Adaptive attitude control of spacecraft using neural networks

Acta Astronautica 64 (2009) 778 – 786 www.elsevier.com/locate/actaastro

Adaptive attitude control of spacecraft using neural networks Henzeh Leeghim∗ , Yoonhyuk Choi, Hyochoong Bang Department of Aerospace Engineering, KAIST, Daejon 305-701, Republic of Korea Received 24 May 2008; received in revised form 11 November 2008; accepted 5 December 2008 Available online 22 January 2009

Abstract An adaptive control technique can be applicable to reorient spacecraft with uncertain properties such as mass, inertial and various misalignments. A nonlinear quaternion feedback controller is chosen as a baseline attitude controller. A linearly added adaptive input supported by neural networks to the baseline controller can estimate and eliminate the uncertain spacecraft property adaptively. The normalized input neural networks (NINNs) are examined for reliable computation of the adaptive input. The newly defined learning rules of the neural networks are established appropriately for a spacecraft. To prove the stability of the closed-loop dynamics with the control law, Lyapunov stability theory is considered. As a result, the proposed approach results in the uniform ultimate boundedness in tracking error and robustness of the chattering and the singularity problems. © 2008 Elsevier Ltd. All rights reserved. Keywords: Adaptive control; Neural networks; Spacecraft attitude control

1. Introduction High resolution earth observation spacecraft usually requires high precision attitude control capability to accomplish given mission objectives. However, the uncertainties in the spacecraft system actuated by conventional fixed-gain PID controller and external disturbances in general introduce steady state error and undesirable phase during large angle maneuvers [1]. The error ultimately affects precise attitude control performance. In practical applications, for example, the system inertia can be considered as uncertainty to account for changes in overall system configuration, for example, fuel consumption, out-gassing, etc. ∗ Corresponding author.

E-mail addresses: [email protected] (H. Leeghim), [email protected] (Y. Choi), [email protected] (H. Bang). 0094-5765/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2008.12.004

One potential approach to handle the model uncertainty is adaptive control. Adaptive control parameterizes the uncertainty in terms of certain unknown parameters, and attempts to employ feedback strategy to learn these parameters during the operation of the system. Slotine and DiBenedetto developed an adaptive controller including the Gibbs parameters to compensate model uncertainty [2]. A nonlinear adaptive control algorithm in the presence of inertia uncertainties was developed [3]. This algorithm has a singularity problem so that the nonlinear controller forces the output to zero when the denominator of the control law is zero. Naturally, it induces a chattering problem. As an enabling nonlinear control theory, the feedback linearization technique has been applied to a wide variety of systems. A direct adaptive control strategy for a spacecraft with uncertain moment of inertia based on the feedback linearization was proposed by Sheen and Bishop [4]. Most adaptive control methods have been restricted to

H. Leeghim et al. / Acta Astronautica 64 (2009) 778 – 786

the systems with linear unknown parameters. Furthermore, the feedback linearization-based attitude control and momentum management of spacecraft is dependent on the initial conditions due to singularity problem [5]. Mathematically, the singularity problem corresponds to the case when the denominator of the control law becomes zero. The denominator in the nonlinear control form, for example, includes angular rate or attitude [6]. If angular rate or attitude is zero, then the denominator may become zero. If the absolute value of the denominator goes down to a certain level, then most of the controller produces no control signal. Unfortunately, this approach introduces the chattering problem. The uncertainties in the spacecraft are in general nonlinear caused by various sources [1]. The system considered in this paper is subject to uncertain function, not parametric uncertainty. Since the neural network was demonstrated as a universal smooth function approximator [7], extensive studies have been conducted for diverse applications, especially pattern recognition, identification, estimation, and control of dynamic systems [8–12]. Application was made to adaptive control using neural networks for a general serial-link rigid robot arm [13]. The structure of the neural network controller is derived by filtered error approach. Calise et al. have extensively worked on the control and estimation of aircrafts and helicopters using neural network [14–16]. Adaptive output feedback control using a high-gain observer and radial basis function neural network were proposed for nonlinear systems represented by input–output models [17]. Also, a nonlinear adaptive flight control system was designed by backstepping [11] and neural network controller. In this paper, to avoid singularity and chattering problem, the linearization technique is not considered, and the well-known quaternion feedback law [18] is chosen as the baseline controller. The three-layered neural network is augmented to the baseline controller to estimate and remove the unknown terms. The learning rules of the neural network are based on new weight updates, which are specifically designed for a spacecraft attitude control system. This is a new result different from the general update law. Therefore, the adaptive neural network-based quaternion control law can be claimed to provide much flexibility for implementation in generic spacecraft systems. Furthermore, for more reliable computation, normalized input neural network (NINN) is considered. The importance of the input data normalization is emphasized because of various benefits gained for the function approximation [19,20]. This paper is organized in five sections. First, the equations of motion of a rigid spacecraft with

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multiple CMGs are introduced. Then, the NINN is introduced. An adaptive control law with a new update law for spacecraft with uncertainty is followed. In the next section, the stability analysis is presented. Finally, the proposed method is demonstrated using numerical simulation study. 2. Spacecraft dynamics model Consider a spacecraft installed with one variable speed control moment gyro which has been widely studied in the past couple of decades (see Fig. 1). The gimbal inertia matrix and flying wheel inertia matrix based on the gimbal frame orientation are denoted as Ig and Id , respectively. The angular velocity vectors of the gimbal frame and the flying wheel are defined as c˙ g , -g with respect to the gimbal frame orientation, respectively. The angular velocity vector of the spacecraft with respect to the body frame can be derived from the gimbal frame coordinate system such that x = Cxg

(1)

where xg is the angular vector based on the gimbal frame coordinate, and C denotes the associated direction cosine matrix. The total angular momentum of the rigid spacecraft is then represented by h = hs + hg + hd

(2)

where hs corresponds to the angular momentum vector of the spacecraft, and hg , hd represent the angular momentum vectors of the gimbal frame and the flying wheel, respectively. Equations of motion of the spacecraft system from Euler’s equation are described as h˙ = ue

(3) Gimbal frame . γg

Spacecraft body

ϖg

x

Spinning wheel z

Inertial frame

y

Fig. 1. Coordinate definition of spacecraft with one VSCMG.

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where the vector ue is the sum of external torques experienced by the spacecraft. The total time derivative of the spacecraft angular momentum vector is given by [21] h˙ = J x ˙ + x× J x ˙ g + h× ˙ g) − C(h× t c  xg − It c¨ g − Id -

(4)

where the skew-symmetric matrix is defined as ⎡ 0 −x x ⎤ 3

x× ≡ ⎣ x 3 −x2

0

2

−x1 ⎦

(5)

the multiple CMGs, respectively. c, - ∈ Rn are vectors composed of the gimbal angles and the wheel speeds, respectively. hw ∈ Rn is a series of the angular momentum of the flying wheels with respect to the gimbal frame coordinate. Hw ∈ Rn×n is a diagonal matrix formed by the angular momentum of the flying wheels. Finally, (x, ue ) denotes the uncertain term including external disturbances which is regulated by the momentum of inertia of the spacecraft. The time derivatives of the unit vectors of the gimbal frame orientation are expressed as C˙ w = C G

(10)

and J is the total inertia matrix with respect to the spacecraft body frame. In addition,

C˙  = −Cw G

(11)

It = I g + Id

(6)

ht = It xg + It c˙ g + Id -g

where G =diag[˙1 ˙ 2 · · · ˙ n ] and i represents the gimbal angle of i-th CMG.

(7)

h = It c˙ g + Id -g

(8)

x1

0

where It is the inertia matrix including the gimbal frame and the flying wheel with respect to the gimbal frame coordinate. 2.1. Simplification of spacecraft model The momentum of a rigid body with multiple CMGs can be expressed as summation of several CMG momenta. For this reason, equations of motion of a rigid spacecraft with several CMGs are readily obtained using Eq. (4). Designing a control law with the whole governing equations can make control law complex. Fortunately, the governing equation of motion, in Eq. (4) embodies some negligible terms. The angular momentum produced by c˙ g and x exerted to the inertias of gimbal frames and flying wheels are negligible such that ht = h ≈ Id -g . The multiplication of CMGs inertia and c¨ g is negligible, because both terms are small enough such that It c¨ g ≈ 0[21]. It would not generate enough level of torque to influence the spacecraft attitude. One can easily design a control law by disregarding such small quantities. Arrangement based on the combination of the unit vectors of the gimbal frame orientation is useful for designing a control logic. Therefore, the simplified spacecraft dynamics with n CMGs is found to be [21,22] Jx ˙ + x× J x + x× Cw hw = −C Hw c˙ + J (x, ue ) = u + J (x, ue )

(9)

where Cw (c), C (c) ∈ R3×n are matrices spanned by the unit vectors of spinning axis and transverse axis of

2.2. Attitude kinematics Quaternion sometimes called Euler parameters is defined in terms of the principal rotation components. The kinematic differential equations for the quaternion are given by [18] q˙ = − 21 x× q + 21 q4 x

(12)

q˙4 = − 21 xT q

(13)

Note that the vector denoted as q = [q1 q2 q3 ]T is called the quaternion vector. 3. Nonlinear adaptive controller design 3.1. Normalized input neural networks (NINNs) The normalization is a linear scale conversion process that assigns the same absolute value to the corresponding relative variation. The effect of input data pre-treatment prior to the neural network training is demonstrated by a systematic analysis [19,20]. The importance of the input data normalization is emphasized due to several advantages for function approximation. One advantage is that the estimation error can be reduced. Another merit lies in the computation time reduced in the order of magnitude for the training process. This approach provides also an improved capability in discriminating high-risk software [23]. A general three-layered neural network architecture illustrated in Fig. 2 consists of a large number of parallel interconnections of neural processors. The neural network characterized by the output vector (y ∈ Rl )

H. Leeghim et al. / Acta Astronautica 64 (2009) 778 – 786

1 1

y x

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more neurons would better approximate the function, but tend to require more learning time. That is, careful consideration on the significance of the nonlinearity should be investigated to satisfy the assumptions. Significant assumptions and several properties should be investigated to utilize the NINN effectively, and to rigorously prove the stability of an NINN-based control law. Assumption 1. There are ultimately converged ideal weighting matrices at the end of the learning process such that following bounds hold [6,9,11,14]:

Fig. 2. A three-layered neural network architecture.

V  V¯ , W  W¯

(18)

and the input vector (v ∈ Rn+1 ) is mathematically described as

where V¯ and W¯ are the upper bounds.

y = W T r(V T v)

Assumption 2. The NINN is utilized to compensate for system uncertainties. The nonlinear function can be expressed as

(14)

where r(·) denotes an activation function vector defined as r(m) = [1 (1 ) · · · (m )] ∈ R T

m+1

(15)

and i denotes the i-th element of the input vector,  is the activation function, and the two weighting matrices are represented as ⎡ ⎤ bv1 . . . bvm ⎢v ⎥ ⎢ 11 . . . v1m ⎥ ⎢ ⎥, V =⎢ . . . . ... ⎥ ⎣ .. ⎦ vn1 ⎡ bw1 ⎢w ⎢ 11 W =⎢ ⎢ .. ⎣ . wm1

. . . vnm

(19)

where  represents the function reconstruction error. About a constant real number ¯ > 0, the residual norm of y is within ¯ range of the neural network [6,9,11,14]. For this reason, the representation holds with  < ¯. Next, Taylor series expansion of the activation function vector about Vˆ T z, represented as the estimate of V T z, yields ˆ V˜ T z + O(V˜ T z)2 r = rˆ + 

⎤

. . . bwl . . . w1l ⎥ ⎥ .. ⎥ .. ⎥ . . ⎦ . . . wml

y = W T r(V T z) + 

(16)

The bias terms (bvi , bwi ) are absorbed into V and W for notational compactness, respectively. The input vector is defined as v=[1 xT ]T , and x is the variables of the function to be estimated. The NINN can be readily implemented by defining the normalized input vector as v (17) z=s v where s is a positive scaling constant. Note that the norm of v is always nonzero due to the bias input, and the magnitude of the normalized vector is equal to s. A suitable choice of neurons is essential to construct a nonlinear function with converged weighting matrices. The issue of how many neurons are needed to approximate a function is out of scope of this study. Generally,

(20)

where O(V˜ T z)2 denotes terms of order two, and () is Jacobian matrix in terms of the derivatives of the activation vector such that ⎡ 0 ... 0 ⎤ ⎢ j(1 ) . . . ⎢ j1 (m) = ⎢ ⎢ .. .. ⎣ . . 0

...

0 .. .

⎥ ⎥ ⎥ ⎥ ⎦

(21)

j(m ) jm

and the differences of the weighting matrices are represented as V˜ = V − Vˆ , W˜ = W − Wˆ

(22)

Some abbreviations such as Rˆ = R(Vˆ T z), rˆ = r(Vˆ T z), and r = r(V T z) are used for notational simplicity, and Wˆ represents the estimate of W. Lemma 1. For sigmoid, RBF, and tanh functions as the activation functions of the NINN, higher order terms in

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where d = d2 + ¯ is an additional positive constant. Corollary 1 is used to determine the boundedness of d.

Taylor series are bounded by [20] O(V˜ T z)2  c1 + c2 sV˜ 

(23) 3.2. Neural network quaternion feedback law

where c1 , c2 are positive constants. Proof. From Eq. (20) and some norm inequality with the fact that the activation function and associated derivatives are bounded by constants, higher order terms are also bounded by ˆ V˜ T z O(V˜ T z)2  = (r − r) ˆ − ˆ V˜ T z  (r − r) ˆ +  = c1 + c2 V˜ z

At first, a theorem is presented to show that q, x → 0 as t → ∞ under a feedback attitude control command. Theorem 1. Let us consider a sliding surface in the state space q, x defined as e = x + q

(24)

Finally, Lemma 1 is derived by substituting the norm property in Eq. (17) into Eq. (24). The high order terms of the activation functions—sigmoidal, RBF, tanh functions—based on the NINN are simply bounded, since the activation function vector and the time derivative matrix of activation function vector are bounded. It is a result different from that of previous studies [9,11,12,16]. Consequently, this property could lead to a simple condition for the tracking error representation.  Corollary 1. By using the first assumption and Lemma 1, the necessary conditions for the stability analysis are given by [20]

(29)

where is a positive constant. If e approaches zero, then lim q = 0,

t→∞

lim x = 0

t→∞

(30)

This theorem is given by Cristi et al. [24]. Generally, the gyroscopic terms in Eq. (4) are not significant for a majority of practical small angle attitude maneuvers of a spacecraft. However, by simply eliminating those gyroscopic terms, undesirable phase during the large angle maneuver can be prevented. A nonlinear adaptive quaternion feedback law using the NINN to regulate the system in Eq. (9) is described as u = −K e + x× (J x + Cw hw ) − J uad

(31)

ˆ T z d1 sW˜  W˜ T V

(25)

where the control input u is defined by −C Hw c˙ . The controller gain K is a positive definite matrix, and the adaptive input is given by

W T O(V˜ T z)2  d2 + d3 sV˜ 

(26)

uad = Wˆ T rˆ

where di are computable positive constants. Corollary 2. Consider a nonlinear function vector y to be estimated. The approximated function vector based on the NINN is denoted as yˆ . By using Eqs. (20), (22), the difference between the two functions is expressed as [20] y − yˆ = W T r +  − Wˆ T rˆ ˆ V˜ T z + d = W˜ T (rˆ − ˆ Vˆ T z) + Wˆ T 

(27)

where

It can be shown that the last term in the function difference is bounded by d d + d1 sW˜  + d3 sV˜ 

To approximate and eliminate the spacecraft unknown components, the weighting matrices for the adaptive output are updated by learning rules defined as ˆ − k L Vˆ V˙ˆ = LzeT J Wˆ T 

(33)

˙ˆ = M(rˆ −  ˆ Vˆ T z)eT J − k M Wˆ W

(34)

where L, M are symmetric gain matrices, and k is referred to as learning rate. The closed-loop error dynamics with the control input is expressed as J e˙ + K e = J ( − uad )

ˆ T z + W T O(V˜ T z)2 +  d = W˜ T V

(28)

(32)

(35)

Note that the second term of the uncertainty can be directly cancelled out by an explicit control input. Nevertheless, the task is given to the NINN for the compact control law by regarding the term as uncertainty. Furthermore, the gyroscopic term in the governing equation

H. Leeghim et al. / Acta Astronautica 64 (2009) 778 – 786

783

can be regarded as internal disturbances. Motivated by such analysis, an adaptive feedback law can be simply defined as

Let us introduce some useful notations, for simplification, such as

u = −K e − J uad

 = max[J d, k W¯ , k V¯ ]

(36)

Zero matrices as initial values of Wˆ , Vˆ are in general enough to update the weighting matrices of the neural network. 3.3. Stability analysis Let us examine a candidate Lyapunov function defined as VL = 21 eT J e+ 21 tr(W˜ T M −1 W˜ )+ 21 tr(V˜ T L −1 V˜ )

(37)

Note that the uncertain term in Eq. (35) can be replaced with the neural network by Assumption 2 such that  = W T r + . By using Eqs. (33)– (35), the time derivative of VL is derived as V˙ L = − eT K e ˆ Vˆ T z) + Wˆ T  ˆ V˜ T z + d} + eT J {W˜ T (rˆ −  ˆ Vˆ T z)eT J + k W˜ T Wˆ } + tr{−W˜ T (rˆ −  ˆ + k V˜ T Vˆ } + tr{−V˜ T zeT J Wˆ T 

(38)

where the difference between the output from the neural network and the model uncertainty is replaced with Eq. (27). To manipulate the Lyapunov function, some trace equalities are utilized: tr( AB) = tr(B A) when the matrices have compatible dimensions, and aT b = tr(aT b) = tr(abT ) where the vectors are in the same dimension. Therefore, the time derivative of VL can be further simplified as V˙ L = −eT K e + eT J d + tr(k W˜ T Wˆ )+tr(k V˜ T Vˆ )

(39)

The second term except d is removed identically by the trace terms. By replacing with Eqs. (22), (28), and applying the upper bounds of the weighting matrices in Eq. (18), the time derivative of VL becomes V˙ L  − min (K )e2 + J e(d + d1 sW˜  + d3 sV˜ ) − k(W˜ 2 + V˜ 2 ) + k W¯ W˜  + k V¯ V˜ 

(40)

where min (K ) is the minimal eigenvalue of K . For further analysis, the above equation can be rewritten as V˙ L  − min (K )e2 + J e(d1 sW˜  + d3 sV˜ ) − k(W˜ 2 + V˜ 2 ) + J de + k W¯ W˜  + k V¯ V˜ 

(41)

20 = max[d1 , d3 ] n˜ = [V˜  W˜ ]T Therefore, Eq. (41) results in ˜ V˙ L  − min (K )e2 + 2J 0 sen ˜ 2 + (e + n) ˜ − 21 kn  T   e

min (K ) −J 0 s e = − ˜ ˜ n −J 0 s 21 k n ˜ + (e + n)

(42)

Note that a norm property such that 1 a a  2 a is utilized, where i (i = 1, 2) are positive constants, and the subscripts denote different metric systems, respectively. Once again, one can redefine a matrix and a vector as 

min (K ) −J 0 s ˜ T Q= , g˜ = [e n] −J 0 s 21 k The symmetric matrix (Q) can be a positive definite matrix under the following condition: √ (43) ( min (K )21 k) > J 0 s The constraint is possible by a proper selection of K , k and s. Consequently, Eq. (42) is simplified as V˙ L  − min (Q)˜g2 + ˜g and the following condition  ∀˜g >

min (Q)

(44)

(45)

renders V˙ L < 0 outside the compact set. According to the Lyapunov stability theory, this result verifies the uniform ultimate boundedness of ˜g [9]. The adaptive control theories, based upon the neural network studied during the past several years, in general need boundedness of the desired trajectories and their derivatives [9,11,12,16]. This assumption is directly related to the tracking error boundedness. The larger boundedness of the desired trajectory produces the larger tracking and weighting errors. Of course, the trajectory is bounded for all t 0 in practical applications. However, the assumption may be unusual when designing a controller and proving the stability for most control theories. The adaptive nonlinear attitude control laws based on feedback linearization sometimes suffer from the

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singularity problem. As mentioned in the Introduction, the singularity arises when the denominator of the control law is zero. In this paper, to avoid singularity, the feedback linearization technique is not considered, instead the well-known quaternion feedback law is chosen as the baseline controller. Consequently, the suggested neural network-based quaternion feedback law in Eqs. (31) is far from the singularity problem. It is a salient feature from the previous studies. Of course, the chattering problem is also resolved. Next, for the purpose of selecting proper gain matrices for the sliding surface, the approximated secondorder system is given by [18] J h¨ + K h˙ + K h = 0

(46)

where h is the small Euler’s angle vector. The damping ratio and the natural frequency n satisfy the following relationship: K = 2 n J, =

n 2

(47)

Accordingly, proper selection of , n naturally defines the sliding surface gains, and also guarantees the stability of the system. A preferred nominal level of the weighting matrices can be chosen as L=

1 In+1 , J 

M=

1 Im+1 J 

(48)

Finally, zero matrices for initial values are in general enough to update the weighting matrices (Wˆ , Vˆ ) of neural network.

Table 2 Numerical simulation parameters of the NINN. Symbol

Value

Number of hidden neurons Number of input neurons k s L M

5 6 1.0 3.0 × 10−3 2.0 × 10−3 In+1 2.0 × 10−3 Im+1

a 0.50 q1 q2 q3

0.25 0.00 0 50 100 150 Classical quaternion feedback law

200

250

300

200

250

300

b

0.50 0.25 0.00 0

50

100

150 Time(sec) Neural quaternion feedback law

Fig. 3. Comparison of the error quaternion.

4. Simulation study In this section, simple performance comparisons between the proposed adaptive neural network-based quaternion feedback controller and the classical quaternion feedback control law are made through numerical simulation under realistic environment. The simulation environment for the spacecraft system is given in Table 1. A pyramid-type CMG cluster widely studied Table 1 Spacecraft properties for numerical simulations. Symbol

Value

J

diag[100, 200, 300] kg m2 [20, 20, 20, 20]T kg m2 /s 0.707 0.2 0.6 rad/s

hw

n Maximum gimbal rate

in previous studies is employed and 3◦ errors in each alignment axis are added as system uncertainty. The maximum gimbal rate of the CMGs is limited to account for the realistic constraint. In order to illustrate the performance of the neural network-based quaternion feedback laws in Eqs. (31), (36), sinusoidal external disturbances and about 10% severe model uncertainties are randomly added to the numerical simulation. The design parameters of the NINN for the adaptive controller are also listed in Table 2. The output of the NINN limited to an absolute value of 1.0 is used for this simulation. The attitude histories of the spacecraft controlled by the quaternion feedback law and proposed neural adaptive control law are compared. The attitude history by the general quaternion feedback controller is illustrated in Fig. 3(a) under the external disturbances. The attitude history by the proposed neural controller plotted

H. Leeghim et al. / Acta Astronautica 64 (2009) 778 – 786

a 0.2

200

250

300

0.00 -0.05 -0.10

0 50 100 Gimbal angular rate of CMGs

0.0

b

-0.8 50

100

150

200

250

200

250

300

150 200 Time(sec)

250

300

25

0.3

0 -25

0.0

-0.3 Yaw 0

150

50

300

(deg)

Torque(Nm)

150

0.8

Pitch 0

c

100

-50

50

100

150 Time(sec)

200

250

0 50 100 Gimbal angle of CMGs

300

Fig. 4. Uncertainty estimation using neural networks.

in Fig. 3(b) maintains quite a small error in spite of the disturbances. As discussed in the previous section, it is shown from the simulation result that the suggested approach is absolutely free of the singularity and chattering problem. The improvement of the attitude response is owing to the application of the NINN. Plots of the neural network output for uncertainty estimation are shown in Fig. 4. The disturbances/uncertainties of the system are estimated by the NINN as plotted in Fig. 4, and then removed by opposite torque produced by the neural network. The initial neural network’s output is shown to oscillate before the learning process converges. However, the output history produces quite reliable estimation result after the completion of the learning process. Next, the gimbal angle and angular velocity of CMGs are presented to demonstrate the proposed method for a possibility of actual implementation. Firstly, the time history of the CMGs commanded by general quaternion feedback law is displayed in Fig. 5. One can see that the overall gimbal angle and angular rate are in reasonable range to produce control torque. Response of the CMGs controlled by the proposed method are also displayed in Fig. 6. Although the initial history looks undesirable for CMGs to generate torque due to rapid angular rate command, the overall plots are acceptable to construct more control torque for cancelling out uncertainties. Consequently, it results in no chattering as well.

Fig. 5. Time history of the CMGs controlled by quaternion feedback law.

a

0.10

CMG #1 CMG #2 CMG #3 CMG #4

0.05

(rad/s)

Torque(Nm)

b

50

CMG #1 CMG #2 CMG #3 CMG #4

0.05

(rad/s)

0.0

-0.2 Roll 0

0.10

Actual disturbance Estimate by neural networks

0.00 -0.05 -0.10

0 50 100 Gimbal angular rate of CMGs

b

150

200

250

300

150 Time(sec)

200

250

300

50 25

(deg)

Torque(Nm)

a

785

0 -25 -50

0 50 100 Gimbal angle of CMGs

Fig. 6. Time history of the CMGs controlled by proposed method.

5. Conclusions An adaptive control design approach using NINNs for spacecrafts with uncertainty under external disturbances was demonstrated with simulation study. It can be a reliable method by proving that the uniform ultimate boundedness in tracking error is guaranteed. More importantly, the proposed neural quaternion feedback law is free of chattering and singularity problem.

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Furthermore, the suggested law can be regarded reasonable, as it relies on the technically proven quaternion feedback law as a baseline controller. The numerically stable NINN to reject uncertainty and external disturbance were employed. Moreover, it is relatively in a simple form compared with some previous nonlinear adaptive control laws for spacecraft, which have very complex structure and been suffering from singularity or chattering problem as well. Acknowledgements The authors acknowledge several anonymous reviewers for their suggestions on improving this paper, in particular one reviewer who provided many useful and constructive suggestions. This research was supported by NSL (National Space Lab, S1080100012308A0100-12310) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology. References [1] A.Y. Lee, J.W. Yu, P.B. Kahn, R.L. Stoller, Space interferometry mission spacecraft pointing error budgets, IEEE Transactions on Aerospace and Electronic Systems 38 (2) (2002) 502–852. [2] J.J.E. Slotine, M.D. DiBenedetto, Hamiltonian adaptive control of spacecraft, IEEE Transactions on Automatic Control 35 (7) (1990) 848–852. [3] J.L. Junkins, M.R. Akella, R.D. Robinett, Nonlinear adaptive control of spacecraft maneuvers, Journal of Guidance, Control, and Dynamics 20 (6) (1997) 1104–1110. [4] J.J. Sheen, R.H. Bishop, Adaptive nonlinear control of spacecraft, Journal of the Astronautical Sciences 42 (4) (1994) 451–472. [5] S.J. Paynter, R.H. Bishop, Adaptive nonlinear attitude control and momentum management of spacecraft, Journal of Guidance, Control, and Dynamics 20 (5) (1997) 1025–1032. [6] I.-H. Seo, H. Leeghim, H. Bang, Nonlinear momentum transfer control of a gyrostat with a discrete damper using neural networks, Acta Astronautica 62 (2008) 357–373. [7] K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (5) (1989) 359–366. [8] W.Li. Slotine, Neural network control of unknown nonlinear controller, in: Proceedings of American Control Conference, vol. 2, Pittsburgh, PA, 1989, pp. 1136–1141.

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