Adaptive nonlinear robust relative pose control of spacecraft autonomous rendezvous and proximity operations

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Adaptive nonlinear robust relative pose control of spacecraft autonomous rendezvous and proximity operations Liang Sun a,n, Wei Huo a, Zongxia Jiao b a The Seventh Research Division Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191 P.R. China b Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191 P.R. China

art ic l e i nf o

a b s t r a c t

Article history: Received 8 March 2016 Received in revised form 20 July 2016 Accepted 28 November 2016

This paper studies relative pose control for a rigid spacecraft with parametric uncertainties approaching to an unknown tumbling target in disturbed space environment. State feedback controllers for relative translation and relative rotation are designed in an adaptive nonlinear robust control framework. The element-wise and norm-wise adaptive laws are utilized to compensate the parametric uncertainties of chaser and target spacecraft, respectively. External disturbances acting on two spacecraft are treated as a lumped and bounded perturbation input for system. To achieve the prescribed disturbance attenuation performance index, feedback gains of controllers are designed by solving linear matrix inequality problems so that lumped disturbance attenuation with respect to the controlled output is ensured in the L2-gain sense. Moreover, in the absence of lumped disturbance input, asymptotical convergence of relative pose are proved by using the Lyapunov method. Numerical simulations are performed to show that position tracking and attitude synchronization are accomplished in spite of the presence of couplings and uncertainties. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Spacecraft control Rendezvous and proximity Robust control Adaptive control Disturbance attenuation

1. Introduction Many space programs such as rendezvous and docking, capturing, salvage and repair, refueling, and debris removal in orbit have been received increasing attention in recent years. A key enabling technology in these missions is autonomous rendezvous and proximity that requires precise relative pose control between two spacecraft. However, the spacecraft relative kinematics and dynamics are highly nonlinear and coupled, thus traditional linear control designs are unsuitable, especially when close distance and large angle relative maneuvers are required. A robust controller based on state dependent Riccati equation technology was designed by [1] for spacecraft relative pose motion in rendezvous and proximity. Relative position control based on phase plane control technique and relative attitude control based on relative quaternion feedback scheme were described in [2], but the uncertainties of the pursue spacecraft were not considered. The integrated position and attitude control problem was considered for spacecraft rendezvous and proximity with parametric uncertainties, bounded disturbances, and measurement noises in n Corresponding author. Present address: The Seventh Research Division, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191 P. R. China E-mail addresses: [email protected] (L. Sun), [email protected] (W. Huo), [email protected] (Z. Jiao).

[3], and ultimate boundedness of the errors was achieved. Liang and Ma [4] proposed a Lyapunov-based adaptive tracking control approach for tracking an arbitrary angular velocity of a tumbling satellite before docking and for stabilizing the rotation of the twosatellite compound system after docking. This method stabilizes the rotational motion only, but the coupled relative translational motion was not considered as in [5]. Singla et al. [6] proposed an output feedback adaptive control to solve the spacecraft autonomous rendezvous and docking problem under measurement uncertainties, but the coupling effect between relative translation and relative rotation was not considered. Xin and Pan [7] addressed a closed-form nonlinear optimal control solution of spacecraft to approach a target spacecraft by using θ-D technique. Although the translational and rotational dynamics coupled with the flexible structure motion were considered in one unified optimal control framework, but the parametric uncertainties and external disturbances were not considered. Then, they researched the same problem in [8,9] and redesigned new optimal controllers with considering modeling uncertainties. In [10], the relative position control problem in spacecraft rendezvous and proximity was converted into a model predictive control optimization problem with considering constraints on thrust magnitude, constraints on spacecraft positioning within line-of-sight cone, and constraints on approach velocity. A command filter based adaptive backstepping controller was developed for spacecraft rendezvous

http://dx.doi.org/10.1016/j.isatra.2016.11.022 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Sun L, et al. Adaptive nonlinear robust relative pose control of spacecraft autonomous rendezvous and proximity operations. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.11.022i

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2

and proximity in [11], but the amount of online estimate parameters is large to increase the computational burden. The relative navigation, guidance and control algorithms of spacecraft rendezvous and proximity was designed based on the analytical closed-form solution of the Tschauner-Hempel equations in [12], where the methods are general and able to translate the chaser spacecraft in any direction and approach to the target spacecraft. In [13], a nonlinear adaptive position and attitude tracking controller was proposed for spacecraft rendezvous and proximity, and the dual quaternions were used to represent the absolute and relative attitude and position. In [14] and [15], the control problem of multiple spacecraft rendezvous and proximity were studied, and a linear quadratic optimal distributed controller was also formulated. Motivated by aforementioned observations, we consider the problem of driving a chaser spacecraft to a fixed position with respect to the target and reorienting the chaser's attitude along with the attitude of target. The new contributions in this work are as follows.

 Compared with the models presented in [16,17], the external



disturbances in uncontrolled target dynamics are considered in this paper, and unknown external disturbances in two spacecraft dynamics are regarded as a lumped and bounded perturbation in system model. Relative position and relative attitude controllers are developed based on an adaptive robust control method, where the controllers have classical proportionalintegral-derivative structure, and adaptive laws are used to estimate the uncertain mass and inertia parameters of the chaser spacecraft and the unknown inertia of the target spacecraft. Compared with the controllers in [1–15], a simple six-degreesof-freedom adaptive robust relative motion controller is designed to achieve spacecraft rendezvous and proximity, and the performance of relative position tracking and attitude synchronization is evaluated by L2-gain from lumped perturbation to controlled output. The proposed robust state feedback controller has positive definite gain matrices whose condition to be satisfied is given by a linear matrix inequality. The closed-loop system is uniformly ultimately bounded stable with the L2-gain less than any given small level. Meanwhile, in the absence of lumped perturbation, asymptotic stability of the closed-loop system is also proved in the Lyapunov framework.

The rest of this paper is arranged as following. In Section 2, mathematical model of the spacecraft proximity maneuvers is derived, and objective of controller design is stated. A detailed designing procedure of adaptive robust controllers and stability analysis are presented in Section 3. Simulation results are displayed in Section 4. Finally, Section 5 concludes the work. Notations Skew symmetric matrix derived from vector a = [a1, a2, a3 ]T ∈ 3 is defined by

⎧ − 1, ai < 0 ⎪ sgn (ai ) = ⎨ 0, ai = 0, ⎪ ai > 0 ⎩ 1,

i = 1, 2, 3.

2. Problem statement 2.1. Dynamics of the chaser and target We investigate a control problem in which a chaser spacecraft tracks a tumbling space target under the influence of disturbances. Frames and vectors are defined in Fig. 1, where i ≜ {Oxi yi zi } denotes the Earth-centered inertial frame, c ≜ {Cxyz} and t ≜ {Txt yt zt } are chaser and target spacecraft body-fixed frames, respectively. The objective of controller design is to control the chaser so that its mass center C tracks point P and frame c tracks frame t . The position of mass center C and the attitude of c with respect to i are given by following kinematics and dynamics expressed in frame c , if the modified Rodrigues parameters(MRP) are used for attitude parametrization [18].

⎧ r ̇ = v − S (ω) r ⎪ ⎪ mv̇ + mS (ω) v + mμr = f + d c ⎨ ⎪ σ ̇ = G (σ ) ω ⎪ Jω̇ + S (ω) Jω = τ + w ⎩ c where G (σ ) =

1 [(1 4

(1)

− σ Tσ ) I3 + 2S (σ ) + 2σσ T ] , μ = μg /∥ r ∥3 ; r ∈ 3

is the position and σ is the MRP attitude; v, ω ∈ 3 are linear and angular velocities; μg is the gravitational constant of the Earth, f , τ ∈ 3 are the control force and torque; dc , wc ∈ 3 are the disturbance force and torque; m ∈  and J ∈ 3 × 3 are the chaser mass and the positive definite symmetric inertia matrix, respectively. With ignoring the control forces and torques, kinematics and dynamics of a tumbling target can be described in frame t as

⎧ rṫ = vt − S (ωt ) rt ⎪ ⎪ mt vṫ + mt S (ωt ) vt + mt μt rt = dt ⎨ ⎪ σṫ = G (σt ) ωt ⎪ J ω̇ + S (ω ) J ω = w ⎩ t t t t t t where

G (σt ) =

1 [(1 4

−

σtT σt ) I3

(2) + 2S (σt ) +

2σt σtT ],

μt = μg /∥ rt ∥3;

rt ∈ 3 and σt are position and attitude of the target; vt , ωt ∈ 3 are linear and angular velocities of the target; dt , wt ∈ 3 are the disturbance force and torque; mt ∈  and Jt ∈ 3 × 3 are mass and inertial matrix of the target, respectively. Remark 1. When the external disturbances dt = 0 , and wt = 0 in (2), then the undisturbed dynamics of the target can be written as vṫ = − S (ωt ) vt − μt rt and Jt ω̇ t = − S (ωt ) Jt ωt . The target's orbit

⎡ 0 − a3 a2 ⎤ ⎢ ⎥ S (a) = ⎢ a3 0 − a1⎥ ∈ 3 × 3. ⎢⎣ − a2 a1 0 ⎥⎦ it satisfies ∥ S (a )∥ = ∥ a ∥, aTS (a ) = 0 , and S (a ) b = − S (b) a , bTS (a ) b = 0 for any b ∈ 3. ∥ a ∥ ≤ ∥ a ∥1, where ∥ a ∥ and ∥ a ∥1 denote vector 2-norm and 1-norm, respectively. A > 0 denotes A being positive definite, and ∥ A ∥ is the induced matrix 2-norm. I3 and O3 are 3  3 unitary and zero matrices, respectively. sgn (a ) ≜ [sgn (a1) , sgn (a2 ) , sgn (a3 )]T for any a ∈ 3, where the standard signum function is defined by

Fig. 1. Definitions of frames and vectors.

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potential energy is −μg /∥ rt ∥ with defining the potential energy zero at infinity [18], then the total orbit energy of the target is 1 E1 = 2 vtT vt − μg /∥ rt ∥, while the target's angular kinetic energy is 1 E2 = 2 ωtT Jt ωt . It can be directly computed from the target's undisturbed dynamics that E1̇ (t ) = 0 and E2̇ (t ) = 0, this means E1 (t ) ≡ E1 (0) and E2 (t ) ≡ E2 (0), thus the target's velocity and angular velocity satisfy ∥ vt (t )∥ < ∞ and ∥ ωt (t )∥ < ∞ in the absence of external disturbances.

3

Rvṗ t = R [vṫ + S (ω̇ t ) pt ] 1 Rdt − μt Rrt − RS (pt ) ω̇ t mt 1 = − S (Rωt )[Rvpt − RS (ωt ) pt ] + Rdt mt μg (r − re − Rpt ) − − RS (pt ) ω̇ t ∥ R T (r − re ) − pt ∥3 = − RS (ωt ) vt +

= − S (ω − ωe )[v − ve − S (ω − ωe ) Rpt ] + Remark 2. The external disturbances dc , wc , dt , and wt in (1) and (2) mainly include the influence of atmospheric drag, solar radiation, mass distribution of the Earth and third-body perturbing forces [19]. In addition, the vibration of flexible appendages on spacecraft can be also treated as the disturbances inputs for the spacecraft [20].

−

μg (r − re − Rpt ) ∥ R T (r − re ) − pt ∥3

1 Rdt mt

− RS (pt ) ω̇ t

(8)

and ω̇ t is calculated from (2) and (6) as

ω̇ t = − Jt−1 S (ωt ) Jt ωt + Jt−1 wt = − Jt−1 S (R T (ω − ωe )) Jt R T (ω − ωe ) + Jt−1 wt Thus, equation (7) can be rewritten as

2.2. Dynamics of the relative motions Let the rotation matrix from t to c be [18]

R = I3 −

(9)

4 (1 − σeT σe ) 8 S (σe ) + S 2 (σe ) (1 + σeT σe )2 (1 + σeT σe )2

(3)

⎧ rė = ve − S (ω) re ⎪ ⎪ mvė = − mg + n1 + d1 + f ⎨ ⎪ σė = G (σe ) ωe ⎪ Jω̇ = − S (ω) Jω − JS (ω) ω + n + d + τ ⎩ e e 2 2

(10)

where and the MRP relative attitude σe is defined by

σe =

σt

(σ Tσ

− 1) + σ (1 − σtT σt ) − 1 + σtT σt σ Tσ + 2σtT σ

g = S (ω) ve + S 2 (ω − ωe ) Rpt + μr − 2S (σt ) σ (4)

According to Fig. 1, the position and velocity of point P with respect to O represented in frame t are given by

⎧ rpt = rt + pt ⎨ ⎩ vpt = vt + S (ωt ) pt

(5)

where pt ∈ 3 is a constant vector in frame t . The relative position, relative velocity, and relative angular velocity are defined in frame c as

(6)

Substituting (6) and (1) and using identities Ṙ = − S (ωe ) R, rṗ t = vpt − S (ωt ) rpt and R−1 = RT yield the relative motion equations in frame c as ⎧ rė = ve − S (ω) re ⎪ ⎪ mvė = − m [S (ω) v + μr + Rv ̇pt − S (ωe )(v − ve )] + f + d c ⎨ ⎪ σė = G (σe ) ωe ⎪ ⎩ Jω̇ e = − S (ω) Jω − J [Rω̇ t + S (ω) ωe ] + τ + wc

kinematic

,

m Rdt + mRS (pt ) Jt−1 wt , d2 = wc − JRJt−1 wt , mt

n1 = − mRS (pt ) Jt−1 S (R T (ω − ωe )) Jt R T (ω − ωe ), n2 = JR Jt−1 S (R T (ω − ωe )) Jt R T (ω − ωe ). Remark 3. It is worthy noticing that, terms S (ω)re , n1, and d1 in (12) reflect that relative translational motion between two spacecraft is affected greatly by relative rotational motion. This indicates the natural couplings in spacecraft rendezvous and proximity system. d1 and d2 denote the lumped disturbances in the relative position and relative attitude dynamics. In (12), following assumptions are employed in the subsequent development.

⎧ re = r − Rrp t ⎪ ⎨ ve = v − Rvpt ⎪ ⎩ ωe = ω − Rωt

where

d1 = d c −

μg (r − re − Rpt ) ∥ R T (r − re ) − pt ∥3

Jacobian

matrix

G (σe ) =

Assumption 2. External disturbances dc , wc , dt , and wt are unknown but they satisfy d c (t ) ∈ L2 [0, T ], wc (t ) ∈ L2 [0, T ], dt (t ) ∈ L2 [0, T ],

(7) 1⎡ 4⎣

(

1−

σeT σe

)

and

wt (t ) ∈ L2 [0, T ], that is T

T

I3+

T

∫0 ∥ dc (t )∥2 dt < ∞, ∫0 ∥ wc (t )∥2

dt < ∞ , ∫ ∥ dt (t )∥2 dt < ∞ , and ∫0T 0

2S (σe ) + 2σe σeT ⎤⎦ is nonsingular [18]; Rvṗ t is calculated from (5), (2), (6) and RS (a ) = S (Ra ) R for any a ∈ 3 as

Assumption 1. Chaser's mass m is an unknown positive constant ¯ with two known constants and bounded by m0 ≤ m ≤ m ¯ > 0; chaser's inertial matrix J is an unknown positive m0 > 0, m definite symmetric constant matrix, but there exist two known constants λ 0 > 0 and λ¯ > 0 such that λ 0 ≤ ∥ J ∥ ≤ λ¯ ; target's mass mt is an unknown bounded constant, and its inertial matrix Jt is an unknown bounded positive definite symmetric matrix.

∥ wt (t )∥2 dt < ∞

at any time T.

Assumption 3. The chaser can directly measure its motion variables {r , v , σ , ω} and relative motion variables {re, ve, σe, ωe } from the measurement devices mounted on the chaser's body [21–23]. However, the target's motion variables {rt , vt , σt , ωt } are assumed to be unavailable directly for the chaser. Remark 4. Since external disturbances dc , wc , dt , and wt are

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4

bounded and belong to L2 [0, T ] in Assumption 2, then from constant vector pt and ∥ R ∥ = 1, we know lumped disturbances satisfy

1⎡ ⎣ k3 G T (σe ) σe + K4 ωe ⎤⎦ − ki2 ξ2 β2 − Yθ^ − n^2 ∥ ω − ωe ∥2 sgn (β1σe + β2 ωe )

τ= −

T

d1 (t ) ∈ L2 [0, T ] and d2 (t ) ∈ L2 [0, T ] such that ∫ ∥ d1 (t )∥2 dt < ∞ 0 and

T

∫0 ∥ d2

(t )∥2

dt < ∞ at any time T.

where t

ξ1 = ∫ [re + 0

2.3. Control objective In this work, the objective of control design is to achieve relative position tracking and attitude synchronization under Assumptions 1–3, such that the chaser is driven at the desired proximity position pt and its attitude is synchronized with the target. The robust adaptive controller is designed to achieve tracking performance in the L2-gain sense. More specifically, for a given level of disturbance attenuation γ > 0, the L2-gain from the T lumped disturbance input d = ⎡⎣ d1T, d2T ⎤⎦ to the system output T z = ⎡⎣ reT, veT, σeT, ωeT ⎤⎦ is used to evaluate the relative motion control performance, there exists a control law u = [f T , τ T ]T such that the closed-loop system satisfies the attenuation level of lumped disturbance d with respect to system output z is ensured in the L2-gain sense:

∫0

T

∥ z (t )∥2 dt ≤ γ 2

∫0

T

∥ d (t )∥2 dt ,

(13b)

∀T>0

(11)

Moreover, when the external disturbances dc , wc , dt , and wt are absent in the relation motion model, that is d = 0 , then it is guaranteed that the closed-loop system output z converges to zero asymptotically.

α2 S (ω) re ] dt , α1

> 0 (i = 1, 2) , ^ , n^ , > 0; m 1 respectively. Substituting

K4T

t

ξ2 = ∫ [σe + 0

β2 (I β1 3

− G ) ωe ] dt; αi > 0, βi

k1 > 0, k3 > 0, k i1 > 0, k i2 > 0, K2 = K2T > 0, K4 = n^2 , and θ^ are estimations of m, n1, n2, and θ , (13) into (12) yields the closed-loop system

rė = ve − S (ω) re k1 1 re − K2 ve − ki1ξ1 α2 α2 α1 ^ ˜ g + h1 + d1 + mS (ω) re + m α2

(14a)

mvė = −

σė = G (σe ) ωe

Jω̇ e = −

(14b)

(14c)

k3 T 1 G (σe ) σe − K4 ωe β2 β2

− ki2 ξ2 − Yθ˜ + h2 + d2

(14d)

^ − m and θ˜ = θ^ − θ, h1 = n1 − n^1 ∥ p ∥ ∥ ω − ωe ∥2 ˜ =m where m t sgn (α1re + α2 ve ) , h2 = n2 − n^2 ∥ ω − ωe ∥2 sgn (β1σe + β2 ωe ) .

3. Control design and stability analysis Select a function. Denote the chaser inertia matrix

V=

⎡J J J ⎤ ⎢ 11 12 13 ⎥ J = ⎢ J12 J22 J23⎥ ⎥ ⎢J ⎣ 13 J23 J33⎦ With introducing a linear operator a = [a1, a2, a3 ]T as

L (a ) for any vector

α k1 T k re re + 2 mveT ve + α1mreT ve + 3 σeT σe 2 2 2 β 1 2 1 ˜T ˜ ˜ + + 2 ωeT Jωe + β1σeT Jωe + θ θ m 2 2δ 1 2δ 2 α + α2 ki1reT ξ1 + 1 ki1ξ1T ξ1 + β2 ki2 σeT ξ2 2 β1 1 2 1 2 T + ki2 ξ2 ξ2 + n˜1 + n˜ 2 2 2η1 2η2

(15)

⎡ a1 0 0 0 a3 a2 ⎤ ⎢ ⎥ L (a) ≜ ⎢ 0 a2 0 a3 0 a1⎥ ⎢⎣ 0 0 a3 a2 a1 0 ⎥⎦

where δ i > 0, ηi > 0, n˜ i = n^i − ni (i = 1, 2). Obviously, from Assumption 1, we have

we have Ja = L (a ) θ , where θ ≜ [J11 , J22 , J33 , J23 , J13 , J12 ]T . Then, Eq. (12) turns into

V≥

⎧ rė = ve − S (ω) re ⎪ ⎪ mvė = − mg + f + n1 + d1 ⎨ ⎪ σė = G (σe ) ωe ⎪ Jω̇ = Yθ + τ + n + d ⎩ e 2 2

(16)

where z¯ = [reT, veT, ξ1T , σeT, ωeT , ξ2T ]T , Λ = diag {Λ1, Λ2 },

(12)

with Y = L (S (ωe )(ω − ωe )) − S (ω) L (ω). Since the target's inertial matrix Jt is unknown but constant in Assumption 1, then we can define two unknown constants n1 = m ∥ Jt−1 ∥∥ Jt ∥ and n2 = ∥ J ∥∥ Jt−1 ∥∥ Jt ∥. Design control inputs

1⎡ ⎤ ^ ⎣ k1re + K2 ve − α1mS (ω) re ⎦ − ki1ξ1 α2 ^ g − n^ ∥ p ∥∥ ω − ω ∥2 sgn (α r + α v ) +m 1 e 1 e 2 e t

1 T 1 2 1 ˜T ˜ 1 2 1 2 ˜ + z¯ Λz¯ + m θ θ+ n˜1 + n˜ 2 2 2δ 1 2δ 2 2η1 2η2

⎡ k1I3 α1m0 I3 α2 ki1I3⎤ ⎢ ⎥ Λ1 = ⎢ α1m0 I3 α2 m0 I3 0 ⎥, ⎢⎣ α k I α1ki1I3 ⎥⎦ 0 2 i1 3 ⎡ k3 I3 β1λ 0 I3 β2 ki2 I3⎤ ⎢ ⎥ Λ2 = ⎢ β1λ 0 I3 β2 λ 0 I3 0 ⎥. ⎢ ⎥ 0 β1ki2 I3 ⎦ ⎣ β2 ki2 I3

f= −

(13a)

Thus, V ≥ 0 if Λ > 0. ^ ̇ θ˜ ̇ = θ^ ,̇ n˜1̇ = n^1̇ , and n˜ 2̇ = n^2̇ yields the time ˜ ̇ = m, Recalling m derivative of V as

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disturbance attenuation performance and stability for the 6DOF closed-loop system with the proposed controllers will be studied in the following theorem.

V̇ = k1reT rė + α2 mveT vė + α1mrė T ve + α1mreT vė +

k3 σeT σė

β2 ωeT Jω̇ e

+

+

β1σė T Jωe

+

β1σeT Jω̇ e

+ α2 ki1rė T ξ1 + α2 ki1reT ξ1̇ + α1ki1ξ1T ξ1̇ + β2 ki2 σė T ξ2 + β2 ki2 σeT ξ2̇ + β1ki2 ξ2T ξ2̇ +

̇ 1 ^̇ 1 1 1 ˜ + θ˜ Tθ^ + n˜1n^1̇ + n˜ 2T n^2̇ mm δ1 δ2 η1 η2

Substituting (14) into (17) and using the fact to

V̇ = −

(17) reT S (ω) re

= 0 give rise

+

β1

(20)

Proof. By the completion of square from (19), we obtain

V̇ + ∥ z ∥2 − γ 2 ∥ d ∥2 ≤ − zTΣz +

ki2 σeT (I3 − G) ωe

− γ2 d −

̇ 1 ^̇ 1 1 1 ˜ + θ˜ Tθ^ + n˜1n^1̇ + n˜ 2T n^2̇ mm δ1 δ2 η1 η2

1 Wz 2γ 2

2

1 T T z W Wz 4γ 2

⎛ ⎞ 1 + zTz ≤ − zT ⎜ Σ − I12 − W TW ⎟ z 2 4γ ⎝ ⎠

for any d ∈ L2 [0, T ]. Therefore, condition (20) implies

+ (α1re + α2 ve )T h1 + (β1σe + β2 ωe )T h2

V̇ ≤ γ 2 ∥ d ∥2 − ∥ z ∥2 ≤ 0

+ (α1re + α2 ve )T d1 + (β1σe + β2 ωe )T d2

Thus, the control objective (11) is achieved by integrating both sides of inequality (21) with respect to any time interval [0, T ]. This indicates the L2-gain of the closed-loop system is less than or equal to γ [25]. With regard to the case dc = 0, wc = 0, dt = 0 , and wt = 0 , that is d = 0 , equation (19) becomes

Since (α1re + α2 ve )T h1 ≤ − n˜1 ∥ pt ∥∥ ω − ωe ∥2 ∥ α1re + α2 ve ∥1 and (β1σe + β2 ωe )T h2 ≤ − n˜ 2 ∥ ω − ωe ∥2 ∥ β1σe + β2 ωe ∥1, then adaptive updating laws for the unknown parameters can be assigned as

⎧ ^̇ T T ⎪ m = − δ1 [(α1re + α2 ve ) g + α1ve S (ω) re ] ⎪ ^̇ ⎪ θ = δ2 Y T (β1σe + β2 ωe ) ⎨ ⎪ n^ ̇ = η ∥ p ∥∥ ω − ω ∥2 ∥ α r + α v ∥ e 1 e 2 e 1 t 1 ⎪ 1 ⎪ n^ ̇ = η ∥ ω − ω ∥2 ∥ β σ + β ω ∥ ⎩ 2 e 2 1 e 2 e 1 As

1 W TW ≥ 0 4γ 2

Moreover, the state variable of the closed-loop system becomes z → 0 as t → ∞ for any initial state when d = 0 .

˜ [(α1re + α2 ve )T g + α1veT S (ω) re ] +m β − 1 k3 σeT Gσe − ωeT (K4 − β1G TJ ) ωe β2 β − 1 σeT K4 ωe − (β1σe + β2 ωe )T Yθ˜ β2 β22

Theorem 1. Consider the spacecraft proximity maneuvers model (12) under Assumptions 1–3, given the positive design parameters α1, α2, β1, β2, δ1, δ2 and γ, the closed-loop system of (14) with controller (13) and adaptive law (18) satisfies the L2-gain less than or equal to γ from lumped disturbance input d to system output z , if feedback gains satisfy linear matrix inequalities

Λ > 0, Σ > 0, Σ − I12 −

α1 T α k1re re − veT (K2 − α1mI3 ) ve − 1 reT K2 ve α2 α2

+ α2 ki1reT re + β2 ki2 σeT σe +

5

noted in

∥ G (σe )∥ =

1 (1 4

V ̇ ≤ − z T Σz

(18)

[24], Jacobian 1 4

+ σeT σe ) ,

matrix G (σe ) has properties 1 1 ≤ ∥ G (σe )∥ ≤ 2 , σeT G (σe ) = 4 (1 + σeT σe ) σeT ,

3

and ∥ I3 − G (σe )∥ ≤ 2 , then recalling Assumption 1 and adaptive laws (18) results in

α1 T α k1re re − veT (K2 − α1mI3 ) ve − 1 reT K2 ve α2 α2 β β − 1 k3 σeT Gσe − ωeT (K4 − β1G TJ ) ωe − 1 σeT K4 ωe β2 β2

V̇ ≤ −

+ α2 ki1reT re + β2 ki2 σeT σe +

3β22 2β1

(19)

where Σ = diag {Σ1, Σ2 }, α1 ⎡ α1 k I − α k I ⎤ K 2 i1 3 2α 2 2 ⎢ α2 1 3 ⎥ Σ1 = ⎢ α1 ⎥, ¯ K K α mI − 2 1 3⎦ ⎣ 2α 2 2

2

1

β1 K 2β2 4

−

K4 −

3β22

⎤

β1 λ¯I 2 3

⎥, ⎥ ⎦

and V̇ (t ) ≤ 0 if Σ > 0. Since V (t ) ≥ 0 and V̇ (t ) ≤ 0, then V(t) is monotonically decreasing along the closed-loop system trajectory and bounded by zero. Hence, V(t) has a finite limit as t → ∞ and

0 ≤ V (∞) ≤ V (t ) ≤ V (0) < ∞, ∀ t ≥ 0 then from (16), we have

0 ≤ λm (Λ)∥ z¯ ∥2 +

∫0

+ (α1re + α2 ve )T d1 + (β1σe + β2 ωe )T d2

⎡ β1 ⎢ 4β2 k3 I3 − β2 ki2 I3 Σ2 = ⎢ ⎢ β1 K − 3β22 k I ⎣ 2β 4 4β i2 3

(22)

1 ∥ ϑ ∥2 ≤ V (t ) < ∞, ∀ t ≥ 0 2ϵ

where λm (Λ) is the minimum eigenvalue of Λ, ϵ = min {δ1, δ2, η1, η2 } ˜ , θ˜ T , n˜1, n˜ 2 ]T . Thus, z¯ and ϑ are uniformly bounded. and ϑ = [m According to (22, we know

ki2 σeT ωe

≤ − z T Σ z + z T W Td

(21)

k I 4β1 i2 3⎥

⎡ α1I3 α2 I3 O3 O3 ⎤ ⎥. W=⎢ ⎣ O3 O3 β1I3 β2 I3⎦ Although above robust adaptive controllers for relative translation and relative rotation are designed independently, the

∞

∥ z¯ ∥2 dt ≤ −

1 λm (Λ)

∫0

∞

V̇ (t ) dt ≤

V (0) <∞ λm (Λ)

This means z¯ is square integrable, thus z is also square integrable. When dt = 0 and wt = 0 , it is known that ∥ vt ∥ < ∞ and ∥ ωt ∥ < ∞ from Remark 1. Recalling the boundedness of z and ϑ , it follows that ∥ v ∥ = ∥ ve + R [vt + S (ωt ) pt ]∥ < ∞ and ∥ ω ∥ = ∥ ωe + Rωt ∥ < ∞ from (5) and (6), it means v and ω are uniformly bounded. Furthermore, we know f and τ are uniformly bounded from (13), thus z ̇ is uniformly bounded from ((14), this means z is uniformly continuous. According to the Babarlat's Lemma [25], we conclude □ z → 0 as t → ∞. Remark 5. Theorem 1 ensures the uniform ultimate bounded stability of the relative position and relative attitude, and the boundedness of the estimated parameters. However, convergence of the estimated parameters to the real values is not guaranteed. Moreover, in order to achieve better disturbance attenuation performance, theoretically we can select infinitely small γ. However, such a small choice of γ allows excessive large control force and

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torque inputs. Since saturation for actuators is inevitable in practical problems, trade-off is required between choice of smaller γ and practical tracking performance. The proper value of γ will only be founded by trial-and-error through the simulations. Remark 6. It is obvious that the key of the controller development is deriving feasible solution of the LMIs in (20). Generally, the problem of feasible solution of the LMIs in (20) can be formulated as determining the positive definite property of real-symmetric partitioned matrices Λ, Σ , and Σ − I12 −

1 4γ 2

WTW . Before choosing

the proper controller gains k1, K2, k3, K4, k i1, and ki2, we should firstly set parameters α1, α2, β1, β2, γ . Then, with known constants ¯ , λ¯ , we can easily decide the positive definite property of m0 , λ 0 , m matrices Λ1, Λ2 , Σ1, Σ2, and Σ − I12 −

1 4γ 2

WTW to obtain the tuning

rules of controller gains. Namely, if following simple matrices Γi (i = 1, … , 6) are positive definite, then Λ, Σ , and

Σ − I12 −

1 4γ 2

WTW are also positive definite, such as

⎡ k1 α1m0 α2 ki1⎤ ⎢ ⎥ Γ1 = ⎢ α1m0 α2 m0 0 ⎥ ∈ 3 × 3, Γ2 = ⎢⎣ α k α1ki1⎥⎦ 0 2 i1 3 × 3,

Fig. 3. Response of relative position under perturbation-free.

⎡ k3 β1λ 0 β2 ki2 ⎤ ⎢ ⎥ 0 ⎥∈ ⎢ β1λ 0 β2 λ 0 ⎢ ⎥ β1ki2 ⎦ ⎣ β2 ki2 0

α1 ⎡ α1 k − α k λ (K ) ⎤ 2 i1 2α 2 m 2 ⎢ α2 1 ⎥ Γ3 = ⎢ α1 ∈ 2 × 2, ¯⎥ ⎣ 2α λm (K2 ) λm (K2 ) − α1m⎦ 2

⎡ β1 k − β2 ki2 ⎢ 4β2 3 ⎢ Γ4 = ⎢ β1 λ (K ) − 3β22 k ⎣ 2β2 m 4 4β1 i2

β1 λ (K4 ) 2β2 m

−

λm (K4 ) −

3β22

⎤

β1 ¯ λ 2

⎥ ∈ 2 × 2, ⎥ ⎦

k 4β1 i2 ⎥

⎤ ⎡ α1 α1 1 2 λ (K ) − α1α2 ⎥ ⎢ α 2 k1 − α2 ki1 − 1 − 4γ 2 α1 2α 2 m 2 Γ5 = ⎢ ⎥ ∈ 2 × 2, α1 1 2 ¯ − 1 − 2 α2 ⎥ λ α α λ α ( ) − ( ) − K K m ⎢ m m 2 1 2 2 1 2α 2 ⎦ ⎣ 4γ ⎤ ⎡ β1 3β22 β1 1 2 ⎢ 4β2 k3 − β2 ki2 − 1 − 4γ 2 β1 2β2 λm (K4 ) − 4β1 ki2 − β1β2 ⎥ ⎥ ∈ 2 × 2, Γ6 = ⎢ 3β22 ⎢ β1 β1 ¯ 1 2⎥ λ β β λ λ β ( ) − − ( ) − − − K k K 1 m 4 1 2 ⎢⎣ 2β2 m 4 4β1 i2 2 ⎦ 4γ 2 2 ⎥ where λm (K2 ) and λm (K4 ) denotes minimum eigenvalues of matrices K2 and K4, respectively. Thus, we can easily derive order principal minor determinant of matrices Γi to obtain tuning rules of controller gains. Then, based on commercial software Matlab, we can find many feasible solutions for characteristic polynomials to determine the controller gains. Specially, because of the proposed controller has typical structure of proportional-integralderivative controller, then k1, K2, k3, K4, k i1, and ki2 can be chosen to

Fig. 4. Control input (13) under perturbation-free.

satisfy the basic rules of classic proportional-integral-derivative control in the set of feasible solutions.

4. Simulation and discussion The performance of the proposed controller is discussed in this numerical example. For this purpose, we simulate a scenario of autonomous spacecraft proximity mission in orbit. The physical parameters about chaser and target are

m = 58.2 (kg), mt = 1425.6 (kg), ⎡ 598.3 − 22.5 − 51.5⎤ ⎢ ⎥ J = ⎢ − 22.5 424.4 − 27 ⎥ (kgm2), ⎣ − 51.5 − 27 263.6 ⎦ ⎡ 3336.3 − 135.4 − 154.2⎤ ⎢ ⎥ Jt = ⎢ − 135.4 3184.5 − 148.5⎥ (kgm2). ⎣ − 154.2 − 148.5 2423.7 ⎦

Fig. 2. Response of relative attitude under perturbation-free.

The desired proximity position expressed in frame t is pt = [0, 5, 0]T (m). The initial conditions for chaser motion and relative motion are respectively set as r (0) = [1, 1, 1]T × 7.078 ×106 (m) , v (0) = [2, 3, − 2]T × 103 (m/s) , σ (0) = 0, ω (0) = 0 (rad/ s) , re (0) = [50/ 2 , 0, − 50/ 2 ]T (m) , ve (0) = [0.5, − 0.5, 0.5]T (m/ . s) , σe (0) = [0.5, − 0.6, 0.7]T , ωe (0) = [0.02, − 0.02, 0.02]T (rad/s) Prior known parameters about chaser for controller design are ¯ = 60, λ 0 = 3000 and λ¯ = 3500. m0 = 50, m

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Fig. 7. Response of relative attitude under perturbation. Fig. 5. Response of estimation θ^ under perturbation-free.

are definitely influenced by these unknown external disturbances, and the controller with disturbance attenuation performance should be used for the spacecraft rendezvous and proximity. 4.2. Performance under external perturbation Then, we examine the proposed controller (13) performance for closed-loop system with external disturbances. Since the distance between chaser and target are small with respect to the radius of two spacecraft, thus their external disturbances can be modeled similarly. Here the external disturbance inputs are treated as the following form: ⎡ 0.3 − 0.2 sin (ωo t ) − 0.4 cos (ωo t ) ⎤ ⎢ ⎥ d c = d t = ⎢ 0.3 − 0.3 sin (ωo t ) + 0.2 cos (ωo t ) ⎥ + 0.1 × rand (3, 1)(N) , ⎢ 0.2 + 0.4 sin (ω t ) + 0.2 cos (ω t ) ⎥ ⎣ o o ⎦ ⎡ 0.06 − 0.04 sin (ωo t ) + 0.05 cos (ωo t ) ⎤ ⎢ ⎥ wc = wt = ⎢ 0.07 + 0.05 sin (ωo t ) − 0.04 cos (ωo t ) ⎥ + 0.01 × rand (3, 1)(Nm) , ⎢ 0.04 − 0.03 sin (ω t ) + 0.03 cos (ω t ) ⎥ ⎣ o o ⎦

^ , n^1, n^2 under perturbation-free. Fig. 6. Response of estimations m

4.1. Performance under perturbation-free First, we use the proposed controllers (13) to show the effectiveness of controllers when d = 0 . The adaptive controller parameters are chosen to satisfy only the conditions Λ > 0 and Σ > 0 in (20). The controller gains are set as αi = βi = 1 (i = 1, 2) , k1 = 5, K2 = 65I3, k3 = 100, K4 = 180I3, k i1= ^ (0) = n^ (0) = n^ (0)= 0.8, k = 0.3, δ = δ = 0.02, η = η = 0.1, m i2

1

2

1

2

1

2

0, θ^ (0) = 0 . Figs. 2–6 show the responses of relative position and relative attitude motion by using controllers (13), respectively. Figs. 5 and 6 show the bounded estimations for unknown parameters. As can be seen from Fig. 2–3, controller (13) has good transient response and steady-state performance. Thus the proposed controller can guarantee that the chaser tracks the target effectively in the absence of external disturbances. Since the external perturbation is absent, then the solution of linear matrix inequalities (20) is simplified as only choosing positive definite matrices Λ and Σ , so that smaller feedback gains k1, K2, k3, K4, ki1, and ki2 can be set directly to achieve our control objective. Because smaller feedback gains yield smaller control effort in Fig. 4, then the convergence rate in Figs. 2 and 3 is slow. Moreover, in more practical situation, there are various external disturbances in space, such as atmospheric drag, solar radiation, mass distribution of the Earth and third-body perturbing forces. Thus the dynamics of two spacecraft

where ωo = μg /∥ r ∥3 , μg = 3.986 × 1014 (m3/s2), and rand (3, 1) denotes the 3  1 vector of random zero-mean Gauss white noise signal. The adaptive controller parameters are chosen to satisfy conditions in (20 when γ = 0.4 , and the controller design parameters are set as α1 = 0.1, α2 = 1, β1 = 0.1, β2 = 1 and feedback gains of the controllers are derived by solving LMIs in (20) as k1 = 11, K2 = 200I3, k3 = 100, K4 = 200I3, k i1 = 0.9, k i2 = 0.6, δ1 = δ2=

^ (0) = n^ (0) = n^ (0) = 0, θ^ (0) = 0 . 0.02, η1 = η2 = 0.1, m 1 2 The relative attitude time histories are shown in Fig. 7 including the relative attitude and relative angular velocities. Fig. 8 shows the responses of relative position and relative velocity. As can be seen from Fig. 7–10, attitude synchronization and desired position are reached in about 100 (s) by using controller (13), this indicates the good tracking control performance. The control forces and the control torques present in Fig. 9 show that the initial control efforts are large in order to drive the chaser to the desired position and attitude quickly. Figs. 10 and 11 show the bounded estimation of the unknown parameters. The uniformly ultimately bounded of the closed-loop system can be guaranteed. These simulation results demonstrate that the control strategies proposed in this research is able to track the position and attitude of the target. Furthermore, to quantitatively describe the performance of the proposed controller (13), we firstly consider the weak disturbance attenuation capability of the controller under γ = 0.8 to compare with the ones under γ = 0.4 , then we can design the controller parameters are

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^ , n^1, n^2 under perturbation. Fig. 11. Response of estimations m Fig. 8. Response of relative position under perturbation.

Fig. 9. Control input (13) under perturbation.

Fig. 12. Comparison of controller performance under different γ.

Fig. 13. Response of relative attitude with controller in [16]. Fig. 10. Response of estimation θ^ under perturbation.

α1 = 0.05, α2 = 0.5, β1 = 0.05, β2 = 0.5, k1 = 3, K2 = 50I3, k3 = 100, K4 = 100I3, k i1 = 0.45 , and k i2 = 0.3 based on LMIs in (20. Thus, simulation results of the proposed controller with different disturbance attenuation index are shown in Fig. 12. As can be seen in Fig. 12, smaller disturbance attenuation index results in better transient response performance with smaller dynamic overshoot and shorter transient response time. Secondly, to show the advantages of the proposed controller (13), the simulation results

using controller in existing article [16] are presented in Figs. 13 and 14. Then, comparing them with Figs. 7 and 8, we can conclude the proposed controller (13) is better than the controller in [16] with smaller dynamic overshoot and shorter transient response time. 5. Conclusion State feedback robust adaptive controllers are proposed in this paper to solve the control problem of spacecraft proximity

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[3] [4] [5]

[6]

[7] [8] [9] [10]

Fig. 14. Response of relative position with controller in [16].

maneuvers under modeling uncertainties, where parametric uncertainties and unknown couplings are compensated online by adaptive update laws, external disturbances are attenuated to satisfy the prescribed level of L2-gain. It has been concluded that the chaser can tracks the target position and attitude precisely by using the designed controller in spacecraft rendezvous and proximity. The performance of the proposed controller has been discussed through numerical studies. In future, the flexible vibration of two spacecraft is further considered and dealt with the proposed adaptive robust controller to achieve spacecraft rendezvous and proximity mission.

[11] [12]

[13]

[14]

[15]

[16]

[17]

Acknowledgements This work was supported by the China Postdoctoral Science Foundation under Grant No. 2016M590031, the National Key Development Program for Basic Research of China under Grant No. 2012CB821204, and the National Natural Science Foundation of China under Grant Nos. 61134005 and 61327807. The author would like to thank the anonymous reviewers and the associate editor for the helpful comments on the paper.

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