Robust and global attitude stabilization of magnetically actuated spacecraft through sliding mode

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Robust and global attitude stabilization of magnetically actuated spacecraft through sliding mode

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Ahmet Sofyalı

, Elbrous M. Jafarov

a, 2

, Rafael Wisniewski

b,3

a

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79 81 82 83

Istanbul Technical University, Istanbul, 34469, Turkey b Aalborg University, Aalborg, 9220, Denmark

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a, 1

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a r t i c l e

i n f o

Article history: Received 25 November 2016 Received in revised form 18 August 2017 Accepted 25 January 2018 Available online xxxx Keywords: Magnetic spacecraft attitude control Sliding mode control

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a b s t r a c t

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The inertial pointing problem of a rigid satellite by solely magnetic torqueing is considered in this paper. To ensure globally uniformly ultimately bounded motion about the reference in inertial space, a sliding mode attitude control law, which consists of equivalent and reaching control terms, based on a novel time-varying sliding manifold is designed. The originality of the sliding manifold relies on the inclusion of two time-integral terms. The usage of the proposed sliding manifold makes the application of the equivalent control method to the considered problem possible, and it is proven that the state trajectories reach the newly designed sliding manifold in finite time even under the effect of four realistically modeled disturbance components and parametric uncertainty of all inertia matrix entries. For the constructed purely magnetic attitude control system, stability and existence of the sliding mode as well as state trajectories’ finite time convergence to the sliding manifold are demonstrated via Lyapunov function techniques. The results of a simulation example verify the robust stability of the designed attitude control system. The steady state performance of the attitude control system is evaluated in the altitude range of low-Earth-orbits. © 2018 Elsevier Masson SAS. All rights reserved.

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1. Introduction

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SPACECRAFT’S angular motion around their center of gravity, namely their attitude is controlled around their three body axes basically by onboard torque producing actuator triads. The triads consist of three orthogonally placed actuators. There are three types of such actuators: momentum exchange devices (reaction/momentum wheels, control moment gyroscopes), reaction thrusters (gas jets), and magnetic torquers (rods, coils). The members of the first two types produce directly the control torques around the body axis along which they are placed on the spacecraft. The loss of any member renders the attitude control system underactuated around that axis. The asymptotic stabilization problem of such directly torque producing and underactuated systems

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E-mail address: [email protected] (A. Sofyalı). 1 Research Assistant, Department of Astronautical Engineering, Istanbul Technical University, Faculty of Aeronautics and Astronautics Ayazaga, Campus Maslak Sariyer, 34469 Istanbul, Turkey. 2 Professor, Department of Aeronautical Engineering, Istanbul Technical University, Faculty of Aeronautics and Astronautics Ayazaga, Campus Maslak Sariyer, 34469 Istanbul, Turkey. IAA Corresponding Member. 3 Professor, Department of Electronic Systems, Automation and Control Section, Aalborg University, Fredrik Bajers Vej 7, 9220 Aalborg Ø, Denmark. https://doi.org/10.1016/j.ast.2018.01.022 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

have been an important topic of investigation for researchers in automatic control field [1–3]. The conclusion is that it is impossible to stabilize such underactuated rigid spacecraft in even locally asymptotical manner by using continuous time-invariant control laws [4]. The sole usage of the third type of actuators leads to a challenging and therefore interesting control problem. Even if all three magnetic torquers are operational, the control torque lies in a plane, which is orthogonal to the geomagnetic field vector at the satellite’s location. This phenomenon emerges from the fact that, according to physics, the magnetic control torque is the output of the cross-product of the magnetic moment vector, which is what the magnetic actuator triad directly produces, with the local geomagnetic field vector. Thus, attitude control by purely magnetic actuation lacks three-axis control authority intrinsically. However, the controllability of such an underactuated system could be proven thanks to its second challenging property, time-variance, by using a nonrotating dipole model for the geomagnetic field [5]. The time-variance results from the orbital motion of the satellite around the Earth provided that the orbital plane does not coincide with the equatorial plane of the geomagnetic field. As a result, the system can be considered to be instantaneously underactuated [6] because the direction along which there is no control

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Nomenclature

 B b bn CB DB

 d n d unc d f n h I J Ji Jn kint q kq ks k ss  M m n q q⊥ B q B

 [–] q ’s component along B

q4 s T

skew-symmetric matrix of the vectorial quaternion component (3 × 3) [–] scalar quaternion component [–] sliding surface vector (3 × 1) [rad/s] orbital period [s]

q˜

33 34 35 36

local geomagnetic field vector of size 3 × 1 [T] control matrix (7 × 3) nominal control matrix (7 × 3)  (3 × projection matrix onto the plane orthogonal to B 3) [–]  (3 × 3) projection matrix onto the direction along B [–] total disturbance vector (7 × 1) nominal disturbance vector (7 × 1) disturbance vector solely due to satellite model uncertainty (7 × 1) nominal system vector (7 × 1) orbital altitude [km] identity matrix (3 × 3) [–] uncertain inertia matrix (3 × 3) [kg m2 ] principal moments of inertia [kg m2 ], i = 1, 2, 3 nominal inertia matrix (3 × 3) [kg m2 ] dimensionless sliding surface design parameter [–] sliding surface design parameter [rad/s] continuous reaching law design parameter [N ms] discontinuous reaching law design parameter [N m] magnetic control moment vector (3 × 1) [A m2 ] input number of the control system [–] orbital angular velocity (mean motion) [rad/s], order of the control system [–] vectorial quaternion component (3 × 1) [–]  [–] q ’s component orthogonal to B

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T aero Td T gg T mag Tmc T solar T unc ts t0  u  eq u  reach u   B u

 ⊥ B u

ω ω ⊥ B ω  B ω˜ x xN J γ θ λ

ϕ ψ ..2 ..∞ |..| ..i2

aerodynamic drag torque vector (3 × 1) [N m] environmental disturbance torque vector (3 × 1) [N m] gravity-gradient torque vector (3 × 1) [N m] residual magnetic torque vector (3 × 1) [N m] magnetic control torque vector (3 × 1) [N m] solar pressure torque vector (3 × 1) [N m] disturbance torque vector solely due to satellite model uncertainty (3 × 1) [N m] starting moment of the sliding mode [s] starting moment of the control process [s] control vector (3 × 1) [N m] equivalent control vector (3 × 1) [N m] reaching control vector (3 × 1) [N m]  ’s component parallel to B [N m] u

 ’s component orthogonal to B [N m] u absolute angular velocity vector (3 × 1) [rad/s] ω ’s component orthogonal to B [rad/s]

ω ’s component along B [rad/s]

skew-symmetric matrix of the absolute angular velocity vector (3 × 3) [rad/s] state vector (7 × 1) reference state vector for inertial pointing (7 × 1) inertia uncertainty matrix (3 × 3) [kg m2 ] auxiliary torque vector (3 × 1) [N m] attitude (Euler) angle [deg] eigenvalue attitude (Euler) angle [deg] attitude (Euler) angle [deg] L 2 (quadratic) norm of a vectorial signal L ∞ norm of a vectorial signal determinant of a matrix induced L 2 norm of a matrix

37

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

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

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authority varies with respect to the body axes while the satellite moves along its orbit. The controllability of attitude control systems employing gas jets [7] and reaction wheels [8] has also been investigated before for both the cases of full and lacking control authority. Since magnetic actuators suit the small satellite concept well regarding their favorable properties in terms of mass, volume, nominal power consumption, low failure risk due to nonmoving structural elements, the research interest in purely magnetic attitude control problem also serves purposes of engineering application. If the satellite’s mission requires moderate pointing accuracy (>1◦ ), no rapid stabilization and agile maneuvering (in hours), magnetic actuators have the capability to serve as primary actuators. Besides many university pico/nanosatellites (mostly CubeSats) controlled by purely magnetic actuation [9], ORBCOMM [10], and Ørsted [11] are two microsatellites actively controlled by solely magnetic means in addition to passive gravity gradient stabilization assist. A remarkable application of this approach is the GOCE mission by ESA. The satellite GOCE weighing 1052 kg, which has a passively aerodynamically stable structural design, employed only a magnetic rod triad for attitude control and is the first and so far only nonsmall satellite with such an attitude control system [12]. A recently launched minisatellite Proba-V of ESA, which has no passive attitude stabilization assist, utilizes purely magnetic threeaxis stabilization in its safe mode [13]. These real life examples indicate the industrial need for control algorithms that will drive a magnetic torquer triad in more beneficial ways, especially in ways

guaranteeing global and robust stabilization without passive stabilization assist. It is aimed with this paper to present in detail an achievement in global and robust attitude stabilization of a rigid satellite by purely magnetic actuation in a nearly circular orbit with low altitude, which does not lie in the geomagnetic equatorial plane; this achievement has been first presented briefly in [14]. The majority of works in literature dealing with purely magnetic attitude control in three-axis proposed local solutions [15–23]. In [20], which is the first and only literature survey on purely magnetic attitude control, particularly local approaches to the problem are well classified. [23] proposes a robust, but local solution to the problem by designing a control system that has stability robustness against model uncertainty via periodic-state feedback and H ∞ control. There is a limited number of works that propose global solutions to the considered problem, which can be summarized as follows:

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1) A globally asymptotical solution to the Earth (nadir)-pointing problem; valid for satellites with gravity-gradiently stable inertia distribution, based on the periodicity assumption of the geomagnetic field [24,25]. The periodic extension of the Lyapunov’s stability theory is used to derive the state-feedback controller. 2) An almost globally asymptotical solution to the Earth-pointing problem; valid for satellites with gravity-gradiently stable inertia distribution, based on average controllability of the system provided by the – not necessarily periodic – variation of the geomagnetic field during one orbital period [26]. The averag-

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ing theory is used to derive the state-feedback controller. In this solution, initial angular velocities are restricted to be sufficiently lower than the orbital angular velocity. 3) An almost globally asymptotical solution to the inertialpointing problem by state- [27] and output- [28] feedback; based on average controllability as in 2). The averaging theory is used to derive the solutions, in which initial angular velocities are restricted to be sufficiently lower than the orbital angular velocity.

(x, t ) x˙ (t ) = f n (x) + bn (x, t ) u +d

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In none of these works, satellite model (inertia matrix) uncertainty or environmental disturbance torques are taken into account during controller design. To the best knowledge of the authors, in literature, there are two examples of robust efforts that deal with nonlinear attitude dynamics:

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a) The solution in 2) is aided by a dynamic neural network to gain robustness against variations in inertia matrix and disturbance due to nondiagonal inertia matrix [29]. This design has the counteraction capability to disturbing gravitygradient torque that arises in the Earth-pointing problem from the gravity-gradiently unstable inertia distribution. The robustness is seen in simulation results, it is not deduced by theory. b) The solution in 3) is revisited by taking the inertia matrix uncertainty into account during the design. The result is a locally exponentially stable inertial-pointing controller that is also robust to satellite model uncertainty [30]. The averaging theory is used besides the Lyapunov’s stability theory. The upper and lower bounds for the principal moments of inertia are assumed to be known. No environmental disturbance torque is taken into account during controller design.

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The literature survey indicates that no controller that provides the inertial pointing problem with a global and robust stability has been designed yet. Such a solution that is also based on the original time-varying dynamical structure of the problem without requiring a periodic or averaged geomagnetic field seems to be highly welcomed if, in addition, it takes both environmental disturbance torques and inertia matrix uncertainty into account during controller design. With the aforementioned aim, in this work, it is benefited from the opportunity of designing an attracting manifold of reduced order dynamics in state space, which is given to the designer by the sliding mode control method. As now widely known, sliding mode control is a powerful tool for providing a controlled nonlinear system with robustness to parameter uncertainties and external disturbances [31–34]. A sliding manifold is proposed for the purely magnetic attitude control problem to overcome its two challenges: instantaneous underactuation and time-variance. By using the designed manifold, the equivalent control method can be applied to the problem and a magnetic sliding mode attitude controller can be designed. The following parts of the paper are organized as follows: Section 2 formulates the attitude stabilization problem. In Section 3, the magnetic sliding mode attitude controller is designed. Results of a simulation example are presented in Section 4. Section 5 concludes the paper.

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2. Problem formulation

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In this work, the control problem that is aimed to be solved by purely magnetic actuation and in a global and robust manner is inertial pointing of a rigid satellite. As derived in [35], the state equation describing the problem is as follows:

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n (x, t ) + dunc (x, t ).  f n (x) + bn (x, t ) u +d

(1)

 1×3 1 0 1×3 ] T , Here; x = [ q q4 ω ] and x N  [0 T T

T

⎡

1 (q 2 4

⎡ ⎢

72

(2)

⎥

77

(3)

⎤

 3×1 0 0

⎢

J n−1 [ T gg (x) + T aero (x, t ) + T solar (x, t ) + T mag (x, t )]

⎡

⎡

n

x, t ) d (

unc (x, t ) = ⎢ d ⎣

0 J n−1 T unc (x, t )

79 80 81 82 83 84

(4)

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⎤

 3×1 0

78

85

⎥ ⎦,

0 J −1 T

⎥ ⎦

⎤

 3×1 0

⎢ =⎣

74 76

J n−1 C B

n (x, t ) = ⎣ d

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 1×3 ⎦ , bn (x, t ) = ⎣ 0 ⎡

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⎤

03×3

68 70

⎤

ω + q × ω ) ⎥ f n (x) = ⎢ ) − 12 (q · ω ⎣ ⎦, − 1  ) − J n (ω × J n ω

10 11

3

89

⎥ ⎦.

(5)

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As seen in (5), the effect of satellite model uncertainty on system dynamics can be represented by the torque vector

 ×  Jω  −J J T unc (x, t ) = −ω

−1



 × Jω  + Tmc + Td (x, t ) . −ω (6)

The reader should refer to [35] for the derivation of (6). J = J n +  J , and J n  diag( J 1 , J 2 , J 3 ), which means that body axes of the satellite nominally coincide with its principal body axes.

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Definition 1.  J   J 1 +  J 2 , where

⎡

δ1 J 1

 J1  ⎣ 0

0

0 δ2 J 2 0

104

⎤

105

0 0 ⎦; δ3 J 3

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δ1 , δ2 , δ3 ∈ [−δ¯ J 1 ; +δ¯ J 1 ]

(7)

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represents the uncertainty due to imperfect CAD calculations or erroneous measurements of principal moments of inertia, and

⎤

⎡

δ11 δ12 δ13  J 2  (1 + δ¯ J 1 ) max( J 1 , J 2 , J 3 ) ⎣ δ12 δ22 δ23 ⎦ ; δ13 δ23 δ33 ¯ ¯ δi j ∈ [−δ J 2 ; +δ J 2 ], i , j = 1, 2, 3,

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(8)

models the uncertain angular difference between body and principal axis systems of the satellite.

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The inclusion of the defined projection matrix C B in bn emerges from the nature of magnetic actuation and owes its structure to the usage of the projection-based control law, is defined in (9). That law has been widely used in purely magnetic control system design after first proposed in [15].

 (x, t ) × B (x, t ) = Tmc (x, t ) = M =

B˜ (x, t ) T B˜ (x, t )

 B (x, t )2



 B (x, t ) × u  B (x, t )22

  C B (x, t ) u u



× B (x, t )

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⎡ 1 2

=

3

1 B 21 + B 22 + B 23

B 22 + B 23

−B1 B2

−B1 B3

⎤

⎢ ⎣ −B2 B1

B 23 + B 21

⎥ − B 2 B 3 ⎦ u .

−B3 B1

−B3 B2

B 21

4

+

3. Magnetic sliding mode attitude controller

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B 22

5

(9)

6

 to the applied (actual) It relates the desired (ideal) control law u magnetic control torque Tmc through two successive cross products. It also reduces energy consumption by preventing magnetic actu which will not ators from producing magnetic moment along B, contribute to Tmc .

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Remark 1. In a purely magnetic control system utilizing the projection-based law, the applied control torque is equal to the  At any  is orthogonal to B. desired torque only at moments when u  ’s component that is orthogonal to B can other moment, only u  to B is a desiract on the satellite. Thus, the orthogonality of u able situation, which is occasional due the dependency of the ideal feedback law on system’s states that can take arbitrary values according to the attitude motion of the satellite. It can be concluded  ’s component that is parallel to B into account in the that taking u feedback law design is going to provide the control system with the information on how close the instantaneous situation to the desirable one is.

  B be the component of u that is parallel to B Definition 2. Let u as defined in (10).



  B (x, t ) = u − u ⊥ B (x, t )  I − C B (x, t ) u  D B (x, t ) u u.

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 (x, t ) B (x, t )T B  B (x, t )2

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=

40

⎡

1 B 21 + B 22 + B 23

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(10)

Here, a second projection matrix D B is defined:

D B (x, t ) 

B 21

⎢ ⎣ B2 B1 B3 B1

B1 B2 B 22 B3 B2

B1 B3

⎤ ⎥

B2 B3 ⎦ .

43

B 23

A projection matrix pair is known to have the following properties:

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| C B | = | D B | = 0, λ1,2,3 (C B ) = 1, 1, 0,

(12a)

λ1,2,3 ( D B ) = 0, 0, 1,

(12b)

C B i2 =  D B i2 = 1,

(12c)

C B + D B = I,

(12d)

C B D B = 03×3 ,

(12e)

C kB = C B ,

D kB = D B ;

k = 1, 2, 3, . . . .

(12f)

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The considered nonlinear robust control problem is solved by the sliding mode control method, which provides the controller designer with the opportunity of designing an attracting manifold of reduced order dynamics in state space. The equivalent control approach is utilized to obtain a control law consisting of two parts as follows:

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  u eq + u reach . u

76

Remark 2. If the state equation of the control system is examined, its following properties can be written down easily: control-affine with a nonlinear drift term and in regular form. An affine system is said to be in regular form if the state equation can be divided into two blocks, one of which has the dimension of n–m and does not depend on control whereas the dimension of the other block with control is equal to m. C B renders the system both timevarying (non-autonomous) and underactuated by entering the last 3 × 3 block of the control matrix. The underactuation is obvious from the mathematical fact that rank(bn ) = 2 < m = 3. Because the  = bγ cannot be solved for a vector γ due to the singuequation d larity of C B , it can be concluded that disturbances in the control system are unmatched because they do not satisfy the matching condition by entering the system at different points with control signals [36,37].

(13)

⎛

⎞

kq

0

0

s(x) = ω  +⎝ 0

kq

 + K q q; 0 ⎠ q  ω

0

0

kq

03×3

k q > 0.

(14)

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78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 103 104 105

⎤

 1×3 ⎦ , b = ⎣0

71

102

This manifold is first proposed in [39] as the solution of the optimal attitude control problem, where a performance index having  as state and control vectors, respectively, is mini[q T q4 ] T and ω mized. It should be noted that the attitude control problem solved in [39] has the control matrix given in (15)

⎡

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77

Sliding mode control is a powerful tool for providing the controlled system with robustness to parameter uncertainties and ex eq is the conternal disturbances. The equivalent control vector u  on the tinuous equivalent of the discontinuous control vector u intersection of the sliding surfaces, namely on the sliding manifold. This equivalence is an idealization that only holds in the ideal  3×1 and under the assumption that sliding mode, i.e., for s = s˙ = 0  reach the switching of the discontinuous reaching control vector u occurs at theoretically infinite frequency. The reaching control vector is responsible for making state trajectories reach the manifold in finite time under disturbances and model uncertainty effects, and it becomes definitely zero once the ideal sliding begins, where “ideal” indicates that the switching frequency is infinite. Because infinite switching frequency is impossible from the application point of view, in the applications of sliding mode control, discontinuity is still present in control signals even after the sliding mode is entered [31–34]. There have been previous attempts of applying sliding mode control to the purely magnetic attitude control problem. In 1998, the equivalent control method is employed with a continuous reaching control term or reaching law thus the resulting control law is unable to carry the attitude motion into the sliding mode [38]. The sliding manifold used in [38] is

(11)

41 42

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106 107 108 109 110 111 112

(15)

J −1 and it belongs to the case of direct torque production, which is possible with use of reaction wheels or gas jets. In literature, it has become the preferred manifold when dealt with the sliding mode attitude control problem. In the same year with [38], a tracking controller, which closes the inner loop of a two-step control system designed based on backstepping approach, is developed via the equivalent control method with the aim of disturbance rejection. However, the purely magnetic attitude control problem is not considered in terms of global stability and robustness to model uncertainty [40]. In 2010, a passivity-based sliding mode controller is proposed. Even though that controller gives promising simulation results, it is not analyzed if the state trajectories reach the sliding manifold or not, which means that robustness against disturbances is not evaluated [41]. In [42], a second order sliding mode controller is designed by using a nonlinear manifold. The analysis on the reachability of the used manifold under the proposed control effect is

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absent in the study. To the best knowledge of the authors, there is only one more and very recent work, in which a sliding manifold is constructed by iteratively determining its variable design parameters at each step of the control process [43]. The aim of [43] is to obtain a control vector that is always orthogonal to the local geomagnetic field vector by a proper manifold design, which resembles the motivation of [14] and this work. Again, the reachability analysis is not presented. Disturbances are taken into account in simulation trials rather than during the controller design, and model uncertainty is not covered.

11 12

3.1. Newly proposed sliding manifold

15 16 17

Inspired by the conclusion in Remark 1, the following timevarying sliding manifold for the purely magnetic attitude control problem is proposed

t

18 19 20 21 22

−1

24





⎡

kint q

M int q  nK q ⎣ 0

28

0

0

0

kint q 0

with kint q > 0.

kint q (17)

30

33 34 35 36 37 38 39 40 41

  B is included in the sliding surface As suggested in Remark 1, u vector, which is based on the one presented in (14). The following three subsections will respectively show how its inclusion enables the application of the equivalent control method to the problem and leads to a manifold that renders the reference asymptotically stable in the sliding mode and is reachable by the state trajectories in finite time in the reaching mode. The second integral term with q as its integrand guarantees that the angular position converges to its reference in the sliding mode as the angular velocity converges to zero. In subsections 3.3 and 3.4, the need for this second term in the proposed sliding surface vector structure will be clarified.

42 43

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The equivalent control vector is derived under ideal sliding as 3×1 and d = 0 7×1 . If s is derived w.r.t. sumption, i.e., for s = s˙ = 0 time by assuming that the attitude motion is in the sliding mode and no more subject to environmental disturbances and undesired effects due to model uncertainty,

s˙ (x, t ) = G x˙ + M int q q + J n D B (x, t ) u eq  = G f n (x) + bn (x, t ) u eq + M int q q + J n−1 D B (x, t ) u eq  = G f n (x) + M int q q + J n−1 C B (x, t ) + D B (x, t ) u eq = G f n (x) + M int q q + J n−1 u eq = 0 (18) −1

 3×1 I ] . can be obtained thanks to (12d), where G  ∂s1 /∂x = [ K q 0  eq can be solved for from (18) by The equivalent control vector u using also (2):

61 62 63

 eq (x) = − u

1 2

) + ω  × Jnω  + q × ω  − J n M int q q. J n K q (q4 ω

(19)

64 65 66

) + ω  × Jnω  + q × ω . J n K q (q4 ω

(20)

70 71 72

3.3. Stability analysis of sliding mode

73 74

In the sliding phase on the time interval of [t s ,∞), i.e., when  and under ideal sliding assumption, it is straightforward s = s˙ = 0, to write

 eq = 0 3×1 = −C B CB DBu

˙ + K q q˙ + M int q q) J n (ω





 ˙ + kq (q˙ + nkint q q)  B Jn ω

76 77 79 80

By multiplying both hand sides of (21) by C B from left,



75

78

(21)

81 82

(22)

83 84 85

(23)

86 87 88

Remark 3. It can be easily verified that the utilization of the manifold in (14) instead of the proposed one leads to the following

90 91

 asymptotically under the assumption of no disturbances, when 0  reach ≡ − K s s. u

92

 is underactuated, and the The instantaneous direction of B ˙ + kq (q˙ + nkint q q)]} in the plane that component of the vector { J n [ω  is equal to zero. So, it can be is instantaneously orthogonal to B ˙ )⊥ B + kq [( J n q˙ )⊥ B + nkint q ( J n q)⊥ B ] = 0 holds concluded that ( J n ω in the plane where the control system has continuous control authority along the whole control process. Consider the case of attitude control via reaction wheels or  + thrusters through the sliding mode on the manifold s3  ω t dτ , which is what the manifold in (16) would K q q + M int q t q 0 be reduced to if C B ≡ I 3×3 and D B ≡ 03×3 held and, as a re was valid for the magnetic attitude sult, the definition Tmc  u control system. During the ideal sliding phase on this manifold, ˙ )⊥ B + kq [( J n q˙ )⊥ B + nkint q ( J n q)⊥ B ] = 0 together with C B J n s˙ 3 = ( J n ω

95

˙ ) B + kq [( J n q˙ ) B + nkint q ( J n q) B ] = 0 hold. It can D B J n s˙ 3 = ( J n ω

3.2. Derivation of equivalent control term

44 45

2

Note 1. It is verified through simulation that the vector ˙ + kq (q˙ + nkint q q)]} becomes parallel to B as s converges to { J n [ω

⎤

0 ⎦  nK q K int q

68

89

29

32

 eq = − CBu

is true in the sliding mode.

where

27

31

(16)

67 69

1

can be written from (12e). (22) indicates that

s2 [x,t ]

25 26

t

s(x, t ) = ω  + K q q + M int q qdτ + J n D B (x, τ ) u dτ ,    t0 t0 s1 [x]   

23

 eq unless depending on the equation, which cannot be solved for u  eq ≡ u eq : unrealistic assumption of C B u

 eq = − J n ω ˙ − J n K q (q˙ + nK int q q). DBu

13 14

5

be concluded that, in the plane that is instantaneously orthogonal  the case with s3 is equivalent to the case of attitude conto B, trol via magnetic torquers through the sliding mode on the man˙ ) B + kq [( J n q˙ ) B + ifold in (16). On the other hand, because ( J n ω

 does not hold in the case of purely magnetic nkint q ( J n q) B ] = 0 control through (16), the mentioned equivalence is not valid along  However, because there is inthe instantaneous direction of B.  and because B rotates with trinsically no control action along B, respect to body axes while the satellite moves along its orbit, the equality

93 94 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

˙ + K q q˙ + M int q q) = 0 J n (ω

(24)

 in the plane of continuous control auis equivalent to J n s˙ = 0 thority. This equivalence is going to be benefited from in the stability analysis of solely magnetically controlled attitude motion on the proposed manifold. Note that the same equivalent control vector as the one in (19) can be derived from the relation  which is what (24) indicates. ˙ + K q q˙ + M int q q = 0, of ω The result of the stability analysis of the ideal sliding mode under the assumption of no disturbances is formulated as the following lemma.

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6

1 2 3 4 5

 = 0,  the Lemma. Under the assumption of no disturbances, i.e., for d attitude motion of a rigid satellite on such an orbit and on the sliding manifold in (16) is uniformly asymptotically stable at the inertial reference state x N for all values of the sliding surface design parameters kq and kint q .

6 7 8 9 10 11 12 13 14 15 16

Proof. If the satellite’s orbital plane does not coincide with the geomagnetic equator plane then the attitude motion solely controlled by a magnetic actuator triad is controllable according to Theorem 5.1 together with Remark 5.1 in [5], which means that  will cover the whole the plane instantaneously orthogonal to B three dimensional space during the control process. Let us define a positive definite Lyapunov function candidate as

V = q q + (q4 − 1) T

2

17

1

+

18

 t

2

19

t0

 t

20 21

× M int q

22 23 24 25 26 27

−1 −1 qdτ + M int J q n

T

t ( D B u eq )dτ t0

−1 −1 qdτ + M int J q n

t0

= q T q + (q4 − 1)2  T 1 −1 −1  + K q q) M int q − M int  + K q q) > 0. + − M int (ω (ω q q (25)

32 33 34 35

 −1 T ˙ + K q q˙ ) M int q M int  + K q q) V˙ = −2q˙ 4 + M int q (ω (ω q  −1 T  + M int (− M int q q) = q T ω q  t  t

36 38

41

t0



39

= q

T

− K q q − M int q  t

43

+ q T M int q

45

t0

 t

= −q T K q q − q T M int q

49 50

 t

51

+ q T M int q

52 53 55 56 57 58 59

t0

t0

−1 −1 qdτ + M int J q n

t0

−1 −1 qdτ + M int J q n



t

t t0

66

71 72 73 75 76 77 78 79 80 81 82 83 84 85 86

89

0

0

 (s) − J n 0 ⎦ sgn k ss

ks

0

0

0

ks

0 ⎦ s

0

0

ks

k ss , k s > 0.

91 92 93 94 95 96

(29)

97 98 99 100

(30)

101 102 103 104 105

(31)

106

 in (31) is able to carry the state trajectories to the sliding If u manifold under disturbances and model uncertainty effect in a finite time interval after the control process begins at the moment of t 0 , the attitude motion enters sliding mode at a moment of t s . To check if this occurs, the ideality assumption has to be abandoned. The resulting complete time derivative of s is as follows:

107 108 109 110 111 112 113 114

(32)

115 116 117 118 119 120

(26)

which can be obtained by making use of the kinematic equation  and the equain (1) and (2), the sliding manifold equation s = 0, ˙ + K q q˙ + M int q q = 0 that is derived from (24). Thus, as tion of ω

− Jn

65

k ss

−1 ⎣

The existence conditions of the sliding mode, i.e., the stability conditions of the reaching mode arising under external disturbances and model uncertainty in the designed magnetic satellite attitude control system are presented in the following theorem.

( D B u eq )dτ

61

64

0

0

s˙ (x, t ) = G f n (x) + d(x, t ) + M int q q + J n−1 u (x, t ) = 0 3×1 .



= −q T K q q < 0,

 −1 t

0

⎤

1



( D B u eq )dτ t0

k ss

⎡

) + ω  × Jnω  + q × ω  J n K q (q4 ω 2  (s) − K s s. − J n M int q q − K ss sgn

( D B u eq )dτ

 q4 → 1 together with M int q time goes to infinity, q → 0,

63

=− u



t

60 62

s˙ = − J n

−1 ⎣

⎤

The resulting sliding mode attitude control law is

( D B u eq )dτ t0

−1 −1 qdτ + M int J q n

47

54

−1

qdτ − J n

t0

46 48

⎡

  reach (x, t ) = − K ss sgn  s(x, t ) − K s s(x, t ). u



t

70

88

(28)

Then the reaching control vector becomes

( D B u eq )dτ t0

t

42 44

−1

qdτ − M int q J n

× M int q −

37

40

−1

69

87

 (s) − J n−1 K s s;  − J n−1 K ss sgn

−1

68

90

The time derivative of the function V in (25) is given by



(27)

According to the reaching law approach [44], the reaching dynamics can be assigned as

2

29 31

In the reaching phase on the time interval of [t 0 , t s ) preceding the sliding phase (t s ≤ t < ∞), the attitude motion is driven by both the equivalent and reaching controllers. The ideality assumption is still valid, and state trajectories are tried to be carried to the sliding manifold by the complete control law given in (13). Because s, s˙ = 0 3×1 hold in the reaching mode, this time, the equation

 (x, t ) = u eq (x) + J n s˙ (x, t ). u

t0

28 30

3.4. Existence analysis of sliding mode/stability analysis of reaching mode

 as can be solved for u

( D B u eq )dτ

67

74

s˙ (x, t ) = G f n (x) + M int q q + J n−1 u = 0 3×1



t

t

Remark 4. If the integral term M int q t qdτ was absent from the 0 proposed sliding manifold, the control action would continue un As a result of the equality s = 0 that  converge to 0. til only ω holds in the sliding mode, a steady state error being equal to ∞ − K q−1 J n−1 t0 ( D B u )dτ might be observed in q ’s response, which means that the satellite might be stabilized at attitudes different than the reference.

t

t0

qdτ →

 → 0 on the slid( D B u eq )dτ , which indicates that also ω t0

ing manifold. So, it is shown that, under the assumption of no disturbances, x → x N in the sliding mode. Notice that the uniformity claimed in Lemma follows from the fact that the derivation in this subsection is dependent neither on t s nor on the direction  2 of B.

Theorem. Consider the instantaneously underactuated attitude motion (1)–(6) of a rigid satellite with uncertain inertia matrix subjected to environmental disturbances, which is stabilized by the magnetic sliding mode controller (16), (31). Provided that the satellite’s orbital plane does not coincide with the geomagnetic equator plane and the conditions of Lemma are satisfied then the considered motion enters the sliding mode in finite time (at t = t s , t 0 < t s < ∞) if the following two conditions hold: 1) the design parameter k s of the continuous reaching law is positive, which is true for its all values; 2) the design parameter k ss of the discontinuous reaching law satisfies the condition in (33) together with L1 ≥ L2 .

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1 2

k ss = k ss [x, kq , kint q ] >

L1 +

√

L2

3L 2

 × 3  T gg ∞ +  T aero ∞ +  T solar ∞ +  T mag ∞   √  22 [ L 1 + (2 + δ¯ J 1 ) J max ]ω 3L 2 . + L 1 − L 2 + J max kq ω  2 . . . + n J maxkq kint q q2 2

4 5 6 7

10 11 12 13 14 15 16 17 18 19 20

Here;





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

2

s J n s > 0. T

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

(36)



unc

Proceeding with manipulations after the substitution of (37) into (36) leads to

= s











∗ + I −  J J −1 C B u reach Td + T unc



−1





−1

− s K s s + s  J J C B K s s     ∗  (s) − Td + T unc ≤ −s T I −  J J −1 C B K ss sgn T

T

−1

− s K s s +  J i2 J     ∗  (s) − Td + T unc = −s T I −  J J −1 C B K ss sgn T

77

80 81 82 83 84 85 86 87 88 89 90 91

Remark 5. It can be verified that the utilization of the manifold in (14) instead of the proposed one in (16) leads to

 T



( Td + T unc ) + C B u reach     ∗ ∗∗ + C B u reach + T unc = s T Td + T unc     ∗ + I −  J J −1 C B u reach = s T Td + T unc      ∗  (s) + K s s − Td + T unc = −s T I −  J J −1 C B K ss sgn     ∗  (s) − Td + T unc = −s T I −  J J −1 C B K ss sgn   − s T I −  J J −1 C B K s s. (40)

V˙ (s) = s

Because C B is a positive semi-definite matrix according to (12b), the reaching condition is not satisfied in that case.

92 93 94 95 96 97 98 99 100 101 102 103 104 105

C B i2  K s i2 s22 i2

107 109 110 111 112

3 $

113

|si | − σ2 s22

114

i =1

s22 2 

≤ −σ1 s2 − σ

115

  = − σ1 + σ2 s2 s2

116 117

< −σ1 s2

  ∗  (s) + K s s − Td + T unc = −s I −  J J C B K ss sgn     ∗  (s) − Td + T unc = −s T I −  J J −1 C B K ss sgn   − s T I −  J J −1 C B K s s     ∗  (s) − Td + T unc = −s T I −  J J −1 C B K ss sgn T

Note 2. For detailed definitions and derivations of terms in (33) and in the appendix, the reader is recommended to consult to [35].

 (s) − σ2 s22 = −σ1 V˙ < −σ1 s T sgn

∗∗  T unc

 T

to [44] that the manifold is reached in finite time. The derivation of (33) from (39a) is provided in Appendix A, where the inequality L 2 / L 1 < 1 is obtained as a prerequisite, which implies that the second reachability condition in (39b) is satisfied by the designed controller. 2

(37)



76

108



∗ ∗∗ + u reach + T unc Td + T unc

(39a)

It is possible to express the time derivative of the Lyapunov function used in the reachability analysis in the previous subsection as a function of itself through the following manipulations:

 T ∗



,

3.5. Finite time convergence of state trajectories to sliding manifold

 ×  Jω  × Jω  −  J J −1 (−ω  + C B u eq + Td ) T unc = −ω

V˙ (s) = s T

∞

75

106

T unc can be divided into two parts as follows:

48



(34b)

 n + dunc ) + M int q q + J −1 u V˙ (s) = s T J n s˙ = s T J n G ( f n + d n   1 ) − ω  × Jnω  + q × ω  ... J K (q ω 2 n q 4 T = s . . . + ( Td + T unc ) + J n M int q q + u eq + u reach ⎡1 ⎤ ) − ω  × Jnω  + q × ω  + ( Td + T unc ) J K (q ω 2 n q 4 ⎢ ⎥ = s T ⎣ + J n M int q q . . . − 12 J n K q (q4 ω )  + q × ω ⎦

  + − J J −1 C B u reach .   

∗ Td + T unc

74

79

Successive substitution of (32) and (19) into the time derivative of the function V in (35) gives



CB

− 1 

73

If (39) is satisfied, # # there exists a function ε = ε (s) =  σ 3i=1 |si |  (kss − l) 3i=1 |si |, which is always positive for s = 0, and thus V˙ ≤ −ε − s T K s s < −ε < 0 holds. That means according

(35)

 × Jnω  − J n M int q q + u reach +ω  T   = s ( T d + T unc ) + u reach .

I −J J

−1

72

78

Proof. Let us define a positive definite Lyapunov function candidate as

V (s) =



71

(39b)

Thus, there exists a sliding mode in purely magnetically controlled attitude motion. In other words, the finite time stability of the sliding mani which is the equilibrium in the reaching mode, is guaranteed fold s = 0, by the designed magnetic spacecraft attitude control system in a global and robust manner.

1

70

(38)

L 2 / L 1 ≤ 1.

21 22

69

(34a)

L 2  δ¯ J 1 + δ¯ J 2 (1 + δ¯ J 1 ) max( J 1 , J 2 , J 3 )

 (δ¯ J 1 + δ¯ J 2 + δ¯ J 1 δ¯ J 2 ) J max .

68

which indicates the following two conditions that guarantee that the reaching condition s T s˙ < 0 is satisfied:

k ss > l 

L 1  (1 − δ¯ J 1 ) min( J 1 , J 2 , J 3 )  (1 − δ¯ J 1 ) J min ,

67

L1

(33)

8 9

− s T K s s + ks s22 L1     ∗  (s) − Td + T unc = −s T I −  J J −1 C B K ss sgn " ! L2 − 1− k s s22 ,

L1 − L2

√ 

3

7

≤ −√

σ1 √ J max

118

V < 0,

(41)

120 121

where

σ1  kss − l = kss − !

σ2  1 − 3 $

119

L2



"

L1

122

− 1   ∗ I −  J J −1 C B Td + T unc > 0, ∞

k s > 0,

|si | ≥ s2 ,

123 124

(42a)

126

(42b)

127 128

(42c)

i =1

0 < J min s22 ≤ V (s) ≤ J max s22 .

125

129 130 131

(42d)

132

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8

1 2 3 4 5 6 7 8 9

are made use of. As in [45], the following inequality for V (t ) can be obtained from (41):

!

σ1

V (t ) ≤ − √ t+ 2 J max

%

"2 V0

,

V 0 = V (t 0 ).

(43)

It is obvious for√the solution V (t ) in (43) that it vanishes at latest 2

J max √

when t = t s = σ V 0 is reached (due to the fact that dV /dt < 1 0), when the sliding vector s vanishes as well.

10 11

4. A simulation example

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

In this section, the results of a simulation starting from a nonstationary state that is far from the reference will be presented: To numerically verify the existence of the sliding mode (see Fig. 6); to numerically prove that the trajectories are carried to the vicinity of the reference state on the proposed sliding manifold under realistic conditions (see Figs. 1–4); to numerically prove that the trajectories are carried to the reference state on the proposed sliding manifold under ideal conditions (see Fig. 14). The controller is implemented in continuous form. The employed numerical integrator is the fourth order Runge–Kutta solver, and the integration time step is taken constant and as equal to 1 second. Only the plots in Fig. 6 are obtained by taking the time step as equal to 0.01 s to be able to read the time to reach the sliding surfaces from the graphs more accurately. The used simulation orbit and satellite model are the same as the ones in [35] (see Table 1). The reader may refer to [35] for mathematical models and infinity norm calculations of the four environmental torque components, which enable highly realistic simulation of disturbed attitude motion and systematic design of the magnetic sliding mode attitude controller, respectively. The design and simulation of the magnetic attitude control system are carried out under the following two assumptions.

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

Assumption 1. All of the states are available for the controller at each instance. This assumption is reasonable because any modern small satellite is capable of running a state estimator. Please be informed that the proposed control law is expected to take over the authority as the attitude acquisition mode controller after the angular kinetic energy is reduced, e.g., by the widely used B-dot law, which requires only successive magnetometer measurements to drive the magnetic actuator triad, in the detumbling mode of the mission. The initial angular speed used in the simulation example is on the order of the mean motion n (see Table 1), which is quite low compared to possible initial speeds after orbit injection, whereas the initial angular orientation is far from the reference orientation as will be seen in Table 2. The designed controller will be taken into operation to conduct the large angle manoeuvre to the vicinity of the reference when the angular speed is reduced by the detumbling controller to a few times the mean motion. At the beginning of that relatively stationary attitude acquisition mode, the initialization of the state estimator would be possible via utilization of, e.g., a two-vector algorithm to guess the initial angles provided that an inertial measurement unit is in usage. The robust stability of the designed control system can be expected to be preserved even in the presence of such an observer in the controller loop because the purely magnetic attitude control system is intrinsically slow due to its low torque production capability. Therefore, a properly designed observer would converge in the early phase of the transient regime of the controlled attitude motion, and the controller would be capable of dealing with the perturbation the initially high estimation error would have caused. An ongoing study on integrating the control system presented in this paper with a disturbance observer has demonstrated the feasibility of such an observer-controller system in a numerical framework.

Assumption 2. Magnetometer measurements are unaffected from magnetic moments produced by magnetic actuators, therefore uninterrupted magnetic actuation and control is possible, which means simultaneous magnetic sensing and actuation.

67 68 69 70 71

Table 2 presents quantities related with the simulation and the initial state.

72 73 74 75

Note 3. The sliding surface design parameter kq serves the controller designer as a weighting parameter between angular velocity  0 2 is on the orand angular orientation. Because  q2 ≤ 1 and ω der of 1×10−3 rad/s, kq may be selected initially in the interval of 1–10×10−3 1/s and then tuned through simulation trials. The multiplication of the dimensionless sliding surface design parameter kint q with the mean motion can be considered as the inverse of a time constant that determines the rate of q ’s convergence to zero vector. Thus, kint q may be selected initially so high that 1/(nkint q ) is lower than the orbital period T , i.e., it may be selected initially higher than 1/(2π ). The continuous reaching law design parameter k s proportionally accelerates the finite time convergence of s to zero vector. k s may be taken on the order of the ratio k ss (t 0 )/s0 2 initially.

76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

Note 4. The following k ss (t ) that satisfies the related condition in (33) is used in the simulation:

k ss (t ) = 1.1 ×

×

L1 +

√ 

√

91 92 93 94

3L 2

95

L1 − L2

3  T gg ∞ +  T aero ∞ +  T solar ∞ +  T mag ∞

⎧ ⎫  (t )22 ⎪ [ L 1 + (2 + δ¯ J 1 ) J max ]ω √ ⎪ ⎨ ⎬ 3L 2 k + .  (t )2 . . . + J max 2q ω ⎪ L1 − L2 ⎪ ⎩ ⎭ . . . + n J maxkq kint q q(t )2



96 97 98 99

(44)

100 101 102 103

Fig. 1 depicts that the angular orientation of the satellite converges to the vicinity of reference attitude state of x N in nearly one orbit although the initial angular velocity is on the order of the mean motion, which is a superiority to the solutions in [27,28]. On the other hand, the magnetic sliding mode controller is unable to completely counteract the disturbances due to environmental effects and satellite model uncertainty as clearly seen in Fig. 2 as well as in Fig. 1. This undesired result emerges from the fact addressed in Remark 2 that the disturbances are unmatched. Thus, the achievement can be stated from the robust control application view as follows: The attitude motion (1)–(6) of a rigid satellite on an orbit satisfying the controllability condition in [5] and controlled by the magnetic sliding mode controller (16), (31) is globally uniformly ultimately bounded about the inertial reference state x N . Because a system is said to be robustly stable if the system is nominally stable as well, and allows changing in certain specific bounds of perturbation while keeping stability according to Definition 1 in [46], it can be concluded the mentioned result implies stability robustness. To further assess the achievable steady state performance by the designed controller, simulations with the satellite model presented in Table 1 and with the values of δ¯ J 1 , δ¯ J 2 , kq , kint q , k s given in Table 2 was run for altitudes of 600, 500, 450, 400 km; the steady state error margins of attitude angles were obtained as [−23◦ , +17◦ ], [−36◦ , +30◦ ], [−52◦ , +50◦ ], [−180◦ , +180◦ ], respectively whereas that margin is [−20◦ , +11◦ ] for the simulation orbit presented in Table 1 as observed in Fig. 2. The margin corresponding to 400 km means that at that altitude, where T aero becomes the dominant environmental disturbance component by

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9

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. 1. Quaternions.

Fig. 2. Attitude angles.

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

surpassing T gg , the disturbance torque component along the in is so high that the stabilization cannot stantaneous direction of B be accomplished. If a mathematical relation between the disturbance torque com and the error vector of attitude angles or quaterponent along B nions was available and a disturbance observer was employed to estimate Td then at least the offsets of attitude angles from zero could be removed by inputting the compensating values to the control system as reference angles (see Fig. 3). A related simulation study revealed that the nonzero reference of [ϕ0 θ0 ψ0 ] = [−4◦ − 4◦ 9.5◦ ] results in a steady state error margin of [−15◦ , +15◦ ], which is an improvement of 25% if compared to [−20◦ , +11◦ ] as far as the maximum absolute error is concerned.

34 35 36 37

Remark 6. The simulations carried out by restricting the environmental disturbance torque vector Td to instantaneously lie in the plane orthogonal to the local geomagnetic field vector revealed

38 39 40 41 42 43 44

Orbit quantity

Value

Satellite model quantity

Value

h [km] n [deg/s] T [s]

703.463 6.07 × 10−2 5931

J 1 [kg m2 ] J 2 [kg m2 ] J 3 [kg m2 ] M sat [A m2 ]

2.904 3.428 1.275 20

46

86

that the designed magnetic sliding mode attitude controller is capable of completely rejecting disturbances in that plane. This fact implies that the steady state errors in state variables’ responses emerge from the effect ofthe components of Td and T unc along  which cannot be counteracted by the instantaneous direction of B, the control system. This numerical investigation result, which can be seen in Fig. 4, also indicates an important property of the proposed controller: it is capable of completely rejecting the residual magnetic torque vector T mag that lies in the plane instantaneously  which is the first or second dominant environorthogonal to B, mental disturbance torque in small satellite missions in low-Earthorbits. Notice that the attitude response shown in Fig. 4 belongs to the results of the simulation carried out for the altitude of 400 km. The residual steady state error in Fig. 4, which is in the range between circa −1.5◦ and +1.5◦ , is caused by the component of T unc   When the same simulation was repeated for T unc = 0, along B. asymptotic convergence to the zero vector can be observed in responses. The discontinuous control effect, which carries the state trajectories to the sliding manifold in finite time and keeps them on the manifold thereafter (see Fig. 6), causes high frequency oscillations of insignificant amplitude in angular velocity responses, which are shown in Fig. 5. The time for the attitude motion to enter the sliding mode, i.e., t s was read from the second graph in Fig. 6 as circa 40.5 seconds.

47 48 49

85 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105

Table 1 Quantities belonging to simulation orbit and used satellite model.

45

84

106 107 108 109 110 111 112 113

Table 2 Quantities related with the simulation and initial conditions.

114 115

50

Quantity

Value

Initial conditions

Value

116

51

δ¯ J 1 [–] δ¯ J 2 [–]

0.1

[ϕ0 θ0 ψ0 ] [deg]

[100 80 90]

117

0.06

[q1 q2 q3 q4 ]0 [–]

[0.123 0.707 0 0.696]

118

ω 0T [×10−2 deg/s] s0T [×10−3 1/s]  0 2 [n] ω

[5 7 6]

119

[1.18 2.99 1.047]

120

1.728

121

52 53

L 2 / L 1 [–]

0.496 < 1

54

kq [1/s]

2.5 × 10−3

kint q [–]

7.5 × 10−1

55 56 57 58 59 60

 T gg ∞ [N m]  T aero ∞ [N m]  T solar ∞ [N m]  T mag ∞ [N m]  T unc ∞ [N m]

61 62

65 66

122

7.633 × 10−8

123

2.948 × 10−7

124

2.819 × 10−6

125

5.854 × 10−6 + 2.439 ×  22 + 4.96 × 10−1 ×  u 2 ω

126

k ss (t ) [N m]

4.788 × 10−5 + 1.422 ×  22 + 7.301 × 10−3 × 101 × ω  2 + 1.16 × 10−5 × q2 ω

k ss (t 0 ) [N m]

1.172 × 10−4

k s [N ms]

1 × 10−1

63 64

3.625 × 10−6

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Fig. 5. (Absolute) Angular velocities.

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Fig. 4. Attitude angles under restricted environmental disturbances (at 400 km).

Fig. 6. Sliding surface vector components.

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Because the discontinuous reaching law design parameter k ss is computed by the condition in (44) as being adjusted also according to the infinity norms of the environmental torque components, which except  T solar ∞ are functions of altitude, finite time converge to the sliding manifold was observed in all of the simulation runs for lower altitudes. t s was read as 39.6, 37.4, 34.9, and 30.3 seconds for 600, 500, 450, 400 km, respectively.

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Note 5. Note that, because the employed actuators and sliding mode control method match well together, the designed controller has the potential to be utilized onboard after necessary modifications and improvements with regard to application. Such suitability has been rarely encountered in engineering application of variable structure control since its first example for being used in the main loop of a control system back in the Soviet era where diaphragm actuators driven by a pneumatic variable structure controller [47] operated in the sliding mode for petrochemical process control [48,49].

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Fig. 7. Magnetic control moments.

58 59 60 61 62 63 64 65 66

104

123 124

Fig. 7 indicates that attitude acquisition is achieved without exceeding the saturation limit of onboard magnetic actuators (see Table 1), which is also true for lower altitudes. The following six figures show all torques acting on the satellite along the control process. The magnetic control torque’s order is 10−4 (see Fig. 8), which is comparable to the varying bound of disturbance torque due to model uncertainty (see Fig. 13). As expectable in regard to the used simulation orbit and satellite

model, the dominant environmental torque is the gravity-gradient one (see Fig. 9), which is followed by the residual magnetic torque (see Fig. 12). As a result of the nearly inertially pointing final attitude, aerodynamic torque components oscillate almost T -periodically (see Fig. 10) whereas solar torque components seem to tend to constant values (see Fig. 11) in steady-state. The results in Figs. 7–11 verify that all undesirable torques are bounded and their bounds can be calculated realistically.

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Fig. 10. Aerodynamic drag torque components.

 = 0,  the plots in the following If the simulation is repeated for d two figures can be drawn. Fig. 14 depicts asymptotic convergence of Euler angles to their reference values that are zero while the magnitude of the sliding surface vector tends to zero as presented in Fig. 15.

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5. Conclusion

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In this paper, a robust and global solution was developed for the purely magnetic attitude control problem by proposing a sliding manifold for the problem. All of four main environmental disturbance torque components as well as parametric model uncertainty in its most general form were taken into account during the controller design. The proposed magnetic sliding mode attitude controller guarantees a rigid satellite’s attitude motion’s stability robustness against the mentioned undesirable effects by carrying state trajectories to the vicinity of the reference state x N , which results in a globally ultimately bounded motion about x N , provided that the plane of the satellite’s nearly circular orbit does not coincide with the geomagnetic equator plane. If external disturbances are assumed to instantaneously lie in the plane orthogonal to the local geomagnetic field vector and model uncertainty is assumed to be absent, the designed controller ensures global asymptotic stabilization of attitude motion at the reference state. If external disturbances are assumed to instantaneously lie in the plane orthogonal to the local geomagnetic field vector while model uncertainty exists, the designed controller ensures global convergence of state trajectories to the very close vicinity of the reference state. To the best knowledge of the authors, the existence of a sliding mode in nonlinear attitude motion that is controlled by solely magnetic actuation was shown for the first time in this paper thanks to the proposed sliding manifold, which makes its usage essential for fur-

ther sliding mode based robust control attempts to the considered problem. The aim of the study was designing a controller that will guarantee that the controlled state variables are not going to diverge from the vicinity of their references, which is accomplished by forcing the spacecraft’s attitude to enter a stable sliding motion and to keep that motion. Because there is no control action along the instantaneous geomagnetic field vector, there is also no counteraction to the disturbances along that direction, which results in a residual steady state error. Thus, performance robustness cannot be expected from a purely magnetically actuated system; however, stability robustness was achieved, which is a contribution both to the literature on magnetic spacecraft attitude control and to the library of available attitude control laws. The designed controller is believed to possess an application potential thanks to the suitability of magnetic actuators to be driven by signals with rapidly changing direction and even magnitude in case of proportional drive. From the practical point of view, the proposed controller can serve small satellite missions in low-Earth-orbits as an attitude acquisition mode or three-axis safe mode controller that has guaranteed stability under realistic conditions. Moreover, it can serve missions with fine pointing requirement as a baseline controller to be aided by a relatively low capacity reaction wheel or thruster triad that will be utilized solely to provide the lacking control torque component along the local geomagnetic field vector. Developing such hybrid actuation schemes that possess the property of stability robustness is a topic of ongoing research, whose first results presented in [50] indicated that the steady state performance of the control system introduced in this paper can be significantly improved via a very limited augmentation by tiny momentum exchange devices while the stability robustness is preserved. The proposed sliding manifold structure made the derivation of the equivalent control term possible for an instantaneously underactuated system and realized a sliding mode in that system, which may point out to a potential for generalization of this structure to another systems with singular blocks in their control matrices. The projection matrix pair plays a significant role in the presented design. Taking those two matrices as uncertain may increase the robustness of the magnetic sliding mode attitude control system against measurement and modeling errors of the local geomagnetic field vector.

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Conflict of interest statement

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None declared.

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Acknowledgements

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This work is supported by the Science Fellowships and Grant Programs Department (BIDEB) of the Scientific and Technological Research Council of Turkey (TUBITAK).

(A.1)

If (A.1) is applied on the right hand side of the inequality in (39a) and the result is manipulated further,

I −J J

≤



CB

− 1 

Td + T unc

I −  J J −1 C B

− 1 

119

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a2 / 3 ≤ a∞ ≤ a2 .

−1

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Let a be an arbitrary 3 × 1 vectorial signal. Then the following inequalities hold between its infinity and 2-norms:



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Appendix A. Derivation of (33) from (39a)

√

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∗



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∞ ∗

Td + T unc

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≤

1

√

2

≤

3 4 5

8 9 10



 T gg ∞ +  T aero ∞ +  T solar ∞ L1 − L2  ∗ +  T mag ∞ + T unc

− 1 I −  J J −1 C B ≤

13

16 17 18 19 20 21 22 23 24

 2

(A.2)

can be obtained by using also the following inequality for the induced 2-norm of the inverse matrix ( I −  J J −1 C B )−1 [51]:

12

15

 ∗  Td 2 + T unc

3L 1 

11

14

i2

∞

6 7

− 1 I −  J J −1 C B

1

≤ i2

1 1 −  J J −1 C B i2 1

1 −  J i2  J −1 i2 C B i2

=

1 − L2/L1

=

L1 L1 − L2

(A.3)

.

Note that (12c) together with  J i2 = L 2 and  J −1 i2 = 1/ L 1 leads to (A.3), which implies that

 J J −1 C B

i2

≤ L2/L1 < 1

(A.4)

should be always satisfied. ∗ ’s definition in (37) and the similar derivation of  T  unc ∞ T unc in [35] allows writing of

L 2 

∗

T unc ∞ = L1

25

+

26

√



L 1 + (1 + δ¯ J 1 ) J max   22

ω

3( T gg ∞ +  T aero ∞



+  T solar ∞ +  T mag ∞ ) +  u eq 2 .

27

(A.5)

28 29

In the next step,

30 31

1

  u eq 2 ≤ − J n K q T ( q, q4 )ω 2

32 34

 22 + J max ≤ J max ω

35 36

39 41

∗ T unc

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

kq 2

 2 + n J maxkq kint q q2 ω

(A.6)

is substituted into (A.5) to arrive at

38 40

2

+  − n J n K q K int q q2

33

37

 2 + ω˜ J n ω

⎧ ⎫  22 [ L 1 + (2 + δ¯ J 1 ) J max ]ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎨ ⎬   + 3( T gg ∞ +  T aero ∞ L2 ≤ . (A.7) ∞ ⎪ L1 ⎪ +  T solar ∞ +  T mag ∞ ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ k  2 + n J maxkq kint q q2 + J max 2q ω

˜ i2 = Note that the equations  T ( q , q4 )i2  q4 I + q˜ i2 = 1 and ω  2 are used in (A.6). Then the maximum value the limit in (A.2) ω can take during the control process becomes

√

3L 1 + 3L 2 

 T gg ∞ +  T aero ∞ +  T solar ∞ +  T mag ∞

L1 − L2

√

+

! 

3L 2

L1 − L2

+ J max

kq 2





 22 L 1 + (2 + δ¯ J 1 ) J max ω "

 2 + n J maxkq kint q q2 , ω

(A.8)

which is the right hand side of the inequality in (33). References [1] C.I. Byrnes, A. Isidori, On the attitude stabilization of rigid spacecraft, Automatica 27 (1) (1991) 87–95, https://doi.org/10.1016/0005-1098(91)90008-P. [2] J.-M. Coron, E.-Y. Keraï, Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two control torques, Automatica 32 (5) (1996) 669–677, https://doi.org/10.1016/0005-1098(95)00194-8. [3] P. Morin, C. Samson, Time-varying exponential stabilization of a rigid spacecraft with two control torques, IEEE Trans. Autom. Control 42 (4) (1997) 528–534, https://doi.org/10.1109/9.566663.

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