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Research Article Improved Path Planning and Attitude Control Method for Agile Maneuver Satellite with Double-Gimbal Control Moment Gyros

Peiling Cui,1,2 Jingxian He,1,2 Jian Cui,1,2 and Haitao Li1,2

1School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing 100191, China 2Science and Technology on Inertial Laboratory, Beijing 100191, China

Correspondence should be addressed to Peiling Cui; [email protected]

Received 4 November 2014; Revised 2 February 2015; Accepted 8 February 2015

Academic Editor: Mohamed Abd El Aziz

Copyright © 2015 Peiling Cui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Double-gimbal control moment gyros can implement the satellite attitude maneuver efficiently. In order to reduce the energy consumption of double-gimbal control moment gyros and avoid the singularity state, an attitude maneuver path planning method is proposed by using the improved Fourier basis algorithm. Considering that the choice of the Fourier coefficients is important for the Fourier basis algorithm to converge quickly, a choosing method of the initial Fourier coefficients which can reduce the computational time of the path planning algorithm notably is proposed. Moreover, an attitude-tracking feedback controller based on the Fourier basis path planning algorithm is designed to acquire robustness. Simulation results show that the proposed path planning algorithm can implement attitude maneuver path planning in shortened time. Meanwhile, the feasibility of the attitude-tracking feedback controller which is based on the above path planning algorithm is verified in terms of the low energy consumption, high attitude-tracking precision, and the safe use of double-gimbal control moment gyros.

1. Introduction

High-resolution earth observation demands the satellite to have the ability of rapid attitude maneuver. Therefore, control moment gyros (CMGs) which can output large and precise torque are ideal attitude control actuator for the agile maneu- ver satellite. The main problem of using CMGs is the existence of singularity state, at which CMGs cannot produce torque in a certain direction. According to the gimbal numbers, CMGs can be divided into two classes, including single-gimbal CMGs (SGCMGs) and double-gimbal CMGs (DGCMGs). The singularity avoidance steering law is very important for SGCMGs to fulfill the requirement of the rapid attitude maneuver. Compared with SGCMG, DGCMG has more than one degree of freedom, and its angular momentum envelop is nearly spherical. Therefore, it is believed that DGCMGs can implement the rapid attitude maneuver with high efficiency.

Existing attitude control logics with CMGs are composed of attitude control laws and CMG steering laws [1, 2]. According to attitude tracking errors, the desired attitude control torque is computed by the attitude control laws.

The objective of the CMG steering laws is to avoid singularity states. A large-angle attitude maneuver control law based on nonlinear methods is proposed in [1], and a singularity robustness plus null motion steering law is designed to ensure DGCMGs against singularity. References [3, 4] investigate on the singularity avoidance steering law specially. A modified singular-direction avoidance steering law of SGCMGs is introduced in [3]. Kennel presents the vector distribution steering law for parallel-mounted DGCMGs in [4], which makes use of redundant degrees of freedom of DGCMGs.

However, when the CMG steering law is separated from the attitude control law, CMG singularity problem and energy consumption cannot be considered in the computing process of attitude control torque. In addition, there exist singularity states which cannot be avoided by CMG steering laws. To deal with the above problems, the attitude control and CMG steering law should be integrated together as a whole control system.

An effective method of integrating the attitude control law and CMG steering law together is to plan the attitude maneu- ver path according to the satellite maneuver characteristic.

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 878724, 11 pages http://dx.doi.org/10.1155/2015/878724

http://dx.doi.org/10.1155/2015/878724

2 Mathematical Problems in Engineering

Recent path planning algorithms mainly include Legendre pseudospectral algorithm [5, 6], generic algorithm (GA) [7, 8], and Fourier basis algorithm (FBA) [9, 10]. By using an integral cost function consisting of energy consumption and time cost index, [5] applies the Legendre pseudospectral algorithm to the optimal path planning. Reference [6] reviews key theoretical results of the pseudospectral optimal method which are important for successful flights. An energy optimal attitude maneuver path is planned by using the improved GA [7].

The Legendre pseudospectral algorithm and generic algo- rithm have many optimal parameters, thus resulting in com- plex computation. Moreover, the two algorithms generate time-discrete control inputs, which need to be approximated by continuous functions after the path planning. The rapid attitude maneuver control systems of agile satellites are real-time systems. Therefore, the Legendre pseudospectral algorithm and generic algorithm are not very suitable for the rapid attitude maneuver control. Regarding the gimbal rates of SGCMGs as control inputs, [9, 10] proposes a continuous optimal path planning method based on the FBA. The idea of FBA is to approximate control inputs by applying a Fourier series. The path planning method based on FBA is advantageous in terms of the computational time, and the path planning results are continuous. However, the Fourier coefficient vector is initialized by some random process in [9, 10], and the algorithm cannot converge rapidly. The choosing method of the initial Fourier coefficient has not been introduced yet. Moreover, there exists no Fourier basis path planning method which is proposed to investigate the attitude control of agile maneuver satellite with DGCMGs. Besides, since the disturbances and attitude tracking errors have not been considered in the path planning, the attitude tracking controller which can achieve the system robustness needs to be improved [5, 9].

This paper fills the above gap in the existing literature. The path planning and attitude control law of the agile maneuver satellite with DGCMGs are integrated together. The energy consumption and terminal state errors of double- gimbal control moment gyroscopes are considered at the

same time. An improved FBA is proposed for the attitude maneuver path planning for agile maneuver satellite with DGCMGs. In order to shorten the computational time of the path planning, a choosing method of the initial Fourier coefficient is introduced. Meanwhile, the attitude tracking feedback controller based on the path planning method is proposed by applying a sliding mode controller. Simulations are carried out to verify the availability of the introduced path planning method and the corresponding attitude tracking controller.

2. Model of the Satellite Attitude Control System

As shown in Figure 1, the two parallel-mounted DGCMGs are used as the attitude control actuators. x

𝑏 , y 𝑏 , and z

𝑏 are

unit vectors aligned along the axes of the satellite body frame coordinate. The orthonormal set of unit vectors x

𝑖 , y 𝑖 , z 𝑖 is

used to describe the orientation of the 𝑖th DGCMG relative to the spacecraft body frame, where x

𝑖 is fixed along the

inner gimbal axis, y 𝑖 is fixed along the outer gimbal axis,

and z 𝑖 is fixed along the wheel spin axis. 𝛼

𝑖 and 𝛽

𝑖 denote

the inner gimbal angle and outer gimbal angle of the 𝑖th DGCMG, respectively. �̇�

𝑖 and ̇𝛽

𝑖 represent the inner gimbal

rate and outer gimbal rate of the 𝑖th DGCMG, respectively. Assuming that ℎ

0 denote the wheel angular momentum of

the DGCMG, the total angular momentum of two parallel- mounted DGCMGs expressed in the satellite body frame coordinate can be written as

hcmg = ℎ0 [

[

cos 𝛼 1 cos 𝛽 1 + cos 𝛼

2 cos 𝛽 2

− cos 𝛼 1 sin 𝛽 1 − cos 𝛼

2 sin 𝛽 2

− sin 𝛼 1 − sin 𝛼

]

. (1)

The output torque of DGCMGs can be obtained by using the time derivative of hcmg as follows:

Tcmg = − ̇hcmg

= −ℎ 0

[

− sin 𝛼 1 cos 𝛽 1

− cos 𝛼 1 sin 𝛽 1

− sin 𝛼 2 cos 𝛽 2

− cos 𝛼 2 sin 𝛽 2

sin 𝛼 1 sin 𝛽 1

− cos 𝛼 1 cos 𝛽 1

sin 𝛼 2 sin 𝛽 2

− cos 𝛼 2 cos 𝛽 2

− cos 𝛼 1

0 − cos 𝛼 2

]

̇𝛿= −ℎ 0 C (𝛿) ̇𝛿,

(2)

where𝛿 = [𝛼 1 𝛽 1 𝛼 2 𝛽 2 ]

𝑇 denotes the gimbal angle vector of two parallel-mounted DGCMGs and ̇𝛿 is the time deriva- tive of𝛿with respect to the satellite body frame. C is a 3 × 4 matrix, which can describe the three-dimensional control authority of DGCMGs. As the singularity index det(CCT) approaches zero, DGCMGs will encounter singularity.

Considering the angular momentum of each component in the satellite attitude control system, the total angular momentum is

Hb = Ib𝜔+ hcmg, (3)

Mathematical Problems in Engineering 3

�̇�1

̇𝛽1

�̇�2

̇𝛽2

Figure 1: Illustration of the two parallel-mounted DGCMGs.

where Ib represents the satellite moment of inertia and 𝜔 represents the attitude angular velocity. Supposing that the satellite body is rigid, the dynamic equation is

̇H 𝑏 +𝜔× H

𝑏 = T 𝑑 , (4)

where ̇H 𝑏 is the time derivative of H

𝑏 with respect to

the satellite body frame and T 𝑑 is the disturbance toque.

According to (2) and (3), (4) can be rewritten as

Ib�̇�+𝜔× Ib𝜔+ ℎ0C (𝛿) ̇𝛿+𝜔× hcmg = T𝑑. (5)

Aiming at describing the large-angle attitude maneuver, the quaternion kinematics equation yields

q̇ = 1

[

0 −𝜔 𝑥

−𝜔 𝑦

−𝜔 𝑧

𝜔 𝑥

0 𝜔 𝑧

−𝜔 𝑦

𝜔 𝑦

−𝜔 𝑧

0 𝜔 𝑥

𝜔 𝑧

𝜔 𝑦

−𝜔 𝑥

]

q = E (𝜔) q, (6)

where q = [𝑞 0 q]𝑇 is unit quaternion. 𝑞

0 and q are scalar and

vector components of q. 𝜔 𝑥 , 𝜔 𝑦 , and 𝜔

𝑧 denote elements of𝜔

expressed in the satellite body frame coordinate. E is a 4 × 4 matrix associated with the attitude angular velocity.

Selecting x = [q𝑇 𝜔𝑇]𝑇 as the state variable, the attitude control model of the satellite equipped with DGCMGs can be described as

ẋ = f (𝛿) u + g (x,𝛿) , (7)

g (x,𝛿) = [ E (𝜔) q

−J−1(𝜔× J𝜔+𝜔× hcmg) ] , (8)

f (𝛿) = [ 0

−J−1C (𝛿) ℎ0 ] , (9)

where the control input vector u is the DGCMG gimbal rate vector ̇𝛿. The boundary conditions of (7) are

x (0) = x0, x (𝑡𝑓) = x𝑑, (10)

where 𝑡 𝑓 is the desired attitude maneuver time. x

0 is the initial

state variable vector. x 𝑑 is the desired state variable vector.

3. Path Planning Method Based on the Improved Fourier Basis Algorithm

3.1. Introduction of the Improved Fourier Basis Algorithm. The path planning method based on the improved FBA adopts a Fourier series to approximate the control inputs. The searching methods, such as gradient descent method and Newton method, are applied to search for the near optimal Fourier coefficients, which make the cost function of the attitude path planning converge to its minimal value. In order to achieve the safe use of DGCMGs, the path planning objective is to obtain the smooth attitude maneuver path which keeps the maximal gimbal rate low, avoiding vibrating motions. Therefore, the energy consumption of DGCMGs should be low enough to meet the requirement of the path planning objective. By considering the terminal state errors and the energy consumption of DGCMGs, the cost function of the attitude maneuver path planning yields

𝐽 (u) = ∫ 𝑡𝑓

(u𝑇u) 𝑑𝑡 + (x (𝑡 𝑓 ) − x 𝑑 )

𝑇

G (x (𝑡 𝑓 ) − x 𝑑 ) , (11)

where G is a weighting matrix, which should be large enough to reduce the terminal state errors. The first term in the cost function is the energy cost of control inputs, and the second term depends on terminal state errors. If the control inputs are expressed by an orthonormal basis for L

2 ([0, 𝑡 𝑓 ]), the

search for control inputs can be converted into solving for optimal parameters of the expression. By choosing a Fourier basis to describe the control input vector and restricting control inputs to the first three terms of the Fourier basis, the following can be obtained:

u =𝜙𝛼, (12)

where

𝜙= [0.5I 4 sin (𝜔

𝑓0 𝑡) I 4 sin (2𝜔

𝑓0 𝑡) I 4 sin (3𝜔

𝑓0 𝑡) I 4 cos (𝜔

𝑓0 𝑡) I 4 cos (2𝜔

𝑓0 𝑡) I 4 cos (3𝜔

𝑓0 𝑡) I 4 ] , (13)

where 𝜙 denotes the Fourier base term matrix, 𝛼 denotes the Fourier coefficient vector, I

4 is a 4 × 4 identity matrix,

and 𝜔 𝑓0

= 2𝜋/𝑡 𝑓 is the basic frequency of the Fourier basis

approximation. Inserting (12) into (11), the cost function of

the path planning comes to depend on the Fourier coefficients as follows:

𝐽 (𝛼) =𝛼 𝑇 𝛼+ (x (𝑡

𝑓 ) − x 𝑑 )

𝑇

G (x (𝑡 𝑓 ) − x 𝑑 ) . (14)

4 Mathematical Problems in Engineering

In this way, the problem of the path planning based on FBA is equivalent to a nonlinear search problem for the near optimal coefficients that makes the cost function approach its minimal value.

3.2. Choosing Method of the Initial Fourier Coefficients. The choice of the initial Fourier coefficients is very important to make the path planning method based on improved FBA converge rapidly, which can improve the convergence speed of the algorithm. The initial Fourier coefficients are computed by the initial path of DGCMG gimbal rates. Moreover, as can be seen in (7), gimbal rates can be generated by the quaternion. Therefore, the choice of the initial Fourier coef- ficients can start with the determination of initial quaternion paths. (q

1 , q 2 , . . . , q

𝑛 ) is used to denote the initial quaternion

path, where q 1 and q

𝑛 are the vector components of the

initial quaternion and the target quaternion, respectively. q 𝑗 (𝑗 = 2, . . . 𝑛 − 1) represents the vector components of

the intermediate nodes of the quaternion path. The path length 𝑛 = 𝑡

𝑓 /𝑇 𝑠 is calculated by the simulation time 𝑡

𝑓 and

simulation step 𝑇 𝑠 . According to (6), q

2 can be computed

by the initial quaternion q 1 and initial attitude angular

velocity𝜔 0 . To meet the requirement of satellite attitude agile

maneuver, intermediate nodes (q 3 , . . . , q

𝑛−1 ) in the initial

quaternion path are produced based on the sigmoid function:

q 𝑗 =

q 𝑛 − q 2

1 + exp𝑛 [− (𝑐𝑗𝑇 𝑠 + (𝑎 − 2) 𝑇𝑠

− 𝑚)]

+ q 2

𝑗 ∈ (3, 4, . . . , 𝑛 − 1) ,

(15)

where 𝑚 is an appropriate constant. 𝑎 and 𝑐 can be computed as follows:

f 2 ( 𝑡) =

q 𝑛 − q 2

1 + exp𝑛 [− (𝑡 − 𝑚)] + q 2 ,

Δf 2 (𝑗) = f

2 ((𝑗 + 1) 𝑇

𝑠 ) − f 2 (𝑗𝑇 𝑠 ) ,

𝑎 = min (find (Δf 2 > 10

−3 )) ,

𝑐 =

𝑏 − 𝑎 + 1

𝑛 − 2

𝑏 = max (find (Δf 2 > 10

−3 )) ,

(16)

where 𝑡 ∈ (0, 𝑇 𝑎 ) and 𝑇

𝑎 is a constant larger than 𝑡

𝑓 .

When the quaternion path (q 1 , q 2 , . . . , q

𝑛 ) is obtained,

the attitude angular velocity can be achieved by the time derivative of quaternion. Furthermore, DGCMG gimbal rates can be calculated by the attitude angular velocity according to (5). Finally, the Fourier series shown in (13) is applied to approximating the evaluating gimbal rates, thus generating the initial Fourier coefficient vector.

3.3. Search Method for the Fourier Basis Algorithm. The New- ton method based on the first and second partial derivatives of the cost function can improve the convergence property of the path planning algorithm. However, it is difficult to compute the second partial derivative of the cost function, and the above choosing method of the initial Fourier coef- ficient vector has improved the convergence speed already. Therefore, the gradient descent method which only based on the first partial derivative of the cost function is applied to searching for the near optimal Fourier coefficient vector. The computation of the first-order derivative of the cost function at the searching point𝛼

𝑛 is shown as follows:

𝜕J (𝛼 𝑛 )

𝜕𝛼 𝑛

= −𝜇 (𝛼 𝑛 − y𝑇(𝑡

𝑓 ) G (x − x

𝑑 )) , (17)

where 𝜇 is a positive constant and y(𝑡 𝑓 ) is described by

y (𝑡 𝑓 ) =

𝜕x (𝑡 𝑓 )

𝜕𝛼 .

(18)

y(𝑡 𝑓 ) has no simple mathematic expression, and it can be

numerically evaluated by integrating its time derivative ẏ(𝑡) from 𝑡 = 0 to 𝑡 = 𝑡

𝑓 . ẏ(𝑡) and y(𝑡

0 ) are given as

ẏ (𝑡) = ( 4

∑

𝑖=1

𝜕f 𝑖

𝜕𝛿 u 𝑖 ) z + f𝜙+

𝜕g 𝜕x

y + 𝜕g 𝜕𝛿

z, y (𝑡 0 ) = 0,

(19)

where z is the partial derivative of gimbal rates with respect to the Fourier coefficient vector. Its time derivative and initial value are

ż =𝜙, z (𝑡 0 ) = 0. (20)

By applying the gradient descent method, the Fourier coefficient variation Δ𝛼about the searching point𝛼

𝑛 is given

Δ𝛼= −𝜇 (𝛼 𝑛 − y𝑇(𝑡

𝑓 ) G (x − x

𝑑 )) . (21)

Thus, the updating equation of the Fourier coefficient vector can be obtained:

𝛼 𝑛+1

=𝛼 𝑛 + Δ𝛼. (22)

The updating operation of the Fourier coefficient vector should be continued, until the near optimal solution of the path planning satisfies the requirement of DGCMG energy consumption and terminal state accuracy at the same time. Results of the proposed path planning method are composed of the varying path of the DGCMG gimbal rates, quaternion, and attitude angular velocity.

4. Attitude Tracking Feedback Controller Based on the Improved Path Planning

4.1. Model of Attitude Tracking Error. The error quaternion q 𝑒 = [𝑞 𝑒0

q 𝑒 ]

𝑇 can be computed by the desired quaternion

Mathematical Problems in Engineering 5

Path planning based on improved

FBA

DGCMG system

Satellite system

Attitude tracking feedback controller

Computing of attitude tracking error

+ Gimbal rate command

Control torque

Tracking error

Disturbance

Planning path of quaternion and attitude angular velocity

Δu

unorm

Figure 2: Block diagram of the attitude tracking feedback controller.

q 𝑑 = [𝑞 𝑑0

q 𝑑 ]

𝑇 and the current quaternion q = [𝑞 0 q]𝑇 as

follows:

𝑞 𝑒0 = 𝑞 0 𝑞 𝑑0 + q𝑇q

𝑑 ,

q 𝑒 = 𝑞 𝑑0 q − 𝑞 0 q 𝑑 + q × q

𝑑 ,

(23)

where 𝑞 𝑒0 and q 𝑒 have the constraint equation: 𝑞2

𝑒0 +q 𝑒

𝑇q 𝑒 = 1.

Assuming that 𝜔 𝑑 is the desired attitude angular velocity, then

the error of the attitude angular velocity is defined as

𝜔 𝑒 =𝜔− C𝑏

𝑑 𝜔 𝑑 , (24)

where C𝑏 𝑑 = (𝑞

𝑒0 −q 𝑒

𝑇q 𝑒 )I 3 +2q 𝑒 q 𝑒

𝑇 −2𝑞 𝑒0 q 𝑒

×, q 𝑒

× is the cross matrix of q

𝑒 , and I

3 denotes 3 × 3 identity matrix.

By inserting (24) into (5), the following can be obtained:

Ib�̇�𝑒 = Γ(Ib,𝜔𝑒) − ℎ0C (𝛿) u + T𝑑, (25)

whereΓ(Ib,𝜔𝑒) = −(𝜔𝑒 + C 𝑏

𝑑 𝜔 𝑑 ) × [Ib(𝜔𝑒 + C

𝑏

𝑑 𝜔 𝑑 ) + hcmg] +

Ib(𝜔𝑒 × C 𝑏

𝑑 𝜔 𝑑 − C𝑏 𝑑 �̇� 𝑑 ).

According to the quaternion kinematics (6), the kinemat- ics equation based on error quaternion can be written as

̇𝑞 𝑒0 = −

q 𝑒

𝑇 𝜔 𝑒 ,

̇q 𝑒 =

Z (q 𝑒 )𝜔 𝑒 ,

(26)

where Z(q 𝑒 ) = 𝑞 𝑒0 I 3 + q 𝑒

×.

4.2. Design of the Attitude Tracking Controller Based on the Improved Path Planning. The path planning method can acquire the continuous path of DGCMG gimbal rates which are effective in smooth action. The feedforward controller using the planning path of gimbal rates directly can track the desired quaternion path accurately without considering the disturbances. However, there always exist several dis- turbances, and the feedforward controller cannot maintain precise attitude control. In order to acquire the robustness against disturbances, the attitude tracking feedback controller is introduced based on the attitude maneuver path planned by

the improved FBA. Since sliding mode controller has strong robustness and favorable dynamic response performance, the sliding mode controller is applied to the design of attitude tracking feedback controller. Moreover, the planning paths of the quaternion and attitude angular velocity are regarded as the tracking paths of the attitude maneuver. The objective of the feedback controller design is to compensate for the attitude tracking error by adding Δu to unorm, where unorm represents the planning gimbal rate of the proposed path planning method and Δu is the added control input vector which can eliminate errors. The block diagram of the attitude tracking feedback controller is illustrated by Figure 2.

The sliding surface S is chosen as

S = 𝜔 𝑒 + Kq 𝑒 , (27)

where K is a positive diagonal matrix, 𝜔 𝑒 is the error between

the real angular velocity and the planning angular velocity, and q

𝑒 is the vector component of error quaternion.

In order to verify the stability of the sliding mode S = 0, the Lyapunov function 𝑉

1 is selected as follows:

𝑉 1 =

q 𝑒

𝑇q 𝑒 +

(1 − 𝑞 𝑒0 )

2 = 1 − 𝑞

𝑒0 . (28)

It is obvious that 𝑉 1 is positive, and its time derivative is

̇ 𝑉 1 = − ̇𝑞 𝑒0 =

q 𝑒

𝑇 𝜔 𝑒 . (29)

Inserting S = 0 into (29), the following can be obtained:

̇ 𝑉 1 = −

(K−1) 𝑇

𝜔 𝑒

𝑇 𝜔 𝑒 . (30)

Since K is a positive matrix, there exists ̇𝑉 1

< 0, thus guaranteeing the stability of the sliding mode. The time derivative of S is

̇S = I−1b (Γ(Ib,𝜔𝑒) − ℎ0C (𝛿) ul + T𝑑) + 1

KZ (q 𝑒 )𝜔 𝑒 , (31)

where ul = unorm + Δu is the total control input vector. In order to generate a proper control law, another Lya-

punov function 𝑉 2 is chosen as

𝑉 2 =

S𝑇IbS. (32)

6 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7

0.2

0.4

0.6

0.8

1.2 Q

ua te

rn io

−0.2

t (s)

qd0 qd1

qd2 qd3

(a)

0 1 2 3 4 5 6 7

0.2

0.4

0.6

0.8

1.2

Q ua

te rn

io n

−0.2

t (s)

qd0 qd1

qd2 qd3

(b)

Figure 3: Time histories of quaternion. (a) Based on the path planning method in [7]. (b) Based on the path planning method in this paper.

The time derivative of 𝑉 2 can be written as

̇ 𝑉 2 = S𝑇Ib ̇S

= S𝑇(Γ(Ib,𝜔𝑒) − ℎ0C (𝛿) ul + T𝑑 + 1

IbKZ (q𝑒)𝜔𝑒) . (33)

Let

ul = (ℎ0C (𝛿)) −1

⋅ (Γ(Ib,𝜔𝑒) + T𝑑 + 1

IbKZ (q𝑒)𝜔𝑒 + K𝑔S

+ K 𝛽 sign (S)) .

(34)

Then (33) can be rewritten as

̇ 𝑉 2 = −𝑔S𝑇S − 𝛽S𝑇 sign (S) , (35)

where K 𝑔 and K

𝛽 are both positive diagonal matrices. It is

obvious that ̇𝑉 2 < 0, and the sliding surface can be reachable

with the control law. The saturation function is defined as

sat (S, 𝜉) = { { { {

{ { { {

{

1 S > 𝜉 S 𝜉

|S| ≤ 𝜉

−1 S < −𝜉,

(36)

where 𝜉 is a small positive constant. Aiming at avoiding the vibrating motions, the sign function in (34) is substituted by the saturation function as follows:

ul = (ℎ0C (𝛿)) −1

⋅ (Γ(Ib,𝜔𝑒)+ T𝑑+ 1

IbKZ (q𝑒)𝜔𝑒+ K𝑔S + K𝛽 sat (S)) .

(37)

Then the added control input vector can be calculated by

Δu = ul − unorm. (38)

As can be seen from (37) and (38), the added control input vector Δu is associated with the attitude tracking errors, thus enabling it to compensate for the existing errors. It is confirmed that the added control inputs can improve the accuracy of the attitude control system.

5. Simulations and Discussions

Simulations for the path planning method based on improved FBA and the associated attitude tracking feedback controller are performed by using two parallel-mounted DGCMGs. The satellite moment of inertia is

J = [[ [

12 0.12 0.12

0.13 13 0.13

0.15 0.15 15

]

kg ⋅ m2. (39)

Related parameters of the satellite and DGCMGs are shown as follows: the desired attitude maneuver time 𝑡

𝑓 =

7 s, the simulation step 𝑇 𝑠

= 0.1 s, the initial quater- nion q

1 = [1 0 0 0]

𝑇, the desired quaternion q 𝑑

Mathematical Problems in Engineering 7

0 1 2 3 4 5 6 7

8 A

ng ul

ar v

el oc

ity (d

eg /s

)

t (s)

−1

(a)

0 1 2 3 4 5 6 7

A ng

ul ar

v el

oc ity

(d eg

/s )

t (s)

−2

(b)

Figure 4: Time histories of attitude angular velocity. (a) Based on the path planning method in [7]. (b) Based on the path planning method in this paper.

[0.97 0.26 0 0]

𝑇, the initial attitude angular velocity𝜔 0 =

[1.1 1.2 1.6]

𝑇 deg /s, the desired attitude angular velocity 𝜔 𝑑

= [0 0 0]

𝑇 deg /s, the wheel angular momentum of DGCMGs ℎ

0 = 6 Nms, the initial gimbal angle vector 𝛿

0 =

[0 0 0 −90]

𝑇 deg, and the maximal gimbal rate is 10 deg/s.

5.1. Path Planning Results Based on the Improved FBA. Path planning results based on the improved FBA are compared with that in [7]. During the comparison, the related parame- ters of the satellite and DGCMGs are all the same. Parameters of the initial Fourier coefficient choosing method are 𝑚 = 8.05, 𝑛 = 1.96. Parameters of searching method are G = diag(0, 4, 4, 4, 6, 6, 6)×102, 𝜇 = 2.2×10−4. The stop condition of the Fourier coefficient update is ‖q

𝑒 ‖ ∞

≤ 0.001, ‖𝜔 𝑒 ‖ ∞

≤

0.05 deg /s. The DGCMG energy consumption in the path planning method is calculated by the following equation:

∫

𝑡𝑓

u𝑇u 𝑑𝑡 = ∫ 𝑡𝑓

̇𝛿 𝑇 ̇𝛿𝑑𝑡. (40)

Figures 3–7 show the attitude maneuver path planning results based on the path planning method in [7] and that proposed in this paper. Figure 3 illustrates the varying path of the quaternion. Figure 4 illustrates the varying path of the attitude angular velocity. As can be seen from Figure 4, the attitude angular velocity of the both methods can converge to the desired value in the given maneuver time. Figure 4 illustrates the varying path of the attitude angular velocity.

Figure 5 gives the planning result of the DGCMG gimbal rates. As shown in Figure 5(a), DGCMG gimbal rates of the path planning method in [7] nearly reach the limit of 10 deg/s. In contrast, the proposed improved path planning method can keep DGCMG gimbal rates under 8.2 deg/s; thus, the

safe use of DGCMGs is maintained, as shown in Figure 5(b). Furthermore, the initial gimbal rates of the path planning method in [7] reach 8.2 deg/s. However, the initial gimbal rates of the proposed improved path planning are no more than 0.8 deg/s. Since the tracking speed of the gimbal rate controller is limited, vibration cannot be avoided when the initial gimbal rates are large. The DGCMG gimbal angles are shown in Figure 6. All of the gimbal angles vary smoothly.

Figure 7 describes the singularity index of the two path planning methods. The minimal value of the singularity index in Figure 7(b) is larger than that in Figure 7(a). It indicates that the proposed method can plan effective paths which prevent DGCMGs from encountering singularity states. The computational time of the path planning method in [7] is about 30.4 s. In contrast, the proposed method based on FBA can implement the path planning within 3.2 s, and its computational time is 90% shorter than that of the method in [7]. Moreover, the DGCMG energy consumption of the proposed method is 0.0805, which is 20% lower than that of the method in [7]. In a word, the proposed path planning method based on FBA is more effective than the method in [7] in the aspect of the computational time and DGCMG energy consumption.

5.2. Attitude Tracking Feedback Controller Based on the Improved FBA. Simulations for the attitude tracking con- trollers are carried out in this section. Results of the feedback controller have been compared with that of the feedforward controller which uses the path planning directly. Because the simulation time is much shorter than the varying period of the environmental disturbance, the disturbance torque vector is regarded as constant: T

𝑑 = [0.02 0.04 0.01]

𝑇Nm.

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7

10 G

im ba

l r at

e (d

eg /s

)

−5

−10

t (s)

Inner 1 Outer 1

Inner 2 Outer 2

(a) G

im ba

l r at

e (d

eg /s

)

0 1 2 3 4 5 6 7

−5

−10

t (s)

Inner 1 Outer 1

Inner 2 Outer 2

(b)

Figure 5: Time histories of gimbal angle velocity. (a) Based on the path planning method in [7]. (b) Based on the path planning method in this paper.

0 1 2 3 4 5 6 7

G im

ba l a

ng le

(d eg

)

t (s)

−20

−40

−60

−80

−100

−120

Inner 1 Outer 1

Inner 2 Outer 2

(a)

G im

ba l a

ng le

(d eg

)

0 1 2 3 4 5 6 7

t (s)

−20

−40

−60

−80

−100

−120

Inner 1 Outer 1

Inner 2 Outer 2

(b)

Figure 6: Time histories of gimbal angle. (a) Based on the path planning method in [7]. (b) Based on the path planning method in this paper.

Parameters of the attitude tracking feedback controller are chosen as K = diag(2, 1.2, 1.2), Kg = diag(5, 8, 8), and K𝛽 = diag(1.0, 1.2, 1.2), 𝜉 = 0.1.

Figures 8–11 give the simulation results of the feedforward and feedback controllers. Figure 8 illustrates the attitude quaternion trajectory. Figure 9 illustrates the attitude angular velocity trajectory. Figure 9(a) shows that the attitude angular velocity of the feedforward controller is still larger than 2 × 10

−2 rad/s at the terminal time, and the attitude maneuver

has not been finished in the desired maneuver. On the other hand, the attitude angular velocity in Figure 9(b) is within 10

−3 rad/s at the terminal time; thus, the feedback controller implements the attitude maneuver task.

Figure 10 illustrates the DGCMG gimbal rates. As can be seen in Figure 10, the safe use of DGCMGs can be guaranteed by both the feedforward and feedback attitude tracking controllers. Because the feedforward controller makes use of the planning results directly, the gimbal rate trajectory in

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 1.5

1.6

1.7

1.8

1.9

2.1

2.2

t (s)

de t( C C T )

(a)

0 1 2 3 4 5 6 7 1.5

1.6

1.7

1.8

1.9

2.1

2.2

t (s)

de t( C C T )

(b)

Figure 7: Time histories of singularity index. (a) Based on the path planning method in [7]. (b) Based on the path planning method in this paper.

0 1 2 3 4 5 6 7 0

0.2

0.4

0.6

0.8

Q ua

te rn

io n

t (s)

qd0 qd1

qd2 qd3

(a)

0 1 2 3 4 5 6 7 0

0.2

0.4

0.6

0.8

Q ua

te rn

io n

t (s)

qd0 qd1

qd2 qd3

(b)

Figure 8: Time histories of quaternion. (a) Angular velocity with feedforward control. (b) Angular velocity with feedback control.

Figure 10(a) is the same as the planning path of gimbal rates. As shown in Figure 10(b), the DGCMG energy consumption increases owing to the feedback compensation. It is suggested that the robustness of the control system is obtained at the cost of increasing DGCMG energy consumption. Figure 11 illustrates gimbal angles of DGCMGs. The trajectories of gimbal angles vary smoothly during the whole maneuver time.

6. Conclusions

Considering the energy consumption and terminal state errors of double-gimbal control moment gyroscopes at

the same time, the path planning method based on an improved FBA is proposed for planning a near optimal path with low energy consumption and smoothly varying DGCMG gimbal rates. The feedforward control using the above path can prevent the DGCMGs from encountering the singularity, and the rapid actuation of DGCMGs is guaranteed. In order to improve the search efficiency of the Fourier coefficients and reduce the computational time of the path planning, a choosing method of the Fourier coefficients is put forward according to the satellite maneuver characteristic. Owing to the fact that the feedforward control cannot acquire the robustness, a feedback controller based on the path planning is introduced. The forward control inputs

10 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7

10 A

ng ul

ar v

el oc

ity (d

eg /s

)

t (s)

−2

(a) A

ng ul

ar v

el oc

ity (d

eg /s

)

0 1 2 3 4 5 6 7

t (s)

−2

(b)

Figure 9: Time histories of attitude angular velocity. (a) Angular velocity with feedforward control. (b) Angular velocity with feedback control.

0 1 2 3 4 5 6 7

G im

ba l r

at e

(d eg

/s )

−5

−10

t (s)

Inner 1 Outer 1

Inner 2 Outer 2

(a)

G im

ba l r

at e

(d eg

/s )

0 1 2 3 4 5 6 7

−5

−10

t (s)

Inner 1 Outer 1

Inner 2 Outer 2

(b)

Figure 10: Time histories of gimbal rate. (a) Gimbal rates with feedforward control. (b) Gimbal rates with feedback control.

of the feedback controller are the planning gimbal rates, and the slide mode control method is used to compute the feedback control inputs. Thus, the accuracy of attitude control system is improved notably. Finally, the effectiveness of the proposed path planning method and attitude tracking feed- back controller is validated by simulations. Comparing with the path planning method in [7], the proposed path planning method is advantageous in terms of the initial gimbal rates, maximal gimbal rates, and energy consumption of DGCMGs.

Moreover, the computational time of path planning is much shorter than that of [7] and thus is benefiting to the satellite attitude agile maneuver.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Mathematical Problems in Engineering 11

0 1 2 3 4 5 6 7

20 G

im ba

l a ng

le (d

eg )

t (s)

−20

−40

−60

−80

−100

−120

Inner 1 Outer 1

Inner 2 Outer 2

(a) G

im ba

l a ng

le (d

eg )

0 1 2 3 4 5 6 7

t (s)

−20

−40

−60

−80

−100

−120

Inner 1 Outer 1

Inner 2 Outer 2

(b)

Figure 11: Time histories of gimbal angle. (a) Gimbal angles with feedforward control. (b) Gimbal angles with feedback control.

Acknowledgments

This research has been supported by National Natural Science Foundation of China under Grant 61121003, the National Basic Research Program (973 Program) of China under Grant 2009CB2400101C, and National Civil Aerospace Preresearch Project.

References

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[7] P. L. Cui and F. Y. Liu, “Attitude path planning for agile satellite based on improved genetic algorithm,” Electronics Optics & Control, vol. 19, no. 12, pp. 23–28, 2012 (Chinese).

[8] R. Kala, “Multi-robot path planning using co-evolutionary genetic programming,” Expert Systems with Applications, vol. 39, no. 3, pp. 3817–3831, 2012.

[9] Y. Kusuda and M. Takahashi, “Feedback control with nominal inputs for agile satellites using control moment gyros,” Journal of Guidance, Control, and Dynamics, vol. 34, no. 4, pp. 1209–1218, 2011.

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