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%28ASCE%29AS.1943-5525.0000791

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New Sliding Mode Attitude Controller Design Based on
Lumped Disturbance Bound Equation
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A. Sofyalı, Ph.D. 1; and E. M. Jafarov, Ph.D., D.Sc. 2
Abstract: In this paper, first the mathematical models of four major environmental disturbance torque components and corresponding bound
equations are presented. Then the effect of the inertia matrix uncertainty on rigid satellite’s attitude dynamics is defined as an external torque
in the derived state equation, which together with environmental torques forms the so-called lumped disturbance torque. After obtaining the
complete equation for the model uncertainty-induced torque vector, its exact bound equation is derived by using matrix-vector norm relations.
To validate the significance of these preliminary results for use in robust attitude controller design, a new modification of the classical sliding
mode attitude controller present in literature is proposed, which is the primary contribution of this paper. The new design that is based on
comprehensive knowledge of the lumped disturbance’s bounded variation leads to a decision rule on the switching control gain that is not
excessively conservative. After verifying the accuracy of the bound equations in a simulation under no control, a second simulation is carried
out with control input from the designed sliding mode controller to show that the proposed design works. The superiority of the new design is
discussed in comparison with another design from literature that does not exploit the complete model of the inertia matrix uncertainty-induced
torque through a comparative simulation’s result. The conclusion is that the modified controller design results in an attitude control system
that has guaranteed robust stability in addition to reasonable conservativeness thanks to the newly obtained comprehensive decision rule on
the switching control gain. DOI: 10.1061/(ASCE)AS.1943-5525.0000791. © 2017 American Society of Civil Engineers.
Author keywords: Environmental disturbance; Inertia matrix uncertainty; Lumped disturbance; Matrix-vector norm; Sliding mode attitude
control.
Introduction
There are two main classes of uncertainty sources in satellite attitude dynamics: (1) environmental effects and (2) uncertainty in the
information on moments and products of inertia. Controlling attitude under uncertain conditions requires either mathematical models of disturbing effects on attitude dynamics or information on
their bounds if they are bounded.
Although some robust control techniques such as sliding mode
control does not necessitate modeling of unwanted effects, the
maximum values they can take during the whole operation, namely
the bounds, must be somehow known at the controller design step
because, e.g., in the sliding mode control, the so-called reaching
condition is satisfied only if the switching control gain can be
selected to be higher than the bound of the lumped disturbance
for the whole control process. If such information is absent or not
reliable, which may be the case in the practical application of the
sliding mode control method, the preferred solution is mostly the
easier one, namely to decide on a gain value that is expected to
be sufficiently high. This easier solution results in excessive conservativeness that leads to unnecessarily high control effort and
rise in the undesired phenomenon called chattering, which means
1
Control System Design Engineer, EDS Aerospace, ARI-1 Teknokent,
Istanbul Technical Univ., Ayazaga Campus, Maslak, Sariyer, Istanbul
34469, Turkey (corresponding author). E-mail: sofyali@hotmail.com
2
Professor, Faculty of Aeronautics and Astronautics, Dept. of
Aeronautical Engineering, Istanbul Technical Univ., Ayazaga Campus,
Maslak, Sariyer, Istanbul 34469, Turkey.
Note. This manuscript was submitted on February 8, 2017; approved
on May 26, 2017; published online on October 25, 2017. Discussion
period open until March 25, 2018; separate discussions must be submitted
for individual papers. This paper is part of the Journal of Aerospace Engineering, © ASCE, ISSN 0893-1321.
© ASCE
high-frequency oscillation in a system’s dynamic response to discontinuous control effect (Cong et al. 2012). Another solution is to
estimate the unknown bound relying on an adaptation scheme,
which is known in literature as the adaptive sliding mode control
approach and is so far highly studied on (Hu et al. 2007). The adaptive solution may lead to boundless increase in the estimated gain
value due to the fact that the sliding surface vector becomes not
exactly equal to zero even if the sliding mode is entered, which
is named as the parameter drift problem and dealt with by using
the dead-zone technique, the σ-modification, or the boundary layer
scheme (Cong et al. 2012; Hu et al. 2007). In addition, there is
another problem called the over-adaptation emerging from overestimation of the gain due to, e.g., an unrelated reaching phase adaptation, which triggered another line of studies in literature (Cong
et al. 2012). As shown, besides being more complex than the conventional sliding mode control, the adaptive version has also issues
to be dealt with.
This paper deals with uncertain attitude dynamics of a rigid satellite in a circular low-Earth orbit (LEO) under environmental disturbances and inertia matrix uncertainty by relying on the fact that
those uncertainty effects are bounded. With the aim of relieving the
need to estimate the switching control gain, this paper presents the
infinity norm of the disturbance torque vector due to model uncertainty, which is calculated rigorously from the derived torque equation and the definitions for inertia matrix uncertainty by employing
matrix-vector norm relations. The torque vector equation is derived
as an explicit function of only the time variable and the states,
whereas it is presented in literature as a function of also the angular
velocity’s rate to the best knowledge of the authors (Chen et al.
2014; Zhao et al. 2012). Because equations of motions are not expressed in state space in Chen et al. (2014) and Zhao et al. (2012) or
many others, the derivation of that vector is not complete. This work
also proposes bound equations on all of four major environmental
04017082-1
J. Aerosp. Eng., 2018, 31(1): 04017082
J. Aerosp. Eng.
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disturbance torque components, namely the gravity-gradient, the
aerodynamic drag, the solar pressure, and the residual magnetic
torque vectors. Studies such as Yang and Sun (2002) also present
the maximum possible values of external torque components, even
though not of all the four; the contribution of this study is numerical
verification of the given bounds’ validity through high-fidelity simulation. In simulations, exact or realistic mathematical models of
all four external disturbance components are used. All of those four
models are also included in the text, which is preferred by few
papers such as Meng et al. (2012).
The first purpose of this paper is to derive the bound of the
lumped disturbance torque that consists of the external disturbance
torque components and the undesirable torque due to satellite inertia matrix uncertainty. The bound of the lumped disturbance is
equal to the sum of the infinity norms of the four environmental
disturbance torque vectors and the satellite model uncertaintyinduced torque vector. Equations that give the infinity norms of the
gravity-gradient, aerodynamic drag, solar pressure, and residual
magnetic torques are provided, which require knowledge of the
mission orbit and satellite’s mass and dimensional properties in addition to access to some well-known books on attitude dynamics
and control. Relying on the relation for the inverse of a sum of
matrices, the torque due to model uncertainty is derived as a function of the states, the control input, and the resultant external disturbance torque. The inertia matrix uncertainty is modeled by two
additive matrices: the diagonal one corresponding only to the parametric uncertainty in principal moments of inertia; and the full one
representing solely the difference due to the angular deviation of
the body axis system from the principal axis system. By using those
two definitions, the infinity norm of the model uncertainty-induced
torque is calculated; the result is a function of two respective parametric uncertainty bounds, minimum and maximum principal moments of inertia, two norms of the angular velocity and control
vectors, and infinity norms of the four environmental torque vectors. All derivations are carried out for both inertial and Earth
(nadir) pointing cases. In addition, three different types of actuators
are considered for the definitions.
Sliding mode control is a powerful tool for providing the controlled system with robustness to parameter uncertainties and external disturbances, especially if the undesired effects are bounded
and their bounds are known (Utkin 1992; Edwards and Spurgeon
1998; Jafarov 2009; Shtessel et al. 2014). Thus a new sliding mode
attitude controller is designed by modifying a classical one from
literature using the previously obtained results, which is the second
and main purpose of this work. The resulting controller is compared to another one that is designed based on incomplete knowledge of the model uncertainty-induced torque, which is mostly the
case in literature to the best knowledge of the authors. Simulations
are carried out by employing exact or realistic mathematical models
for torques and the Sun position vector together with high fidelity
atmosphere and geomagnetic field models.
The paper is organized as follows. First, the state equation of
attitude motion under environmental disturbances and inertia matrix uncertainty is derived, which is manipulated to obtain a structure convenient for robust control application. The terms in the state
equation are defined for both inertial and Earth pointing cases.
Second, the mathematical models of four environmental torques
and the equations for bound values of the signals in their three
channels are presented. The bound equations are written regarding
the fact that those torques are of infinity norm-bounded nature.
Then, the infinity norm of the satellite model uncertainty torque
is obtained by using matrix algebra and norm relations. In the following section, a new sliding mode attitude controller design is carried out. Finally, the results of three exemplary simulations are
© ASCE
presented. The first one is without control, the second one belongs
to the designed control system, and in the third one, the result of a
design approach that is common in literature is simulated for comparison purposes with the new controller.
Equations of Attitude Motion under Disturbances
and Parametric Model Uncertainty
The attitude motion of a rigid satellite orbiting the Earth is described by a set of dynamic and kinematic differential equations,
which are derived step-by-step in, for example, Wie (1998). To
avoid trigonometric expressions leading to singularity, quaternions
are preferred to attitude (Euler) angles to represent the angular orientation of the satellite body frame with respect to the reference
frame. Then the orthogonal direction cosine matrix C between
those two frames can be written as function of the vectorial [q ¼
ð q1 q2 q3 ÞT ] and the scalar (q4 ) quaternion components:
2
1 − 2ðq22 þ q23 Þ
6
C ¼ 4 2ðq1 q2 − q3 q4 Þ
2ðq1 q3 þ q2 q4 Þ
2ðq1 q2 þ q3 q4 Þ
2ðq1 q3 − q2 q4 Þ
3
1 − 2ðq21 þ q23 Þ
7
2ðq2 q3 þ q4 q1 Þ 5 ð1Þ
2ðq2 q3 − q4 q1 Þ
1 − 2ðq21 þ q22 Þ
Dynamic and Kinematic Equations
The three dynamic equations can be written as the following vectorial differential equation:
J ω̇ þ ω × Jω ¼ Td þ Tc
ð2Þ
In this paper, J ≜ J n þ ΔJ is the 3 × 3 uncertain inertia matrix
defined as the sum of the nominal inertia matrix J n and the inertia
uncertainty matrix ΔJ; ω is the 3 × 1 angular velocity vector of the
body reference frame B (Fig. 1) with respect to the inertial reference frame N (i.e., ECI)
Td ¼ Tgg þ Taero þ Tsolar þ Tmag
ð3Þ
is the 3 × 1 environmental disturbance torque; and Tc is the 3 × 1
control torque. The four components of Td will be covered
extensively.
The vectorial
1
q̇ ¼ ðq4 ω þ q × ωÞ
2
ð4aÞ
1
q̇4 ¼ − ðq · ωÞ
2
ð4bÞ
and the scalar
differential equations describe the kinematics. Eqs. (4a) and (4b)
are valid if the reference frame is N; in other words, inertial pointing (IP) is the case of analysis/synthesis. In that case, the direction
cosine matrix is defined as C ≡ CB=N ≜ ½ n1 n2 n3 , where the
vectors ni ði ¼ 1, 2; 3Þ are unit vectors of N written in B. If the
reference frame is selected to be the orbital reference frame A, kinematic equations have to be rewritten as
1
q̇ ¼ ðq4 ωB=A þ q × ωB=A Þ
2
1
n
¼ ðq4 ω þ q × ωÞ þ ðq4 a2 þ q × a2 Þ
2
2
04017082-2
J. Aerosp. Eng., 2018, 31(1): 04017082
ð5aÞ
J. Aerosp. Eng.
1
1
n
q̇4 ¼ − ðq · ωB=A Þ ¼ − ðq · ωÞ − ðq · a2 Þ
2
2
2
ð5bÞ
Case 1: Inertial Pointing
The IP reference state is
xN ≜ ½ 01×3
according to the following relation with n being the angular speed
of the satellite around the Earth, namely the mean motion
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ω ≡ ωB=N ¼ ωB=A þ ωA=N ¼ ωB=A − na2
¼ ½fn ðxÞ þ ΔfðxÞ þ ½bn þ ΔbuðxÞ þ ½dn ðx; tÞ þ Δdðx; tÞ
ð7Þ
The 7 × 1 state vector is defined as
x ¼ ½ qT
q4
ωT T
2
6
ΔfðxÞ ¼ 6
4
03×3
7
7
5
0
−J −1
n ðω
× ΔJωÞ −
J −1
unc ðω
ð13Þ
× JωÞ
is the 7 × 1 uncertain system vector
6
dn ðx; tÞ ¼ 6
4
2
6
¼6
4
03×1
0
J −1
n Td ðx; tÞ
3
7
7
5
3
03×1
0
J −1
n ½Tgg ðxÞ þ Taero ðx; tÞ þ Tsolar ðx; tÞ þ Tmag ðx; tÞ
7
7
5
ð14Þ
is the 7 × 1 nominal disturbance vector, and
2
6
Δdðx; tÞ ¼ 6
4
3
03×1
0
J −1
unc ½Tgg ðxÞ þ Taero ðx; tÞ þ Tsolar ðx; tÞ þ Tmag ðx; tÞ
ð15Þ
is the 7 × 1 uncertain disturbance vector.
3
7
6
bn ¼ 4 01×3 5
3
03×1
ð8Þ
The 7 × 3 control matrix consists of the nominal
2
ð12Þ
is the 7 × 1 nominal system vector
2
ẋðtÞ ¼ fðxÞ þ buðxÞ þ dðx; tÞ
ð11Þ
3
1
6 2 ðq4 ω þ q × ωÞ 7
6
7
6
7
1
fn ðxÞ ¼ 6
7
6 − ðq · ωÞ 7
2
4
5
−J −1
n ðω × J n ωÞ
Attitude State Equation with Parametric Uncertainty
Representation of attitude equations in state space is more convenient to analyze the dynamical system. The properties such as being
control-affine, time-varying, underactuated, being in regular form,
and having matched/unmatched disturbances can be determined by
studying the state equation. These properties also govern the controller synthesis by pointing out control methods that are suitable
for the dynamical system.
The state equation of disturbed attitude motion of a rigid satellite
with uncertain inertia orbiting the Earth can be written as follows:
01×3 T
2
ð6Þ
Then the considered problem is Earth pointing (EP) or
nadir pointing with the corresponding direction cosine matrix
C ≡ CB=A ≜ ½ a1 a2 a3 .
Note 1. In this paper, the nominal inertia matrix is considered to
be equivalent to J when B coincides with the principal reference
frame. Thus J n ≜ diagðJ 1 ; J 2 ; J 3 Þ, where J i ði ¼ 1, 2; 3Þ is satellite’s ith principal moment of inertia. In reality, there is hardly such
a coincidence. In addition, J i have parametric uncertainty due to
imperfect computer-aided design (CAD) calculations or erroneous
measurements even if fuel consumption/sloshing and/or mass
movements are not present. To take these two kinds of uncertainties
into account, ΔJ has to be a full matrix rather than a diagonal one.
Its definition will be given together with the variation intervals of its
uncertain elements based on reasonable assumptions.
1
ð9Þ
Case 2: Earth (Nadir) Pointing
The EP reference state is
J −1
n
xA ≜ ½ 01×3
1
0 −n 0 T
ð16Þ
and the uncertain
2
03×3
3
7
6
Δb ¼ 4 01×3 5
The definitions are as follows:
ð10Þ
J −1
unc
parts. The derivation of J −1
unc seen in Eq. (10) will be provided.
Other terms in Eq. (7) differ for IP and EP problems. Therefore
their definitions will be given separately for those two cases.
© ASCE
3
1
n
ðq
ðq
ω
þ
q
×
ωÞ
þ
a
þ
q
×
a
Þ
2 7
62 4
2 4 2
7
6
7
6
1
n
fn ðxÞ ¼ 6
7
− ðq · ωÞ − ðq · a2 Þ
7
6
2
2
5
4
−1
2
−1
−J n ðω × J n ωÞ þ 3n J n ða3 × J n a3 Þ
04017082-3
J. Aerosp. Eng., 2018, 31(1): 04017082
2
ð17Þ
J. Aerosp. Eng.
7
7
5
2
6
6
6
ΔfðxÞ ¼ 6
6
4
0
−J −1
n ðω
::: þ
× ΔJωÞ −
3n2 ½J −1
n ða3
2
6
dn ðx; tÞ ¼ 6
4
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3
03×1
× ΔJa3 Þ þ
× JωÞ : : :
J −1
unc ða3
× Ja3 Þ
0
J −1
n ½Taero ðx; tÞ þ Tsolar ðx; tÞ þ Tmag ðx; tÞ
ð27Þ
whereas
Tunc ðx; tÞ ¼ −ω × ΔJω þ 3n2 ða3 × ΔJa3 Þ
7
7
5
0
þ Tsolar ðx; tÞ þ Tmag ðx; tÞ
7
7
5
ð20Þ
Remark 1. The definitions in Eqs. (17)–(20) are valid only for a
triinertial rigid satellite in a circular orbit with principal moments of
inertia satisfying
J2 > J1 > J3
ð21Þ
because only then the gravity-gradient torque acts as a restoring
torque about xA given in Eq. (16) (Hughes 2004). Otherwise,
for both IP and EP problems, Eqs. (13)–(15) hold instead of
Eqs. (18)–(20), and the last three rows in Eq. (17) are same with
the ones in Eq. (12).
Improved State Space Representation
The aim of this subsection is to represent all terms in Eq. (7) that are
functions of ΔJ with a single disturbance term, namely a 7 × 1
disturbance vector due to satellite model uncertainty, as
dunc ðx; tÞ ≜ ΔfðxÞ þ ΔbuðxÞ þ Δdðx; tÞ
ð22Þ
As a result, the uncertain state equation takes the following
form:
ẋðtÞ ¼ fn ðxÞ þ bn uðxÞ þ dðx; tÞ
≜ fn ðxÞ þ bn uðxÞ þ dn ðx; tÞ þ dunc ðx; tÞ
ð23Þ
The right-hand side of Eq. (23) has as many terms as the one of
the initial state equation in the first line of Eq. (7). After substitutions and by using the following expression for J −1
unc
−1
−1
J −1
unc ¼ −J n ΔJJ
− ΔJJ −1 ½−ω × Jω þ Tc þ Td ðx; tÞ
ð19Þ
3
03×1
J −1
unc ½Taero ðx; tÞ
Tunc ðx; tÞ ¼ −ω × ΔJω − ΔJJ−1 ½−ω × Jω þ Tc þ Td ðx; tÞ
3
03×1
2
6
Δdðx; tÞ ¼ 6
4
J −1
unc ðω
7
7
7
7 ð18Þ
7
5
ð24Þ
ð28Þ
is valid for the case of EP. Eq. (3) substitutes for Td ðx; tÞ in both
cases.
Remark 2. With derivation of Eq. (27) or Eq. (28), it is achieved
to represent the effect of inertia uncertainty on attitude motion by a
single, newly defined disturbance term, which can be simply added
to Td ðx; tÞ in the right-hand side of Eq. (2) allowing J’s in its lefthand side replaced by J n ’s.
The disturbance torque due to satellite model uncertainty is dependent on the angular velocity, the environmental disturbance
torque, and also the control input. In attitude motion, the environmental disturbance torque components are infinity norm-bounded
as will be shown in the next section, therefore Tunc ðx; tÞ is also
expected to have the same norm property.
Remark 3. Considering the given expressions for the terms in
Eq. (23) together with its structure, it can be concluded that attitude
control systems are control-affine and in regular form regardless of
the control actuation type. Even though the definitions presented
earlier are valid for attitude dynamics actuated by reaction thrusters
(gas jets), the structure of Eq. (23) is preserved for systems with
magnetic actuators or reaction wheels.
The uncertain state equation of attitude control system
with reaction wheels has the extra terms of −J −1
n ðω × hÞ and
ΔJJ −1 ðω × hÞ in the equations of fn ðxÞ’s last three rows and
Tunc ðx; tÞ, respectively. In this paper, h is the 3 × 1 resultant angular momentum vector stored in the reaction wheels placed along
the three body axes (Wie 1998). Because there is no difference in
definition of the control matrix, it can be stated that attitude control
systems with either thrusters or wheels have matched disturbances.
This property together with being control-affine and in regular f
orm is required by robust control methods such as conventional
(classical) or integral sliding mode control.
If attitude dynamics are controlled by magnetic actuators
via interaction with local geomagnetic field, the control matrix
becomes b ¼ bðx; tÞ ¼ ½ 03×3 03×1 ðJ −1 CB ÞT T , where CB ¼
~
tÞ=kBðx; tÞk22 with Bðx; tÞ being the 3 × 1 local geoB~ T ðx; tÞBðx;
~
magnetic field vector and Bðx;
tÞ its skew-symmetric matrix. Such
a difference renders the attitude control system time-varying and
underactuated. In addition, disturbances are unmatched in the attitude control system with magnetic actuators (Sofyalı et al. 2015).
which is derived from Henderson and Searle (1981)
−1
J −1 ¼ ðJ n þ ΔJÞ−1 ¼ J −1
n ðI − ΔJJ Þ
−1
−1
¼ ðI − J −1 ΔJÞJ −1
n ≜ J n þ J unc
Environmental Disturbances
ð25Þ
and dunc ðx; tÞ can be obtained in the same form with dn ðx; tÞ
3
2
03×1
7
6
7
ð26Þ
0
dunc ðx; tÞ ¼ 6
5
4
−1
J n Tunc ðx; tÞ
In this paper, Tunc ðx; tÞ is the 3 × 1 disturbance torque due to
satellite model uncertainty. When the case is IP
© ASCE
There are four major components of the environmental disturbance
torque vector as shown in Eq. (3): gravity-gradient torque Tgg , aerodynamic drag torque Taero , solar pressure torque Tsolar , and residual
magnetic torque Tmag .
Gravity-Gradient Torque
The gravity-gradient torque emerges from spatially varying interaction of satellite’s mass distributed through its volume with Earth’s
gravity field. It has the following simple formula (Wie 1998):
04017082-4
J. Aerosp. Eng., 2018, 31(1): 04017082
J. Aerosp. Eng.
Tgg ðxÞ ¼ 3n2 ða3 × J n a3 Þ
ð29Þ
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Remark 4. Vector a3 is the third column of CB=A . In the EP problem, C ≡ CB=A and a3 is simply the third column of C, which is
given in Eq. (1). Conversely, CB=A has to be calculated in the IP
problem due to the equivalence of C ≡ CB=N . For circular orbits,
CB=N ½ v̂ ðv̂ × x̂Þ −x̂ is the necessary transformation formula,
where x̂ and v̂ are respectively the unit vectors of satellite’s position
and velocity vectors written in N.
The mean motion for a circular orbit can be calculated by
Eq. (30)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3.986 × 1014
rad
nðhÞ ¼
ð6.378 × 106 þ hÞ3 s
Fig. 1. Reference frames of attitude motion
ð30Þ
as a function of the orbital altitude h, which is in meters.
The highest absolute value that one of the three components
of Tgg , which is a vector signal, can get during the process, namely
Tgg ‘s infinity norm is (Fortescue et al. 2011; Larson and Wertz
1999)
gravity center along b1 , b2 , and b3 , respectively. In the utilized
simulation environment, ρ is calculated by the high fidelity atmosphere model NRLMSISE-00 (Wertz 1978). The velocity vector in
Eq. (32) is written in B, which is the result of the following matrixvector multiplication: vB ¼ CB=N v.
The infinity norm of the vector signal Taero can be accepted to be
o
n
kTgg ½xðtÞk∞ ¼ sup maxðjT gg ji ðτ ÞjÞ
τ
i
3
¼ n2 maxðjJ 3 − J 2 j; jJ 3 − J 1 j; jJ 1 − J 2 jÞ;
2
i ¼ 1,2; 3
kTaero ½xðtÞ; tk∞ ¼ CD Aρv2
The aerodynamic drag torque is exerted to satellite’s body by
atmospheric particles that impact on its ram surfaces during its
orbital motion. This torque acting on satellites with shape of rectangular prism can be modeled by (Wertz 1978; Gregory 2004)
3
X
The distance of the geometric center to the gravity center generally does not exceed one-fifth of the shortest edge length of
satellites with shape of rectangular prism and with no retractable
appendages. The CD is taken as equal to 2.5 both in Eqs. (32)
and (33), which corresponds to the possibly worst case. The A in
Eq. (33) is calculated as 50% higher than the product of two highest
edge lengths. Satellite’s orbital velocity in a circular orbit is a function of h, which is in meters, as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3.986 × 1014 m
vðhÞ ¼
6.378 × 106 þ h s
Tai ;
i¼1
Ta1
Ta2
Ta3
82
9
3
1
>
>
>
>
sgnðv
ÞX
−
ΔX
>6
>
1
<
=
7
2
1
6
7
B
;
¼ − CD YZρjv1 j 6
×
v
7
−ΔY
>
>
4
5
2
>
>
>
>
:
;
−ΔZ
82
9
3
−ΔX
>
>
>
>
<6 1
=
7
1
B
6
7
sgnðv
ÞY
−
ΔY
¼ − CD ZXρjv2 j 4
×v ;
2
5
>
>
2
2
>
>
:
;
−ΔZ
82
9
3
−ΔX
>
>
>
>
<6
=
7
1
B
−ΔY
7×v
¼ − CD XYρjv3 j 6
4
5
>
>
2
>
>
: 1 sgnðv ÞZ − ΔZ
;
3
2
ð34Þ
and ρ in Eq. (33) is calculated by the exponential atmosphere model
(Vallado 1997)
h−h0
H
ρðhÞ ¼ ρ0 e−
ð32Þ
where CD = aerodynamic drag coefficient of the satellite; X, Y, and
Z = edge lengths along b1 , b2 , b3 (Fig. 1), respectively; ρ = local
atmospheric density or the local total mass density as defined by the
employed atmosphere model; vi = component of satellite’s velocity
vector along bi ; and ΔX, ΔY, and ΔZ are the components of the
position vector of satellite’s geometric center with respect to its
© ASCE
ð33Þ
ð31Þ
Aerodynamic Drag Torque
Taero ¼
minðX; Y; ZÞ
5
ð35Þ
which is valid for the altitude interval of 0 to 1,000 km. In this
paper, ρ0 , h0 , and H are read for the orbital altitude from Table 7.4
in Vallado (1997).
Solar Pressure Torque
The solar pressure torque is applied to satellite’s body as a result of
solar radiation pressure on its illuminated surfaces. This torque acting on satellites with shape of rectangular prism can be calculated
by (Wertz 1978; Montenbruck and Gill 2000)
04017082-5
J. Aerosp. Eng., 2018, 31(1): 04017082
J. Aerosp. Eng.
Tsolar ¼
3
X
T⊙i ;
i¼1
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T⊙1
T⊙2
T⊙3
0
2 31
9
82
3
1
>
>
>
>
>
2 3 >
1
>
6 sgnBr̂B · 6 0 7CX − ΔX 7
>
>
1 >
@
4
5
A
>
>
6
7
s=c⊙
<6 2
=
7
B
6 7
B
6
7
¼ −CRP YZP⊙ r̂s=c⊙ · 4 0 5 × 6
× r̂s=c⊙ ;
0
7
>
>
>
>6
7
>
>
>
4
5
−ΔY
0 >
>
>
>
>
;
:
−ΔZ
9
82
3
−ΔX
>
>
>
>
2 31
0
>
2 3 >
>
>6
7
0
>
0 >
>
>
6
7
=
< 61
7
6
7
C
B
B
6 7
B
B
7
·
1
Y
−
ΔY
sgn
r̂
4
5
A
@
;
¼ −CRP ZXP⊙ r̂s=c⊙ · 4 1 5 × 6
×
r̂
s=c⊙
s=c⊙
6
7
>
>
>
>
62
7
>
>
>
>
5
0
0
>
>4
>
>
;
:
−ΔZ
9
82
3
−ΔX
>
>
>
>
>
>
2 3 >6
>
7
>
0 >
−ΔY
>
7
>
=
<6
6
7
0
2
3
1
B
6 7
B
6
7
0
¼ −CRP XYP⊙ r̂s=c⊙ · 4 0 5 × 6
×
r̂
s=c⊙ >
7
>
>6 1
B
6 7C
>
7
>
>
>
4 sgn@r̂Bs=c⊙ · 4 0 5AZ − ΔZ 5
1 >
>
>
>
>
2
;
:
1
which is derived in a similar fashion to the derivation of Taero in
Gregory (2004). In this paper, CRP is the radiation pressure coefficient of the satellite; P⊙ ¼ 4.56 × 10−6 N=m2 is the solar radiation pressure value at 1 astronomical unit (AU) distance from the
Sun (Montenbruck and Gill 2000); r̂Bs=c⊙ ¼ CB=N r̂s=c⊙ is the unit
position vector of the Sun with respect to the satellite obtained after
the subtraction r̂s=c⊙ ¼ r̂⊙ − x̂ is carried out where r̂⊙ is the unit
position vector of the Sun with respect to the Earth and is calculated
by the algorithm available in Vallado (1997).
In this case Tsolar ’s infinity norm can be obtained by
(Montenbruck and Gill 2000)
kTsolar ½x; tk∞
1.496 2 minðX; Y; ZÞ
¼ CRP AP⊙
1.470
5
ð37Þ
kTmag ½xðtÞ; tk∞ ¼ M res
ð36Þ
2 × 8.1 × 1015 Tm3
ð6.378 × 106 þ hÞ3
ð39Þ
Both in Eqs. (38) and (39), M res ¼ m × 1 × 10−3 A m2 =kg is
used, which is the relation read from Table 8.1 in Chobotov
(1991) regarding the fact that residual magnetic torque may dominate the other three environmental torques in small satellite missions, which is the dominant mission type in LEO. In this paper,
m is satellite’s mass.
Remark 5. If the mission orbit and the satellite’s dimensions and
mass together with its nominal inertia matrix are known a priori, the
bounds on environmental disturbance torques can be evaluated by
Eqs. (31), (33), (37), and (39) during the design process of a robust
control system provided that the utilized robust control approach
does not necessitate instantaneous knowledge of disturbances during the controlled process.
The variable CRP is taken as equal to 2, which is its maximum
possible value, both in Eqs. (36) and (37). Again, A in Eq. (37) is
150% of the product of two highest edge lengths.
Satellite Model with Parametric Uncertainties
Residual Magnetic Torque
Modeling
The residual magnetic torque emerges from the interaction of the
residual magnetic dipole moment M res of the satellite with the local
geomagnetic field B according to
Definition. In this paper, the inertia uncertainty matrix ΔJ is
defined as follows:
ΔJ ≜ ΔJ 1 þ ΔJ 2
Tmag ¼ Mres × B
ð38Þ
where both vectors are written in the body reference frame. The
M res is inducted in the satellite body mainly by current loops.
The B is computed in the simulation environment by the high
fidelity spherical harmonic geomagnetic field model IGRF2015
(Davis 2004).
The infinity norm of Tmag is the result of the multiplication of
the local geomagnetic field’s magnitude calculated according to the
simple dipole model of the field (Sidi 1997) with M res
© ASCE
ð40Þ
The first term
2
δ1 J1
6
ΔJ 1 ≜ 4 0
0
0
δ2 J2
0
0
3
7
0 5
ð41Þ
δ3 J3
represents the parametric uncertainty due to imperfect CAD calculations or erroneous measurements of principal moments of inertia
of a rigid satellite. The second term
04017082-6
J. Aerosp. Eng., 2018, 31(1): 04017082
J. Aerosp. Eng.
2
δ11
6
ΔJ 2 ≜ ð1 þ δ̄ J1 ÞmaxðJ 1 ; J 2 ; J 3 Þ4 δ12
δ13
δ 12
δ 13
3
δ 22
7
δ 23 5
δ 23
δ 33
ðkTgg k2 þ kTaero k2 þ kTsolar k2 þ kTmag k2 Þ
pffiffiffi
≤ 3ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞ þ kTmag k∞ Þ
ð42Þ
ð45Þ
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From matrix algebra
models the difference between the actual principal inertia matrix
and the actual inertia matrix that is calculated with respect to
the body reference frame B by six uncertain parameters.
Assumption 1. The actual principal moments of inertia differ
from their corresponding nominal values at most by 10%:
δ 1 ; δ 2 ; δ 3 ∈ ½−0.1; þ0.1 ≜ ½−δ̄ J1 ; þδ̄ J1 .
Assumption 2. The absolute value of the elements of ΔJ 2
does not exceed 15% of the value of ð1 þ δ̄J1 ÞmaxðJ 1 ; J 2 ; J 3 Þ:
δ ij ∈ ½−0.15; þ0.15 ≜ ½−δ̄J2 ; þδ̄J2 ;i; j ¼ 1,2; 3. To obtain this
bound, the principal frame is assumed to coincide with B after
rotated
diagonal
axis, which has the unity vector of
pffiffiffi around
pffiffiffi its p
ffiffiffi
½ 1= 3 1= 3 1= 3 T , for 10 degrees. A conservative approach
is preferred to count for uneven inertia distribution, which is
the case with elongated (e.g., gravity-gradiently stabilized) or flat
(e.g., spin-stabilized) satellites. For satellites with evenly distributed inertia, i.e., with moments of inertia values close to each other,
a much lower δ̄J2 value such as 0.06 is acceptable.
k − Jki2 ¼ kJki2 ¼ ð1 þ δ̄J1 ÞJ max
kJ −1 ki2 ¼
1
1
1
≜
≜
ð1 − δ̄ J1 ÞminðJ 1 ; J 2 ; J 3 Þ ð1 − δ̄ J1 ÞJ min L1
k − ΔJki2 ¼ kΔJki2 ¼ ½δ̄ J1 þ δ̄ J2 ð1 þ δ̄ J1 ÞmaxðJ 1 ; J 2 ; J 3 Þ
≜ ðδ̄ J1 þ δ̄J2 þ δ̄ J1 δ̄ J2 ÞJ max ≜ L2
(
L2
½L1 þ ð1 þ δ̄J1 ÞJ max kωk22
kTunc ½x; u; tk∞ ¼
L1
!
~ i2 k − Jki2 kωk2 þ kuk2 : : :
kωk
: : : þ kTgg k2 þ kTaero k2 þ kTsolar k2 þ kTmag k2
ð43Þ
Knowing that the induced two norm of a vector’s skewsymmetric matrix is equal to the two norm of that vector,
~ i2 ¼ kωk2 , Eq. (43) can be rewritten as
i.e., kωk
kTunc k2 ≤
L2
L1
½L1 þ ð1 þ δ̄ J1 ÞJ max kωk22
kTgg k∞ þ kTaero k∞ : : :
: : : þ kTsolar k∞ þ kTmag k∞
!
)
þ kuk2
þ kuk2
ð49Þ
Case 2: Earth Pointing
By starting with Eq. (28)
(
L2
kTunc ½x; u; tk∞ ¼
½L1 þ ð1 þ δ̄J1 ÞJ max kωk22 þ 3n2 L1
L1
!
)
kTgg k∞ þ kTaero k∞ : : :
pffiffiffi
þ kuk2
þ 3
: : : þ kTsolar k∞ þ kTmag k∞
is derived, which’s only difference from Eq. (49) is the presence of
the extra term 3n2 L2 .
Remark 6. Eq. (49) or Eq. (50) has to be computed onboard at
each step of the control process to adjust the controller according to
varying ω and u. The derivation results in this subsection are convenient for use in robust attitude controller design.
Sliding Mode Attitude Controller Design
2
ð44Þ
kq
6
sðxÞ ¼ ω þ 4 0
where the following norm inequality valid for vectors of size 3 × 1
is employed:
© ASCE
: : : þ kTsolar k∞ þ kTmag k∞
)
In this section, the design of a new sliding mode attitude controller
will be carried out. The originality will be the usage of the derived
infinity norms of the unwanted torque components in the design to
obtain a decision rule on the switching control gain value that is not
too conservative. The used sliding surface vector is
(
pffiffiffi
þ 3
!
kTgg k∞ þ kTaero k∞ : : :
ð50Þ
~ i2 k − ΔJki2 kωk2 þ k − ΔJki2 kJ −1 ki2
kTunc k2 ≤ kωk
×
ð48Þ
could be p
derived
or defined, which led to Eq. (44). The relation
ffiffiffi
kTunc k2 = 3 ≤ kTunc k∞ ≤ kTunc k2 holds between the two different norm types of 3 × 1 vectors, which allows taking the right-hand
side of Eq. (44) as equal to kTunc k∞
pffiffiffi
þ 3
Case 1: Inertial Pointing
To derive the infinity norm of Tunc , an upper bound for its two
(Euclid) norm is sought first because the induced two norms of
the matrices in Eq. (27) are obtainable as follows:
ð47Þ
and
Derivation of Bound on Disturbance Torque due to
Inertia Matrix Uncertainty
Note 2. In the following derivation, the norm inequality kTc k2 ≤
kuk2 is utilized. The equality is valid for control systems with
gas jets or reaction wheels whereas the inequality holds for the
torque Tc ¼ CB u induced by magnetic actuators because of the
fact that the induced two norm of CB is equal to 1 (Sofyalı
et al. 2015): kCB ki2 ¼ 1.
ð46Þ
04017082-7
J. Aerosp. Eng., 2018, 31(1): 04017082
0
kq
0
3
07
5q;
kq > 0
ð51Þ
0 0 kq
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
≜K q
J. Aerosp. Eng.
which has become the preferred sliding vector in literature on attitude control after first proposed in Vadali (1986), where its structure is proved to guarantee that the sliding motion ends at the
reference. In this case kq is the sliding surface design parameter.
The utilized design approach is the equivalent control method,
which results in a control vector that is the sum of the equivalent
control vector ueq and the discontinuous reaching control vector
ureach
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Tc ≡ u ≜ ueq þ ureach
¼ sT ½ðTd þ Tunc Þ þ ureach ð52Þ
The following reaching dynamics are assigned (Hung et al.
1993) as in Vadali (1986) and Lo and Chen (1995) to
ureach ðxÞ ¼ −K ss ðxÞsgn½sðxÞ − K s sðxÞ;
V̇ðsÞ ¼ sT J n ṡ ¼ sT J n ½Gðfn þ dn þ dunc Þ þ J −1
n u
2
3
1
J
K
ðq
ω
þ
q
×
ωÞ
−
ω
×
J
ω
:
:
:
n q 4
n
5
¼ sT 4 2
: : : þ ðTd þ Tunc Þ þ ueq þ ureach
2
3
1
J
K
ðq
ω
þ
q
×
ωÞ
−
ω
×
J
ω
þ
ðT
þ
T
Þ
:
:
:
n q 4
n
d
unc
5
¼ sT 4 2
1
: : : − 2 J n K q ðq4 ω þ q × ωÞ þ ω × Jn ω þ ureach
which becomes
V̇ðsÞ ¼ sT ½ðTd þ Tunc Þ þ ðureach þ T
unc Þ
¼ sT ½ðTd þ Tunc Þ þ ðI − ΔJJ −1 Þureach K ss ; K s > 0 ð53Þ
where K ss ¼ diagðkss Þ and K s ¼ diagðks Þ are the continuous and
discontinuous reaching control gain matrices of size 3 × 3, respectively, thus kss corresponds to the aforementioned switching control
gain. Note also that the considered problem is IP, and the system is
actuated by a reaction thruster triad, both the same with the cases in
Vadali (1986).
Derivation of the Equivalent Control Vector
The equivalent control vector is derived under ideal sliding
assumption, i.e., for s ¼ ṡ ¼ 03×1 and d ¼ 07×1 . If s is derived with
respect to 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
¼ −sT fðI − ΔJJ −1 Þ½K ss sgnðsÞ þ K s s − ðTd þ Tunc Þg
¼ −sT ½ðI − ΔJJ −1 ÞK ss sgnðsÞ − ðTd þ Tunc Þ
− sT ðI − ΔJJ −1 ÞK s s
Derivation of the Decision Rule on kss
Eq. (59) indicates that kss > kðI − ΔJJ −1 Þ−1 ðTd þ Tunc Þk∞ is the
first condition for the satisfaction of the reaching condition sT ṡ < 0.
As known, for an arbitrary 3 × 1 vectoral signal a, the following
inequalities hold between its infinity and two norms:
pffiffiffi
kak2 = 3 ≤ kak∞ ≤ kak2
ð60Þ
Using Eq. (60)
kðI − ΔJJ −1 Þ−1 ðTd þ Tunc Þk∞
ð54Þ
≤ kðI − ΔJJ −1 Þ−1 ðTd þ Tunc Þk2
can be obtained, where the 3 × 7 matrix G ≜ ∂s=∂x ¼
½ K q 03×1 I and Gbn ¼ J −1
n according to Eq. (9). ueq can be
solved for from Eq. (54) by using Eq. (12) as
≤ kðI − ΔJJ −1 Þ−1 ki2 ðkTd k2 þ kTunc k2 Þ
L1 pffiffiffi
3ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞
≤
L1 − L2
þ kTmag k∞ Þ þ kTunc k2
ð55Þ
In the reaching mode that precedes the sliding mode, s; ṡ ≠ 03×1
hold. If the ideality assumption is abandoned, the time derivative of
s is derived as
ṡðxÞ ¼ G½fn ðxÞ þ dðx; tÞ þ J −1
n uðxÞ ≠ 03×1
1
1
≤
−1
1 − kΔJJ ki2 1 − kΔJki2 kJ −1 ki2
1
L1
≤
¼
1 − L2 =L1 L1 − L2
kðI − ΔJJ −1 Þ−1 ki2 ≤
ð56Þ
At this step, it is necessary to divide Tunc in Eq. (27) into continuous and discontinuous parts as follows:
kTunc k2 ≤
≜Tunc
ð57Þ
≜T
unc
Differentiating the chosen Lyapunov function candidate VðsÞ ¼
ð1=2ÞsT J n s gives
© ASCE
ð62Þ
Eq. (62) relies on the definitions in Eqs. (47) and (48), and
implies that kΔJJ −1 ki2 ≤ L2 =L1 < 1 has always to be satisfied
Tunc ¼ −ω × ΔJω − ΔJJ −1 ð−ω × Jω þ ueq þ Td Þ
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
þ ð−ΔJJ ureach Þ
|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}
ð61Þ
can be obtained by using also the following inequality for the
induced two norm of the inverse matrix ðI − ΔJJ −1 Þ−1 (Meyer
2000):
Reaching Analysis
−1
ð59Þ
after the substitution of Eq. (57) into Eq. (58).
ṡðxÞ ¼ Gẋ ¼ G½fn ðxÞ þ bn ðx; tÞueq ¼ Gfn ðxÞ þ J −1
n ueq ¼ 03×1
1
ueq ðxÞ ¼ − J n K q ðq4 ω þ q × ωÞ þ ω × J n ω
2
ð58Þ
L2
f½L1 þ ð1 þ δ̄ J1 ÞJ max kωk22
L1
pffiffiffi
þ 3ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞
þ kTmag k∞ Þ þ kueq k2 g
ð63Þ
can be written in the same way with kTunc k2 by using its definition
in Eq. (57). If the following inequality for ueq, which utilizes the
~ i2 ¼ 1 and kωk
~ i2 ¼ kωk2
equations kTðq; q4 Þki2 ≜ kq4 I þ qk
04017082-8
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J. Aerosp. Eng.
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1
~ n ωk2 ≤ J max kωk22
kueq k2 ≤ k− J n K q Tðq; q4 Þωk þ kωJ
2
2
kq
þ J max kωk2
ð64Þ
2
is used in Eq. (63), then the following inequality for kTunc k2 is
arrived at:
L
kTunc k2 ≤ 2 ½L1 þ ð2 þ δ̄ J1 ÞJ max kωk22
L1
pffiffiffi
þ 3ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞
kq
ð65Þ
þ kTmag k∞ Þ þ J max kωk2
2
Finally, Eq. (65) is substituted into Eq. (61) to obtain
kðI − ΔJJ −1 Þ−1 ðTd þ Tunc Þk∞
pffiffiffi L þ L2
≤ 3 1
ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞ þ kTmag k∞ Þ
L1 − L2
kq
L2
2
þ
½L1 þ ð2 þ δ̄ J1 ÞJ max kωk2 þ J max kωk2
L1 − L2
2
ð66Þ
pffiffiffi
L2
L2
kT
unc k2 ≤ L1 ðkureach k2 Þ ≤ L1 ð 3kss þ ks ksk2 Þ, has to be added to
the right-hand side of the inequality in Eq. (65). Then, that upper
bound can be taken as equal to
kTunc k∞ ¼
L2
½L1 þ ð2 þ δ̄J1 ÞJ max kωk22
L1
pffiffiffi
þ 3ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞ þ kTmag k∞ Þ
pffiffiffi
kq
þ J max kωk2 þ 3kss þ ks ksk2
ð69Þ
2
Derivation of the Decision Rule on ks
The second reaching condition can be read from Eq. (59) as
ðI − ΔJJ −1 ÞK s ≥ 0. It is straightforward to rewrite the condition as
ðI − ΔJJ −1 ÞK s ¼ ½I − ðJ − J n ÞJ −1 K s ¼ ½ðI − IÞ þ J n J −1 K s
¼ J n J −1 K s ≥ 0
ð70Þ
where J n , J −1 , and K s are symmetric and positive definite matrices
by definition. Thus, J n J −1 K s is symmetric and positive definite.
In conclusion, J n J −1 K s > 0 and the second reaching condition
is satisfied for every value of ks .
which renders the first reaching condition be written as
kss ðtÞ ¼ kss ½xðtÞ
pffiffiffi L þ L2
ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞ þ kTmag k∞ Þ
> 3 1
L1 − L2
kq
L2
½L1 þ ð2 þ δ̄J1 ÞJ max kωðtÞk22 þ J max kωðtÞk2
þ
L1 − L2
2
ð67Þ
Remark 7. The derivation of the rule in Eq. (67) is the primary
contribution of this work, which is structured on complete modeling of the lumped disturbance and rigorous derivation of its bound.
To the best knowledge of the authors, such a comprehensive mathematical relation between the switching control gain and the
bounds on external disturbance components and inertia matrix uncertainty has not been presented yet in line of studies on sliding
mode attitude control, which is started by the pioneering work of
Vadali (1986) and followed by classical works such as Lo and
Chen (1995).
Remark 8. The rule in Eq. (67) is state dependent, so it enables
adjusting kss according to the instant Euclid norm of the angular
velocity vector during the operation as long it is available to the
control system. This results in a reasonably conservative switching
control action.
Note 3. The following kss ðtÞ that satisfies the inequality in
Eq. (67) can be used for the controller:
pffiffiffi L þ L2
kss ðtÞ ¼ 1.1 × 3 1
ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞
L1 − L2
L2
þ kTmag k∞ Þ þ
½L1 þ ð2 þ δ̄ J1 ÞJ max kωðtÞk22
L1 − L2
kq
þ J max kωðtÞk2
ð68Þ
2
Note 4. To obtain the upper bound of the two norm of the model
uncertainty-induced torque for the controlled case, the right-hand
side of the inequality for kT
unc k2 ‘s Euclid norm, which is equal to
© ASCE
Simulation Results and Superiority Discussion of
the Proposed Design
Simulation without Control
The first exemplary simulation is carried out with u ¼ 0. The simulation orbit belongs to ITUpSAT-1, the picosatellite of Istanbul
Technical University, which was launched on September 23, 2009.
The model of the Danish microsatellite Ørsted is used. Table 1
presents the quantities belonging to the nearly circular simulation
orbit (CelesTrak 2015) and Ørsted’s model (Wisniewski 1996).
Note 5. The close values of its principal moments of inertia seen
in Table 1 indicates that the used Ørsted model has an even inertia
distribution, thus δ̄ J2 ¼ 0.06 is used in the evaluation of kTunc k∞
according to Assumption 2 together with δ̄ J1 ¼ 0.1 according to
Assumption 1.
The values of ΔX, ΔY, and ΔZ are chosen as 0.035, 0.025,
and 0.05 m, respectively, which are lower than ð1=5Þ × 0.34 m.
Table 1. Quantities Belonging to Simulation Orbit and Used Satellite
Model
Quantity
Orbit
h (km)
n (degrees=s)
v (km=s)
ρ (kg=m3 )
Satellite model
X (m)
Y (m)
Z (m)
A (m2 )
m (kg)
J 1 (kg m2 )
J 2 (kg m2 )
J 3 (kg m2 )
04017082-9
J. Aerosp. Eng., 2018, 31(1): 04017082
Value
703.463
6.07 × 10−2
7.503
3.476 × 10−14
0.45
0.34
0.68
0.459
61.8
2.904
3.428
1.275
J. Aerosp. Eng.
Table 2. Bounds on Disturbance Torques
Distance torque components (Nm)
3,625 × 10−6
kTaero k∞ Eq. (33)
7,633 × 10−8
kTsolar k∞ Eq. (37)
2,948 × 10−7
kTmag k∞ Eq. (39)
2,819 × 10−6
kTunc ðtÞk∞ Eq. (49)
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Value
kTgg k∞ Eq. (31)
5,854 × 10−6 þ 2,439×kωðtÞk22
Fig. 3. Angular velocities
Fig. 2. Quaternions
Ørsted’s residual magnetic dipole moment vector is Mres ¼
½ 0 3 × 10−2 0 T Am2 according to Bak (1999), which corresponds to an M res value lower than the one computed by M res ¼
m × 1 × 10−3 A m2 =kg.
In the simulation, ΔJ 1 ≜ diagð0.1J 1 ; −0.1J 2 ; 0.05J 3 Þ is used,
and ΔJ 2 is calculated by assuming that the angular difference
between the body and the principal reference frames is 10 degrees around the diagonal axis of the principal frame. It is verified
that Assumption 2 would not be violated even if Ørsted’s model
with deployed gravity-gradient boom, which is ½J 1 ; J 2 ; J 3 ¼
½181.25; 181.78; 1.28 kg m2 (Wisniewski 1996), would be used.
The infinity norms of five disturbance torque components are
tabulated in Table 2.
The simulation is initiated at the reference state of the IP problem, which is given in Eq. (11). Figs. 2 and 3 show how the state
variables are carried out of the reference state by the disturbance
torques due to environmental effects and satellite model uncertainty during the first five orbital periods, each equal to 5,931 s or
circa 99 min.
The disturbance torques are shown in the following five figures
as upper-bounded and lower-bounded by their infinity norm values.
Remark 9. Only for demonstrational purposes, the gravitygradient torque in Fig. 4 is calculated by using J instead of J n :
Tgg ðxÞ ¼ 3n2 ða3 × Ja3 Þ
ð71Þ
Thus the following related bound is superimposed on the graphs
of Tgg ’s components.
© ASCE
Fig. 4. Gravity-gradient torque
3
kTgg k∞ ¼ n2 max½jð1 þ δ 3 ÞJ 3 − ð1 þ δ 2 ÞJ 2 j; jð1 þ δ3 ÞJ 3
2
− ð1 þ δ 1 ÞJ 1 j; jð1 þ δ 1 ÞJ 1 − ð1 þ δ2 ÞJ 2 j
ð72Þ
Figs. 2 and 3 point out that a control system that is robust
against disturbances is highly necessitated to achieve IP or EP.
Disturbances are observed to vary in the obtained bounds in
Figs. 4–8, which means that the derivations in this paper are validated. The dependency of the varying bound in Fig. 8 on the two
norms of the angular velocity is clear from a comparison between
Figs. 3 and 8.
The formulation of the uncertainty effect as a disturbance torque
allows the comparison of its order with the orders of environmental
disturbance torques. Under given simulation conditions, it is comparable to the dominant gravity-gradient torque and the residual
magnetic torque.
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Fig. 5. Aerodynamic drag torque
Fig. 7. Residual magnetic torque
Fig. 8. Model uncertainty torque
Fig. 6. Solar pressure torque
Table 3. Quantities Related to Simulation with Control and Initial
Conditions
Simulation with Control
The second exemplary simulation is carried out with u ¼ ueq þ
ureach . Table 3 presents the quantities related to the simulation.
Note that kTgg k∞ , kTaero k∞ , kTsolar k∞ , and kTmag k∞ are the
same in cases without and with control, which means that the
mentioned four values in Table 1 are also valid for the second
simulation.
By using the same values for other quantities mentioned in the
previous subsection, the following controlled attitude responses are
obtained.
As expected from a sliding mode controller, the disturbances
due to environmental effects and model uncertainty are rejected
(Fig. 9). In return, oscillations with high frequency and low amplitude are observable in angular velocity responses as shown in
Fig. 10, which is induced by the discontinuous control signals.
Fig. 11 depicts that the sliding mode is entered rapidly. Because
the simulation step size is taken as 1 s, which corresponds to a
© ASCE
Quantity
Control conditions
δ̄J1
δ̄J2
L2 =L1
kq (1=s)
ks (Nm s)
kss ðtÞ (Nm)
kTunc ðtÞk∞ (Nm)
Initial conditions
ðq1 ; q2 ; q3 ; q4 Þ0
ωT0 (×10−2 degrees=s)
sT0 (×10−3 1=s)
kω0 k2 (n)
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Value
0.1
0.06
0.496 < 1
2.5 × 10−3
1 × 10−1
3.853 × 10−5 þ 8.211×kωðtÞk22
þ 4.215 × 10−3 ×kωðtÞk2
5.854 × 10−6 þ 8.211×kωðtÞk22 þ 4.215 × 10−3
× kωðtÞk2 þ 4.959 × 10−2 × ks½xðtÞk2
þ 8.589 × 10−1 × kss ðtÞ
(0.123 0.707 0 0.696)
(5 7 6)
(1.18 2.99 1.047)
1.728
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Fig. 9. Controlled attitude angles
Fig. 11. Sliding surface vector components
Fig. 10. Controlled angular velocities
Fig. 12. Model uncertainty torque under control
switching frequency of 1 Hz that is a realizable value in application,
the ideal sliding does not occur as observed in the third graph of
Fig. 11. Ideal sliding requires infinite switching frequency, which
is not realizable. In nonideal sliding the switching occurs not on
the sliding manifold but in the some vicinity of the manifold. The
higher the frequency, the narrower that vicinity.
Fig. 12 shows that as angular velocity components converge to
zero in steady state the bound on the model uncertainty-induced
torque converges to their minimum values, which is constant as
can be deduced from Table 3. Similarly, the switching control gain
oscillates around a constant value in steady state (Fig. 13). Finally,
the control torque components are presented in Fig. 14, which are
in the order of 10−4 Nm.
upper bounds on the disturbances due to environmental effects and
inertia matrix uncertainty. In this subsection, a second rule on kss
will be derived by using the Lyapunov function candidate VðsÞ ¼
ð1=2ÞsT Js instead of VðsÞ ¼ ð1=2ÞsT J n s together with the dynamic equations given in Eq. (2) as done in Chen and Lo
(1993) and Lo and Chen (1995). Note that the term ΔJ ω̇ appears
in the right-hand side of V̇ ’s equality if V ¼ ð1=2ÞsT J n s is employed, which renders the reaching analysis complicated.
It is straightforward to obtain
2
6
C7
B
1
C7 − sT K s s
B
~
V̇ ¼ −sT 6
4K ss sgnðsÞ − @Td þ 2 ΔJK q Tω − ωΔJω
A5
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
≜TΔJ
Superiority Evaluation through a Comparative Design
and Simulation
If the design would not be based on the state space model, the exact
equation of Tunc , and its derived infinity norm, it might not be guaranteed that the derived decision rule on kss exceeds the total actual
© ASCE
13
0
ð73Þ
by using Eq. (2), Eq. (4a), the definition of the uncertain inertia
matrix, Eqs. (53) and (55). Eq. (73) implies that the reaching
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Fig. 13. Switching control gain
Fig. 15. Continuous part of model uncertainty torque under control
with Eq. (74) instead of Eq. (68)
Fig. 14. Control torque
condition is satisfied for kss > kTgg k∞ þ kTaero k∞ þ kTsolar k∞ þ
kTmag k∞ þ kTΔJ k∞ . Following the necessary steps gives
kss ðtÞ ¼ 1.1 × ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞ þ kTmag k∞ Þ
kq
þ L2 kωðtÞk22 þ kωðtÞk2
2
¼ 7.497 × 10−6 þ 5.691 × 10−1 ×kωðtÞk22
þ 7.113 × 10−4 × kωðtÞk2
ð74Þ
which, regardless if ΔJ is zero matrix or not, reduces to kss ðtÞ ¼
1.1 × ðkTgg k∞ þ kTaero k∞ þ kTsolar k∞ þ kTmag k∞ Þ in steady state,
i.e., when ω → 0 as t → ∞. Thus the rule in Eq. (74) does not
account for the disturbance due to inertia matrix uncertainty in
steady state, which prevents the controller from producing discontinuous control action with adequate amplitude to counteract also
the disturbance due to inertia matrix uncertainty besides the environmental ones. Conversely, the rule in Eq. (67) takes the nonzero
© ASCE
Fig. 16. Switching control gain with Eq. (74) instead of Eq. (68)
Tunc ðx; ∞Þ in steady state into account thanks to the constant coefficient of ðL1 þ L2 Þ=ðL1 − L2 Þ, which is greater than one and
becomes zero only if ΔJ is zero matrix.
The same simulation as in the previous subsection is carried out
by only using Eq. (74) instead of Eq. (68) for comparison purposes.
The unwanted torque due to ΔJ is defined as TΔJ in Eq. (73). If
even only the continuous part of Tunc in Eq. (27), namely Tunc
according to Eq. (57), is plotted against the infinity norm of TΔJ ,
it can be numerically observed that kTΔJ ½xðtÞk∞ ¼ L2 ½kωðtÞk22 þ
ðkq =2ÞkωðtÞk2 cannot serve as the bound equation on the model
uncertainty-induced torque Tunc because kTΔJ ½xð∞Þk∞ ¼ 0
although ΔJ is not zero matrix in steady state (Fig. 15).
Because the rule in Eq. (74) leads to a lower steady state value of
kss compared to the one by the rule in Eq. (68) as clear from the
comparison of Figs. 13–16, the steady state error margin of the attitude angles, which is approximately between −0.05° and þ0.05°
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Conclusions
Fig. 17. Controlled attitude angles with Eq. (74) instead of Eq. (68)
In this work, a sliding mode attitude controller was designed on the
basis of a newly derived decision rule on the discontinuous reaching law design parameter kss. In the derivation of the new rule, the
lumped disturbance bound equation was utilized. To be able to
combine the environmental disturbances and the model uncertainty
effect into a lumped expression, the exact mathematical formula of
the torque due to inertia matrix uncertainty was derived by using a
novel complete uncertainty model. The primary contribution of this
study is the structure of the aforementioned new decision rule,
which mathematically guarantees that kss exceeds the maximum
lumped disturbance torque component in any channel along the
whole control process. This theoretically means that the reaching
condition is satisfied and the sliding mode is preserved. It also renders kss variable according to the angular velocity vector and is thus
not too conservative.
Both inertial and Earth-pointing cases were covered throughout
the text. Moreover, the differences in terms of the state equation in
cases of employment of reaction thrusters, reaction wheels, or magnetic actuators were pointed out where necessary.
A simulation under no control effect was carried out to verify the
validity of all the derivations and assumptions presented in the text.
Its results numerically verified that the bounds calculated by the
related equations are neither exceeded by the corresponding vector
signal components nor turned out to be excessively conservative. A
second simulation numerically verified the functionality of the control system and the derived new rule. It showed that the classical
sliding mode control method based on the equivalent control approach could be modified to robustly control the uncertain attitude
dynamics in a reasonably conservative way thanks to the derived
bound value of lumped disturbance, which is available to the robustly stabilizing controller at each instance when the angular
velocity is known. Finally, the superiority of the new controller was
clarified by comparing its decision rule on kss to the one of another
controller that was designed without using the complete model of
Tunc . A comparative simulation depicted the insufficiency of the
other controller to produce discontinuous control signals with adequate amplitude in steady state.
Acknowledgments
Fig. 18. Control torque with Eq. (74) instead of Eq. (68)
(Fig. 17), is slightly narrower than the margin obtained by the proposed controller and shown in Fig. 9, which is circa ½−0.2°; þ0.1°.
Fig. 18 depicts that the initial magnitudes of the control torque’s
components are lower by the controller based on the rule in
Eq. (74), which can be expected in regard to the comparison of
Figs. 13 and 16.
As a result, the superiority of the proposed design can be summarized as follows. It leads to a decision rule on kss that guarantees
rejection of disturbances due to environmental effects and inertia
matrix uncertainty without unnecessarily high discontinuous control action because the complete effect of the inertia matrix uncertainty on the system is taken into account in the design. The
proposed design does not necessitate trial-and-error to decide on
kss . Because it uses the exact or realistic information on disturbances, which results in reasonably conservative operation of the
control system, the need for utilizing adaptive sliding mode control
is alleviated.
© ASCE
This work is supported by the Science Fellowships and Grant Programmes Department (BIDEB) of the Scientific and Technological
Research Council of Turkey (TUBITAK). The authors would like
to thank the dear editor and the dear reviewers for their comments
and suggestions that improved the quality of the paper.
References
Bak, T. (1999). “Spacecraft attitude determination: A magnetometer
approach.” Ph.D. thesis, Aalborg Univ., Aalborg, Denmark.
CelesTrak. (2015). “NORAD two-line element sets.” 〈http://celestrak.com
/NORAD/elements/〉 (May 2, 2015).
Chen, Y. P., and Lo, S. C. (1993). “Sliding-mode controller design
for spacecraft attitude tracking maneuvers.” IEEE Trans. Aerospace
Electronic Syst., 29(4), 1328–1333.
Chen, Z., Cong, B. L., and Liu, X. D. (2014). “A robust attitude
control strategy with guaranteed transient performance via modified
Lyapunov-based control and integral sliding mode control.” Nonlinear
Dyn., 78(3), 2205–2218.
Chobotov, V. A. (1991). Spacecraft attitude dynamics and control, Krieger
Publishing Company, Malabar, FL.
04017082-14
J. Aerosp. Eng., 2018, 31(1): 04017082
J. Aerosp. Eng.
Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved.
Cong, B., Chen, Z., and Liu, X. (2012). “Disturbance observer-based
adaptive integral sliding mode control for rigid spacecraft attitude
maneuvers.” J. Aerosp. Eng., 227(10), 1660–1671.
Davis, J. (2004). “Mathematical modeling of Earth’s magnetic field.”
Technical Note, Virginia Polytechnic Institute and State Univ.,
Blacksburg, VA.
Edwards, C., and Spurgeon, S. K. (1998). Sliding mode control: Theory
and applications, Taylor & Francis, London.
Fortescue, P., Swinerd, G., and Stark, J. (2011). Spacecraft systems
engineering, 4th Ed., Wiley, Hoboken, NJ.
Gregory, B. S. (2004). “Attitude control system design for ION, the Illinois
observing nanosatellite.” M.Sc. thesis, Univ. of Illinois at UrbanaChampaign, Urbana, IL.
Henderson, H. V., and Searle, S. R. (1981). “On deriving the inverse of a
sum of matrices.” SIAM Rev., 23(1), 53–60.
Hu, Q., Xie, L., and Gao, H. (2007). “Adaptive variable structure and active
vibration reduction for flexible spacecraft under input nonlinearity.”
J. Vib. Control, 13(11), 1573–1602.
Hughes, P. C. (2004). Spacecraft attitude dynamics, Dover Publications,
Mineola, NY.
Hung, J. Y., Gao, W., and Hung, J. C. (1993). “Variable structure control:
A survey.” IEEE Trans. Electron. Mag., 40(1), 2–8.
Jafarov, E. M. (2009). Variable structure control and time-delay systems,
WSEAS Press, Athens, Greece.
Larson, W. J., and Wertz, J. R. (1999). Space mission analysis and
design, 3rd Ed., Microcosm Press and Kluwer Academic Publishers,
El Segundo, CA.
Lo, S. C., and Chen, Y. P. (1995). “Smooth sliding-mode control for spacecraft attitude tracking maneuvers.” J. Guidance Control Dyn., 18(6),
1345–1349.
Meng, Q., Zhang, T., and Song, J. Y. (2012). “Modified model-based faulttolerant time-varying attitude tracking control of uncertain flexible
satellites.” J. Aerosp. Eng., 227(11), 1827–1841.
© ASCE
Meyer, C. D. (2000). Matrix analysis and applied linear algebra, SIAM,
Philadelphia.
Montenbruck, O., and Gill, E. (2000). Satellite orbits: Models, methods and
applications, Springer, Berlin.
Shtessel, Y., Edwards, C., Fridman, L., and Levant, A. (2014). Sliding mode
control and observation, Springer, New York.
Sidi, M. J. (1997). Spacecraft dynamics and control: A practical engineering approach, Cambridge University Press, New York.
Sofyalı, A., Jafarov, E. M., and Wisniewski, R. (2015). “Time-varying
sliding mode in rigid body motion controlled by magnetic torque.”
Proc., Int. Workshop on Recent Advances in Sliding Modes (RASM
2015), Istanbul, Turkey.
Utkin, V. I. (1992). Sliding modes in control and optimization, Springer,
London.
Vadali, S. R. (1986). “Variable-structure control of spacecraft large-angle
maneuvers.” J. Guidance Control Dyn., 9(2), 235–239.
Vallado, D. A. (1997). Fundamentals of astrodynamics and applications,
McGraw-Hill, New York.
Wertz, J. R. (1978). Spacecraft attitude determination and control, Reidel
Publishing Company, Dordrecht, Netherlands.
Wie, B. (1998). Space vehicle dynamics and control, American Institute of
Aeronautics and Astronautics, Reston, VA.
Wisniewski, R. (1996). “Satellite attitude control using only electromagnetic actuation.” Ph.D. thesis, Aalborg Univ., Aalborg,
Denmark.
Yang, C. D., and Sun, Y. P. (2002). “Mixed H2 =H∞ state-feedback design
for microsatellite attitude control.” Control Eng. Practice, 10(9),
951–970.
Zhao, J., Jiang, B., Shi, P., Gao, Z., and Xu, D. (2012). “Fault-tolerant
control design for near-space vehicles based on a dynamic terminal
sliding mode technique.” J. Systems and Control Eng., 226(6),
787–794.
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