- Email: [email protected]
Robust formation flying control for a team of satellites subject to nonlinearities and uncertainties
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.1 (1-9)
Aerospace Science and Technology ••• (••••) ••••••
1
67
Contents lists available at ScienceDirect
68
2 3
Aerospace Science and Technology
4
69 70 71
5
72
6
www.elsevier.com/locate/aescte
7
73
8
74
9
75
10
76
11 12 13
Robust formation flying control for a team of satellites subject to nonlinearities and uncertainties
16 17 18 19 20
Hao Liu
a,b,∗
, Yu Tian
a, b
, Frank L. Lewis
c ,d
c
, Yan Wan , Kimon P. Valavanis
e
a
School of Astronautics, Beihang University, Beijing 100191, PR China b Key Laboratory of Spacecraft Design Optimization and Dynamic Simulation Technologies of Ministry of Education, Beihang University, Beijing 100191, PR China c University of Texas at Arlington Research Institute, University of Texas at Arlington, Fort Worth, TX 76118, USA d State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, PR China e Department of Electrical and Computer Engineering, University of Denver, Denver, CO 80208, USA
a r t i c l e
i n f o
27 28
31 32 33 34
83 84 85 86
90
Article history: Received 15 December 2018 Received in revised form 17 January 2019 Accepted 3 October 2019 Available online xxxx
29 30
82
89
a b s t r a c t
24 26
81
88
22
25
79
87
21 23
78 80
14 15
77
Keywords: Formation control Robust control Nonlinear system Uncertain system Satellite
This paper addresses the problem of robust formation controller design for a group of satellites the dynamics of which involve nonlinearities and uncertainties. A formation flying controller is proposed for the satellite group, which includes a position controller to form the desired accurate formation and an attitude controller to align the satellite attitudes. It is shown that the trajectory and attitude tracking errors of the overall closed-loop control system can converge into a given neighborhood of the origin in a finite time. Numerical simulation studies are provided to demonstrate the advantages of the proposed formation control scheme. © 2019 Published by Elsevier Masson SAS.
91 92 93 94 95 96 97 98 99 100
35
101
36
102
37
103
38
104 105
39 40
1. Introduction
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57
In recent years, formation flying of a team of satellites has attracted attention in the field of aerospace. The use of a multisatellite formation flying system, instead of a single satellite, can effectively improve the overall system performance and benefit a range of space missions, such as large-scale space environment monitoring, deep space exploration, electronic reconnaissance, and commercial communications, as illustrated in [1–4]. With the trend on continuous improvement of aerospace mission efficiency, satellite formation flying has become an important application prospect in cooperative tasks, due to advantages such as high reliability, low cost, and strong adaptability. However, the dynamics of each satellite is highly nonlinear, and the uncertainties involved in the satellite dynamic model and the corresponding complex environments can influence the performance of the flight control system. The satellites are subject to parameter uncertainties and external disturbances - such as the Earth’s non-spherical gravity, the solar
58 59 60 61 62 63 64 65 66
*
Corresponding author at: School of Astronautics, Beihang University, Beijing 100191, PR China. E-mail addresses: [email protected] (H. Liu), [email protected] (Y. Tian), [email protected] (F.L. Lewis), [email protected] (Y. Wan), [email protected] (K.P. Valavanis). https://doi.org/10.1016/j.ast.2019.105455 1270-9638/© 2019 Published by Elsevier Masson SAS.
radiation pressure, and the third body gravity, which should not be ignored when high performance control is required - that are difficult to measure accurately, as illustrated in [5]. It is for this reason that robust formation control problems with the aim to minimize or limit the influence of uncertainties associated with the functionality of a group of satellites witness increased interest in recent years. In the past two decades, several classical formation control strategies including the leader-follower approach, the behavioral strategy, and the virtual structure method have been proposed (see, e.g., [6–13]), despite reported limitations. Moreover, the consensus-based strategy, which may unify the previous three approaches under a general framework, has gained interest in addressing the formation control problem as shown in [14]. Recently reported research relates to the formation control of a group of satellites: In [15–18], the attitude synchronization problem was studied for multiple satellites with three degrees-offreedom (3DOF). In [15], a predictive smooth variable structure filtering method was presented to achieve attitude synchronization during satellite formation flying. The relative attitude pointing control of formation flying for multiple satellites was studied in [16]. A coordinated attitude control algorithm using the state-dependent Riccati equation approach was developed in [17] for satellite formation flying. Attitude synchronization control for
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.2 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
satellite formation flying was discussed in [18]. Furthermore, the formation control problems for multiple satellites involving 3DOF position dynamics were studied in [19–23]. In [19,20], the satellite position models were developed based on the Clohessy-Wiltshire (C-W) equation and the attitude dynamics was ignored in the linear quadratic regulator design. In [21], the dynamics of each spacecraft was described via a linear time-periodic approximation, and a distributed controller was proposed for a formation of multiple spacecrafts following a periodic orbit. An asymptotic second-order sliding mode control method applied to satellite formation flying was discussed in [22]. Nonlinear control methods were studied in [4,23]. In [4], an adaptive control approach was developed for the relative position tracking problem of satellite formation flying. In [23], a nonlinear feedback control law was developed by using a low-and-high gain control scheme to solve a relative position keeping problem for a team of satellites, but the attitude control problems were not further discussed in the flying controller design. A nonlinear model predictive control augmented with a disturbance observer was proposed in [24–27]. Formation control problems for a group of satellites with 6DOF position and attitude dynamics were investigated in [28,29]. Nonlinear adaptive control laws were applied to the formation maintenance and reconfiguration in [28], but the gravity potential term was ignored, and the leader satellite followed a circular reference orbit. A synchronization framework was presented in [29] to deal with the spacecraft formation flying problem, but the formation center was assumed to be in a circular reference orbit. Therefore, robust formation control for a team of satellites considering full complex dynamics models with nonlinearities and uncertainties is still open. This paper centers on addressing the robust control problem for a group of satellites to achieve formation flying. Since satellite formation flying requires simultaneous relative orbit and attitude to accomplish required tasks, both the trajectory formation and attitude alignment control problems must be addressed. The proposed controller includes a position controller to form the desired formation trajectories and patterns, and an attitude controller to align the satellite attitudes. Compared to previous results, main contributions of this paper are summarized as follows: First, the highly nonlinear model of each satellite with 3DOF position dynamics and 3DOF attitude dynamics is studied in the context of the formation control problem. In [19–21], simplified dynamic models were used to describe the satellite motion, but it is not accurate enough in complex aerospace. The trajectory formation and attitude coordination problems were discussed separately in [19–23]. Second, the effects of parametric uncertainties and external disturbances are considered while designing the robust formation controller. Uncertainty rejection problems were ignored in satellite formation control problems reported in [16,17], while limited types of uncertainties were discussed in stability analysis of the constructed global closed-loop control systems [3,15,20,23]. In addition, the formation control protocol discussed in [4,15,23,28] was not distributed. In this paper, the robust formation control problem is solved through graph theory, and the resulted control law is distributed, which is comparatively easy to be implemented in practical applications. Compared to previously published research in [30], novelties in the current paper include: The rotation model is described by the modified Rodrigues parameter (MRP) without redundancy instead of the quaternions. The controller structures are different, and the attitude controller designed in the current paper is a consensus-based controller. It is proven that the tracking errors of the global closed-loop control system can converge into a given neighborhood of the origin in a finite time. The rest of this paper is organized as follows. Preliminaries on the satellite model and the graph theory, and the problem description are presented in Section 2. The formation control protocol de-
67 68 69 70 71 72 73 74 75 76 77 78 79 80
Fig. 1. Schematic representation of satellites.
81 82
sign is developed in Section 3. Robustness analysis of the proposed global closed-loop control system is given in Section 4. Numerical simulation results for multi-satellite formation flying are shown in Section 5. Section 6 concludes the paper.
83 84 85 86 87
2. Preliminaries and problem description
88
Throughout the paper, I N × N ∈ R N × N is an identity matrix, 0 N × N ∈ R N × N is a zero matrix, c N , j ∈ R N ×1 is a column vector with 1 on the j-th element and 0s elsewhere, and ⊗ is
the Kronecker product. For a = a1 a2 symmetric matrix S (·) is defined as:
⎡
0 S (a) = ⎣ a3 −a2
−a3 0 a1
a3
T
∈ R3×1 , the skew-
89 90 91 92 93 94
⎤
95
a2 −a1 ⎦ . 0
96 97 98 99
2.1. Graph theory
100 101
In this paper, finite satellites form a team, labeled from 1 to N. Let = {1, 2, · · · , N }. A directed graph G = { V , E , W } is used to describe the interaction relationship among the satellites in the group (Fig. 1). A set of nodes V = { v 1 , v 2 , . . . , v N } indicates the satellite group, where the node
v i represents satellite i. A set of edges E ∈ v i , v j : v i , v j ∈ V shows the influence relationship in
the satellite group, where an edge e i j = v i , v j indicates satellite j can receive information from satellite i and satellite i is called the neighbor of satellite j. The set
of all neighbors of node v i is denoted as N i = v j ∈ V : v j , v i ∈ E . The mutual influence intensity among the satellites can be described by an adjacency matrix W = w i j ∈ R N × N and each element w i j indicates the impact strength of satellite j on satellite i. When there is no information flow from satellite j to satellite i, w i j = 0. Besides, w ii = 0 (i ∈ ).
102
The in-degree of each node is denoted as cdi ( v i ) =
and
116
the in-degree matrix is denoted as D = diag {cdi ( v i )} ∈ R . The graph Laplacian matrix is defined as L = D − W . If there exists a node in the graph G, which has directed paths to all other nodes, the graph has a spanning tree and the node is called the root of the tree.
117
2.2. Satellite mathematical model
123
N
j =1 w i j N ×N
M
Eˆ M = eˆ 1M , eˆ 2M , eˆ 3
the chief satellite-fixed frame with the origin
104 105 106 107 108 109 110 111 112 113 114 115
118 119 120 121 122 124
Consider each satellite in the team as a rigid body with 3DOF dynamics and 3DOF attitude dynamics. Let Eˆ I = I I position eˆ 1 , eˆ 2 , eˆ 3I be the inertial system attached to the Earth,
103
B ˆB ˆB in its mass center, and Eˆ Bi = eˆ 1i , e 2i , e 3i the body-fixed frame attached to each satellite. The chief satellite is selected as the reference benchmark for the entire formation flying system, where the
125 126 127 128 129 130 131 132
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.3 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
1 2 3 4 5 6 7 8
relative states of the satellite team can be measured, and the formation tasks can be planned and assigned. The chief satellite can also be regarded as a virtual leader for the team. These relative trajectories have been widely used in [2,4,20,23,28] for satellite formation control, which can also result distributed control laws. As depicted in [20], the satellite relative position dynamics can be described by the following equation as:
10
p¨ i = A p1i p˙ i + A p2i p i + A p3i + u p i + d pi ,
11
where
9
12 13 14
A p1i
15
⎡
⎡
16 17 18
A p2i = ⎣
c˙ 2f aL − μ g h pi
−¨c f aL
22 23 24 25 26 27 28 29 30 31
and A p3i 3×1
∈R
34 35 36 37 38 39 40 41 42
0
− μ g h pi
0
⎤ ⎦,
0 0
T
. pi =
p xi
p yi
p zi
T
represents the position vector expressed in Eˆ M , μ g the
geocentric gravitational constant, ro L the orbit radius of the chief
T
satellite, c f aL the true anomaly, u pi = u xi u yi u zi ∈ R3×1 the relative control acceleration vector, and h pi = ((r0L + p xi )2 + p 2yi +
T
p 2zi )−3/2 . Let d pi = dxi d yi d zi ∈ R3×1 denote the disturbance torque caused by external disturbances such as solar radiation, atmospheric drag, unknown constants and harmonics with unknown phase and magnitude as described in [5]. The derivatives of the true anomaly in (2) can be obtained by the following equations as:
2 3 c˙ f aL = H p 1 + 2ceL cc1 + (1 + 5cc2 ) ceL /2 + O ceL
c¨ f aL = −2H p c˙ 2f aL
=
H 2p
2 ceL n L c s1 + ceL
,
(n L c s2 + 3n L cc2 c s1 ) + O 2 3 1 + 4ceL cc1 + (3 + 7cc2 ) ceL /2 + O ceL ,
where H p =
μ
2 g a L 1 − c eL
−1
M (σi ) σ¨ i + C (σi , σ˙ i ) σ˙ i = u τ i + P (σi )
dτ i ,
(5)
−1
a2L ,
3 ceL
(2)
c s1 = sin c f M L , c s2 = sin(2c f M L ),
c c1 = cos c f M L , c c2 = cos(2c f M L ), a L and c eL are the orbital semilong axis and the eccentricity of the chief satellite respectively,
55 56 57 58 59 60 61 62 63 64 65 66
where the invertible matrix
P (σi ) = 2S (σi ) + 2σi σi T + (1 − σi T σi ) I 3×3 /4,
diag J xi , J yi , J zi
T
∈ R3×1 is the angular velocity vector, J i = ∈ R3×3 indicates the inertia matrix, and τi =
ωi = ωxi ω yi ω zi
85
89
u τ i = C (σi , σ˙ i ) σ˙ i + M (σi ) u σ i ,
101
N
(6)
one can rewrite the position model (1) and attitude model (5) as:
(7)
for satellite i, where u σ i is the virtual attitude control input to be designed, and
⎡
0
2c˙ NfaL
0
0 0
⎣ −2c˙ N AN p1i = f aL ⎡
N
0
⎤
σ i = M
(σi ) P
(σi )dτ i − M
− M −1 (σi ) M (σi )u σ i .
93 94 95 96 97 98 99 100 102 103 105 106 107 108 109
112
−1
92
111
c¨ NfaL
c˙ 2f aL
0
−1
91
110
0 ⎦, 0
3 2 ⎢ c˙ f aL + 2μ g r0L ⎢ AN p2i = ⎢ −¨c NfaL ⎣
90
104
p¨ i = A N p˙ + A N p + u pi + p i , p1i i p2i i σ¨ = u σ i + σ i , i ∈ ,
(4)
54
84
88
˙ i = − S (ωi ) J i ωi + τi + dτ i , Jiω
53
83
87
˙ i + A pi = A p1i p p2i p i + A p3i + d pi ,
52
μ g /a3L denotes the average orbital angular velocity of the
82
For the satellite motion model shown in (1), (2), and (5), the real parameter can be split into two parts: the nominal part represented by the superscript N and the uncertain part represented by the superscript , which satisfy, for example, J i = J iN + J i and M (σi ) = M N (σi ) + M (σi ). The parameter uncertainties external disturbances are bounded and assumed to satisfy −and M 1 (σi ) M (σi ) < 1 to guarantee that a positive control torque 1 can yield a positive angular acceleration of the satellite. Similarly to [33], for the attitude model (5), by using the full state feedback and the straightforward feedback linearization methods based control law as: N
,
81
86
2.3. Problem description
(3)
51
74
79
σ˙ i = P (σi ) ωi ,
50
73
78
The terms pi ∈ R3×1 and σ i ∈ R3×1 are named the equivalent disturbances, including nonlinear dynamics, parametric uncertainties, and external disturbances, and satisfy
49
72
(σi )( J i P˙ (σi ) + S ( P (σi )σ˙ i ) J i P (σi )).
chief satellite, c f M L = n L t indicates the mean anomaly of the chief satellite with the initial mean anomaly 0◦ , and the function O (·) means the infinitesimal of higher order. The equations in (2) are approximate solutions of the derivatives of the true anomaly obtained after the Lagrange series expansion of the terms c eL and c f ML. As shown in [31], the satellite attitudes can be represented by the MRP σ . For satellite i, σ is a parameter vector satisfying σi = [σ1i , σ2i , σ3i ]T = n i tan (θi /4) ∈ R3×1 , where n i is the Euler axis and θi indicates the three Euler angle. As illustrated in [32], the attitude kinematics and dynamics can be described by the following nonlinear equations,
48
71
77
44
47
70
76
nL =
46
69
and
43 45
68
75
where
Remark 1. It can be observed that the satellite model expressed in (1) and (2) does not impose any restrictions on the eccentricity of the orbit. Furthermore, one can see that the satellite model involves highly nonlinear and coupled dynamics, and the rotation motion can significantly affect the translational dynamics.
−μ g h pi
0
= μ g (1 − h pi r03L ) r02L
32 33
c¨ f aL
dτ i = dτ xi ing as the additional torque on each satellite. Define the control input as u τ i = P −1 (σi )τi . Combining (3) and (4), one can rewrite the Lagrangian expression as:
67
80
20
∈ R3×1 represents the control torque vector, T dτ yi dτ zi ∈ R3×1 is the external disturbance act-
C (σi , σ˙ i ) = P
0 0 ⎦, 0 c˙ 2f aL
T
τxi τ yi τzi
M (σi ) = P −1 (σi ) J i P −1 (σi )
⎤
0
19 21
2c˙ f aL 0 0
0 = ⎣ −2c˙ f aL 0
(1)
3
N
0
− μ g r03L 0
−1
0
−μ g r03L
⎤
113
⎥ ⎥ ⎥. ⎦
115
114 116 117 118 119 120 121 122 123 124 125
(σi )C (σi , σ˙ i ) σ˙ i
126
(8)
127 128
Remark 2. By ignoring the equivalent disturbances pi and σ i , the real dynamic model (7) becomes the nominal model. The real model (7) can be regarded as the nominal model added with the equivalent disturbances pi and σ i .
129 130 131 132
JID:AESCTE
AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.4 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
4
1 2 3 4 5 6 7 8 9 10 11 12
The objective of this paper consists of two parts: first, to achieve the desired formation trajectories and patterns for the team of satellites; second, to align the satellite attitudes. Define the desired trajectory as p r ∈ R3×1 expressed in Eˆ M , and p¨ r = 0.
T
Let ξi j = ξ1,i j ξ2,i j ξ3,i j ∈ R3×1 (i , j ∈ ) denote the desired position deviation between satellite i and satellite j, which also determine the formation pattern of the satellite team. Let ξi j = ξi − ξ j (i , j ∈ ), where ξi indicates the desired position deviation between the virtual leader and satellite i, and satisfies that N i =1 ξi = 03×1 .
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
p
p
p
p
z˙ 1i = − f pi z1i −
In this section, the robust formation controller for a team of satellites in the communication network is designed, which includes a position controller and an attitude controller to form the desired formation trajectories and to align the satellite attitudes to zero respectively. 3.1. Position controller design
u RC ∈ R3×1 as follows: pi
FC u pi = u pi + u RC pi , i ∈ .
(9)
FC The nominal control input u pi is designed for the nominal po-
sition model and the robust compensating control input u RC is pi constructed to restrain the effects of the equivalent disturbance pi on the real position system. Design the nominal position conFC (i ∈ ) based on LQR control method as follows: trol inputs u pi
= −η F
w i j K p p i − p j − ξi j + K v p˙ i − p˙ j
j∈N i
− η F αli K p p i − ξi − pr + K v p˙ i − p˙ r ,
(10)
where η F represents a scalar coupling gain, and K p , K v ∈ R3×3 are diagonal nominal controller parameter matrices. αli is a constant representing the connection weight between the virtual leader and satellite i: αli = 1 indicates the information can flow from the virtual leader to satellite i, and αli = 0 otherwise. is constructed Furthermore, the robust compensating input u RC pi based on the robust filter F pi (s) as follows:
u RC pi = − F pi (s) pi ,
i ∈ ,
(11)
where s represents the Laplace operator and
72
p
73 74
p
2 2 u RC pi = − f pi p i + f pi z2i , i ∈ .
(13)
75 76
3.2. Attitude controller design
77
Similarly to the position controller, the attitude control input u σ i also includes two parts, the nominal part u σF Ci ∈ R3×1 and the robust compensating part u σRCi ∈ R3×1 as follows:
79
u σ i = u σF Ci + u σRCi ,
83
2
The robust filter F pi (s) is set as F p j ,i (s) = s + f p j ,i (j = 1, 2, 3) with the positive robust filter parameter f p j ,i to be determined. As elaborated in [35–37], the robust filter F pi (i ) would have a wider frequency bandwidth by selecting larger positive parameters f p j ,i ( j = 1, 2, 3). In this case, the filter gain of F pi (s) would approximate the unit matrix and the robust compensating input u RC would approximate − pi . Therefore, the effects of the pi equivalent disturbances can be restrained by selecting robust filter parameters appropriately. However, it is difficult to measure pi directly in practical applications, as shown in (11). From (7), one can have that f p2j ,i /
(12)
i ∈ .
(14)
wij
80 81 82 84
The nominal part u σF Ci is designed as:
85
σi − σ j + K σ σ˙ i − σ˙ j
86 87
j∈N i
88
− αli (σi + K σ σ˙ i ) ,
(15)
where K σ ∈ R3×3 represents a diagonal nominal controller parameter matrix. The robust compensating input u σRCi is constructed as:
89 90 91 92 93
u σRCi = − F σ i (s)σ i ,
(16)
94
where F σ i (s) = diag { F σ 1,i (s), F σ 2,i (s), F σ 3,i (s)} and F σ j ,i (s) =
95
f σ2 j ,i / s + f σ j ,i ( j = 1, 2, 3). The robust compensating part u σRCi can be realized in a similar way to u RC . pi
97
2
96 98 99
Remark 3. A relative motion model involving the chief satellite and the deputy satellites was studied in [38], and the resulted formation control law was distributed based on the model. In the current paper, the proposed attitude and position controllers for the similar relative motion model are distributed, because the designed controller of each satellite only depends on the information of its neighbors and itself.
100 101 102 103 104 105 106 107 108
4. Robustness analysis
109
For satellite i, define the trajectory tracking error epi = e p j ,i = p i − ξi − p r ∈ R3×1 and the linear velocity error evi = e v j ,i = p˙ i − p˙ r ∈ R3×1 . Define the attitude errors e = e = σi ∈ R3×1 (i ∈ σ i σ j , i ). Let K pz = K p K v , K σ z = I 3×3 K σ , and
A pz =
F pi (s) = diag { F p1,i (s), F p2,i (s), F p3,i (s)}.
pi = p¨ i − A Np1i p˙ i − A Np2i p i − u pi .
69 71
z˙ 2i = − f pi z2i + 2 f pi + A N p1 p i + z1i ,
u σF Ci = −
For satellite i, the position control input u pi includes two parts, FC the nominal part u pi ∈ R3×1 and the robust compensating part
FC u pi
68 70
2 f pi + A Np1 f pi − A Np2 p i + u pi ,
67
78
65 66
3. Formation protocol design
13 14
Let f pi = diag f p1,i , f p2,i , f p3,i . Then, substituting (12) to (11), one can realize the robust compensating input u RC by the followpi ing equations:
03×3 03×3
I 3×3 03×3
, Bz =
03×3 I 3×3
, Aσ z =
03×3 − I 3×3
I 3×3 −Kσ
z˙ pi = A pz z pi − η F B z
w i j K p e pi − e p j + K v e vi − e v j
j∈N i
where z pi =
e Tpi
e Tvi
T
112 113 114 116 117
118 119 120 121 122
123
− η F αli B z K p e pi + K v e vi + B z u RC pi + pi , z˙ σ i = A σ z zσ i + B z u σRCi + σ i ,
111
115
.
Then, one can have the node error system for each satellite i as follows:
110
124 125 126
(17)
127 128
= z p j ,i ∈ R6×1 and zσ i = e σ iT
e˙ σT i
T
129
= zσ j ,i ∈ R6×1 . Now, the global closed-loop system can be de-
130
scribed as:
132
131
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.5 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
˜ p, z˙ p = A pc z p + B ˜ σ, z˙ σ = A σ c zσ + B
(18)
˜ p = u RC + pi where z p ∈ R6N ×1 , zσ ∈ R6N ×1 , pi
∈ R3N ×1 ,
˜ σ = u RC + σ i ∈ R3N ×1 , B L = diag {αli } ∈ R N × N , B = I N × N ⊗ σi B z , A pc = I N × N ⊗ A pz − η F ( L + B L ) ⊗ B z K pz , and A σ c = I N × N ⊗ A σ z .
Furthermore, consider the design of symmetric and positive definite matrices Q k = Q kT ∈ R6×6 , k = kT ∈ R3×3 , where k = pz, σ z. Then, the controller parameter matrix K k can be given as: −1 T
K k = k B z P k ,
(19)
26 27 28 29 30 31 32 33
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
max z p (t ) ≤ max c T j
j
π + ς f p ε p / f p . p0 6N , j
71
Furthermore, the attitude tracking performance is analyzed. From (8), a positive constant ςdσ i can be obtained such that the equivalent disturbance σ i satisfies that
72
(25)
76
σ i ∞ ≤ M −1 (σi ) M (σi )1 u σ i ∞ + ςdσ i .
Proof. From (11), (16), and (18), one can obtain that
π p0 = e A pc t z p (0), πσ 0 = e A σ c t zσ (0), and χ Bp = (sI 6N ×6N − A pc )−1 B ( I 3N ×3N − F p (s))1 , χ B σ = (sI 6N ×6N − A σ c )−1 B ( I 3N ×3N − F σ (s))1 .
(21)
According to [35], if A pc and A σ c are asymptotically stable, there exists a positive constant f 1∗ such that: for any f p ≥ f 1∗ and f σ ≥
(26)
54 55 56 57 58 59 60 61 62 63 64 65 66
79 80 81 82 83
(27)
where ςσ z and ςσ d are positive constants depending on f p j ,i . If the constant f σ satisfies f σ ≥ f 1∗ and f σ > ςσ z ς f σ , substituting (27) to (20), one can obtain the following inequality as:
ςσ z πσ 0 ∞ + ςσ d σ ∞ ≤ . 1 − ςσ z ς f σ / f σ
84 85 86 87 88 89
(28)
90 91
Because A σ c is asymptotically stable and the initial state zσ (0) is bounded, πσ 0 is also bounded. Now, from (28) and (20), it can be observed that σ and zσ are bounded and satisfy
92
σ ∞ ≤ εσ , zσ ∞ ≤ εzσ ,
96
(29)
93 94 95 97
where εσ and εzσ are positive constants. Similarly, from (16), (18), and (29), one can obtain that
T max | zσ (t )| ≤ max c 6N , j πσ 0 + ς f σ εσ / f σ . j
and f σ = min f σ j , one can ς f σ , which are independent of
χ Bp ≤ ς f p / f p , χBσ ≤ ς f σ / f σ .
(22)
Therefore, it can be seen from that there exist finite positive ∗ ∗ constants t ∗ and f ∗ satisfying f p j ,i ≥ f and f σ j ,i ≥ f (i ∈ ), and the errors satisfying max j z pi (t ) ≤ εz and max j | zσ i (t )| ≤ εz , ∀t ≥ t ∗ . 2
98 99 100 101 103 104 105 106 107
Remark 4. It should be noted that the theoretical values of the robust filter parameters f pi and f σ i (i ∈ ) derived from Theorem 1 may be conservative. The real values of the robust filter parameters in practical applications may be much smaller than their theoretical values. Therefore, the following unidirectional tuning method can be applied to determine the robust filter parameters as follows. First, the robust filter parameters f pi and f σ i (i ∈ ) are set with certain initial positive values. Second, the robust filter parameters with larger values are adjusted until the desired tracking performance can be obtained.
108 109 110 111 112 113 114 115 116 117 118
52 53
75
102
(20)
where
f 1∗ (i ∈ ), where f p = min f p j obtain positive constants ς f p and f p j and f σ j , and satisfy
74
78
σ ∞ ≤ ςσ z zσ ∞ +ςσ d ,
j
z p ≤ π p0 + χ Bp p , ∞ ∞ ∞ zσ ∞ ≤ πσ 0 ∞ + χ B σ σ ∞ ,
73
77
From (14), (15), and (16), one has that
A Tpz P k + P k A pz + Q k − P k B z k−1 B zT P k = 0.
Theorem 1. Consider the satellite dynamics given by (7), the formation control protocol designed in Section 3, and the scalar coupling gain η F , and K pz and K σ z determined by (19). If the directed graph G has a spanning tree, the root can obtain information from the virtual leader, and the initial states z p (0) and zσ (0) are bounded, then for a given positive constant εz , there exist finite positive constants t ∗ and f ∗ such that: for any f p j ,i ≥ f ∗ and f σ j ,i ≥ f ∗ (i ∈ ), all states involved in the closed-loop control system are bounded, and the tracking errors sat isfy max j z p j ,i (t ) ≤ εz and maxi | zσ i (t )| ≤ εz , ∀t ≥ t ∗ .
68 70
where ςσ zi is a positive constant depends on f p j ,i . Combining (25) and (26), one can obtain that
According to Theorem 1 in [34], if the directed graph G has a spanning tree and the root can obtain information from the virtual leader, then, if η F ≥ λmin pr /2, A pc is asymptotically stable, where λ pi (i ∈ ) represent the eigenvalues of ( L + B L ) and λmin pr = mini ∈ Re(λ pi ). Besides, A σ c is also asymptotically stable, since all its eigenvalues have negative real parts.
67 69
u σ i ∞ ≤ ςσ zi zσ i ∞ + σ i ∞ ,
34 35
all states involved in the transitional dynamics are bounded. Now, from (8), (18), (23), and (24), one can have that
where P k is the positive definite solution of the following Riccati equation
24 25
5
Since the parameter uncertainties and external disturbances are bounded, there exists a positive constant ε pi such that the equivalent disturbance pi in (8) satisfies that
pi ≤ ε pi . ∞
(23) f 1∗ , and be-
Let ε p = maxi ε pi . If the constant f p satisfies f p ≥ cause A pc is asymptotically stable and the initial state z p (0) is bounded, then, from (20) and (23), it can be seen that z p is bounded. In this case, one can obtain that
z p
∞
≤ εzp .
(24) p
p
The bounds of the states of the robust filter z1i and z2i depend on the robust compensator parameters. It can be observed that
5. Simulation results
119 120
This section presents a series of simulation results of five satellites to verify the effectiveness of the proposed robust formation control method, and thus = {1, 2, 3, 4, 5}. Each satellite is modeled as described in Section 2 with J iN = diag {4.34, 4.33, 3.66} kg · m2 and μ g = 3.986 × 1014 m3 /s2 . The trajectory of the virtual
T
leader is given by p = 0.1t 0.1t 0.1t and the five satelare required to form a pentagon formation lites pattern as ξ1= T T T 0 1020 0 , ξ2 = 971 315 0 , ξ3 = 600 −825 0 , r
121 122 123 124 125 126 127 128
T T ξ4 = −600 −825 0 , and ξ5 = −971 315 0 with inter-
129
national units. The upper limit of the simulation time is 300 s. The topological relationship of the five satellites showed in Fig. 2 is
131
130 132
JID:AESCTE
AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.6 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
6
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
Fig. 2. Topological relationship of satellites.
20
86
21
87
22
88
23
89
24
90
25
Fig. 4. Linear velocity response by the proposed controller.
26
91 92
27
93
28
94 95
29
disturbances introduced in [38] are time-invariant to simulate the effects of multiple disturbances including the Earth’s non-spherical gravity, the solar radiation pressure, and the third body gravity. In the current paper, time-varying and non-vanished disturbances are considered, which are more realistic to validate the advantages of the proposed robust formation controller in complex aerospace environments. The real satellite parameters are chosen to be 10% larger than the nominal parameters and the external disturbances T are selected as: d p = 5 sin(2t ) sin(2t ) sin(2t ) for the posi-
30 31 32 33 34 35 36 37 38
39 40 41 42
Fig. 3. Three-dimensional trajectory by the proposed controller. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
⎡
0 0 0 0 1 0 0 0 ⎢ ⎢ wij = ⎢ 0 1 0 0 ⎣1 0 0 0 0 0 0 1
⎤
0 0⎥ ⎥ 0 ⎥. ⎦ 0 0
Only satellite 1 can obtain the information from the virtual leader, and thereby αl1 = 1, αl2 = 0, αl3 = 0, αl4 = 0, and αl5 = 0. The initial orbital elements of the chief satellite are shown as: semimajor axis a L = 7400 km, eccentricity c eL = 0.1, inclination i L = 30◦ , longitude of the ascending node L = 100◦ , argument of periapsis ω L = 0◦ , and the mean anomaly c f M L = 0◦ . The initial values of the relative orbital motion are given as: p 1 (0) = T T T 0 50 0 , p 2 (0) = 50 10 0 , p 3 (0) = 10 −10 0 , p 4 (0) =
T
T
T
−10 −10 0 , p 5 (0) = −20 0 0 , p˙ 1 (0) = 0.1 0.05 0.1 , T T p˙2 (0) = 0 0.1 0.05 , p˙ 3 (0) = −0.1 −0.08 0 , p˙ 4 (0) = T T 0 0.08 0.1 , and p˙ 5 (0) = 0.2 0.1 0.08 . The initial MRP of T the five satellites are chosen as follows: σ1 (0) = 0.2 0.2 0.2 , T T σ2 (0) = 0.15 0.15 0.15 , σ3 (0) = 0.05 0.05 0.05 , σ4 (0) = T T −0.1 −0.1 −0.1 , and σ5 (0) = 0.1 0.1 0.1 . The external
T
for the attition control input and dτ = 0.05 sin t sin t sin t tude torque. The scalar coupling gain is set to be η F = 1. The nominal controller parameters are chosen as: K p = diag {10, 10, 10}, K v = diag {400, 400, 100}, and K σ = diag {9, 9, 9}, to guarantee that the matrices A pc and A σ c are Hurwitz. The robust filer parameters f pi and f σ i are selected as shown in Remark 4. Figs. 3, 4, 5, and 6 depict the three-dimensional trajectories p i , linear velocity responses p˙ i , trajectory tracking error e pi , and the attitude simulation response σi of the five satellites, respectively. The red, blue, pink, green, and black solid lines indicate Satellites 1-5, and the red dotted line represents the formation pattern. One can observe that the trajectory tracking error e pi can converge into a neighborhood of the origin bounded by εz in 200 s, the steadystate trajectory tracking error is approximate 0.05 m, the satellites align their attitudes in 60 s, and the steady-state attitude error is approximately 0.01. Furthermore, the proposed robust formation controller is compared to a leader-following formation controller in [8] for the team of satellites. Fig. 7 shows the formation trajectory of the satellite group. Linear velocity response, trajectory tracking error, and attitude response are illustrated in Figs. 8, 9, and 10 respectively, by the leader-following formation control method. In contrast, the steady-state trajectory tracking error is approximately 0.8 m and the steady-state attitude error is nearly 0.03. From these figures, it can be observed that the proposed closed-loop control system can achieve better trajectory and attitude tracking performances under the influence of nonlinear dynamics, parameter uncertainties, and external disturbances.
96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.7 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
7
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
Fig. 7. Three-dimensional trajectory by the leader-following controller.
19
84 85
20
86
21
87
22
88
23
89
24
90 91
25 26
Fig. 5. Trajectory tracking error by the proposed controller.
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111 112
46 47
Fig. 8. Linear velocity response by the leader-following controller.
48
113 114
49
115
50
116
51
origin in a finite time. Simulation results are given to demonstrate the effectiveness of the designed robust controller.
52 53 54
Fig. 6. Attitude response by the proposed controller.
55
6. Conclusion
62 63 64 65 66
121
None declared.
123 124 125
59 61
119
122
57
60
118 120
Declaration of competing interest
56 58
117
A robust controller is presented for satellite formation flying subject to nonlinearity, parametric uncertainties, and external disturbances. The proposed robust formation controller yields a position controller to form designed formation trajectories and patterns and an attitude controller to align satellite attitudes, for each satellite. Theoretical analysis guarantees that the trajectory and attitude tracking errors can converge into a given neighborhood of
Acknowledgements
126 127
This work was supported by the National Natural Science Foundation of China under Grants 61873012, 61503012, and 61633007, the National Science Foundation under Grants 1730675 and 1714519, and the Office of Naval Research under Grant N00014-17-1-2239.
128 129 130 131 132
JID:AESCTE
AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.8 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Fig. 9. Trajectory tracking error by the leader-following controller.
27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
Fig. 10. Attitude response by the leader-following controller.
56 57 58 59 60 61 62 63 64 65 66
References [1] W. Wang, G. Mengali, A.A. Quarta, J. Yuan, Distributed adaptive synchronization for multiple spacecraft formation flying around Lagrange point orbits, Aerosp. Sci. Technol. 74 (2018) 93–103. [2] X. Huang, Y. Yan, Y. Zhou, Y. Yang, Dual-quaternion based distributed coordination control of six-DOF spacecraft formation with collision avoidance, Aerosp. Sci. Technol. 67 (2017) 443–455. [3] C. Wei, S.-Y. Park, C. Park, Optimal H ∞ robust output feedback control for satellite formation in arbitrary elliptical reference orbits, Adv. Space Res. 54 (6) (2014) 969–989.
[4] H.-C. Lim, H. Bang, Adaptive control for satellite formation flying under thrust misalignment, Acta Astronaut. 65 (1) (2009) 112–122. [5] H. Sun, S. Li, S. Fei, A composite control scheme for 6DOF spacecraft formation control, Acta Astronaut. 69 (7) (2011) 595–611. [6] D. Lee, S. Kim, J. Suk, Formation flight of unmanned aerial vehicles using track guidance, Aerosp. Sci. Technol. 76 (2018) 412–420. [7] F. Morbidi, G.L. Mariottini, D. Prattichizzo, Observer design via immersion and invariance for vision-based leader-follower formation control, Automatica 46 (1) (2010) 148–154. [8] A. Mahmood, Y. Kim, Leader-following formation control of quadcopters with heading synchronization, Aerosp. Sci. Technol. 47 (1) (2015) 68–74. [9] G. Antonelli, F. Arrichiello, S. Chiaverini, Experiments of formation control with multirobot systems using the null-space-based behavioral control, IEEE Trans. Control Syst. Technol. 17 (5) (2009) 1173–1182. [10] T. Balch, R.C. Arkin, Behavior-based formation control for multirobot teams, IEEE Trans. Autom. Control 14 (6) (1998) 926–939. [11] Y. Cao, W. Ren, Distributed coordinated tracking with reduced interaction via a variable structure approach, IEEE Trans. Autom. Control 57 (1) (2011) 33–48. [12] T. Broek, N. Wouw, H. Nijmeijer, Formation control of unicycle mobile robots: a virtual structure approach, Int. J. Control 84 (11) (2010) 1886–1902. [13] M. Xin, S.N. Balakrishnan, H.J. Pernicka, Position and attitude control of deepspace spacecraft formation flying via virtual structure and θ -D technique, J. Dyn. Syst. Meas. Control 129 (5) (2007) 689–698. [14] W. Ren, Consensus strategies for cooperative control of vehicle formations, IET Control Theory Appl. 1 (2) (2007) 505–512. [15] L. Cao, Y. Chen, Z. Zhang, Predictive smooth variable structure filter for attitude synchronization estimation during satellite formation flying, IEEE Trans. Aerosp. Electron. Syst. 53 (3) (2017) 1375–1383. [16] Z. Zhang, J. Li, H. Bao, Relative attitude pointing tracking control of satellite formation flying with considering the orientation of solar panel, J. Astronaut. 128 (28) (2008) 2786–2787. [17] I. Chang, S.-Y. Park, K.-H. Choi, Decentralized coordinated attitude control for satellite formation flying via the state-dependent Riccati equation technique, Int. J. Non-Linear Mech. 44 (8) (2009) 891–904. [18] J. Zhou, Q. Hu, M. Friswell, Decentralized finite time attitude synchronization control of satellite formation flying, J. Guid. Control Dyn. 36 (1) (2013) 185–195. [19] J. Su, Y. Dong, Sliding mode variable structure control for electromagnetic satellite formation station-tracking, J. Astronaut. 32 (5) (2011) 1093–1099. [20] S.-H. Mok, Y-H. Choi, H.-C. Bang, Collision avoidance using linear quadratic control in satellite formation flying, Int. J. Aeronaut. Space Sci. 11 (4) (2010) 351–359. [21] M. Paolo, K. Tamas, G. Eberhard, V. Michel, A decomposition-based approach to linear time-periodic distributed control of satellite formations, IEEE Trans. Control Syst. Technol. 19 (3) (2011) 481–492. [22] J. Li, Y. Pan, K.D. Kumar, Design of asymptotic second-order sliding mode control for satellite formation flying, J. Guid. Control Dyn. 35 (1) (2012) 309–316. [23] Y.-H. Lim, H.-S. Ahn, Relative position keeping in satellite formation flying with input saturation, J. Franklin Inst. 351 (2) (2014) 1112–1129. [24] J. Yang, J. Sun, W.X. Zheng, S. Li, Periodic event-triggered robust output feedback control for nonlinear uncertain systems with time-varying disturbance, Automatica 94 (2018) 324–333. [25] J. Yang, Z. Ding, Global output regulation for a class of lower triangular nonlinear systems: a feedback domination approach, Automatica 76 (2017) 65–69. [26] J. Yang, W.X. Zheng, Offset-free nonlinear MPC for mismatched disturbance attenuation with application to a static var compensator, IEEE Trans. Circuits Syst. II, Express Briefs 61 (1) (2014) 49–53. [27] J. Yang, H. Wu, L. Hu, S. Li, Robust predictive speed regulation of converterdriven DC motors via a discrete-time reduced-order GPIO, IEEE Trans. Ind. Electron. 66 (10) (2018) 7893–7903, https://doi.org/10.1109/TIE.2018.2878119. [28] A. Umair, M.W. David, R.L. Jaime, Control of electromagnetic satellite formations in near-Earth orbits, J. Guid. Control Dyn. 33 (6) (2010) 1883–1891. [29] S.-J. Chung, U. Ahsun, J.E. Slotine, Application of synchronization to formation flying spacecraft: Lagrangian approach, J. Guid. Control Dyn. 32 (2) (2009) 512–526. [30] H. Liu, Y. Tian, F.L. Lewis, Y. Wan, K.P. Valavanis, Robust formation control for a team of satellites, in: International Conference on Unmanned Aircraft Systems, 2018, pp. 30–35. [31] P. Tsiotras, Further passivity results for the attitude control problem, IEEE Trans. Autom. Control 43 (11) (1998) 1597–1600. [32] O. Egelend, Passivity-based adaptive attitude control of a rigid spacecraft, IEEE Trans. Autom. Control 39 (4) (1994) 842–846. [33] S. Nair, K. Subbarao, Attitude control of spacecraft formations subject to distributed communication delays, in: AAS/AIAA Space Flight Mechanics Meeting, 2017, pp. 1707–1722. [34] H. Zhang, F.L. Lewis, A. Das, Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback, IEEE Trans. Autom. Control 56 (8) (2011) 1948–1952. [35] H. Liu, X. Wang, Y. Zhong, Quaternion-based robust attitude control for uncertain robotic quadrotors, IEEE Trans. Ind. Inform. 11 (4) (2015) 406–415.
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.9 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
1 2 3 4 5
[36] H. Liu, F. Peng, F.L. Lewis, Y. Wan, Robust tracking control for tail-sitters in flight mode transitions, IEEE Trans. Aerosp. Electron. Syst. 55 (4) (2019) 2023–2035, https://doi.org/10.1109/TAES.2018.2880888. [37] H. Liu, T. Ma, F.L. Lewis, Y. Wan, Robust formation control for multiple quadrotors with nonlinearities and disturbances, IEEE Trans. Cybern. (2019), in press, https://doi.org/10.1109/TCYB.2018.2875559.
9
[38] H. Min, S. Wang, F. Sun, Z. Gao, Y. Wang, Distributed six degree-of-freedom spacecraft formation control with possible switching topology, IET Control Theory Appl. 5 (9) (2010) 1120–1130.
67 68 69 70 71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64
130
65
131
66
132
[m5G; v1.261; Prn:8/10/2019; 15:25] P.1 (1-9)
Aerospace Science and Technology ••• (••••) ••••••
1
67
Contents lists available at ScienceDirect
68
2 3
Aerospace Science and Technology
4
69 70 71
5
72
6
www.elsevier.com/locate/aescte
7
73
8
74
9
75
10
76
11 12 13
Robust formation flying control for a team of satellites subject to nonlinearities and uncertainties
16 17 18 19 20
Hao Liu
a,b,∗
, Yu Tian
a, b
, Frank L. Lewis
c ,d
c
, Yan Wan , Kimon P. Valavanis
e
a
School of Astronautics, Beihang University, Beijing 100191, PR China b Key Laboratory of Spacecraft Design Optimization and Dynamic Simulation Technologies of Ministry of Education, Beihang University, Beijing 100191, PR China c University of Texas at Arlington Research Institute, University of Texas at Arlington, Fort Worth, TX 76118, USA d State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, PR China e Department of Electrical and Computer Engineering, University of Denver, Denver, CO 80208, USA
a r t i c l e
i n f o
27 28
31 32 33 34
83 84 85 86
90
Article history: Received 15 December 2018 Received in revised form 17 January 2019 Accepted 3 October 2019 Available online xxxx
29 30
82
89
a b s t r a c t
24 26
81
88
22
25
79
87
21 23
78 80
14 15
77
Keywords: Formation control Robust control Nonlinear system Uncertain system Satellite
This paper addresses the problem of robust formation controller design for a group of satellites the dynamics of which involve nonlinearities and uncertainties. A formation flying controller is proposed for the satellite group, which includes a position controller to form the desired accurate formation and an attitude controller to align the satellite attitudes. It is shown that the trajectory and attitude tracking errors of the overall closed-loop control system can converge into a given neighborhood of the origin in a finite time. Numerical simulation studies are provided to demonstrate the advantages of the proposed formation control scheme. © 2019 Published by Elsevier Masson SAS.
91 92 93 94 95 96 97 98 99 100
35
101
36
102
37
103
38
104 105
39 40
1. Introduction
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57
In recent years, formation flying of a team of satellites has attracted attention in the field of aerospace. The use of a multisatellite formation flying system, instead of a single satellite, can effectively improve the overall system performance and benefit a range of space missions, such as large-scale space environment monitoring, deep space exploration, electronic reconnaissance, and commercial communications, as illustrated in [1–4]. With the trend on continuous improvement of aerospace mission efficiency, satellite formation flying has become an important application prospect in cooperative tasks, due to advantages such as high reliability, low cost, and strong adaptability. However, the dynamics of each satellite is highly nonlinear, and the uncertainties involved in the satellite dynamic model and the corresponding complex environments can influence the performance of the flight control system. The satellites are subject to parameter uncertainties and external disturbances - such as the Earth’s non-spherical gravity, the solar
58 59 60 61 62 63 64 65 66
*
Corresponding author at: School of Astronautics, Beihang University, Beijing 100191, PR China. E-mail addresses: [email protected] (H. Liu), [email protected] (Y. Tian), [email protected] (F.L. Lewis), [email protected] (Y. Wan), [email protected] (K.P. Valavanis). https://doi.org/10.1016/j.ast.2019.105455 1270-9638/© 2019 Published by Elsevier Masson SAS.
radiation pressure, and the third body gravity, which should not be ignored when high performance control is required - that are difficult to measure accurately, as illustrated in [5]. It is for this reason that robust formation control problems with the aim to minimize or limit the influence of uncertainties associated with the functionality of a group of satellites witness increased interest in recent years. In the past two decades, several classical formation control strategies including the leader-follower approach, the behavioral strategy, and the virtual structure method have been proposed (see, e.g., [6–13]), despite reported limitations. Moreover, the consensus-based strategy, which may unify the previous three approaches under a general framework, has gained interest in addressing the formation control problem as shown in [14]. Recently reported research relates to the formation control of a group of satellites: In [15–18], the attitude synchronization problem was studied for multiple satellites with three degrees-offreedom (3DOF). In [15], a predictive smooth variable structure filtering method was presented to achieve attitude synchronization during satellite formation flying. The relative attitude pointing control of formation flying for multiple satellites was studied in [16]. A coordinated attitude control algorithm using the state-dependent Riccati equation approach was developed in [17] for satellite formation flying. Attitude synchronization control for
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.2 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
satellite formation flying was discussed in [18]. Furthermore, the formation control problems for multiple satellites involving 3DOF position dynamics were studied in [19–23]. In [19,20], the satellite position models were developed based on the Clohessy-Wiltshire (C-W) equation and the attitude dynamics was ignored in the linear quadratic regulator design. In [21], the dynamics of each spacecraft was described via a linear time-periodic approximation, and a distributed controller was proposed for a formation of multiple spacecrafts following a periodic orbit. An asymptotic second-order sliding mode control method applied to satellite formation flying was discussed in [22]. Nonlinear control methods were studied in [4,23]. In [4], an adaptive control approach was developed for the relative position tracking problem of satellite formation flying. In [23], a nonlinear feedback control law was developed by using a low-and-high gain control scheme to solve a relative position keeping problem for a team of satellites, but the attitude control problems were not further discussed in the flying controller design. A nonlinear model predictive control augmented with a disturbance observer was proposed in [24–27]. Formation control problems for a group of satellites with 6DOF position and attitude dynamics were investigated in [28,29]. Nonlinear adaptive control laws were applied to the formation maintenance and reconfiguration in [28], but the gravity potential term was ignored, and the leader satellite followed a circular reference orbit. A synchronization framework was presented in [29] to deal with the spacecraft formation flying problem, but the formation center was assumed to be in a circular reference orbit. Therefore, robust formation control for a team of satellites considering full complex dynamics models with nonlinearities and uncertainties is still open. This paper centers on addressing the robust control problem for a group of satellites to achieve formation flying. Since satellite formation flying requires simultaneous relative orbit and attitude to accomplish required tasks, both the trajectory formation and attitude alignment control problems must be addressed. The proposed controller includes a position controller to form the desired formation trajectories and patterns, and an attitude controller to align the satellite attitudes. Compared to previous results, main contributions of this paper are summarized as follows: First, the highly nonlinear model of each satellite with 3DOF position dynamics and 3DOF attitude dynamics is studied in the context of the formation control problem. In [19–21], simplified dynamic models were used to describe the satellite motion, but it is not accurate enough in complex aerospace. The trajectory formation and attitude coordination problems were discussed separately in [19–23]. Second, the effects of parametric uncertainties and external disturbances are considered while designing the robust formation controller. Uncertainty rejection problems were ignored in satellite formation control problems reported in [16,17], while limited types of uncertainties were discussed in stability analysis of the constructed global closed-loop control systems [3,15,20,23]. In addition, the formation control protocol discussed in [4,15,23,28] was not distributed. In this paper, the robust formation control problem is solved through graph theory, and the resulted control law is distributed, which is comparatively easy to be implemented in practical applications. Compared to previously published research in [30], novelties in the current paper include: The rotation model is described by the modified Rodrigues parameter (MRP) without redundancy instead of the quaternions. The controller structures are different, and the attitude controller designed in the current paper is a consensus-based controller. It is proven that the tracking errors of the global closed-loop control system can converge into a given neighborhood of the origin in a finite time. The rest of this paper is organized as follows. Preliminaries on the satellite model and the graph theory, and the problem description are presented in Section 2. The formation control protocol de-
67 68 69 70 71 72 73 74 75 76 77 78 79 80
Fig. 1. Schematic representation of satellites.
81 82
sign is developed in Section 3. Robustness analysis of the proposed global closed-loop control system is given in Section 4. Numerical simulation results for multi-satellite formation flying are shown in Section 5. Section 6 concludes the paper.
83 84 85 86 87
2. Preliminaries and problem description
88
Throughout the paper, I N × N ∈ R N × N is an identity matrix, 0 N × N ∈ R N × N is a zero matrix, c N , j ∈ R N ×1 is a column vector with 1 on the j-th element and 0s elsewhere, and ⊗ is
the Kronecker product. For a = a1 a2 symmetric matrix S (·) is defined as:
⎡
0 S (a) = ⎣ a3 −a2
−a3 0 a1
a3
T
∈ R3×1 , the skew-
89 90 91 92 93 94
⎤
95
a2 −a1 ⎦ . 0
96 97 98 99
2.1. Graph theory
100 101
In this paper, finite satellites form a team, labeled from 1 to N. Let = {1, 2, · · · , N }. A directed graph G = { V , E , W } is used to describe the interaction relationship among the satellites in the group (Fig. 1). A set of nodes V = { v 1 , v 2 , . . . , v N } indicates the satellite group, where the node
v i represents satellite i. A set of edges E ∈ v i , v j : v i , v j ∈ V shows the influence relationship in
the satellite group, where an edge e i j = v i , v j indicates satellite j can receive information from satellite i and satellite i is called the neighbor of satellite j. The set
of all neighbors of node v i is denoted as N i = v j ∈ V : v j , v i ∈ E . The mutual influence intensity among the satellites can be described by an adjacency matrix W = w i j ∈ R N × N and each element w i j indicates the impact strength of satellite j on satellite i. When there is no information flow from satellite j to satellite i, w i j = 0. Besides, w ii = 0 (i ∈ ).
102
The in-degree of each node is denoted as cdi ( v i ) =
and
116
the in-degree matrix is denoted as D = diag {cdi ( v i )} ∈ R . The graph Laplacian matrix is defined as L = D − W . If there exists a node in the graph G, which has directed paths to all other nodes, the graph has a spanning tree and the node is called the root of the tree.
117
2.2. Satellite mathematical model
123
N
j =1 w i j N ×N
M
Eˆ M = eˆ 1M , eˆ 2M , eˆ 3
the chief satellite-fixed frame with the origin
104 105 106 107 108 109 110 111 112 113 114 115
118 119 120 121 122 124
Consider each satellite in the team as a rigid body with 3DOF dynamics and 3DOF attitude dynamics. Let Eˆ I = I I position eˆ 1 , eˆ 2 , eˆ 3I be the inertial system attached to the Earth,
103
B ˆB ˆB in its mass center, and Eˆ Bi = eˆ 1i , e 2i , e 3i the body-fixed frame attached to each satellite. The chief satellite is selected as the reference benchmark for the entire formation flying system, where the
125 126 127 128 129 130 131 132
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.3 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
1 2 3 4 5 6 7 8
relative states of the satellite team can be measured, and the formation tasks can be planned and assigned. The chief satellite can also be regarded as a virtual leader for the team. These relative trajectories have been widely used in [2,4,20,23,28] for satellite formation control, which can also result distributed control laws. As depicted in [20], the satellite relative position dynamics can be described by the following equation as:
10
p¨ i = A p1i p˙ i + A p2i p i + A p3i + u p i + d pi ,
11
where
9
12 13 14
A p1i
15
⎡
⎡
16 17 18
A p2i = ⎣
c˙ 2f aL − μ g h pi
−¨c f aL
22 23 24 25 26 27 28 29 30 31
and A p3i 3×1
∈R
34 35 36 37 38 39 40 41 42
0
− μ g h pi
0
⎤ ⎦,
0 0
T
. pi =
p xi
p yi
p zi
T
represents the position vector expressed in Eˆ M , μ g the
geocentric gravitational constant, ro L the orbit radius of the chief
T
satellite, c f aL the true anomaly, u pi = u xi u yi u zi ∈ R3×1 the relative control acceleration vector, and h pi = ((r0L + p xi )2 + p 2yi +
T
p 2zi )−3/2 . Let d pi = dxi d yi d zi ∈ R3×1 denote the disturbance torque caused by external disturbances such as solar radiation, atmospheric drag, unknown constants and harmonics with unknown phase and magnitude as described in [5]. The derivatives of the true anomaly in (2) can be obtained by the following equations as:
2 3 c˙ f aL = H p 1 + 2ceL cc1 + (1 + 5cc2 ) ceL /2 + O ceL
c¨ f aL = −2H p c˙ 2f aL
=
H 2p
2 ceL n L c s1 + ceL
,
(n L c s2 + 3n L cc2 c s1 ) + O 2 3 1 + 4ceL cc1 + (3 + 7cc2 ) ceL /2 + O ceL ,
where H p =
μ
2 g a L 1 − c eL
−1
M (σi ) σ¨ i + C (σi , σ˙ i ) σ˙ i = u τ i + P (σi )
dτ i ,
(5)
−1
a2L ,
3 ceL
(2)
c s1 = sin c f M L , c s2 = sin(2c f M L ),
c c1 = cos c f M L , c c2 = cos(2c f M L ), a L and c eL are the orbital semilong axis and the eccentricity of the chief satellite respectively,
55 56 57 58 59 60 61 62 63 64 65 66
where the invertible matrix
P (σi ) = 2S (σi ) + 2σi σi T + (1 − σi T σi ) I 3×3 /4,
diag J xi , J yi , J zi
T
∈ R3×1 is the angular velocity vector, J i = ∈ R3×3 indicates the inertia matrix, and τi =
ωi = ωxi ω yi ω zi
85
89
u τ i = C (σi , σ˙ i ) σ˙ i + M (σi ) u σ i ,
101
N
(6)
one can rewrite the position model (1) and attitude model (5) as:
(7)
for satellite i, where u σ i is the virtual attitude control input to be designed, and
⎡
0
2c˙ NfaL
0
0 0
⎣ −2c˙ N AN p1i = f aL ⎡
N
0
⎤
σ i = M
(σi ) P
(σi )dτ i − M
− M −1 (σi ) M (σi )u σ i .
93 94 95 96 97 98 99 100 102 103 105 106 107 108 109
112
−1
92
111
c¨ NfaL
c˙ 2f aL
0
−1
91
110
0 ⎦, 0
3 2 ⎢ c˙ f aL + 2μ g r0L ⎢ AN p2i = ⎢ −¨c NfaL ⎣
90
104
p¨ i = A N p˙ + A N p + u pi + p i , p1i i p2i i σ¨ = u σ i + σ i , i ∈ ,
(4)
54
84
88
˙ i = − S (ωi ) J i ωi + τi + dτ i , Jiω
53
83
87
˙ i + A pi = A p1i p p2i p i + A p3i + d pi ,
52
μ g /a3L denotes the average orbital angular velocity of the
82
For the satellite motion model shown in (1), (2), and (5), the real parameter can be split into two parts: the nominal part represented by the superscript N and the uncertain part represented by the superscript , which satisfy, for example, J i = J iN + J i and M (σi ) = M N (σi ) + M (σi ). The parameter uncertainties external disturbances are bounded and assumed to satisfy −and M 1 (σi ) M (σi ) < 1 to guarantee that a positive control torque 1 can yield a positive angular acceleration of the satellite. Similarly to [33], for the attitude model (5), by using the full state feedback and the straightforward feedback linearization methods based control law as: N
,
81
86
2.3. Problem description
(3)
51
74
79
σ˙ i = P (σi ) ωi ,
50
73
78
The terms pi ∈ R3×1 and σ i ∈ R3×1 are named the equivalent disturbances, including nonlinear dynamics, parametric uncertainties, and external disturbances, and satisfy
49
72
(σi )( J i P˙ (σi ) + S ( P (σi )σ˙ i ) J i P (σi )).
chief satellite, c f M L = n L t indicates the mean anomaly of the chief satellite with the initial mean anomaly 0◦ , and the function O (·) means the infinitesimal of higher order. The equations in (2) are approximate solutions of the derivatives of the true anomaly obtained after the Lagrange series expansion of the terms c eL and c f ML. As shown in [31], the satellite attitudes can be represented by the MRP σ . For satellite i, σ is a parameter vector satisfying σi = [σ1i , σ2i , σ3i ]T = n i tan (θi /4) ∈ R3×1 , where n i is the Euler axis and θi indicates the three Euler angle. As illustrated in [32], the attitude kinematics and dynamics can be described by the following nonlinear equations,
48
71
77
44
47
70
76
nL =
46
69
and
43 45
68
75
where
Remark 1. It can be observed that the satellite model expressed in (1) and (2) does not impose any restrictions on the eccentricity of the orbit. Furthermore, one can see that the satellite model involves highly nonlinear and coupled dynamics, and the rotation motion can significantly affect the translational dynamics.
−μ g h pi
0
= μ g (1 − h pi r03L ) r02L
32 33
c¨ f aL
dτ i = dτ xi ing as the additional torque on each satellite. Define the control input as u τ i = P −1 (σi )τi . Combining (3) and (4), one can rewrite the Lagrangian expression as:
67
80
20
∈ R3×1 represents the control torque vector, T dτ yi dτ zi ∈ R3×1 is the external disturbance act-
C (σi , σ˙ i ) = P
0 0 ⎦, 0 c˙ 2f aL
T
τxi τ yi τzi
M (σi ) = P −1 (σi ) J i P −1 (σi )
⎤
0
19 21
2c˙ f aL 0 0
0 = ⎣ −2c˙ f aL 0
(1)
3
N
0
− μ g r03L 0
−1
0
−μ g r03L
⎤
113
⎥ ⎥ ⎥. ⎦
115
114 116 117 118 119 120 121 122 123 124 125
(σi )C (σi , σ˙ i ) σ˙ i
126
(8)
127 128
Remark 2. By ignoring the equivalent disturbances pi and σ i , the real dynamic model (7) becomes the nominal model. The real model (7) can be regarded as the nominal model added with the equivalent disturbances pi and σ i .
129 130 131 132
JID:AESCTE
AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.4 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
4
1 2 3 4 5 6 7 8 9 10 11 12
The objective of this paper consists of two parts: first, to achieve the desired formation trajectories and patterns for the team of satellites; second, to align the satellite attitudes. Define the desired trajectory as p r ∈ R3×1 expressed in Eˆ M , and p¨ r = 0.
T
Let ξi j = ξ1,i j ξ2,i j ξ3,i j ∈ R3×1 (i , j ∈ ) denote the desired position deviation between satellite i and satellite j, which also determine the formation pattern of the satellite team. Let ξi j = ξi − ξ j (i , j ∈ ), where ξi indicates the desired position deviation between the virtual leader and satellite i, and satisfies that N i =1 ξi = 03×1 .
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
p
p
p
p
z˙ 1i = − f pi z1i −
In this section, the robust formation controller for a team of satellites in the communication network is designed, which includes a position controller and an attitude controller to form the desired formation trajectories and to align the satellite attitudes to zero respectively. 3.1. Position controller design
u RC ∈ R3×1 as follows: pi
FC u pi = u pi + u RC pi , i ∈ .
(9)
FC The nominal control input u pi is designed for the nominal po-
sition model and the robust compensating control input u RC is pi constructed to restrain the effects of the equivalent disturbance pi on the real position system. Design the nominal position conFC (i ∈ ) based on LQR control method as follows: trol inputs u pi
= −η F
w i j K p p i − p j − ξi j + K v p˙ i − p˙ j
j∈N i
− η F αli K p p i − ξi − pr + K v p˙ i − p˙ r ,
(10)
where η F represents a scalar coupling gain, and K p , K v ∈ R3×3 are diagonal nominal controller parameter matrices. αli is a constant representing the connection weight between the virtual leader and satellite i: αli = 1 indicates the information can flow from the virtual leader to satellite i, and αli = 0 otherwise. is constructed Furthermore, the robust compensating input u RC pi based on the robust filter F pi (s) as follows:
u RC pi = − F pi (s) pi ,
i ∈ ,
(11)
where s represents the Laplace operator and
72
p
73 74
p
2 2 u RC pi = − f pi p i + f pi z2i , i ∈ .
(13)
75 76
3.2. Attitude controller design
77
Similarly to the position controller, the attitude control input u σ i also includes two parts, the nominal part u σF Ci ∈ R3×1 and the robust compensating part u σRCi ∈ R3×1 as follows:
79
u σ i = u σF Ci + u σRCi ,
83
2
The robust filter F pi (s) is set as F p j ,i (s) = s + f p j ,i (j = 1, 2, 3) with the positive robust filter parameter f p j ,i to be determined. As elaborated in [35–37], the robust filter F pi (i ) would have a wider frequency bandwidth by selecting larger positive parameters f p j ,i ( j = 1, 2, 3). In this case, the filter gain of F pi (s) would approximate the unit matrix and the robust compensating input u RC would approximate − pi . Therefore, the effects of the pi equivalent disturbances can be restrained by selecting robust filter parameters appropriately. However, it is difficult to measure pi directly in practical applications, as shown in (11). From (7), one can have that f p2j ,i /
(12)
i ∈ .
(14)
wij
80 81 82 84
The nominal part u σF Ci is designed as:
85
σi − σ j + K σ σ˙ i − σ˙ j
86 87
j∈N i
88
− αli (σi + K σ σ˙ i ) ,
(15)
where K σ ∈ R3×3 represents a diagonal nominal controller parameter matrix. The robust compensating input u σRCi is constructed as:
89 90 91 92 93
u σRCi = − F σ i (s)σ i ,
(16)
94
where F σ i (s) = diag { F σ 1,i (s), F σ 2,i (s), F σ 3,i (s)} and F σ j ,i (s) =
95
f σ2 j ,i / s + f σ j ,i ( j = 1, 2, 3). The robust compensating part u σRCi can be realized in a similar way to u RC . pi
97
2
96 98 99
Remark 3. A relative motion model involving the chief satellite and the deputy satellites was studied in [38], and the resulted formation control law was distributed based on the model. In the current paper, the proposed attitude and position controllers for the similar relative motion model are distributed, because the designed controller of each satellite only depends on the information of its neighbors and itself.
100 101 102 103 104 105 106 107 108
4. Robustness analysis
109
For satellite i, define the trajectory tracking error epi = e p j ,i = p i − ξi − p r ∈ R3×1 and the linear velocity error evi = e v j ,i = p˙ i − p˙ r ∈ R3×1 . Define the attitude errors e = e = σi ∈ R3×1 (i ∈ σ i σ j , i ). Let K pz = K p K v , K σ z = I 3×3 K σ , and
A pz =
F pi (s) = diag { F p1,i (s), F p2,i (s), F p3,i (s)}.
pi = p¨ i − A Np1i p˙ i − A Np2i p i − u pi .
69 71
z˙ 2i = − f pi z2i + 2 f pi + A N p1 p i + z1i ,
u σF Ci = −
For satellite i, the position control input u pi includes two parts, FC the nominal part u pi ∈ R3×1 and the robust compensating part
FC u pi
68 70
2 f pi + A Np1 f pi − A Np2 p i + u pi ,
67
78
65 66
3. Formation protocol design
13 14
Let f pi = diag f p1,i , f p2,i , f p3,i . Then, substituting (12) to (11), one can realize the robust compensating input u RC by the followpi ing equations:
03×3 03×3
I 3×3 03×3
, Bz =
03×3 I 3×3
, Aσ z =
03×3 − I 3×3
I 3×3 −Kσ
z˙ pi = A pz z pi − η F B z
w i j K p e pi − e p j + K v e vi − e v j
j∈N i
where z pi =
e Tpi
e Tvi
T
112 113 114 116 117
118 119 120 121 122
123
− η F αli B z K p e pi + K v e vi + B z u RC pi + pi , z˙ σ i = A σ z zσ i + B z u σRCi + σ i ,
111
115
.
Then, one can have the node error system for each satellite i as follows:
110
124 125 126
(17)
127 128
= z p j ,i ∈ R6×1 and zσ i = e σ iT
e˙ σT i
T
129
= zσ j ,i ∈ R6×1 . Now, the global closed-loop system can be de-
130
scribed as:
132
131
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.5 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
˜ p, z˙ p = A pc z p + B ˜ σ, z˙ σ = A σ c zσ + B
(18)
˜ p = u RC + pi where z p ∈ R6N ×1 , zσ ∈ R6N ×1 , pi
∈ R3N ×1 ,
˜ σ = u RC + σ i ∈ R3N ×1 , B L = diag {αli } ∈ R N × N , B = I N × N ⊗ σi B z , A pc = I N × N ⊗ A pz − η F ( L + B L ) ⊗ B z K pz , and A σ c = I N × N ⊗ A σ z .
Furthermore, consider the design of symmetric and positive definite matrices Q k = Q kT ∈ R6×6 , k = kT ∈ R3×3 , where k = pz, σ z. Then, the controller parameter matrix K k can be given as: −1 T
K k = k B z P k ,
(19)
26 27 28 29 30 31 32 33
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
max z p (t ) ≤ max c T j
j
π + ς f p ε p / f p . p0 6N , j
71
Furthermore, the attitude tracking performance is analyzed. From (8), a positive constant ςdσ i can be obtained such that the equivalent disturbance σ i satisfies that
72
(25)
76
σ i ∞ ≤ M −1 (σi ) M (σi )1 u σ i ∞ + ςdσ i .
Proof. From (11), (16), and (18), one can obtain that
π p0 = e A pc t z p (0), πσ 0 = e A σ c t zσ (0), and χ Bp = (sI 6N ×6N − A pc )−1 B ( I 3N ×3N − F p (s))1 , χ B σ = (sI 6N ×6N − A σ c )−1 B ( I 3N ×3N − F σ (s))1 .
(21)
According to [35], if A pc and A σ c are asymptotically stable, there exists a positive constant f 1∗ such that: for any f p ≥ f 1∗ and f σ ≥
(26)
54 55 56 57 58 59 60 61 62 63 64 65 66
79 80 81 82 83
(27)
where ςσ z and ςσ d are positive constants depending on f p j ,i . If the constant f σ satisfies f σ ≥ f 1∗ and f σ > ςσ z ς f σ , substituting (27) to (20), one can obtain the following inequality as:
ςσ z πσ 0 ∞ + ςσ d σ ∞ ≤ . 1 − ςσ z ς f σ / f σ
84 85 86 87 88 89
(28)
90 91
Because A σ c is asymptotically stable and the initial state zσ (0) is bounded, πσ 0 is also bounded. Now, from (28) and (20), it can be observed that σ and zσ are bounded and satisfy
92
σ ∞ ≤ εσ , zσ ∞ ≤ εzσ ,
96
(29)
93 94 95 97
where εσ and εzσ are positive constants. Similarly, from (16), (18), and (29), one can obtain that
T max | zσ (t )| ≤ max c 6N , j πσ 0 + ς f σ εσ / f σ . j
and f σ = min f σ j , one can ς f σ , which are independent of
χ Bp ≤ ς f p / f p , χBσ ≤ ς f σ / f σ .
(22)
Therefore, it can be seen from that there exist finite positive ∗ ∗ constants t ∗ and f ∗ satisfying f p j ,i ≥ f and f σ j ,i ≥ f (i ∈ ), and the errors satisfying max j z pi (t ) ≤ εz and max j | zσ i (t )| ≤ εz , ∀t ≥ t ∗ . 2
98 99 100 101 103 104 105 106 107
Remark 4. It should be noted that the theoretical values of the robust filter parameters f pi and f σ i (i ∈ ) derived from Theorem 1 may be conservative. The real values of the robust filter parameters in practical applications may be much smaller than their theoretical values. Therefore, the following unidirectional tuning method can be applied to determine the robust filter parameters as follows. First, the robust filter parameters f pi and f σ i (i ∈ ) are set with certain initial positive values. Second, the robust filter parameters with larger values are adjusted until the desired tracking performance can be obtained.
108 109 110 111 112 113 114 115 116 117 118
52 53
75
102
(20)
where
f 1∗ (i ∈ ), where f p = min f p j obtain positive constants ς f p and f p j and f σ j , and satisfy
74
78
σ ∞ ≤ ςσ z zσ ∞ +ςσ d ,
j
z p ≤ π p0 + χ Bp p , ∞ ∞ ∞ zσ ∞ ≤ πσ 0 ∞ + χ B σ σ ∞ ,
73
77
From (14), (15), and (16), one has that
A Tpz P k + P k A pz + Q k − P k B z k−1 B zT P k = 0.
Theorem 1. Consider the satellite dynamics given by (7), the formation control protocol designed in Section 3, and the scalar coupling gain η F , and K pz and K σ z determined by (19). If the directed graph G has a spanning tree, the root can obtain information from the virtual leader, and the initial states z p (0) and zσ (0) are bounded, then for a given positive constant εz , there exist finite positive constants t ∗ and f ∗ such that: for any f p j ,i ≥ f ∗ and f σ j ,i ≥ f ∗ (i ∈ ), all states involved in the closed-loop control system are bounded, and the tracking errors sat isfy max j z p j ,i (t ) ≤ εz and maxi | zσ i (t )| ≤ εz , ∀t ≥ t ∗ .
68 70
where ςσ zi is a positive constant depends on f p j ,i . Combining (25) and (26), one can obtain that
According to Theorem 1 in [34], if the directed graph G has a spanning tree and the root can obtain information from the virtual leader, then, if η F ≥ λmin pr /2, A pc is asymptotically stable, where λ pi (i ∈ ) represent the eigenvalues of ( L + B L ) and λmin pr = mini ∈ Re(λ pi ). Besides, A σ c is also asymptotically stable, since all its eigenvalues have negative real parts.
67 69
u σ i ∞ ≤ ςσ zi zσ i ∞ + σ i ∞ ,
34 35
all states involved in the transitional dynamics are bounded. Now, from (8), (18), (23), and (24), one can have that
where P k is the positive definite solution of the following Riccati equation
24 25
5
Since the parameter uncertainties and external disturbances are bounded, there exists a positive constant ε pi such that the equivalent disturbance pi in (8) satisfies that
pi ≤ ε pi . ∞
(23) f 1∗ , and be-
Let ε p = maxi ε pi . If the constant f p satisfies f p ≥ cause A pc is asymptotically stable and the initial state z p (0) is bounded, then, from (20) and (23), it can be seen that z p is bounded. In this case, one can obtain that
z p
∞
≤ εzp .
(24) p
p
The bounds of the states of the robust filter z1i and z2i depend on the robust compensator parameters. It can be observed that
5. Simulation results
119 120
This section presents a series of simulation results of five satellites to verify the effectiveness of the proposed robust formation control method, and thus = {1, 2, 3, 4, 5}. Each satellite is modeled as described in Section 2 with J iN = diag {4.34, 4.33, 3.66} kg · m2 and μ g = 3.986 × 1014 m3 /s2 . The trajectory of the virtual
T
leader is given by p = 0.1t 0.1t 0.1t and the five satelare required to form a pentagon formation lites pattern as ξ1= T T T 0 1020 0 , ξ2 = 971 315 0 , ξ3 = 600 −825 0 , r
121 122 123 124 125 126 127 128
T T ξ4 = −600 −825 0 , and ξ5 = −971 315 0 with inter-
129
national units. The upper limit of the simulation time is 300 s. The topological relationship of the five satellites showed in Fig. 2 is
131
130 132
JID:AESCTE
AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.6 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
6
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
Fig. 2. Topological relationship of satellites.
20
86
21
87
22
88
23
89
24
90
25
Fig. 4. Linear velocity response by the proposed controller.
26
91 92
27
93
28
94 95
29
disturbances introduced in [38] are time-invariant to simulate the effects of multiple disturbances including the Earth’s non-spherical gravity, the solar radiation pressure, and the third body gravity. In the current paper, time-varying and non-vanished disturbances are considered, which are more realistic to validate the advantages of the proposed robust formation controller in complex aerospace environments. The real satellite parameters are chosen to be 10% larger than the nominal parameters and the external disturbances T are selected as: d p = 5 sin(2t ) sin(2t ) sin(2t ) for the posi-
30 31 32 33 34 35 36 37 38
39 40 41 42
Fig. 3. Three-dimensional trajectory by the proposed controller. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
⎡
0 0 0 0 1 0 0 0 ⎢ ⎢ wij = ⎢ 0 1 0 0 ⎣1 0 0 0 0 0 0 1
⎤
0 0⎥ ⎥ 0 ⎥. ⎦ 0 0
Only satellite 1 can obtain the information from the virtual leader, and thereby αl1 = 1, αl2 = 0, αl3 = 0, αl4 = 0, and αl5 = 0. The initial orbital elements of the chief satellite are shown as: semimajor axis a L = 7400 km, eccentricity c eL = 0.1, inclination i L = 30◦ , longitude of the ascending node L = 100◦ , argument of periapsis ω L = 0◦ , and the mean anomaly c f M L = 0◦ . The initial values of the relative orbital motion are given as: p 1 (0) = T T T 0 50 0 , p 2 (0) = 50 10 0 , p 3 (0) = 10 −10 0 , p 4 (0) =
T
T
T
−10 −10 0 , p 5 (0) = −20 0 0 , p˙ 1 (0) = 0.1 0.05 0.1 , T T p˙2 (0) = 0 0.1 0.05 , p˙ 3 (0) = −0.1 −0.08 0 , p˙ 4 (0) = T T 0 0.08 0.1 , and p˙ 5 (0) = 0.2 0.1 0.08 . The initial MRP of T the five satellites are chosen as follows: σ1 (0) = 0.2 0.2 0.2 , T T σ2 (0) = 0.15 0.15 0.15 , σ3 (0) = 0.05 0.05 0.05 , σ4 (0) = T T −0.1 −0.1 −0.1 , and σ5 (0) = 0.1 0.1 0.1 . The external
T
for the attition control input and dτ = 0.05 sin t sin t sin t tude torque. The scalar coupling gain is set to be η F = 1. The nominal controller parameters are chosen as: K p = diag {10, 10, 10}, K v = diag {400, 400, 100}, and K σ = diag {9, 9, 9}, to guarantee that the matrices A pc and A σ c are Hurwitz. The robust filer parameters f pi and f σ i are selected as shown in Remark 4. Figs. 3, 4, 5, and 6 depict the three-dimensional trajectories p i , linear velocity responses p˙ i , trajectory tracking error e pi , and the attitude simulation response σi of the five satellites, respectively. The red, blue, pink, green, and black solid lines indicate Satellites 1-5, and the red dotted line represents the formation pattern. One can observe that the trajectory tracking error e pi can converge into a neighborhood of the origin bounded by εz in 200 s, the steadystate trajectory tracking error is approximate 0.05 m, the satellites align their attitudes in 60 s, and the steady-state attitude error is approximately 0.01. Furthermore, the proposed robust formation controller is compared to a leader-following formation controller in [8] for the team of satellites. Fig. 7 shows the formation trajectory of the satellite group. Linear velocity response, trajectory tracking error, and attitude response are illustrated in Figs. 8, 9, and 10 respectively, by the leader-following formation control method. In contrast, the steady-state trajectory tracking error is approximately 0.8 m and the steady-state attitude error is nearly 0.03. From these figures, it can be observed that the proposed closed-loop control system can achieve better trajectory and attitude tracking performances under the influence of nonlinear dynamics, parameter uncertainties, and external disturbances.
96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.7 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
7
1
67
2
68
3
69
4
70
5
71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
Fig. 7. Three-dimensional trajectory by the leader-following controller.
19
84 85
20
86
21
87
22
88
23
89
24
90 91
25 26
Fig. 5. Trajectory tracking error by the proposed controller.
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111 112
46 47
Fig. 8. Linear velocity response by the leader-following controller.
48
113 114
49
115
50
116
51
origin in a finite time. Simulation results are given to demonstrate the effectiveness of the designed robust controller.
52 53 54
Fig. 6. Attitude response by the proposed controller.
55
6. Conclusion
62 63 64 65 66
121
None declared.
123 124 125
59 61
119
122
57
60
118 120
Declaration of competing interest
56 58
117
A robust controller is presented for satellite formation flying subject to nonlinearity, parametric uncertainties, and external disturbances. The proposed robust formation controller yields a position controller to form designed formation trajectories and patterns and an attitude controller to align satellite attitudes, for each satellite. Theoretical analysis guarantees that the trajectory and attitude tracking errors can converge into a given neighborhood of
Acknowledgements
126 127
This work was supported by the National Natural Science Foundation of China under Grants 61873012, 61503012, and 61633007, the National Science Foundation under Grants 1730675 and 1714519, and the Office of Naval Research under Grant N00014-17-1-2239.
128 129 130 131 132
JID:AESCTE
AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.8 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Fig. 9. Trajectory tracking error by the leader-following controller.
27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
Fig. 10. Attitude response by the leader-following controller.
56 57 58 59 60 61 62 63 64 65 66
References [1] W. Wang, G. Mengali, A.A. Quarta, J. Yuan, Distributed adaptive synchronization for multiple spacecraft formation flying around Lagrange point orbits, Aerosp. Sci. Technol. 74 (2018) 93–103. [2] X. Huang, Y. Yan, Y. Zhou, Y. Yang, Dual-quaternion based distributed coordination control of six-DOF spacecraft formation with collision avoidance, Aerosp. Sci. Technol. 67 (2017) 443–455. [3] C. Wei, S.-Y. Park, C. Park, Optimal H ∞ robust output feedback control for satellite formation in arbitrary elliptical reference orbits, Adv. Space Res. 54 (6) (2014) 969–989.
[4] H.-C. Lim, H. Bang, Adaptive control for satellite formation flying under thrust misalignment, Acta Astronaut. 65 (1) (2009) 112–122. [5] H. Sun, S. Li, S. Fei, A composite control scheme for 6DOF spacecraft formation control, Acta Astronaut. 69 (7) (2011) 595–611. [6] D. Lee, S. Kim, J. Suk, Formation flight of unmanned aerial vehicles using track guidance, Aerosp. Sci. Technol. 76 (2018) 412–420. [7] F. Morbidi, G.L. Mariottini, D. Prattichizzo, Observer design via immersion and invariance for vision-based leader-follower formation control, Automatica 46 (1) (2010) 148–154. [8] A. Mahmood, Y. Kim, Leader-following formation control of quadcopters with heading synchronization, Aerosp. Sci. Technol. 47 (1) (2015) 68–74. [9] G. Antonelli, F. Arrichiello, S. Chiaverini, Experiments of formation control with multirobot systems using the null-space-based behavioral control, IEEE Trans. Control Syst. Technol. 17 (5) (2009) 1173–1182. [10] T. Balch, R.C. Arkin, Behavior-based formation control for multirobot teams, IEEE Trans. Autom. Control 14 (6) (1998) 926–939. [11] Y. Cao, W. Ren, Distributed coordinated tracking with reduced interaction via a variable structure approach, IEEE Trans. Autom. Control 57 (1) (2011) 33–48. [12] T. Broek, N. Wouw, H. Nijmeijer, Formation control of unicycle mobile robots: a virtual structure approach, Int. J. Control 84 (11) (2010) 1886–1902. [13] M. Xin, S.N. Balakrishnan, H.J. Pernicka, Position and attitude control of deepspace spacecraft formation flying via virtual structure and θ -D technique, J. Dyn. Syst. Meas. Control 129 (5) (2007) 689–698. [14] W. Ren, Consensus strategies for cooperative control of vehicle formations, IET Control Theory Appl. 1 (2) (2007) 505–512. [15] L. Cao, Y. Chen, Z. Zhang, Predictive smooth variable structure filter for attitude synchronization estimation during satellite formation flying, IEEE Trans. Aerosp. Electron. Syst. 53 (3) (2017) 1375–1383. [16] Z. Zhang, J. Li, H. Bao, Relative attitude pointing tracking control of satellite formation flying with considering the orientation of solar panel, J. Astronaut. 128 (28) (2008) 2786–2787. [17] I. Chang, S.-Y. Park, K.-H. Choi, Decentralized coordinated attitude control for satellite formation flying via the state-dependent Riccati equation technique, Int. J. Non-Linear Mech. 44 (8) (2009) 891–904. [18] J. Zhou, Q. Hu, M. Friswell, Decentralized finite time attitude synchronization control of satellite formation flying, J. Guid. Control Dyn. 36 (1) (2013) 185–195. [19] J. Su, Y. Dong, Sliding mode variable structure control for electromagnetic satellite formation station-tracking, J. Astronaut. 32 (5) (2011) 1093–1099. [20] S.-H. Mok, Y-H. Choi, H.-C. Bang, Collision avoidance using linear quadratic control in satellite formation flying, Int. J. Aeronaut. Space Sci. 11 (4) (2010) 351–359. [21] M. Paolo, K. Tamas, G. Eberhard, V. Michel, A decomposition-based approach to linear time-periodic distributed control of satellite formations, IEEE Trans. Control Syst. Technol. 19 (3) (2011) 481–492. [22] J. Li, Y. Pan, K.D. Kumar, Design of asymptotic second-order sliding mode control for satellite formation flying, J. Guid. Control Dyn. 35 (1) (2012) 309–316. [23] Y.-H. Lim, H.-S. Ahn, Relative position keeping in satellite formation flying with input saturation, J. Franklin Inst. 351 (2) (2014) 1112–1129. [24] J. Yang, J. Sun, W.X. Zheng, S. Li, Periodic event-triggered robust output feedback control for nonlinear uncertain systems with time-varying disturbance, Automatica 94 (2018) 324–333. [25] J. Yang, Z. Ding, Global output regulation for a class of lower triangular nonlinear systems: a feedback domination approach, Automatica 76 (2017) 65–69. [26] J. Yang, W.X. Zheng, Offset-free nonlinear MPC for mismatched disturbance attenuation with application to a static var compensator, IEEE Trans. Circuits Syst. II, Express Briefs 61 (1) (2014) 49–53. [27] J. Yang, H. Wu, L. Hu, S. Li, Robust predictive speed regulation of converterdriven DC motors via a discrete-time reduced-order GPIO, IEEE Trans. Ind. Electron. 66 (10) (2018) 7893–7903, https://doi.org/10.1109/TIE.2018.2878119. [28] A. Umair, M.W. David, R.L. Jaime, Control of electromagnetic satellite formations in near-Earth orbits, J. Guid. Control Dyn. 33 (6) (2010) 1883–1891. [29] S.-J. Chung, U. Ahsun, J.E. Slotine, Application of synchronization to formation flying spacecraft: Lagrangian approach, J. Guid. Control Dyn. 32 (2) (2009) 512–526. [30] H. Liu, Y. Tian, F.L. Lewis, Y. Wan, K.P. Valavanis, Robust formation control for a team of satellites, in: International Conference on Unmanned Aircraft Systems, 2018, pp. 30–35. [31] P. Tsiotras, Further passivity results for the attitude control problem, IEEE Trans. Autom. Control 43 (11) (1998) 1597–1600. [32] O. Egelend, Passivity-based adaptive attitude control of a rigid spacecraft, IEEE Trans. Autom. Control 39 (4) (1994) 842–846. [33] S. Nair, K. Subbarao, Attitude control of spacecraft formations subject to distributed communication delays, in: AAS/AIAA Space Flight Mechanics Meeting, 2017, pp. 1707–1722. [34] H. Zhang, F.L. Lewis, A. Das, Optimal design for synchronization of cooperative systems: state feedback, observer and output feedback, IEEE Trans. Autom. Control 56 (8) (2011) 1948–1952. [35] H. Liu, X. Wang, Y. Zhong, Quaternion-based robust attitude control for uncertain robotic quadrotors, IEEE Trans. Ind. Inform. 11 (4) (2015) 406–415.
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
JID:AESCTE AID:105455 /FLA
[m5G; v1.261; Prn:8/10/2019; 15:25] P.9 (1-9)
H. Liu et al. / Aerospace Science and Technology ••• (••••) ••••••
1 2 3 4 5
[36] H. Liu, F. Peng, F.L. Lewis, Y. Wan, Robust tracking control for tail-sitters in flight mode transitions, IEEE Trans. Aerosp. Electron. Syst. 55 (4) (2019) 2023–2035, https://doi.org/10.1109/TAES.2018.2880888. [37] H. Liu, T. Ma, F.L. Lewis, Y. Wan, Robust formation control for multiple quadrotors with nonlinearities and disturbances, IEEE Trans. Cybern. (2019), in press, https://doi.org/10.1109/TCYB.2018.2875559.
9
[38] H. Min, S. Wang, F. Sun, Z. Gao, Y. Wang, Distributed six degree-of-freedom spacecraft formation control with possible switching topology, IET Control Theory Appl. 5 (9) (2010) 1120–1130.
67 68 69 70 71
6
72
7
73
8
74
9
75
10
76
11
77
12
78
13
79
14
80
15
81
16
82
17
83
18
84
19
85
20
86
21
87
22
88
23
89
24
90
25
91
26
92
27
93
28
94
29
95
30
96
31
97
32
98
33
99
34
100
35
101
36
102
37
103
38
104
39
105
40
106
41
107
42
108
43
109
44
110
45
111
46
112
47
113
48
114
49
115
50
116
51
117
52
118
53
119
54
120
55
121
56
122
57
123
58
124
59
125
60
126
61
127
62
128
63
129
64
130
65
131
66
132