinate system x y z, as shown in Fig. 1, where the origin is fixed at the center of mass of the
(4)
In order to determine the target orbit, the following two augmented equations are required [28].
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μ R€ ¼ Rω2 2 R
(5)
R_ ω_ ¼ 2 ω R
(6)
2 6 6 6 f ðq; vÞ ¼ 6 6 6 4
The control interest here is to design a rendezvous law using the relative position information only to drive arbitrary initial states of system (4) to a small compact region in finite time.
ω2 x þ 2ωz_ þ ω_ z μy 3 R ω2 z 2ωx_ ω_ x þ
μx 3 R3 7 7 7 7 7 7 2μz 5 R3
b, b Denote q v as the estimated information of q, v, respectively. The proposed observer for relative velocity estimation is formulated as
2.2. Some fundamental facts
~ Þα1 b_ ¼ b q v þ ϑβ1 sigðq _b ~Þα2 v ¼ f ðq; b v Þ þ u þ ϑ2 β2 sigðq
Notations. Throughout the paper the following notations will be used. Let 0m and I m denote the m m zero matrix and m m identity matrix, respectively. For any positive definite matrix P, let λmax ðPÞ and λmin ðPÞ be the maximum and minimum eigenvalues of P, respectively.
~ ¼qq b , ϑ, β1 , β2 , α1 2 ð0:5; 1Þ and α2 ¼ 2α1 1 are positive where q design parameters. ~ and v~ ¼ v b Let the observer estimation errors be q v . Then, it follows from (12) and (13) that
For a given vector x ¼ ½x1 ; x2 ; …; xn T 2 ℝn and a given constant a 2 ℝ,
x_ ¼ f ðxÞ;
xa ¼ ½x1a ; x2a ; …; xna T ,
α1 α2 e e v_ e ¼ f ðq; vÞ f q; b v ϑ2 β2 sig qe þw q_ e ¼ v ϑβ1 sig q
sigðxÞa ¼ ½sigðx1 Þa ; sigðx2 Þa ; … pffiffiffiffiffiffiffiffiffi a T ; sigðxn Þ , sgnðxÞ ¼ ½sgnðx1 Þ; sgnðx2 Þ; …; sgnðxn ÞT and jjxjj ¼ xT x. Lemma 1. [29] Consider the following system define
f ð0Þ ¼ 0;
x⊂ℝn ;
and
xð0Þ ¼ x0
(14) Consider the following coordinate transformation
(7)
ε_ 2 ¼ ϑα2 β2 sigðε1 Þα2 þ
!c n X c xi j i¼1
v~ ϑ2
(15)
f ðq; vÞ f ðq; b vÞ þ w ϑ2
(16)
Define A as
(8)
A¼
β1 I 3 β2 I 3
I3 03
(17)
It is easy to verify that the matrix A is Hurwitz. Therefore, there exists a unique matrix P ¼ PT > 0 such that AT P þ PA ¼ Q for any given matrix Q ¼ QT > 0. Theorem 1. For the estimation error system (14) under Assumptions 1, ~; v~Þ can if the observer is designed as (13), the estimation errors ðq converge to a nonzero bounded residual around zero in finite time.
(9)
Lemma 3. [31] For any xi 2 ℝ, i ¼ 1; 2; …; n and a real number c 2 ð0; 1, the following inequality holds !c n n X X c xi xi j n1c i¼1 i¼1
ε2 ¼
ε_ 1 ¼ ϑε2 ϑα1 β1 sigðq~ Þα1
where Vðx0 Þ is the initial value of VðxÞ. Lemma 2. [30] For any x 2 ℝ, y 2 ℝ and c > 0, d > 0, the following inequality holds c d xj yj c xjcþd þ d yjcþd cþd c þ d
~ q ϑ
ε1 ¼ ;
Then, the error system (14) can be rewritten as
domain of attraction of the origin and the settling time Treach satisfies Treach
T
ε ¼ εT1 ; εT2 ;
where f : U→ℝn is continuous on an open neighborhood U of the origin x ¼ 0. Suppose there exists a Lyapunov function VðxÞ, defined on a neigh_ borhood N⊂ℝn of the origin, and VðxÞ l½VðxÞa þ kVðxÞ; 8x 2 N\f0g, þ where a 2 ð0; 1Þ, l; k 2 ℝ . Then, the origin of system (7) is locally finite time stable. The set Ω ¼ fx½VðxÞ1a < l=kg \ N is contained in the 1 k ln 1 ½Vðx0 Þ1a ka k l
(13)
Proof. For system (16), if the term ðf ðq; vÞ f ðq; b v Þ þ wÞ=ϑ2 is omitted, then system (16) reduces to
(10)
ε_ 1 ¼ ϑε2 ϑα1 β1 sigðq~ Þα1 ε_ 2 ¼ ϑα2 β2 sigðε1 Þα2
Lemma 4. [31] For any x 2 ℝ, y 2 ℝ and 0 < ρ ¼ ρ1 =ρ2 1, where ρ1 , ρ2 are two positive odd integers, the following inequality holds
(18)
(11)
For α1 2 ð0:5; 1Þ and α2 ¼ 2α1 1 2 ð0; 1Þ, it is clear that system (18) is homogeneous of degree α1 1 < 0 with respect to the weights ð1; α1 Þ. ~; v~Þ ¼ ζ T Pζ as a Lyapunov function candidate for system Consider Vα ðq
In this section, we first propose a novel finite-time convergence observer to estimate the relative velocity. Then, a new robust output feedback guidance law for spacecraft rendezvous is synthesized.
(18), where ζ ¼ ½ζ T1 ; ζ T2 , ζ 1 ¼ sigðε1 Þ1=r , ζ 2 ¼ sigðε2 Þ1=ðrα1 Þ , r ¼ α1 α2 . Furthermore, denote fα as the vector field of system (18), and let ~ ; v~Þ be the Lie bracket of the vector fields q ~ and v~. Thus, it can be Lfα Vðq ~ ; v~Þ and Lfα Vðq ~; v~Þ are homogeneous of degree 2=r easily verified that Vðq and 2=r þ α1 1, respectively. According to Lemma 4.2 in [32], there exist
jxρ yρ j 21ρ x yjρ
T
3. Robust output feedback rendezvous scheme
c1 ðα1 ; ϑÞ ¼ min Lfα VðzÞ fz:VðzÞ¼1g
3.1. Finite-time observer design for relative velocity estimation
c2 ðα1 ; ϑÞ ¼ max Lfα VðzÞ
(19)
fz:VðzÞ¼1g
_ T , then, (4) can then be rewritten in _ y; _ z Denoting q ¼ ½x; y; zT , v ¼ ½x; a compact form as
such that
q_ ¼ v;
~ ; v~Þβ Lfα Vðq ~ ; v~Þ c2 ðα1 ; ϑÞ½Vðq ~ ; v~Þβ c1 ðα1 ; ϑÞ½Vðq
v_ ¼ f ðq; vÞ þ u þ w
(12)
where β ¼ rð2=r þ α1 1Þ=2 < 1.
where 54
(20)
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Acta Astronautica 144 (2018) 52–60
Substituting (28), (29) to (25) yields
Using the same approach as shown in [32,33], one can obtain that lim c2 ðα1 ; ϑÞ
α1 →1
ϑ λmax ðPÞ
pffiffiffi 6 3λmax ðPÞL 2 2 3ðrα1 þ1Þ=2 δλmax ðPÞ 2rα1 _ q ~; v~Þ c2 Vðq ~ ; v~Þβ þ Vð kζk þ kζk r α1 rα1 ϑ2 c ~; v~Þβ þ c3 Vðq ~ ; v~Þ þ 42 Vðq ~ ; v~Þ1rα1 =2 c2 Vðq ϑ (30)
(21)
In view of the estimation error system (16), consider the following Lyapunov function candidate ~; v~Þ ¼ ζ T Pζ Vðq
pffiffi ðr α1 þ1Þ=2 ðPÞL ρ2 þδÞλmax ðPÞ with c3 ¼ 6r α31λλmax , c4 ¼ 23 r α1 λðmin . ðPÞ min ðPÞ
(22)
~ ; v~Þ along system (16) Then, evaluating the time derivative of Vðq gives
For further analysis, the following two cases are considered. ~ ; v~Þ > 1. Then, it follows from (30) that Case 1: Vðq
2 3 0 d 7 _ q ~; v~Þ þ 2zetaP6 ~ ; v~Þ ¼ Vα ðq f ðq;vÞ f ðq; b Vð 4 vÞ þ w5 dt diag ε2 j1=ðrα1 Þ1 2 r α1 ϑ 3 2 0 7 ~ ; v~Þβ þ 2ζP6 f ðq;vÞ f ðq; b c2 Vðq 4 vÞ5 diag ε2 j1=ðrα1 Þ1 rα1 ϑ2 3 2 0 6 w 7 þ2ζP4 5 diag ε2 j1=ðrα1 Þ1 rα1 ϑ2 X3 1=ðrα1 Þ1 2λmax ðPÞjjζjjjjf ðq;vÞ f ðq; b v Þjj ε2i j i¼1 β ~ ; v~Þ þ c2 Vðq r α1 ϑ2 X3 1=ðrα1 Þ1 2δλmax ðPÞkζk ε 2i j i¼1 þ rα1 ϑ2 (23)
c _ q ~ ; v~Þβ þ c3 þ 42 Vðq ~ ; v~Þ c2 Vðq ~; v~Þ Vð ϑ
~; v~Þ 1 can be According to Lemma 1, one can conclude that Vðq reached in finite time. ~ ; v~Þ 1. Inequality (30) can be further rewritten as Case 2: Vðq c _ q ~; v~Þ c2 Vðq ~ ; v~Þβ þ c3 Vðq ~; v~Þβ þ 42 Vðq ~; v~Þ1r=2 Vð ϑ
c ~ ; v~Þβ1þr=2 c3 Vðq ~ ; v~Þ1r=2 ~; v~Þβ1þr=2 þ 42 Vðq c2 ð1 ϑ0 ÞVðq ϑ
where ϑ0 2 ℝþ . If the inequality ~ ; v~Þβ1þr=2 c3 Vðq ~ ; v~Þβ1þr=2 þ c2 ð1 ϑ0 ÞVðq
_ q ~ ; v~Þ Vð
~; v~Þ þ c2 Vðq
2λmax ðPÞjjζjjL
(24)
~ ; v~Þ Vðq
X3 X3 1=ðrα1 Þ1 ε2i ε2i j i¼1 i¼1
r α1 X3 1=ðrα1 Þ1 2δλmax ðPÞjjζjj ε2i j i¼1 þ rα1 ϑ2
c4 ϑ2
2
2β2þr
c2 ð1 ϑ0 Þ c3
¼
c4 c2 ϑ2 ð1 ϑ0 Þ c3 ϑ2
2
2β2þr
1
2β2þr 1 c5 kζk μ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 λmin ðPÞ c2 ϑ ð1 ϑ0 Þ c3 ϑ
Using Lemma 2 gives ! 3 X 1=ðrα1 Þ þ rα1 ε2i 3ð1 rα1 Þε2j j i¼1
(33)
(34)
To guarantee the feasibility of inequality (34), one can choose ϑ such that c2 ð1 ϑ0 Þ > c3 holds. From the preceding analysis, the residual set of the estimation error ζ can be obtained from inequality (34) as
(25) 1=ðrα1 Þ1 ε2j j
c4 0 ϑ2
holds, then, from inequality (33), the system tends to be finite-time ~; v~Þ can be calculated as attractive and the residual set of Vðq
Substituting inequality (24) into (23) yields β
(32)
~ ; v~Þβ c2 ϑ0 Vðq
Because of the Lipschitz property of function f , there exists a positive constant L such that v Þjj Lkv~k ¼ Lϑ2 jjε2 jj jjf ðq; vÞ f ðq; b
(31)
(35)
Furthermore, using Lemma 3 one can imply that ! 3 X 1=ðrα1 Þ ε2i j i¼1
(26)
~ ; v~Þk kðq
jε1i j þ ϑ2 !
3 X 2=r r=2 ¼ϑ ε1i j i¼1
2
þϑ
3 X
jε2i j
i¼1
3 X 2=ðrα1 Þ rα1 =2 ε2i j i¼1
!r=2 3 X 2=r þ 31rα1 =2 ϑ2 31r=2 ϑ ε1i j i¼1
(27)
31r=2 ϑjjζjjr þ 31rα1 =2 ϑ2 jjζjjrα1
!
!rα1 =2 2=ðrα1 Þ ε2i j i¼1
(36)
3 X
Substituting (35) into (36) one can finally derive the residual set of ~; v~Þ as the estimation error ðq
Applying the inequality ða1 þ a2 Þ2 2a21 þ 2a22 to (27) yields ! ! 3 3 X X pffiffiffi pffiffiffi 1=ðrα1 Þ1 ε2j j ε2i 3 3jjζ 2 jj 3 3kζk j¼1 i¼1
3 X i¼1
where j ¼ 1; 2; 3. Then, one can imply that ! ! ! 3 3 3 X X X 1=ðrα1 Þ1 1=ðrα1 Þ ε2j j ε2i 3 ε2j j j¼1 i¼1 j¼1
ϑ
~ ; v~Þjj 31r=2 ϑμr þ 31rα1 =2 ϑ2 μrα1 jjðq
(28)
(37)
This completes the proof. QED. By Lemma 3, one has 3 X 1=ðrα1 Þ1 3rα1 ε2i j i¼1
!1rα1 3 X 1=ðrα1 Þ 3ðrα1 þ1Þ=2 kζk1rα1 ε1i j i¼1
Remark 1. If one sets α1 ¼ 1, then observer (13) reduces to the linear state observer and therefore only guarantees asymptotical stability, which means that the settling time is infinite. If α1 ¼ 0:5, the proposed observer (13) becomes the well-known super-twisting algorithm for
(29)
55
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Acta Astronautica 144 (2018) 52–60
system state identification. Obviously, the velocity estimation with α1 ¼ 0:5 involves the undesired discontinuous sign function and thus suffers from the chattering problem. Consequently, α1 2 ð0:5; 1Þ can be viewed as the link between the continuous linear control and the discontinuous nonlinear control.
Substituting (46) into (45) gives V_ i
v þvb v Þ ¼ qT v~ þ qT ηd þ qT ðη ηd Þ V_ 0 ¼ qT q_ ¼ qT ð b
ηd ¼ k1 qυ
Proof. For the closed-loop system, consider Ω ¼ V þ W as a Lyapunov function candidate. Computing the time derivative of Ω gives
(39)
_ Ω
þ
where k1 > 0, 1=2 < υ ¼ υ1 =υ2 < 1, where υ1 and υ2 are two positive odd integers. Using (40), one has V_ 0 ¼ k1 qT qυ þ qT v~ þ qT ðη ηd Þ
i¼1
i¼1
1þυ 2 qi j þ υσ i j1þυ 1þυ
3 X 21υ 1þυ qi j þ υσ i j1þυ 1 þ υ i¼1
3 X
Vi ;
Vi ¼
i¼1
1
2υ η υ ∫ i s1=υ η1= ds di 1þ1=υ η
ð2 υÞ21υ k1
di
V_ i
_ c2 Vðq ~; v~Þβ k3 Ω
_ c2 Vðq ~; v~Þβ k3 Ω
(44)
(50)
3 3 X X 1þυ 1þυ qi j k3 σ i j þ M1 i¼1 i¼1
(51) P3 vi σ i =k1 . i¼1 ~
3 3 X X 1þυ 1þυ qi j k3 σ i j þ cm i¼1 i¼1
(52)
where cm is a positive constant. Inequality (52) shows that the closed-loop system is UUB by 8 < Θ ¼ ζ; q; σ k ζ k :
(45)
cm
!1=ð2βÞ
c2 λβmin ðPÞ
9 3 3 X cm X cm = 1þυ 1þυ ; qi j k ; σ i j k ; 3 i¼1 3 i¼1 (53)
Next, it follows from Lemma 4 that
21υ jηi jjσ i jυ jσ i j1υ
ðjσ i jjηi ηdi j þ jσ i jjηdi jÞ 2 21υ σ i σ i jυ þ k1 σ i qi jυ ! ½k1 þ 21υ ð1 þ υÞ σ i j1þυ k1 υσ i j1þυ 21υ þ 1þυ 1þυ 1υ
jσ i j1þυ
i¼1
~; v~; q; σ ÞjV þ W Vmax g. Then, M1 Consider the compact set Θ1 ¼ fðq is bounded in the compact set Θ1 . With this in mind, it follows from (51) that
1 2υ 2 Vi ηi ηdi σ i j σ i j 1þ1= υ 1þ1= υ ð2 υÞ21υ k1 ð2 υÞk1 1
1υ
2
3 X
k1 þ 21υ ð1 þ υÞ þ k3 ð1 þ υÞk1
~ ; v~Þ þ c4 Vðq ~; v~Þ1r α1 =2 =ϑ2 þ qT v~ þ where M1 ¼ c3 Vðq
(43)
It follows from Lemmas 3 and 4 that jηi jjηi ηdi jjσ i j1υ
!
with k3 > 0. Then, one can imply that
(42)
Differentiating Vi with respect to time yields 1υ ~vi þ η η 1 1=υ υ ∂ηi ¼ 1υ i ∫ ηi s1=υ ηdi ds þ σ 2 i di ∂t 2 k1 ð2 υÞ21υ k11þ1=υ 1 1 ∂η 1υ ðj~vi j þ jηi jÞηi ηdi σ i j1υ þ σ 2υ i 1þ1=υ i ∂t 2 k1 ð2 υÞ21υ k1
21υ þ υ þ k3 1þυ 1þ1=υ
To determine the real control law, consider the following Lyapunov function W ¼ V0 þ
(49)
k1 þ 21υ ð1 þ υÞ k2 1þ1=υ ð1 þ υÞk1 ð2 υÞ21υ k1
k2 ð2 υÞ21υ k1
Substituting (42) into (41) gives V_ 0 k1 qT qυ þ qT v~ þ
3 X jv~i jjσ i j þ k1 i¼1
k1
υ Denoting σ ¼ η1=υ η1= d , then, it follows from Lemmas 3 and 4 that
1υ
c4 ~ ; v~Þ1rα1 =2 Vðq ϑ2
Choose k1 , k2 such that
(41)
3 X 3 P υ 1υ qi η η 2 qi σ i j di i
~; v~Þβ þ c3 Vðq ~; v~Þ þ c2 Vðq
3 21υ þ υ X k1 jqi j1þυ þ qT v~ 1þυ i¼1
(40)
(48)
Theorem 2. For system (12) under control law (48) with observer (13), the closed-loop system is uniformly ultimately bounded (UUB). Moreover, the relative distance and its rate between two neighboring spacecraft can converge to a small compact region around zero in finite time.
where η ¼ b v and ηd is the virtual control law to be determined. For system (39), the virtual control law ηd is designed as
3 P
σ
(47)
∂ηi ∂t
where k2 > 0 is a design parameter.
In order to achieve autonomous rendezvous, consider V0 ¼ 0:5qT q as a Lyapunov function. Differentiating V0 with respect to time yields
i¼1
2υ 1þ1=υ i Þ21υ k1
~Þα2 v Þ ϑ2 β2 sigðq u ¼ k2 σ 2υ1 f ðq; b
(38)
3.2. Output feedback rendezvous scheme design
1
Then, the real control law is designed as
to accelerate the convergence rate.
qT ðη ηd Þ
1þυ υσ i j1þυ 1 ½k1 þ 21υ ð1 þ υÞ σ i j þ ~vi σ i þ k1 ð1 þ υÞk1 1þυ þ ð2 υ
Remark 2. Moreover, for large initial estimation errors, that is, ~ð0Þk≫1, one can add linear terms as kq ~ b_ ¼ b ~Þα1 þ ϑβ1 q q v þ ϑβ1 sigðq _v ¼ f ðq; b ~ ~Þα2 þ ϑ2 β2 q b v Þ þ u þ ϑβ2 sigðq
(46)
(54)
P P Then, one can imply that W c5 ð 3i¼1 q2i þ 3i¼1 σ 2i Þ, where c5 is a 1þ1=υ
constant and is defined as c5 ¼ maxf1=2; 1=½ð2 υÞk1 follows from Lemma 3 that
56
g. Finally, it
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W_
Acta Astronautica 144 (2018) 52–60
3 3 3 X X X j~vi jjσ i j 1þυ 1þυ k3 qi j k3 σ i j þ qT v~ þ k1 i¼1 i¼1 i¼1 ! 3 3 3 X X j~vi jjσ i j 1þυ X 1þυ k3 þ qT v~ þ qi j þ σ i j k1 i¼1 i¼1 i¼1
Table 1 Summary of required model parameters for simulation.
(55)
3 X k3 W ð1þυÞ=2 j~vi jjσ i j ð1þυÞ=2 þ qT v~ þ k1 c5 i¼1
Parameter
Value
Target orbital parameters
the semi-major axis a ¼ 2:4616 107 m, the eccentricity e ¼ 0:73074, and the gravity constant μ ¼ 3:986 1014 m3 =s2
Initial relative kinematics
initial relative position: ½ 50m 70m 60m T , initial relative velocity: ½ 0
0
0 T , initial true anomaly: θð0Þ ¼ 0
ϑ ¼ 5, β1 ¼ 1, β2 ¼ 0:2, α1 ¼ 0:8, k1 ¼ 0:05, k2 ¼ 0:15, υ ¼ 17=23. Considering the control input of real spacecraft is limited in practice, the maximum controlled acceleration of the chaser is assumed to be bounded by umax ¼ 0:5m=s2 . The required model parameters for simulation are summarized in Table 1. Since the external perturbations from other objects in space may cause periodic variations [34], these effects can be formulated as periodic disturbances as
Since the closed-loop system is UUB, (55) becomes P ð1þυÞ=2 W_ c6 W ð1þυÞ=2 þ cn , where c6 ¼ k3 =c5 > 0, cn ¼ qT v~ þ 3i¼1 ~vi σ i =k1 is bounded. As 0 < ð1 þ υÞ=2 < 1, Lemma 1 reveals that the relative distance and its rate can converge to a small region around zero in finite time. This completes the proof. QED. Remark 3. From the guidance command (48), it is clear that the control input only requires the position measurement in practical implementation, whereas most previous rendezvous guidance laws require full state feedback information [11,14–16]. This means that the cost of the proposed approach related to the onboard sensors can be reduced and, therefore, the low-cost autonomous rendezvous is achievable.
wx ¼ 0:005sinð0:2tÞ; wy ¼ 0:005cosð0:2tÞ wz ¼ 0:005½0:6sinð0:2tÞ þ 0:6cosð0:2tÞ To show the superiority of finite-time stable control law, the asymptotical convergence high-gain observer-based output feedback rendezvous control law [24] is also performed in simulations for the purpose of comparison. This rendezvous control law is defined as
Remark 4. In Ref. [24], the authors proposed an output feedback law for spacecraft rendezvous by utilizing a high-gain observer to reconstruct the relative velocity information. Although the effect of the external disturbances can be rejected to a specified level by the high-gain feedback, only asymptotical convergence can be guaranteed by this approach. As a comparison, the proposed rendenzvous law leverages the geometric homogeneity concept to provide finite-time convergence, which gives better performance in terms of the control precision, as shown in the simulations.
8 μx a b > ux ¼ ω2 x þ 2ωbz_ þ ω_ z 3 2 x bx_ > > > ε R ε 0 > 0 > > > < μy a bb uy ¼ 3 2 y y_ ε0 ε0 R > > > > > > 2μz a b > > : uz ¼ ω2 z 2ωbx_ ω_ x þ 3 2 z bz_ ε0 R ε0
3.3. Design procedure of the proposed rendezvous law Although the proposed output feedback rendezvous guidance law seems to be quite complicated, it only requires simple design procedures and low computational burdens. This feature is of great importance for the online implementation of the proposed rendezvous law when the computation power is limited. The detailed design procedure of the proposed output-feedback guidance law is summarized as follows.
(56)
_ respectively, and are governed _ y, _ z, _ b _ b z_ are the estimations of x, where b x, y, by the following high-gain observer 8 l1 > > bx_ ¼ bx_ þ ðx bx Þ > > > ε > > > > > a b b l2 _ > b > > > x_ ¼ ε2 x ε0 x_ þ ε2 ðx bx Þ > 0 > > > > > > _ b l1 > > < by ¼ y_ þ ε ðy by Þ
(1) Choose a suitable homogeneity power α1 to achieve desired convergence time of the velocity estimator. Obviously, smaller α1 provides faster convergence rate of the estimation error, since it is more close to the discontinuous controller. This, however, also requires larger control effort and thus trade-off design is required under certain conditions. Once α1 is fixed, calculating α2 as α2 ¼ 2α1 1. (2) Select a suitable observer gain ϑ and the scaling factors β1 ; β2 . As shown in (35), these three parameter mainly determine the estimation accuracy of the proposed state observer since ϑ directly affects the convergence region and β1 ; β2 determine P. (3) Design a suitable controller power term υ, which mainly affects the convergence time of the proposed output-feedback rendezvous law. Once υ is fixed, the guidance gains k1 and k2 can be properly designed according to condition (50).
> > by__ ¼ a y b by_ þ l2 ðy by Þ > > > ε0 ε20 ε2 > > > > > > > _ b l1 > bz ¼ z_ þ ðz bz Þ > > > ε > > > > a b l2 > b_ > > : z_ ¼ 2 z ε bz_ þ 2 ðz bz Þ
ε0
0
(57)
ε
The design parameters for control law (56) and observer (57) are all chosen from Ref. [24] without any further modifications. It follows from the results in Ref. [24] that control law (56) ensures that the relative distance and its rate converge to a small region around zero asymptotically.
4. Simulation results 4.2. Comparison results without actuator faults In this section, the effectiveness and robustness of the proposed rendezvous law is validated through numerical simulations by an example of close-range approaching phase in an elliptical spacecraft rendezvous mission.
The response curves of the relative distance and its rate between two spacecraft under both guidance laws are depicted in Figs. 2 and 3, respectively. The control input power of these two control laws are presented in Fig. 4. From these three figures, one can observe that the relative position and its rate between these two neighboring flying vehicles converge to a small region around zero with desired transient performance and the control input efforts under these two laws are
4.1. Simulation setup The design parameters of the proposed rendezvous law are selected as 57
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Acta Astronautica 144 (2018) 52–60
Fig. 2. Relative distance. Fig. 5. Relative distance estimations.
Fig. 3. Relative velocity.
Fig. 6. Relative velocity estimations.
the real system states in a finite time. Since the proposed output feedback rendezvous guidance law only utilizes the position measurement and is given in a closed-form, it requires low computational loads, with the recorded computational time of each step being less than 0:5ms in our computer simulations. This feature is of great importance for the online implementation of the proposed rendezvous law when the computation power is limited. As we consider the periodic external disturbances, the maximum steady-state position error eq and velocity error ev are leveraged as the performance index for evaluations. The mean position error eq and velocity error ev of 200 Monte-Carlo runs are summarized in Table 2. From this table, one can note that the proposed guidance law exhibits smaller steady-state tracking errors and thus is much more suitable for high-value spacecraft rendezvous missions. Fig. 4. Control input power.
4.3. Comparison results with actuator faults comparable. However, the zoomed-in graphs in Figs. 2 and 3 show that the proposed law exhibit more accurate response, demonstrating the superiority of disturbance rejection performance of finite-time convergence compared with asymptotical convergence. The estimation performance of the proposed observer in the considered scenario is presented in Figs. 5 and 6. It can be seen that the observed system states converge to
In practice, actuator fault is an important issue encountered in real spacecraft. In this subsection, it will be shown that, due to the inherent robustness property of the proposed rendezvous law, it can actually achieve the same objective even for time-varying actuator faults. As shown in Ref. [35], there usually exist two types of actuator faults in 58
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Table 2 Mean position error eq and velocity error ev of 200 Monte-Carlo runs. Parameter
eq
ev
Rendezvous law (56) [24] Proposed law (48)
0:12m 0:01m
0:024m=s 0:001m=s
practical systems. One is the additive fault, which means that the bounded faults enter the system control channel in an additive way, and the other one is the losing of control effectiveness. Consider the uncertain rendezvous dynamics with actuator fault as q_ ¼ v;
v_ ¼ f ðq; vÞ þ ðlu þ FÞ þ w
(58)
where F ¼ ½F1 ; F2 ; F3 T with kFk lF represents the additive fault, l ¼ ½l1 ; l2 ; l3 T with 0 li 1 stand for losing of control effectiveness. Note that li ¼ 1, li ¼ 0 and 0 < li < 1 mean that the ith actuator works normally, has completely failed and has partially lost its effectiveness, respectively. Define a new vector as Δ ¼ ðlu þ FÞ þ w u, then system (58) can be rewritten as q_ ¼ v;
v_ ¼ f ðq; vÞ þ u þ Δ
Fig. 8. Relative velocity.
(59)
Since the actual control input of spacecraft is always bounded due to physical limits, it follows from Assumption 1 that Δ is also bounded. Note that (59) has similar structure as compared with (12) and therefore the proposed rendezvous law is also feasible to faulty system (59). The time-varying actuator faults in simulations are selected as
8 <
Fi ¼ 0; Fi ¼ 0:01 þ 0:05sinð0:2tÞ;
li ¼ 1; li ¼ 0; : li ¼ 0:8 þ 0:1sinð0:2tÞ;
0 < t < 100s otherwise
0 < t < 50s 20s t < 70s otherwise
The response curves of the relative distance and its rate between two spacecraft under both guidance laws without any parameter tunings are depicted in Figs. 7 and 8, respectively. The control input power of these two control laws are presented in Fig. 9. It follows from Figs. 7 and 8 that the proposed law also achieves high-precision rendezvous while control law (56) has relatively larger steady-state error in the presence of timevarying actuator faults. As the actuators completely failed during 20s t < 70s, the convergence time is relatively longer than that of the fault-free case. These results clearly show that the proposed rendezvous law has strong robustness against actuator faults and therefore is more reliable in real applications.
Fig. 9. Control input power.
The mean position error eq and velocity error ev of 200 Monte-Carlo runs are summarized in Table 3. From this table, one can note that the proposed guidance law shows similar performance as the case without actuator failure, whereas the rendezvous law (56) generates much larger steady-state tracking errors. 5. Conclusions The problem of autonomous spacecraft rendezvous during the closerange closing phase in elliptical orbit is investigated based on finite-time control approach. The proposed formulation is essentially an output feedback control approach, which requires no relative velocity and any priori information of the lumped disturbance. Stability analysis shows that the relative distance and its rate between two neighboring spacecraft can converge to a small region around zero in finite time. Numerical simulations reveal that the proposed rendezvous law has better disturbance rejection performance, strong robustness against actuator faults and smaller steady-state tracking errors than the asymptotical rendezvous law. Future work includes considering control input constraint for the future onboard implementation of the proposed control law. Table 3 Mean position error eq and velocity error ev of 200 Monte-Carlo runs.
Fig. 7. Relative distance. 59
Parameter
eq
ev
Rendezvous law (56) [24] Proposed law (48)
4:1m 0:19m
0:26m=s 0:02m=s
S. He, D. Lin
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