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Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties
Chinese Journal of Aeronautics, (2019), 32(3): 674–687
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties Qinglei HU a,*, Li XIAO b, Chenliang WANG a a b
School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China Sino-French Engineer School, Beihang University, Beijing 100191, China
Received 5 January 2018; revised 2 July 2018; accepted 7 September 2018 Available online 5 January 2019
KEYWORDS Adaptive control; Attitude tracking; Changing center of mass; Fault-tolerant; Spacecraft
Abstract This paper is devoted to adaptive attitude tracking control for rigid spacecraft in the presence of parametric uncertainties, actuator faults and external disturbance. Specifically, a dynamic model is established based on one-tank spacecraft, which explicitly takes into account changing Center of Mass (CM). Then, a control scheme is proposed to achieve attitude tracking. Benefiting from explicitly considering the changing CM during the controller design process, the proposed scheme possesses good robustness to parametric uncertainties with less fuel consumption. Moreover, a fault-tolerant control algorithm is proposed to accommodate actuator faults with no need of knowing the actuators’ fault information. Lyapunov-based analysis is provided and the closed-loop system stability is rigorously proved. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed controllers. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The computation method of unsteady transonic flow based on N-S equations should be best accurate, but to threedimensional complex problems, it can be achieved only on large computers, and moreover, the results are not ideal sometimes.1 A viscous/inviscid interaction method is an applicable one and the computation time can be reduced by two orders. * Corresponding author. E-mail address: [email protected] (Q. HU). Peer review under responsibility of Editorial Committee of CJA.
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Attitude tracking is a fundamental mission that most spacecraft must perform during their operations and this issue has been extensively studied in the existing literatures.1–3 In particular, adaptive control techniques have been largely applied to addressing the problem of unknown or uncertain system parameters.4–7 However, the majority of available adaptive attitude tracking control solutions do not consider the changing Center of Mass (CM) caused by fuel consumption. The fuel often provides a significant contribution to the overall mass of the spacecraft to execute station keeping and attitude maneuvering. Therefore, the effect of fuel consumption is important for the determination of CM for the spacecraft. And the changing CM can result in variant body-fixed coordinate frame and time-varying inertia profile, which may degrade the control accuracy or even destroy the system stability. Moreover, fault-tolerant capacity is also required for
https://doi.org/10.1016/j.cja.2018.12.015 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties spacecraft because actuator faults are frequently encountered in practical spacecraft operations. All the above issues create considerable difficulty in the design of attitude control algorithms for achieving accurate, reliable and fast responses, especially when these issues are treated simultaneously. The effect of changing mass properties for spacecraft has been addressed in some of the existing literatures. The interactions among the fuel slosh, the attitude dynamics and the flexible appendages have been studied in Ref.8 via a classical multi-body approach. Jing et al.9 proposed a switching attitude control law by considering the astronomy observation satellite with the sensor of non-negligible mass and length. A robust controller has been designed in Ref.10 to deal with the position changes of CM due to fuel consumption for the micro-satellite. Deng and Yue11 focused on the nonlinearly modelling and attitude dynamics of spacecraft coupled with large amplitude liquid sloshing dynamics and flexible appendage vibration and designed a feedback controller for the attitude maneuvering problem. Another key issue related to spacecraft attitude control involves actuator faults. In practice, actuator faults are frequently encountered due to the harsh working environment. The occurrence of actuator faults might result in mission failure and even system instability. Thus, extensive fault-tolerant control schemes have been carried out for the spacecraft attitude control in the previous literatures. Fuzzy logic has been applied12–14 to design fault-tolerant controllers of the spacecraft. Xiao et al.15 proposed an SMC-based fault-tolerant attitude-tracking control approach with the capability of finite-time convergence with zero tracking error. A faulttolerant control algorithm relying on an ideal reference model has been presented.16 Based on variable structure control, a fault-tolerant tracking control scheme without any requirement of fault identification has been developed.17 Shen et al. proposed a fault-tolerant control law using the control allocation approach to deal with the actuator faults.18 In Ref.19, the integral-type sliding mode control has been employed in the design of the fault-tolerant control algorithm against actuator stuck faults for spacecraft attitude maneuvering. Although significant studies have been conducted to deal with the changing mass properties and the actuator faults, to the best of our knowledge, few of them have treated the changing CM (caused by the mass variation) and actuator faults simultaneously. However, these problems are very likely to happen simultaneously in practical situations, which may degrade the control performance or even result in mission failure. Therefore, it is crucial to take these problems into consideration during the controller design process. In this paper, an adaptive fault-tolerant control scheme is developed to address the spacecraft attitude tracking problem under inertia uncertainty, actuator faults and external disturbance. Based on a one-tank spacecraft, an explicit dynamic model of the timevarying inertia matrix is established, and it is taken into account both in the dynamic equations of the spacecraft and controller design process. The main contribution of this work includes two parts: (A) characterize the time-varying inertia dynamic model with consideration of changing CM to construct the adaptive attitude tracking controller. Hence, the time-varying inertial parameters will not introduce important influence on the control performance with the proposed control scheme; (B) propose an adaptive fault-tolerant control algorithm which can achieve high-precision attitude tracking, despite the presence of time-varying inertia properties, external
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disturbances and actuator faults (including total failures). The system stability is also rigorously analyzed through Lyapunov analysis. The rest of the paper is organized as follows. The mathematical model of a spacecraft with time-varying inertia matrix and the characterization of the time-varying inertia matrix are derived in Section 2. The next section presents the design procedure and stability analysis of the control algorithm. Numerical simulations are provided in Section 4 to demonstrate the performance of the proposed control scheme. Finally, the paper is completed with some concluding comments. Notation. Through this article, Rn is the n-dimensional Euclidean space, and Rnm represents the set of all real matrices of dimension n m. The skew-symmetric matrix operator SðÞ represents the vector cross-product operation between any two vectors, defined by SðaÞb ¼ a b for any a, b 2 R3 . MT is the transpose of M. diagðx1 ; x2 ; . . . ; xn Þ denotes the diagonal matrix of dimension n n with diagonal elements x1 ; x2 ; :::; xn . kmin ðMÞ and kmax ðMÞ are the maximum and minimum eigenvalues of M. jj and k k stand for the usual L1 and L2 norms. 2. Problem statement and preliminaries 2.1. Spacecraft dynamics The governing equations for attitude motion of a spacecraft are formulated in terms of the unit-quaternion representation. The attitude kinematics are given by20: ( q_ 0 ¼ 12 qTv x ð1Þ q_ v ¼ 12 ðq0 I þ Sðqv ÞÞx T
where the unit-quaternion q ¼ ½q0 ; qTv 2 R4 (q0 ðtÞ 2 R and qv ðtÞ 2 R3 are the scalar and vector parts of q respectively) denotes the attitude orientation of the body-fixed frame B with respect to the inertial frame N, and satisfies the constraint qT q ¼ 1; x 2 R3 denotes the inertial angular velocity of the spacecraft expressed in B; I is the 3 3 identity matrix. The corresponding attitude dynamics are written as6 _ _ JðtÞxðtÞ ¼ JðtÞxðtÞ SðxÞJðtÞxðtÞ þ sðtÞ þ dðtÞ
ð2Þ
is the time-varying inertia matrix of the where JðtÞ 2 R spacecraft, and remains symmetric positive-definite for all time; sðtÞ 2 R3 is the control torque input generated by the torque actuators; dðtÞ 2 R3 is the external disturbance. The direction cosine matrix, denoting a transformation from N to B, can be calculated by 33
CðqÞ ¼ ðq20 qTv qv ÞI þ 2qv qTv 2q0 Sðqv Þ Since the desired reference angular velocity is specified in its own reference frame R, the rotation matrix from R to B can be obtained as Cðqe Þ ¼ CðqÞðCðqr ÞÞT
ð3Þ
T ½qe0 ; qTev
where qe ¼ denotes the attitude error quaternion describing the relative attitude of the spacecraft with respect to the reference attitude qr . Then, the angular velocity tracking error is expressed as xe ¼ x Cðqe Þxr
ð4Þ
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where xr 2 R3 represents the bounded reference angular velocity. By differentiating Eqs. (3) and (4) with respect to time, the open-loop attitude-tracking error dynamics are written as ( q_ e0 ¼ 12 qTev xe ð5Þ q_ ev ¼ 12 ðqe0 I þ Sðqev ÞÞxe _ SðxÞJx þ s þ d þ J/ Jx_ e ¼ Jx
ð6Þ
where / ¼ Sðxe ÞCðqe Þxr Cðqe Þx_ r and we note hereafter Q ¼ 0:5ðqe0 I þ Sðqev ÞÞ. The control objective of this paper is to design an adaptive fault-tolerant attitude tracking controller for the spacecraft subject to time-varying inertia matrix, external disturbances and actuator faults so that all closed-loop signals are uniformly bounded and that the attitude tracking errors converge to a small set around the origin. To this end, the following assumption is made13,15,18: Assumption 1. The external disturbance d is bounded by k d k 6 dm , where dm > 0 is an unknown constant.
mðtÞ mð0Þ rf m0 þ mðtÞ
ð8Þ
Define the time-varying inertia matrix as JðtÞ ¼ diagðJ11 ; J22 ; J33 Þ;
Jð0Þ ¼ diagðJ011 ; J022 ; J033 Þ
where Jð0Þ is the inertia matrix at t ¼ 0, and J011 ; J022 and J033 are unknown constants. Using the parallel axis theorem to account for the changing CM location, and considering the fuel consumption, the time-varying inertial matrix at time t can be expressed as21 8 > < J11 ¼ J011 J22 ¼ J022 ðmð0Þ mðtÞÞðr2f þ rðtÞ2 Þ > : J33 ¼ J033 ðmð0Þ mðtÞÞðr2f þ rðtÞ2 Þ
JðtÞ ¼ J0 þ J1 WðtÞ
Generally speaking, our system consists of a rigid spacecraft body with CM initially located at point O (the initial CM, remains fixed to the spacecraft body) and a fuel tank that is offset from point O by a distance of rf in the x-direction. As shown in Fig. 1, the body-fixed frame B remains fixed at the CM and rotates with the spacecraft. The origin Ob is located at the CM, the x-axis, xb , is directed from Ob to the fuel tank, the z-axis, zb , is the initial spin axis, and the y-axis, yb , completes the body-fixed coordinate system. In this paper, both the fuel tank and the mass-invariant part are modeled as mass points, using equation Pthe solving P for CM of a mass point system rr ¼ i mi r= i mi , where rr is the position vector of the system’s CM relative to a fixedpoint; mi and ri are the mass and position vector relative to the same fixed-point of the ith mass point. Then, we can deduce the position vector rðtÞ of the instantaneous CM relative to O as m0 rs þ mðtÞrf m0 þ mðtÞ
rðtÞ ¼
ð9Þ
Then, expanding and rearranging Eq. (9), the inertia matrix can be expressed as
2.2. Characterization of time-varying inertia matrix with changing CM
rðtÞ ¼
vector of the fuel tank relative to O. By definition that rðtÞ has an initial value of zero, rð0Þ ¼ 0, we can obtain rs ¼ mð0Þrf =m0 . Applying this result to Eq. (7), the CM vector can be found by
ð7Þ
where m0 is the total mass of all the mass-invariant parts, rs is the position vector of m0 relative to O, mðtÞ denotes the mass of the remaining fuel at time t, and rf represents the position
ð10Þ
where J0 ¼ diagðJ011 ; J022 mð0Þr2f ; J033 mð0Þr2f Þ is a constant matrix with unknown parameters J0ii ; i ¼ 1; 2; 3;
WðtÞ ¼ mðtÞðr2f þ rðtÞ2 Þ mð0ÞrðtÞ2
is a known and time-dependent matrix; J1 ¼ diagð0; 1; 1Þ. By differentiating Eq. (10) with respect to time, we thus have _ ¼ J1 WðtÞ _ JðtÞ
ð11Þ
The rate of change of the fuel mass is known to be a function of the applied control and actuator hardware characteristics, so that we use the mass relation6 _ mðtÞ ¼ ck u k
ð12Þ
where c > 0 is a known constant. Then, by taking the time derivative of WðtÞ and applying Eq. (12), it follows that " 2 # mðtÞ mð0Þ 2 _ W ¼ ck u krf 1 þ ð1 þ 2mi Þ ð13Þ m0 þ mðtÞ where mi ¼ mð0Þ þ m0 is the initial mass of the spacecraft without any fuel consumption. Remark 1. For the characterization of the inertia matrix, we not only consider the direct influence of the fuel consumption, but also explicitly take into account the impact of the changing CM caused by the fuel consumption. Thus, a more accurate spacecraft dynamic model is obtained. 3. Adaptive controller design
Fig. 1
Body-fixed coordinate frame.
In this section, an adaptive attitude tracking control strategy is presented to solve the current problem under study. We start
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties by synthesizing a basic adaptive controller for the case in which all the actuators work healthily to help illustrate the key idea of accounting for the time-varying inertia properties. The result is then extended to an adaptive fault-tolerant control synthesis. 3.1. Basic adaptive controller design Substituting Eqs. (10) and (11) into Eq. (6), the angular velocity error dynamics can be rewritten as _ þ W1 h þ H1 þ s þ d Jx_ e ¼ J1 Wx
ð14Þ
where H1 ¼ SðxÞ J1 Wx þ J1 W/, and the regressor matrix W1 is constructed in the following fashion:
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_ 1 _ 1 _ 1 ^ V_ ¼ sT J_s þ sT Js h dm d^m þ bxTe qev h 2 g1 g2 _ þ W2 h þ H2 þ s þ d bqev Þ ¼ sT ðJ1 WX _ 1 _ 1 h dm d^m h^ g1 g2 _ þ H2 þ sT s 6 k W2 khk s k þ dm k s k J1 WX 1 ^ _ 1 _ h dm d^m h b2 qTev qev g1 g2 ^ s k þ d^m k s k J1 WX _ þ H2 þsT s ¼ k W2 khk _ 1 1 _ h þ dm k s k d^m b2 qTev qev þ h g1 k W2 kk s k ^ g1 g2 ^ T T ^ ¼ k W2 khk s k þ dm k s k kðtÞs s k1 s s b2 qT q þ bxTe qev
ev ev
h^ þ c1 h þ c2 dm d^m
W1 h ¼ SðxÞ J0 x þ J0 /
ð21Þ
with h ¼ ½J011 ; J022 ; J033 T containing the three elements of J0 . To develop the control scheme, we define the filtered error variable
To proceed, the following key inequality is introduced; specifically, for any scalar e > 0 and x 2 R, we have
s ¼ xe þ bqev
jxj2 jxj2
ð15Þ
where b > 0 is a design parameter. Then, we design the adaptive controller and the accompanying adaptive law as _ ðk1 þ kðtÞÞs s ¼ H2 þ J1 WX
ð16Þ
with
By applying Eq. (22), one can deduce that kW2 k2 ^h ksk2 ^ ffi þe k W2 k hk s k 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
kW2 k2 ^h ksk2 þ e2
d^m k W2 k2 ^h kðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 d^m k s k2 þ e2 k W2 k2 ^h k s k2 þ e2
ð17Þ
^_ ¼ g c ^h k W kk s k h 2 1 1
ð18Þ
_ d^m ¼ g2 c2 d^m k s k
ð19Þ
2
Theorem 1. Consider the attitude tracking error system described by Eqs. (5) and (6) with time-varying inertia matrix characterized by Eqs. (10)–(13). If the control law in Eq. (16), together with the adaptive laws in Eqs. (18) and (19), is applied, then the closed-loop system is globally stable in the sense that all the signals are continuous and bounded, and the tracking errors qev ðtÞ and xe ðtÞ converge to a small set containing the origin. Proof. Consider the following Lyapunov Function Candidate (LFC):
2
d^m ksk þ e2
Further, by completion of squares and using Eq. (23) in Eq. (21), it follows that4
where k1 > 0 is a control gain, X ¼ x ðs=2Þ, W2 is a known regressor matrix satisfying W2 h ¼ W1 h þ bJ0 Qxe , ^ ^ H2 ¼ H1 þ bðJ1 WQxe þ qev Þ, h and dm are the estimations of h (h ¼ k h k) and dm , respectively, and both gi and ci (i ¼ 1; 2) are positive design constants.
2 2 1 1 h þ 1 d V ¼ sT Js þ b½qTev qev þ ð1 qe0 Þ2 þ 2 2g1 2g2 m
ð23Þ
d^m ksk d^m k s k 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þe 2 2 2
2
ð22Þ
ð20Þ
where h ¼ h ^h and dm ¼ dm d^m are the estimation errors. Taking the time derivative of Eq. (20) and substituting Eqs. (5), (10), (11) and (14) into the resulting formula yield
c 2 c2 2 c1 2 c2 2 h dm þ h þ dm V_ 6 k1 sT s b2 qTev qev 1 2 2 2 2 þ 2e 6 aV þ b
ð24Þ
V_ 6 k1 sT s b2 qTev qev þ l1
ð25Þ
V_ 6 k1 sT s b2 qTev qev þ l2
ð26Þ
where a ¼ minf2k1 =kmax ðJÞ; g1 c1 ; g2 c2 ; bg > 0 2 =2Þ þ ðc d2 =2Þ þ 2e < 1 b ¼ bð1 q Þ2 þ ðc h e0
1
2 m
h =2Þ þ ðc2 dm2 =2Þ þ 2e < 1 l1 ¼ ðc1 2 h =4Þ þ ðc d 2 =4Þ þ 2e < b1 < 1 l ¼ ðc 2
2
1
2 m
Note that Eq. (24) implies that V 2 L1 . Then we conclude that and successively. And from Eqs. (25) and (26), one can observe that V_ < 0 when ðs; qev Þ is out of the set pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi B1 ,ððs; qev Þ : k s k 6 l1 =k1 ; k qev k 6 l1 =bÞ or the set pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi Since B2 ,fðs; qev Þ : k s k 6 l2 =k1 ; k qev k 6 l2 =bg. b1 > b2 , the set B1 encloses completely the set B2 . Therefore, once ðs; qev Þ is inside the set B1 , they cannot go out of it because they will be attached back to B2 , which means that the states are confined in the set B1 . Furthermore, the boundedness of ðs; qev Þ indicates that xe is also bounded because of
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Eq. (15). And its bound pffiffiffiffiffi B3 ,fxe : k xe k 6 l2 g.
can
be
determined
as
Remark 2. The time-varying inertia properties with changing CM are considered both in the construction of spacecraft dynamic model and the controller design process. Thus, the change of CM caused by fuel consumption during the spacecraft on-orbit attitude maneuvers will not have a large effect on the control performance under the proposed adaptive controller. This fact can help improve the attitude tracking efficiency and accuracy. 3.2. Adaptive fault-tolerant controller design It is common practice to equip the spacecraft with three or more than three torque actuators for robustness reasons. From a practical viewpoint, despite various efforts to enhance reliability of the components of the torque actuators, multiple anomalies do occur in them during their long operational life. Thus, it is imperative to develop fault-tolerant control algorithms that are capable of accommodating possible actuator faults for spacecraft attitude tracking control. In view of this, in this subsection, we shall present an adaptive fault-tolerant control architecture with a slight modification to the control scheme presented in Section 3.1. To describe actuator faults, the control input of dynamic system Eq. (6) is further expressed as s ¼ DðEu þ uÞ
ð27Þ
where D 2 R (n P 3) is the distribution matrix of the torque actuators; E ¼ diagðe1 ; e2 ; :::; en Þ 2 Rnn with 0 6 ei 6 1 (i ¼ 1; 2; :::; n) is the actuation effectiveness matrix that describes the health condition of the torque actuators; u 2 Rn and u 2 Rn denote the desired control input signals and the unexpected deviation faults, respectively. To proceed, the following assumptions are made. 3n
Assumption 2 15. The additive bias forces u is bounded, and there exists an unknown constant um > 0 such that k u k 6 um .
stant r such that 0 < r 6 kmin ðDEDT Þ holds. Let us define k :¼ 1=r, h ¼ kk h k and c ¼ kc. Choose the smooth control law and the accompanying adaptive laws as u ¼ DT ðk2 þ k ðtÞÞs
ð30Þ
^_ ¼ g k W1 kk s k c ^ h 1 1h
ð31Þ
^c_ ¼ g2 ðUk s k c2^c Þ
ð32Þ
^_ ¼ g k ud kk s k c ^ k 3 3k
ð33Þ
h , ^c and ^ k denote the estimawhere k2 > 0 is the control gain, ^ tions of h , c and k, respectively, and gi > 0 and ci > 0 (i ¼ 1; 2; 3) are design parameters. In Eq. (30), the timevarying gain k ðtÞ is given by 2
k W1 k2 ^ ðc^ UÞ2 h k ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðc^ UÞ2 k s k2 þ e2 k W1 k2 ^ h k s k2 þ e2 ^ k k ud k2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ^ k k ud k2 k s k2 þ e2 2
The main result of the paper is summarized as the following theorem. Theorem 2. Consider the spacecraft attitude dynamics as described by Eqs. (5), (6) and (27). By implementing the proposed control scheme as given in Eqs. (30)–(34), all the closed-loop signals are bounded and the tracking errors qev ðtÞ and xe ðtÞ converge to a small set containing the origin. Proof. Consider the following LFC:
V1 ¼ V0 þ
r 2 r T r 2 h þ c c þ k 2g1 2g2 2g3
Assumption 3 (4;5;18). Assume that the matrix DEDT remains positive definite for all fault scenarios that are under consideration, which indicates that there are at most n 3 actuators that totally fail to work. By applying Eq. (27), Eq. (14) can be rewritten as _ þ W1 h þ H1 þ dðEu þ uÞ þ d Jx_ e ¼ J1 Wx
ð28Þ
We further have 1_ J_s ¼ ud þ W1 h þ DEu þ d Js ð29Þ bqev 2 _ þ bqev and ud ¼ J1 Wxþ _ where d ¼ Du þ d þ Jbq_ ev þ 0:5Js H1 are specified here just for notational simplicity. Inspired by Ref.4, we thus deduce that k d k 6 cT U with T
c ¼ ½c1 ; c2 ; c3 T (ci P 0, i ¼ 1; 2; 3) and U ¼ ½1; k x k; k x k2 . According to Assumption 3, we infer that there exists a con-
ð34Þ
ð35Þ
^ are the estimation where h ¼ h ^ h , c ¼ c ^c , k ¼ k k 1 T T errors, and V0 ¼ 2 s Js þ b½qev qev þ ð1 qev Þ2 . Evaluating the time derivative of Eq. (35) along Eq. (29) and using similar process as in Section 3.1 yield V_ 1 6 r^ kk ud kk s k þ rk W1 k^ h k s k þ r^cT Uk s k r _ h þ sT DEu þ bxTe qev þ h g1 k W1 kk s k ^ g1 r r T þ c g2 Uk s k ^c_ þ g2 g3 _^ k g3 k ud kk s k k
ð36Þ
Similar to Section 3.1, by applying Eq. (22), we have 1 2 2 2 ^ k u k k s k k C B d r^ kk ud kk s k 6 r@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ eA 2 2 2 2 ^ k k ud k k s k þ e 0
ð37Þ
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties 0
2 2
1
C B k W1 k ^h k s k r^ h k W1 kk s k 6 r@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ eA 2 ^2 2 2 k W1 k h k s k þ e
0
2
ð38Þ
1 T
2
2
B ð^c UÞ k s k C r^cT Uk s k 6 r@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ eA 2 2 T 2 ð^c UÞ k s k þ e
ð39Þ
Inserting Eqs. (30)–(34) into Eq. (36) and noting Eqs. (37)– (39), one can easily obtain
T V_ 1 6 k2 rsT s b2 qTev qev þ rc1 h h^ þ rc2 c ^c þ rc3 k ^k þ 3re
ð40Þ Following the similar reasoning as that in the proof of Theorem 1, we further have rc 2 rc t V_ 1 6 k2 rsT s b2 qTev qev 1 h 2 c c 2 2 rc3 2 k þ b1 6 a1 V1 þ b1 2
ð41Þ
and where a1 ,minf2k1 r=kmax ðJÞ; g1 c1 ; g2 c2 ; g3 c3 ; bg 2 2 2 T b1 ¼ bð1 q0 Þ þ ðrc1 h =2Þ þ ðrc2 c c =2Þ þ ðrc3 k =2Þþ 3re. With a similar analysis of Theorem 1, it can be claimed that the state ðs; qev Þ will eventually converge to the set n o pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi B4 , ðs; qev Þ : k s k 6 l3 =k2 r; k qev k 6 l3 =b , in which Furtherl3 ¼ ðrc1 h2 =2Þ þ ðrc2 cT c =2Þ þ ðrc3 k2 =2Þ þ 3re. more, from the boundedness of ðs; qev Þ, we can further infer that xe is also bounded due to Eq. (15), and moreover, the corresponding convergence region can be determined by pffiffiffiffiffi B5 ,fxe : k xe k 6 l4 g, in which l4 ¼ ðrc1 h2 =4Þþ ðrc2 cT c =4Þ þ ðrc3 k2 =4Þ þ 3re < 1. Remark 3. The proposed control algorithm can ensure stable attitude tracking, despite the presence of external disturbances, inertia uncertainties, and actuator faults, as long as the matrix DEDT remains positive definite for all fault scenarios that are under consideration. However, it should be pointed out that an implicit assumption is that the remaining active torque actuators under actuation limits are still able to produce sufficient control torques for spacecraft to perform the given maneuvers. 4. Numerical simulations
and xr ðtÞ ¼ 0:1½cosðt=4Þ; sinðt=5Þ; cosðt=6ÞT rad=s. And the external disturbance is taken as d ¼ 0:06½sinð0:8tÞ; cosð0:5tÞ; cosð0:3tÞT N m. Consider a case in which six two-sided thrusters with the same nominal maximum thrust capability are symmetrically distributed on three axes of the body frame B of the spacecraft in the configuration presented in Fig. 2. Then the distribution matrix can be determined by the distance between the initial CM and the mounting position, and D is expressed as 3 2 d1 d1 0 0 0 0 7 6 D¼40 0 5 0 d2 d2 0 0 0 0 0 d3 d3 And we chose d1 ¼ 0:8 m and d2 ¼ d3 ¼ 0:7 m. The parameters for the characterization of inertia matrix are chosen as c ¼ 0:1, rf ¼ 0:2 m , mð0Þ ¼ 70 kg and m0 ¼ 105 kg. Considering the physical performance of the actuators in practical cases, the available actuation is always limited. Therefore, a saturation function is used for the thrusters in the simulations. The saturation values are set as 10 N, i.e. the actual forces generated by the thrusters remain at 10 N when the command signals get over these values. The controller in Eq. (14) and the controller in Ref.6 are multiplied by the pseudo-inverse of matrix D to accord the dimension and the units in the numerical simulations. 4.1. Healthy actuators In this case, all actuators are assumed to work healthily, and the control scheme in Eqs. (16)–(19) is applied. The main control parameters of the proposed controller are determined as c1 ¼ 4 106 , c2 ¼ 0:01, b ¼ 0:9, e ¼ 0:002, g1 ¼ 3, g2 ¼ 3 and k1 ¼ 20. To show the favorable performance of the proposed control algorithm, comparison between the proposed controller and the control law in Ref.6 and in Ref.4 are presented. For a fair and meaningful comparison, all the controller parameters of Ref.6 and Ref.4 keep their original values, except that b in Ref.4 is chosen as b ¼ 1 for fair comparison. The time responses of relative angular velocity error xe and relative attitude tracking error qev are shown in Fig. 3. One can observe that the proposed controller shows better tracking accuracy than the controller in Ref.6, which is more clearly shown by the norms of the tracking errors in Fig. 4. The degradation of performance of the control scheme in Ref.6 is mainly caused by the CM changes, which are not explicitly taken into account
To illustrate the effectiveness of the proposed control scheme, numerical simulations are conducted in this section. In the simulation, the value of the inertia component J0 is set as 2 3 20 0 0 6 7 J0 ¼ 4 0 17 0 5 kg m2 ð42Þ 0
0
15
For each set of simulations, the initial attitude of the spacepffiffiffiffiffiffiffi T craft is selected as qð0Þ ¼ 0:7; 0:1; 0:5; 0:2 and the angular velocity is xð0Þ ¼ ½0:01; 0:01; 0:01T rad=s. The desired attitude motion is determined by qr ð0Þ ¼ ½1; 0; 0; 0T
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Fig. 2
Distribution schematic of six thrusters.
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Fig. 3
Fig. 4
Time responses of angular velocity and attitude tracking errors with healthy actuators.
Norms of angular velocity error vector and attitude tracking error vector with healthy actuators.
in the controller design of Ref.6. It is also observed from Figs. 3 and 4 that the controller in Ref.4 can achieve favorable control accuracy at the beginning. However, the tracking errors continue to enlarge while the change of CM increases, which means that the controller in Ref.4 has weaker robustness to the time-varying property of CM. In contrast, the changing CM is explicitly taken into consideration during the design process of the proposed control scheme, and thus, the change of CM caused by fuel consumption will not possess an important influence on the control accuracy. The time responses of Euler angles are shown in Fig. 5, and it intuitionally presents the tracking process, which is consistent with the results of tracking errors. The control forces produced by the six thrusters are presented in Fig. 6. It is observed that the controllers in Ref.6 and in Ref.4 command a higher control torque than the proposed controller. For the purpose of verifying the robustness of the proposed controller to various changing velocity of CM, we change the
Fig. 5
Time responses of Euler angles.
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties
Fig. 6
Fig. 7
Time responses of control forces with healthy actuators.
Norms of tracking errors and control signals of proposed controller with healthy actuators.
Fig. 8
Norms of tracking errors and control signals of controller in Ref.6 with different rf .
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Fig. 9
Norms of tracking errors and control signals of controller in Ref.4 with different rf .
Fig. 10
Comparison of CM variation.
Fig. 11 Comparison of fuel consumption for t ¼ 300 s with healthy actuators.
value of rf to 0.4 m and 0.6 m, because larger rf can generate a faster and larger variation of CM as the fuel is consuming. It is clear from the results of Fig. 7 that although the faster variation of CM slightly reduces the convergence rate of the proposed control scheme, the high attitude tracking accuracy is almost not affected. And a larger control effort is required for the faster variation of CM. The norms of angular velocity,
attitude tracking error and control vector with the controller in Ref.6 are illustrated in Fig. 8. It is observed that the controller in Ref.6 suffers from lower accuracy for both attitude and angular velocity tracking due to faster variation of CM, and larger control effort is also demanded. And Fig. 9 shows that the controller in Ref.4 also possesses good robustness to different changing velocities of CM. The results in Fig. 9 and in Fig. 4 indicate that different changing velocities of CM almost do not affect the control performance of the controller in Ref.4 but the time-varying properties of CM produce a degradation of control accuracy. In Figs. 10 and 11, it can be seen that the controllers in Ref.6 and in Ref.4 command higher fuel consumption and thus there is a larger change of CM than the proposed controller for the same rf and the difference increases when rf become larger. The higher fuel consumption stems from the fact that the controllers in Ref.6 and in Ref.4 do not directly take into account the changing CM. 4.2. Faulty actuators Next, simulations are conducted for the case where the spacecraft is subject to actuator faults. The effectiveness matrix EðtÞ is generated by the following equation: ei ðtÞ ¼ 0:7 þ 0:1randðti Þ þ 0:1sinð0:5t þ ip=3Þ
ð43Þ
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties
Fig. 12
Effectiveness factors.
where randðÞ 2 ð1; 1Þ is a random number generator, and the sampling time is T ¼ 13 s ti ¼ mod ðt þ Dti ; TÞ, Dti ¼ 2ði 1Þ, i ¼ 1; 2; :::; 6. The additive fault u is defined as u ¼ 0:003½sinð0:03tÞ; sinð0:05ptÞ; cosð0:15tÞ; sinð0:03ptÞ; sinð0:05tÞ; cosð0:15ptÞT
ð44Þ
The control scheme in Eqs. (30)–(34) is applied and the control parameters are given as b ¼ 0:9; k2 ¼ 30; e ¼ 0:09; g1 ¼ 200; j1 ¼ 3 104 ; g2 ¼ 102; j2 ¼ 0:015; g3 ¼ 100 and j3 ¼ 0:001.
Fig. 13
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Comparison between the proposed controller and those of Ref.6 and Ref.4 is also given in this subsection, and b in Ref.4 is given as b ¼ 1:6 for fair comparison. The effectiveness factors of the six actuators are shown in Fig. 12. Note that the actuators 2, 4 and 6 are shuttled down on purpose at time 11 s, 15 s and 19 s respectively. The angular velocity and attitude tracking errors are presented in Fig. 13 and the norms are shown in Fig. 14. One can observe that the control scheme in Ref.6 cannot achieve satisfactory tracking accuracy anymore due to the actuator
Time responses of angular velocity and attitude tracking errors.
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Fig. 14
Fig. 15
Norms of angular velocity error vector and attitude tracking error vector with faulty actuators.
Snapshots of attitude orientation history.
Fig. 16
faults. In contrast, high control precision is still obtained with the proposed controller. In spite of the reliability of the controller in Ref.4, it still suffers from loss of accuracy as the CM changes. To illustrate the tracking process, a series of snapshots of the actual attitude orientations of four spacecraft with the desired attitude, the attitude generated by the proposed controller and the attitude generated by the controllers in Ref.6 and in Ref.4 are shown in Fig. 15. The x-axis of the bodyfixed frames, xbp ; xb ; xb6 and xb4 ; are identified for the spacecraft (the black dotted lines parallel to xb are constructed to illustrate the attitude tracking errors). It is observed that the spacecraft controlled by the proposed controller and the controller in Ref.4 show almost the same performance to the desired attitude from t ¼ 6 s. In contrast, there is always a significant distinction between the attitude
Time responses of control forces with faulty actuators.
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties
Fig. 17
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Norms of tracking errors and control signals of proposed controller with faulty actuators.
Fig. 18
Norms of tracking errors and control signals of controller in Ref.6 with different rf .
Fig. 19
Norms of tracking errors and control signals of controller in Ref.4 with different rf .
of the spacecraft controlled by the controller in Ref.6 and the desired attitude. The control forces of the thrusters are shown in Fig. 16. One can observe that the control law in Ref.6 commands a minor control input in the initial transient regime. However, for the steady-state regime, it demands a larger control effort
than the proposed controller and the controller in Ref.4, which is more clearly shown by the variation rate of CM and the quantity of fuel consumption in Figs. 20 and 21. Then, we change the value of rf to 0.4 m and 0.6 m, and the results are illustrated in Figs. 17–19. It is shown that in the case of actuator faults, although the control accuracy is slightly affected by different changing velocities of CM, the proposed
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Fig. 20
Comparison of CM variation.
unknown time-varying parameters in inertia matrix to release some assumptions of the system, which is one of the subjects for our future research. Acknowledgement This work was supported partially by the National Natural Science Foundation of China (Nos. 61522301, 61633003). References
Fig. 21 Comparison of fuel consumption for t ¼ 300 s with faulty actuators.
controller and the controller in Ref.4 still hold good robustness to different changing velocities of CM. 5. Conclusions This paper investigated the fault-tolerant attitude tracking control problem for a rigid spacecraft in the presence of external disturbance, actuators faults and time-varying inertia matrix. Changing CM has been directly taken into account during the characterization of the time-varying inertia matrix. Firstly, a basic controller is designed to deal with the attitude tracking mission without actuator fault. Then, two types of actuator faults (effectiveness loss and additive fault) are considered simultaneously, and an auxiliary controller is designed to deal with the actuator faults. The performance of the proposed controller, as well as the necessity of considering the changing CM during the controller design, is illustrated in the numerical simulations. The proposed control algorithm possesses high control accuracy, fault-tolerant capacity and favorable robustness to time-varying inertial parameters. The time-varying part in inertia matrix is modeled as a known parameter with some reasonable simplifications of the system in this paper. However, it is more interesting to consider
1. Costic BT, Dawson DM, Queiroz MS, Kapila V. Quaternionbased adaptive attitude tracking controller without velocity measurements. J Guid Control Dyn 2001;24(6):1214–22. 2. Gao JW, Cai YL. Adaptive finite-time control for attitude tracking of rigid spacecraft. J Aerosp Eng 2016;29(4):04016016. 3. Seo D, Akella MR. High-performance spacecraft adaptive attitude-tracking control through attracing-manifold design. J Guid Control Dyn 2008;31(4):884–91. 4. Cai WC, Liao XH, Song YD. Indirect robust adaptive faulttolerant control for attitude tracking of spacecraft. J Guid Control Dyn 2008;31(5):1456–63. 5. Hu QL, Shao XD. Smooth finite-time fault-tolerant attitude tracking control for rigid spacecraft. Aerosp Sci Technol 2016;55:144–57. 6. Thakur D, Srikant S, Akella MR. Adaptive attitude-tracking control of spacecraft with uncertain time-varying inertia parameters. J Guid Control Dyn 2015;38(1):41–52. 7. Guo Y, Song SM. Adaptive finite-time backstepping control for attitude tracking of spacecraft based on rotation matrix. Chin J Aeronaut 2014;27(2):375–82. 8. Gasbarri P, Sabatini M, Pisculli A. Dynamic modelling and stability parametric analysis of a flexible spacecraft with fuel slosh. Acta Astronaut 2016;127:141–59. 9. Jing WX, Xia XW, Gao CS, Wei WS. Attitude control for spacecraft with swinging large-scale payload. Chin J Aeronaut 2011;24:309–17. 10. Shou HN. Discoid and asymmetrical micro-satellite propulsion mode attitude control with great mass change and without angular rate sensor. Appl Mech Mater 2014;479–480:753–7. 11. Deng ML, Yue BZ. Nonlinear model and attitude dynamics of flexible spacecraft with large amplitude slosh. Acta Astronaut 2017;133:111–20. 12. Huo BY, Xiao YQ, Lu KF, Fu MY. Adaptive fuzzy finite-time fault-tolerant attitude control of rigid spacecraft. J Franklin Instit 2015;352:4225–46.
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties 13. Xu SD, Sun GH, Sun WC. Fuzzy logic based fault-tolerant attitude control for nonlinear flexible spacecraft with sampleddata input. J Franklin Instit 2017;354(5):2125–56. 14. Zou AM, Kumar KD. Adaptive fuzzy fault-tolerant attitude control of spacecraft. Control Eng Pract 2011;19:10–21. 15. Xiao B, Hu QL, Zhang YM. Finite-time attitude tracking of spacecraft with fault-tolerant capability. IEEE Trans Control Syst Technol 2015;23(4):1338–50. 16. Han Y, Biggs JD, Cui NG. Adaptive fault-tolerant control of spacecraft attitude dynamics with actuator failures. J Guid Control Dyn 2015;38(10):2033–42. 17. Bustan D, Pariz N, Sani SKH. Robust fault-tolerant tracking control design for spacecraft under control input saturation. ISA Trans 2014;53:1073–80.
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18. Shen Q, Wang DW, Zhu SQ, Poh EK. Inertia-free fault-tolerant spacecraft attitude tracking using control allocation. Automatica 2015;62:114–21. 19. Hu QL, Zhang YM, Huo X, Xiao B. Adaptive integral-type sliding mode control for spacecraft attitude maneuvering under actuator stuck failures. Chin J Aeronaut 2011;24:32–45. 20. Schaub H, Junkins J. Analytical mechanics of space systems. AIAA Educ Ser; Reston: AIAA 2003;79–120. 21. Mcnair SL, Tragesser S. Dynamics of a thrusting, spinning satellite with changing center of mass position. AIAA/AAS astrodynamics specialist conference, 2012.
Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties Qinglei HU a,*, Li XIAO b, Chenliang WANG a a b
School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China Sino-French Engineer School, Beihang University, Beijing 100191, China
Received 5 January 2018; revised 2 July 2018; accepted 7 September 2018 Available online 5 January 2019
KEYWORDS Adaptive control; Attitude tracking; Changing center of mass; Fault-tolerant; Spacecraft
Abstract This paper is devoted to adaptive attitude tracking control for rigid spacecraft in the presence of parametric uncertainties, actuator faults and external disturbance. Specifically, a dynamic model is established based on one-tank spacecraft, which explicitly takes into account changing Center of Mass (CM). Then, a control scheme is proposed to achieve attitude tracking. Benefiting from explicitly considering the changing CM during the controller design process, the proposed scheme possesses good robustness to parametric uncertainties with less fuel consumption. Moreover, a fault-tolerant control algorithm is proposed to accommodate actuator faults with no need of knowing the actuators’ fault information. Lyapunov-based analysis is provided and the closed-loop system stability is rigorously proved. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed controllers. Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The computation method of unsteady transonic flow based on N-S equations should be best accurate, but to threedimensional complex problems, it can be achieved only on large computers, and moreover, the results are not ideal sometimes.1 A viscous/inviscid interaction method is an applicable one and the computation time can be reduced by two orders. * Corresponding author. E-mail address: [email protected] (Q. HU). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
Attitude tracking is a fundamental mission that most spacecraft must perform during their operations and this issue has been extensively studied in the existing literatures.1–3 In particular, adaptive control techniques have been largely applied to addressing the problem of unknown or uncertain system parameters.4–7 However, the majority of available adaptive attitude tracking control solutions do not consider the changing Center of Mass (CM) caused by fuel consumption. The fuel often provides a significant contribution to the overall mass of the spacecraft to execute station keeping and attitude maneuvering. Therefore, the effect of fuel consumption is important for the determination of CM for the spacecraft. And the changing CM can result in variant body-fixed coordinate frame and time-varying inertia profile, which may degrade the control accuracy or even destroy the system stability. Moreover, fault-tolerant capacity is also required for
https://doi.org/10.1016/j.cja.2018.12.015 1000-9361 Ó 2019 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties spacecraft because actuator faults are frequently encountered in practical spacecraft operations. All the above issues create considerable difficulty in the design of attitude control algorithms for achieving accurate, reliable and fast responses, especially when these issues are treated simultaneously. The effect of changing mass properties for spacecraft has been addressed in some of the existing literatures. The interactions among the fuel slosh, the attitude dynamics and the flexible appendages have been studied in Ref.8 via a classical multi-body approach. Jing et al.9 proposed a switching attitude control law by considering the astronomy observation satellite with the sensor of non-negligible mass and length. A robust controller has been designed in Ref.10 to deal with the position changes of CM due to fuel consumption for the micro-satellite. Deng and Yue11 focused on the nonlinearly modelling and attitude dynamics of spacecraft coupled with large amplitude liquid sloshing dynamics and flexible appendage vibration and designed a feedback controller for the attitude maneuvering problem. Another key issue related to spacecraft attitude control involves actuator faults. In practice, actuator faults are frequently encountered due to the harsh working environment. The occurrence of actuator faults might result in mission failure and even system instability. Thus, extensive fault-tolerant control schemes have been carried out for the spacecraft attitude control in the previous literatures. Fuzzy logic has been applied12–14 to design fault-tolerant controllers of the spacecraft. Xiao et al.15 proposed an SMC-based fault-tolerant attitude-tracking control approach with the capability of finite-time convergence with zero tracking error. A faulttolerant control algorithm relying on an ideal reference model has been presented.16 Based on variable structure control, a fault-tolerant tracking control scheme without any requirement of fault identification has been developed.17 Shen et al. proposed a fault-tolerant control law using the control allocation approach to deal with the actuator faults.18 In Ref.19, the integral-type sliding mode control has been employed in the design of the fault-tolerant control algorithm against actuator stuck faults for spacecraft attitude maneuvering. Although significant studies have been conducted to deal with the changing mass properties and the actuator faults, to the best of our knowledge, few of them have treated the changing CM (caused by the mass variation) and actuator faults simultaneously. However, these problems are very likely to happen simultaneously in practical situations, which may degrade the control performance or even result in mission failure. Therefore, it is crucial to take these problems into consideration during the controller design process. In this paper, an adaptive fault-tolerant control scheme is developed to address the spacecraft attitude tracking problem under inertia uncertainty, actuator faults and external disturbance. Based on a one-tank spacecraft, an explicit dynamic model of the timevarying inertia matrix is established, and it is taken into account both in the dynamic equations of the spacecraft and controller design process. The main contribution of this work includes two parts: (A) characterize the time-varying inertia dynamic model with consideration of changing CM to construct the adaptive attitude tracking controller. Hence, the time-varying inertial parameters will not introduce important influence on the control performance with the proposed control scheme; (B) propose an adaptive fault-tolerant control algorithm which can achieve high-precision attitude tracking, despite the presence of time-varying inertia properties, external
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disturbances and actuator faults (including total failures). The system stability is also rigorously analyzed through Lyapunov analysis. The rest of the paper is organized as follows. The mathematical model of a spacecraft with time-varying inertia matrix and the characterization of the time-varying inertia matrix are derived in Section 2. The next section presents the design procedure and stability analysis of the control algorithm. Numerical simulations are provided in Section 4 to demonstrate the performance of the proposed control scheme. Finally, the paper is completed with some concluding comments. Notation. Through this article, Rn is the n-dimensional Euclidean space, and Rnm represents the set of all real matrices of dimension n m. The skew-symmetric matrix operator SðÞ represents the vector cross-product operation between any two vectors, defined by SðaÞb ¼ a b for any a, b 2 R3 . MT is the transpose of M. diagðx1 ; x2 ; . . . ; xn Þ denotes the diagonal matrix of dimension n n with diagonal elements x1 ; x2 ; :::; xn . kmin ðMÞ and kmax ðMÞ are the maximum and minimum eigenvalues of M. jj and k k stand for the usual L1 and L2 norms. 2. Problem statement and preliminaries 2.1. Spacecraft dynamics The governing equations for attitude motion of a spacecraft are formulated in terms of the unit-quaternion representation. The attitude kinematics are given by20: ( q_ 0 ¼ 12 qTv x ð1Þ q_ v ¼ 12 ðq0 I þ Sðqv ÞÞx T
where the unit-quaternion q ¼ ½q0 ; qTv 2 R4 (q0 ðtÞ 2 R and qv ðtÞ 2 R3 are the scalar and vector parts of q respectively) denotes the attitude orientation of the body-fixed frame B with respect to the inertial frame N, and satisfies the constraint qT q ¼ 1; x 2 R3 denotes the inertial angular velocity of the spacecraft expressed in B; I is the 3 3 identity matrix. The corresponding attitude dynamics are written as6 _ _ JðtÞxðtÞ ¼ JðtÞxðtÞ SðxÞJðtÞxðtÞ þ sðtÞ þ dðtÞ
ð2Þ
is the time-varying inertia matrix of the where JðtÞ 2 R spacecraft, and remains symmetric positive-definite for all time; sðtÞ 2 R3 is the control torque input generated by the torque actuators; dðtÞ 2 R3 is the external disturbance. The direction cosine matrix, denoting a transformation from N to B, can be calculated by 33
CðqÞ ¼ ðq20 qTv qv ÞI þ 2qv qTv 2q0 Sðqv Þ Since the desired reference angular velocity is specified in its own reference frame R, the rotation matrix from R to B can be obtained as Cðqe Þ ¼ CðqÞðCðqr ÞÞT
ð3Þ
T ½qe0 ; qTev
where qe ¼ denotes the attitude error quaternion describing the relative attitude of the spacecraft with respect to the reference attitude qr . Then, the angular velocity tracking error is expressed as xe ¼ x Cðqe Þxr
ð4Þ
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where xr 2 R3 represents the bounded reference angular velocity. By differentiating Eqs. (3) and (4) with respect to time, the open-loop attitude-tracking error dynamics are written as ( q_ e0 ¼ 12 qTev xe ð5Þ q_ ev ¼ 12 ðqe0 I þ Sðqev ÞÞxe _ SðxÞJx þ s þ d þ J/ Jx_ e ¼ Jx
ð6Þ
where / ¼ Sðxe ÞCðqe Þxr Cðqe Þx_ r and we note hereafter Q ¼ 0:5ðqe0 I þ Sðqev ÞÞ. The control objective of this paper is to design an adaptive fault-tolerant attitude tracking controller for the spacecraft subject to time-varying inertia matrix, external disturbances and actuator faults so that all closed-loop signals are uniformly bounded and that the attitude tracking errors converge to a small set around the origin. To this end, the following assumption is made13,15,18: Assumption 1. The external disturbance d is bounded by k d k 6 dm , where dm > 0 is an unknown constant.
mðtÞ mð0Þ rf m0 þ mðtÞ
ð8Þ
Define the time-varying inertia matrix as JðtÞ ¼ diagðJ11 ; J22 ; J33 Þ;
Jð0Þ ¼ diagðJ011 ; J022 ; J033 Þ
where Jð0Þ is the inertia matrix at t ¼ 0, and J011 ; J022 and J033 are unknown constants. Using the parallel axis theorem to account for the changing CM location, and considering the fuel consumption, the time-varying inertial matrix at time t can be expressed as21 8 > < J11 ¼ J011 J22 ¼ J022 ðmð0Þ mðtÞÞðr2f þ rðtÞ2 Þ > : J33 ¼ J033 ðmð0Þ mðtÞÞðr2f þ rðtÞ2 Þ
JðtÞ ¼ J0 þ J1 WðtÞ
Generally speaking, our system consists of a rigid spacecraft body with CM initially located at point O (the initial CM, remains fixed to the spacecraft body) and a fuel tank that is offset from point O by a distance of rf in the x-direction. As shown in Fig. 1, the body-fixed frame B remains fixed at the CM and rotates with the spacecraft. The origin Ob is located at the CM, the x-axis, xb , is directed from Ob to the fuel tank, the z-axis, zb , is the initial spin axis, and the y-axis, yb , completes the body-fixed coordinate system. In this paper, both the fuel tank and the mass-invariant part are modeled as mass points, using equation Pthe solving P for CM of a mass point system rr ¼ i mi r= i mi , where rr is the position vector of the system’s CM relative to a fixedpoint; mi and ri are the mass and position vector relative to the same fixed-point of the ith mass point. Then, we can deduce the position vector rðtÞ of the instantaneous CM relative to O as m0 rs þ mðtÞrf m0 þ mðtÞ
rðtÞ ¼
ð9Þ
Then, expanding and rearranging Eq. (9), the inertia matrix can be expressed as
2.2. Characterization of time-varying inertia matrix with changing CM
rðtÞ ¼
vector of the fuel tank relative to O. By definition that rðtÞ has an initial value of zero, rð0Þ ¼ 0, we can obtain rs ¼ mð0Þrf =m0 . Applying this result to Eq. (7), the CM vector can be found by
ð7Þ
where m0 is the total mass of all the mass-invariant parts, rs is the position vector of m0 relative to O, mðtÞ denotes the mass of the remaining fuel at time t, and rf represents the position
ð10Þ
where J0 ¼ diagðJ011 ; J022 mð0Þr2f ; J033 mð0Þr2f Þ is a constant matrix with unknown parameters J0ii ; i ¼ 1; 2; 3;
WðtÞ ¼ mðtÞðr2f þ rðtÞ2 Þ mð0ÞrðtÞ2
is a known and time-dependent matrix; J1 ¼ diagð0; 1; 1Þ. By differentiating Eq. (10) with respect to time, we thus have _ ¼ J1 WðtÞ _ JðtÞ
ð11Þ
The rate of change of the fuel mass is known to be a function of the applied control and actuator hardware characteristics, so that we use the mass relation6 _ mðtÞ ¼ ck u k
ð12Þ
where c > 0 is a known constant. Then, by taking the time derivative of WðtÞ and applying Eq. (12), it follows that " 2 # mðtÞ mð0Þ 2 _ W ¼ ck u krf 1 þ ð1 þ 2mi Þ ð13Þ m0 þ mðtÞ where mi ¼ mð0Þ þ m0 is the initial mass of the spacecraft without any fuel consumption. Remark 1. For the characterization of the inertia matrix, we not only consider the direct influence of the fuel consumption, but also explicitly take into account the impact of the changing CM caused by the fuel consumption. Thus, a more accurate spacecraft dynamic model is obtained. 3. Adaptive controller design
Fig. 1
Body-fixed coordinate frame.
In this section, an adaptive attitude tracking control strategy is presented to solve the current problem under study. We start
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties by synthesizing a basic adaptive controller for the case in which all the actuators work healthily to help illustrate the key idea of accounting for the time-varying inertia properties. The result is then extended to an adaptive fault-tolerant control synthesis. 3.1. Basic adaptive controller design Substituting Eqs. (10) and (11) into Eq. (6), the angular velocity error dynamics can be rewritten as _ þ W1 h þ H1 þ s þ d Jx_ e ¼ J1 Wx
ð14Þ
where H1 ¼ SðxÞ J1 Wx þ J1 W/, and the regressor matrix W1 is constructed in the following fashion:
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_ 1 _ 1 _ 1 ^ V_ ¼ sT J_s þ sT Js h dm d^m þ bxTe qev h 2 g1 g2 _ þ W2 h þ H2 þ s þ d bqev Þ ¼ sT ðJ1 WX _ 1 _ 1 h dm d^m h^ g1 g2 _ þ H2 þ sT s 6 k W2 khk s k þ dm k s k J1 WX 1 ^ _ 1 _ h dm d^m h b2 qTev qev g1 g2 ^ s k þ d^m k s k J1 WX _ þ H2 þsT s ¼ k W2 khk _ 1 1 _ h þ dm k s k d^m b2 qTev qev þ h g1 k W2 kk s k ^ g1 g2 ^ T T ^ ¼ k W2 khk s k þ dm k s k kðtÞs s k1 s s b2 qT q þ bxTe qev
ev ev
h^ þ c1 h þ c2 dm d^m
W1 h ¼ SðxÞ J0 x þ J0 /
ð21Þ
with h ¼ ½J011 ; J022 ; J033 T containing the three elements of J0 . To develop the control scheme, we define the filtered error variable
To proceed, the following key inequality is introduced; specifically, for any scalar e > 0 and x 2 R, we have
s ¼ xe þ bqev
jxj2 jxj2
ð15Þ
where b > 0 is a design parameter. Then, we design the adaptive controller and the accompanying adaptive law as _ ðk1 þ kðtÞÞs s ¼ H2 þ J1 WX
ð16Þ
with
By applying Eq. (22), one can deduce that kW2 k2 ^h ksk2 ^ ffi þe k W2 k hk s k 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2
kW2 k2 ^h ksk2 þ e2
d^m k W2 k2 ^h kðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 d^m k s k2 þ e2 k W2 k2 ^h k s k2 þ e2
ð17Þ
^_ ¼ g c ^h k W kk s k h 2 1 1
ð18Þ
_ d^m ¼ g2 c2 d^m k s k
ð19Þ
2
Theorem 1. Consider the attitude tracking error system described by Eqs. (5) and (6) with time-varying inertia matrix characterized by Eqs. (10)–(13). If the control law in Eq. (16), together with the adaptive laws in Eqs. (18) and (19), is applied, then the closed-loop system is globally stable in the sense that all the signals are continuous and bounded, and the tracking errors qev ðtÞ and xe ðtÞ converge to a small set containing the origin. Proof. Consider the following Lyapunov Function Candidate (LFC):
2
d^m ksk þ e2
Further, by completion of squares and using Eq. (23) in Eq. (21), it follows that4
where k1 > 0 is a control gain, X ¼ x ðs=2Þ, W2 is a known regressor matrix satisfying W2 h ¼ W1 h þ bJ0 Qxe , ^ ^ H2 ¼ H1 þ bðJ1 WQxe þ qev Þ, h and dm are the estimations of h (h ¼ k h k) and dm , respectively, and both gi and ci (i ¼ 1; 2) are positive design constants.
2 2 1 1 h þ 1 d V ¼ sT Js þ b½qTev qev þ ð1 qe0 Þ2 þ 2 2g1 2g2 m
ð23Þ
d^m ksk d^m k s k 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þe 2 2 2
2
ð22Þ
ð20Þ
where h ¼ h ^h and dm ¼ dm d^m are the estimation errors. Taking the time derivative of Eq. (20) and substituting Eqs. (5), (10), (11) and (14) into the resulting formula yield
c 2 c2 2 c1 2 c2 2 h dm þ h þ dm V_ 6 k1 sT s b2 qTev qev 1 2 2 2 2 þ 2e 6 aV þ b
ð24Þ
V_ 6 k1 sT s b2 qTev qev þ l1
ð25Þ
V_ 6 k1 sT s b2 qTev qev þ l2
ð26Þ
where a ¼ minf2k1 =kmax ðJÞ; g1 c1 ; g2 c2 ; bg > 0 2 =2Þ þ ðc d2 =2Þ þ 2e < 1 b ¼ bð1 q Þ2 þ ðc h e0
1
2 m
h =2Þ þ ðc2 dm2 =2Þ þ 2e < 1 l1 ¼ ðc1 2 h =4Þ þ ðc d 2 =4Þ þ 2e < b1 < 1 l ¼ ðc 2
2
1
2 m
Note that Eq. (24) implies that V 2 L1 . Then we conclude that and successively. And from Eqs. (25) and (26), one can observe that V_ < 0 when ðs; qev Þ is out of the set pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi B1 ,ððs; qev Þ : k s k 6 l1 =k1 ; k qev k 6 l1 =bÞ or the set pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi Since B2 ,fðs; qev Þ : k s k 6 l2 =k1 ; k qev k 6 l2 =bg. b1 > b2 , the set B1 encloses completely the set B2 . Therefore, once ðs; qev Þ is inside the set B1 , they cannot go out of it because they will be attached back to B2 , which means that the states are confined in the set B1 . Furthermore, the boundedness of ðs; qev Þ indicates that xe is also bounded because of
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Q. HU et al.
Eq. (15). And its bound pffiffiffiffiffi B3 ,fxe : k xe k 6 l2 g.
can
be
determined
as
Remark 2. The time-varying inertia properties with changing CM are considered both in the construction of spacecraft dynamic model and the controller design process. Thus, the change of CM caused by fuel consumption during the spacecraft on-orbit attitude maneuvers will not have a large effect on the control performance under the proposed adaptive controller. This fact can help improve the attitude tracking efficiency and accuracy. 3.2. Adaptive fault-tolerant controller design It is common practice to equip the spacecraft with three or more than three torque actuators for robustness reasons. From a practical viewpoint, despite various efforts to enhance reliability of the components of the torque actuators, multiple anomalies do occur in them during their long operational life. Thus, it is imperative to develop fault-tolerant control algorithms that are capable of accommodating possible actuator faults for spacecraft attitude tracking control. In view of this, in this subsection, we shall present an adaptive fault-tolerant control architecture with a slight modification to the control scheme presented in Section 3.1. To describe actuator faults, the control input of dynamic system Eq. (6) is further expressed as s ¼ DðEu þ uÞ
ð27Þ
where D 2 R (n P 3) is the distribution matrix of the torque actuators; E ¼ diagðe1 ; e2 ; :::; en Þ 2 Rnn with 0 6 ei 6 1 (i ¼ 1; 2; :::; n) is the actuation effectiveness matrix that describes the health condition of the torque actuators; u 2 Rn and u 2 Rn denote the desired control input signals and the unexpected deviation faults, respectively. To proceed, the following assumptions are made. 3n
Assumption 2 15. The additive bias forces u is bounded, and there exists an unknown constant um > 0 such that k u k 6 um .
stant r such that 0 < r 6 kmin ðDEDT Þ holds. Let us define k :¼ 1=r, h ¼ kk h k and c ¼ kc. Choose the smooth control law and the accompanying adaptive laws as u ¼ DT ðk2 þ k ðtÞÞs
ð30Þ
^_ ¼ g k W1 kk s k c ^ h 1 1h
ð31Þ
^c_ ¼ g2 ðUk s k c2^c Þ
ð32Þ
^_ ¼ g k ud kk s k c ^ k 3 3k
ð33Þ
h , ^c and ^ k denote the estimawhere k2 > 0 is the control gain, ^ tions of h , c and k, respectively, and gi > 0 and ci > 0 (i ¼ 1; 2; 3) are design parameters. In Eq. (30), the timevarying gain k ðtÞ is given by 2
k W1 k2 ^ ðc^ UÞ2 h k ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðc^ UÞ2 k s k2 þ e2 k W1 k2 ^ h k s k2 þ e2 ^ k k ud k2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ^ k k ud k2 k s k2 þ e2 2
The main result of the paper is summarized as the following theorem. Theorem 2. Consider the spacecraft attitude dynamics as described by Eqs. (5), (6) and (27). By implementing the proposed control scheme as given in Eqs. (30)–(34), all the closed-loop signals are bounded and the tracking errors qev ðtÞ and xe ðtÞ converge to a small set containing the origin. Proof. Consider the following LFC:
V1 ¼ V0 þ
r 2 r T r 2 h þ c c þ k 2g1 2g2 2g3
Assumption 3 (4;5;18). Assume that the matrix DEDT remains positive definite for all fault scenarios that are under consideration, which indicates that there are at most n 3 actuators that totally fail to work. By applying Eq. (27), Eq. (14) can be rewritten as _ þ W1 h þ H1 þ dðEu þ uÞ þ d Jx_ e ¼ J1 Wx
ð28Þ
We further have 1_ J_s ¼ ud þ W1 h þ DEu þ d Js ð29Þ bqev 2 _ þ bqev and ud ¼ J1 Wxþ _ where d ¼ Du þ d þ Jbq_ ev þ 0:5Js H1 are specified here just for notational simplicity. Inspired by Ref.4, we thus deduce that k d k 6 cT U with T
c ¼ ½c1 ; c2 ; c3 T (ci P 0, i ¼ 1; 2; 3) and U ¼ ½1; k x k; k x k2 . According to Assumption 3, we infer that there exists a con-
ð34Þ
ð35Þ
^ are the estimation where h ¼ h ^ h , c ¼ c ^c , k ¼ k k 1 T T errors, and V0 ¼ 2 s Js þ b½qev qev þ ð1 qev Þ2 . Evaluating the time derivative of Eq. (35) along Eq. (29) and using similar process as in Section 3.1 yield V_ 1 6 r^ kk ud kk s k þ rk W1 k^ h k s k þ r^cT Uk s k r _ h þ sT DEu þ bxTe qev þ h g1 k W1 kk s k ^ g1 r r T þ c g2 Uk s k ^c_ þ g2 g3 _^ k g3 k ud kk s k k
ð36Þ
Similar to Section 3.1, by applying Eq. (22), we have 1 2 2 2 ^ k u k k s k k C B d r^ kk ud kk s k 6 r@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ eA 2 2 2 2 ^ k k ud k k s k þ e 0
ð37Þ
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties 0
2 2
1
C B k W1 k ^h k s k r^ h k W1 kk s k 6 r@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ eA 2 ^2 2 2 k W1 k h k s k þ e
0
2
ð38Þ
1 T
2
2
B ð^c UÞ k s k C r^cT Uk s k 6 r@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ eA 2 2 T 2 ð^c UÞ k s k þ e
ð39Þ
Inserting Eqs. (30)–(34) into Eq. (36) and noting Eqs. (37)– (39), one can easily obtain
T V_ 1 6 k2 rsT s b2 qTev qev þ rc1 h h^ þ rc2 c ^c þ rc3 k ^k þ 3re
ð40Þ Following the similar reasoning as that in the proof of Theorem 1, we further have rc 2 rc t V_ 1 6 k2 rsT s b2 qTev qev 1 h 2 c c 2 2 rc3 2 k þ b1 6 a1 V1 þ b1 2
ð41Þ
and where a1 ,minf2k1 r=kmax ðJÞ; g1 c1 ; g2 c2 ; g3 c3 ; bg 2 2 2 T b1 ¼ bð1 q0 Þ þ ðrc1 h =2Þ þ ðrc2 c c =2Þ þ ðrc3 k =2Þþ 3re. With a similar analysis of Theorem 1, it can be claimed that the state ðs; qev Þ will eventually converge to the set n o pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi B4 , ðs; qev Þ : k s k 6 l3 =k2 r; k qev k 6 l3 =b , in which Furtherl3 ¼ ðrc1 h2 =2Þ þ ðrc2 cT c =2Þ þ ðrc3 k2 =2Þ þ 3re. more, from the boundedness of ðs; qev Þ, we can further infer that xe is also bounded due to Eq. (15), and moreover, the corresponding convergence region can be determined by pffiffiffiffiffi B5 ,fxe : k xe k 6 l4 g, in which l4 ¼ ðrc1 h2 =4Þþ ðrc2 cT c =4Þ þ ðrc3 k2 =4Þ þ 3re < 1. Remark 3. The proposed control algorithm can ensure stable attitude tracking, despite the presence of external disturbances, inertia uncertainties, and actuator faults, as long as the matrix DEDT remains positive definite for all fault scenarios that are under consideration. However, it should be pointed out that an implicit assumption is that the remaining active torque actuators under actuation limits are still able to produce sufficient control torques for spacecraft to perform the given maneuvers. 4. Numerical simulations
and xr ðtÞ ¼ 0:1½cosðt=4Þ; sinðt=5Þ; cosðt=6ÞT rad=s. And the external disturbance is taken as d ¼ 0:06½sinð0:8tÞ; cosð0:5tÞ; cosð0:3tÞT N m. Consider a case in which six two-sided thrusters with the same nominal maximum thrust capability are symmetrically distributed on three axes of the body frame B of the spacecraft in the configuration presented in Fig. 2. Then the distribution matrix can be determined by the distance between the initial CM and the mounting position, and D is expressed as 3 2 d1 d1 0 0 0 0 7 6 D¼40 0 5 0 d2 d2 0 0 0 0 0 d3 d3 And we chose d1 ¼ 0:8 m and d2 ¼ d3 ¼ 0:7 m. The parameters for the characterization of inertia matrix are chosen as c ¼ 0:1, rf ¼ 0:2 m , mð0Þ ¼ 70 kg and m0 ¼ 105 kg. Considering the physical performance of the actuators in practical cases, the available actuation is always limited. Therefore, a saturation function is used for the thrusters in the simulations. The saturation values are set as 10 N, i.e. the actual forces generated by the thrusters remain at 10 N when the command signals get over these values. The controller in Eq. (14) and the controller in Ref.6 are multiplied by the pseudo-inverse of matrix D to accord the dimension and the units in the numerical simulations. 4.1. Healthy actuators In this case, all actuators are assumed to work healthily, and the control scheme in Eqs. (16)–(19) is applied. The main control parameters of the proposed controller are determined as c1 ¼ 4 106 , c2 ¼ 0:01, b ¼ 0:9, e ¼ 0:002, g1 ¼ 3, g2 ¼ 3 and k1 ¼ 20. To show the favorable performance of the proposed control algorithm, comparison between the proposed controller and the control law in Ref.6 and in Ref.4 are presented. For a fair and meaningful comparison, all the controller parameters of Ref.6 and Ref.4 keep their original values, except that b in Ref.4 is chosen as b ¼ 1 for fair comparison. The time responses of relative angular velocity error xe and relative attitude tracking error qev are shown in Fig. 3. One can observe that the proposed controller shows better tracking accuracy than the controller in Ref.6, which is more clearly shown by the norms of the tracking errors in Fig. 4. The degradation of performance of the control scheme in Ref.6 is mainly caused by the CM changes, which are not explicitly taken into account
To illustrate the effectiveness of the proposed control scheme, numerical simulations are conducted in this section. In the simulation, the value of the inertia component J0 is set as 2 3 20 0 0 6 7 J0 ¼ 4 0 17 0 5 kg m2 ð42Þ 0
0
15
For each set of simulations, the initial attitude of the spacepffiffiffiffiffiffiffi T craft is selected as qð0Þ ¼ 0:7; 0:1; 0:5; 0:2 and the angular velocity is xð0Þ ¼ ½0:01; 0:01; 0:01T rad=s. The desired attitude motion is determined by qr ð0Þ ¼ ½1; 0; 0; 0T
679
Fig. 2
Distribution schematic of six thrusters.
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Q. HU et al.
Fig. 3
Fig. 4
Time responses of angular velocity and attitude tracking errors with healthy actuators.
Norms of angular velocity error vector and attitude tracking error vector with healthy actuators.
in the controller design of Ref.6. It is also observed from Figs. 3 and 4 that the controller in Ref.4 can achieve favorable control accuracy at the beginning. However, the tracking errors continue to enlarge while the change of CM increases, which means that the controller in Ref.4 has weaker robustness to the time-varying property of CM. In contrast, the changing CM is explicitly taken into consideration during the design process of the proposed control scheme, and thus, the change of CM caused by fuel consumption will not possess an important influence on the control accuracy. The time responses of Euler angles are shown in Fig. 5, and it intuitionally presents the tracking process, which is consistent with the results of tracking errors. The control forces produced by the six thrusters are presented in Fig. 6. It is observed that the controllers in Ref.6 and in Ref.4 command a higher control torque than the proposed controller. For the purpose of verifying the robustness of the proposed controller to various changing velocity of CM, we change the
Fig. 5
Time responses of Euler angles.
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties
Fig. 6
Fig. 7
Time responses of control forces with healthy actuators.
Norms of tracking errors and control signals of proposed controller with healthy actuators.
Fig. 8
Norms of tracking errors and control signals of controller in Ref.6 with different rf .
681
682
Q. HU et al.
Fig. 9
Norms of tracking errors and control signals of controller in Ref.4 with different rf .
Fig. 10
Comparison of CM variation.
Fig. 11 Comparison of fuel consumption for t ¼ 300 s with healthy actuators.
value of rf to 0.4 m and 0.6 m, because larger rf can generate a faster and larger variation of CM as the fuel is consuming. It is clear from the results of Fig. 7 that although the faster variation of CM slightly reduces the convergence rate of the proposed control scheme, the high attitude tracking accuracy is almost not affected. And a larger control effort is required for the faster variation of CM. The norms of angular velocity,
attitude tracking error and control vector with the controller in Ref.6 are illustrated in Fig. 8. It is observed that the controller in Ref.6 suffers from lower accuracy for both attitude and angular velocity tracking due to faster variation of CM, and larger control effort is also demanded. And Fig. 9 shows that the controller in Ref.4 also possesses good robustness to different changing velocities of CM. The results in Fig. 9 and in Fig. 4 indicate that different changing velocities of CM almost do not affect the control performance of the controller in Ref.4 but the time-varying properties of CM produce a degradation of control accuracy. In Figs. 10 and 11, it can be seen that the controllers in Ref.6 and in Ref.4 command higher fuel consumption and thus there is a larger change of CM than the proposed controller for the same rf and the difference increases when rf become larger. The higher fuel consumption stems from the fact that the controllers in Ref.6 and in Ref.4 do not directly take into account the changing CM. 4.2. Faulty actuators Next, simulations are conducted for the case where the spacecraft is subject to actuator faults. The effectiveness matrix EðtÞ is generated by the following equation: ei ðtÞ ¼ 0:7 þ 0:1randðti Þ þ 0:1sinð0:5t þ ip=3Þ
ð43Þ
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties
Fig. 12
Effectiveness factors.
where randðÞ 2 ð1; 1Þ is a random number generator, and the sampling time is T ¼ 13 s ti ¼ mod ðt þ Dti ; TÞ, Dti ¼ 2ði 1Þ, i ¼ 1; 2; :::; 6. The additive fault u is defined as u ¼ 0:003½sinð0:03tÞ; sinð0:05ptÞ; cosð0:15tÞ; sinð0:03ptÞ; sinð0:05tÞ; cosð0:15ptÞT
ð44Þ
The control scheme in Eqs. (30)–(34) is applied and the control parameters are given as b ¼ 0:9; k2 ¼ 30; e ¼ 0:09; g1 ¼ 200; j1 ¼ 3 104 ; g2 ¼ 102; j2 ¼ 0:015; g3 ¼ 100 and j3 ¼ 0:001.
Fig. 13
683
Comparison between the proposed controller and those of Ref.6 and Ref.4 is also given in this subsection, and b in Ref.4 is given as b ¼ 1:6 for fair comparison. The effectiveness factors of the six actuators are shown in Fig. 12. Note that the actuators 2, 4 and 6 are shuttled down on purpose at time 11 s, 15 s and 19 s respectively. The angular velocity and attitude tracking errors are presented in Fig. 13 and the norms are shown in Fig. 14. One can observe that the control scheme in Ref.6 cannot achieve satisfactory tracking accuracy anymore due to the actuator
Time responses of angular velocity and attitude tracking errors.
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Q. HU et al.
Fig. 14
Fig. 15
Norms of angular velocity error vector and attitude tracking error vector with faulty actuators.
Snapshots of attitude orientation history.
Fig. 16
faults. In contrast, high control precision is still obtained with the proposed controller. In spite of the reliability of the controller in Ref.4, it still suffers from loss of accuracy as the CM changes. To illustrate the tracking process, a series of snapshots of the actual attitude orientations of four spacecraft with the desired attitude, the attitude generated by the proposed controller and the attitude generated by the controllers in Ref.6 and in Ref.4 are shown in Fig. 15. The x-axis of the bodyfixed frames, xbp ; xb ; xb6 and xb4 ; are identified for the spacecraft (the black dotted lines parallel to xb are constructed to illustrate the attitude tracking errors). It is observed that the spacecraft controlled by the proposed controller and the controller in Ref.4 show almost the same performance to the desired attitude from t ¼ 6 s. In contrast, there is always a significant distinction between the attitude
Time responses of control forces with faulty actuators.
Adaptive fault-tolerant attitude tracking control for spacecraft with time-varying inertia uncertainties
Fig. 17
685
Norms of tracking errors and control signals of proposed controller with faulty actuators.
Fig. 18
Norms of tracking errors and control signals of controller in Ref.6 with different rf .
Fig. 19
Norms of tracking errors and control signals of controller in Ref.4 with different rf .
of the spacecraft controlled by the controller in Ref.6 and the desired attitude. The control forces of the thrusters are shown in Fig. 16. One can observe that the control law in Ref.6 commands a minor control input in the initial transient regime. However, for the steady-state regime, it demands a larger control effort
than the proposed controller and the controller in Ref.4, which is more clearly shown by the variation rate of CM and the quantity of fuel consumption in Figs. 20 and 21. Then, we change the value of rf to 0.4 m and 0.6 m, and the results are illustrated in Figs. 17–19. It is shown that in the case of actuator faults, although the control accuracy is slightly affected by different changing velocities of CM, the proposed
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Q. HU et al.
Fig. 20
Comparison of CM variation.
unknown time-varying parameters in inertia matrix to release some assumptions of the system, which is one of the subjects for our future research. Acknowledgement This work was supported partially by the National Natural Science Foundation of China (Nos. 61522301, 61633003). References
Fig. 21 Comparison of fuel consumption for t ¼ 300 s with faulty actuators.
controller and the controller in Ref.4 still hold good robustness to different changing velocities of CM. 5. Conclusions This paper investigated the fault-tolerant attitude tracking control problem for a rigid spacecraft in the presence of external disturbance, actuators faults and time-varying inertia matrix. Changing CM has been directly taken into account during the characterization of the time-varying inertia matrix. Firstly, a basic controller is designed to deal with the attitude tracking mission without actuator fault. Then, two types of actuator faults (effectiveness loss and additive fault) are considered simultaneously, and an auxiliary controller is designed to deal with the actuator faults. The performance of the proposed controller, as well as the necessity of considering the changing CM during the controller design, is illustrated in the numerical simulations. The proposed control algorithm possesses high control accuracy, fault-tolerant capacity and favorable robustness to time-varying inertial parameters. The time-varying part in inertia matrix is modeled as a known parameter with some reasonable simplifications of the system in this paper. However, it is more interesting to consider
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