Rigid spacecraft attitude tracking using finite time sliding mode control

Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India

Rigid spacecraft attitude tracking using finite time sliding mode control ⋆ Pyare Mohan Tiwari ∗ S. Janardhanan ∗∗ Mashuq-un-Nabi ∗∗∗ ∗

Electrical Engineering Department, Amity University, Noida , India (e-mail: pmtiwari@ amity.edu). ∗∗ Electrical Engineering Department, Indian Institute of Technology, New Delhi , India, (e-mail: [email protected]) ∗∗∗ Electrical Engineering Department, Indian Institute of Technology, New Delhi, India, (e-mail: [email protected]) Abstract: In this work, using the finite time sliding mode control, the design of attitude tracking control of a rigid spacecraft has been discussed. The proposed control law has been designed by using the novel fast terminal sliding mode surface with a reaching law. The proposed control law , avoids the singularity possibility, and offers the faster convergence speed in the reaching phase and in the sliding phase. In addition, the applied control input is continuous in nature. The finite time stability has been proved using the Lyapunov theory. In last, to show the effectiveness of the proposed control method, simulations have been conducted under the conditions of mass inertia uncertainty and external disturbances, and the results, in comparison with the existing methods have been demonstrated . Keywords: Rigid spacecraft, Attitude tracking , Finite time convergence, Non-singular fast terminal sliding mode, Lyapunov stability. 1. INTRODUCTION

efficacious in eliminating the singularity problem as well as in enhancing the convergence speed.

Over the last decade, the terminal sliding mode (TSM) (Venkataraman and Gulati, 1991; Yu and Man, 1996) control has drawn the substantial attention in the finite time control design. In the TSM based control design, in contrary to the linear sliding surface used in SMC (John et al., 1993), a nonlinear sliding surface is being adopted. The main attraction with TSM control over the conventional sliding mode control (SMC) is the finite time convergence and the improved steady state accuracy. However, the originally proposed TSM was started with the two shortcomings. Firstly, the singularity in control, and secondly, the slow convergence speed for the remotely located system states.To address the singularity issue, initially the indirect approach was adopted (Yu and Man , 1997, 1998). Then, in (Feng et al., 2002; Yu et al., 2005) , the direct approach, nonsingular terminal sliding mode control (NTSM) was proposed, and its efficacy was tested on robotic manipulator problem. In contrast to the indirect approach, the direct approach modifies the structure of sliding surface to remove the singularity in control. Further, to address the convergence speed issue, in (Yu and Man, 2002) , fast terminal sliding mode (FTSM) control has been applied . This has been observed that alone the FTSM or the NTSM will not be able to eliminate both the shortcomings of TSM. Hence, to eliminate the both of the limitations of the originally proposed TSM; recently, in (Yang et al., 2011), the different structures of the nonsingular fast terminal sliding mode (NFTSM) have been proposed and discussed about performances. The NFTSM control has been found

Recently, inspired by the TSM and its variant properties, considerable research in the area of spacecraft attitude control design has been attracted . In this, the very first contribution was made by Erdong and Zhaowei (2008), and in their work, the authors have proposed two different attitude tracking control laws by using TSM with a discontinuous reaching law. The associated main drawback with these control laws, were the singularity problem , the chattering in control input, and the slow convergence speed for the states starting from the remote point . Ding et al. (2009) designed a attitude stabilization control law by using the feedback linearization and the NTSM control together. Subsequently , Tiwari et al. (2010), by using the FTSM theory, proposed a fast converging attitude control law. In this, to ensure the chattering free control, the control law was designed with the power rate reaching law; although; the singularity issue was not addressed. Since then, sufficient number of attitude control strategies have been designed by applying the FTSM (Zheng et al., 2011; Hu et al., 2012; Zeng and Hu, 2012; Liang et al., 2012; Xiao et al. , 2013) and NTSM (Li et al., 2011; Hu et al., 2012; Cao et al., 2013) control theory . In this series, more recently, Lu and Xia (2013) proposed an attitude stabilization control law by using the nonsingular terminal sliding mode surface .

⋆ This work is supported by the fund of Amity University, India

978-3-902823-60-1 © 2014 IFAC

263

On the basis of the above discussion and the available literature , it can be concluded that by addressing together the convergence speed and the singularity, the finite- time attitude control design is still an open problem. So, here, our endeavour is to propose a finite time attitude control 10.3182/20140313-3-IN-3024.00168

2014 ACODS March 13-15, 2014. Kanpur, India

strategy; which not only guarantees the singularity free control, but also ensures the quick convergence speed irrespective of the initial conditions. The proposed control design has been actualized by proposing a novel NFTSM. The paper is organized as follows. In Section 2, spacecraft mathematical model for tracking control problem is presented, and the control objective is also defined. In Section 3, a finite time control law has been developed by proposing a novel non-singular fast convergent sliding surface; and the finite time stability is proved. Further, in Section 4, the effectiveness of the proposed method is shown by the simulation results; and its comparison with the existing controllers is also discussed . Finally, Section 5 concludes this paper.

[ωd1 ωd2 ωd3 ]T ∈ R3 are the desired attitude frame vector quaternion , scalar quaternion, and angular velocity, respectively. Both qe and qd = [qd0 qd1 qd2 qd3 ]T T 2 2 T satisfy the constraint qev = 1, qdv +qd0 qev +qe0 = 1 and qdv 2 T T × respectively. C = (qe0 −2qev )I +2qev qev −2qe0 qev ∈ R3×3 × R represents the rotation matrix between body fixed reference frame and desired reference frame. Then, using (3), the attitude kinematics and the dynamics equation for the tracking problem could be written as 1 × )ωe q˙ev = (qe0 I + qev 2 1 T ωe q˙e0 = − qev 2

2. PROBLEM FORMULATION In this section, the mathematical model of a rigid spacecraft is discussed. In addition, the control objective is also defined.

(4)

J ω˙e = −(ωe +Cωd )× J(ωe +Cωd )+J(ωe× Cωd −C ω˙d )+u+ud . (5) 2.2 Control Objective

2.1 Mathematical Model The kinematics of a rigid spacecraft attitude could be well represented by the unit quaternion, and is defined as follows (Erdong and Zhaowei, 2008) 1 q˙v = (q0 I + q × )ω 2 1 q˙0 = − q T ω 2

(1)

t→tf

where tf = reaching time(t1 )+sliding time(t2 ).

here, ω = [ω1 ω2 ω3 ]T ∈ R3 is the body angular velocity, qv = [q1 q2 q3 ]T ∈ R3 and q0 ∈ R are the vector and the scalar components of the unit quaternion q = [q0 qvT ]T that satisfy the constraint qvT qv + q02 = 1. The attitude dynamics of a rigid spacecraft is defined by the following equation J ω˙ = −ω × Jω + u + ud (2) 3×3 here, J ∈ R is the symmetric inertia matrix of the considered spacecraft, u = [u1 u2 u3 ] ∈ R3 and ud = [ud1 ud2 ud3 ] ∈ R3 are the control input and total external disturbance input, respectively. Attitude tracking mathematical model To define the attitude kinematics and dynamics equation for tracking control problem, the relative attitude error between reference frame and a desired reference frame is required to be T T established. The error quaternion qe = [qe0 , qev ] ∈ R×R3 3 and the angular velocity error ωe ∈ R are measured from body fixed reference frame to the desired reference frame , and the defining equations are as follows (Lu et al., 2013) × qev = qd0 qv − qdv qv − q0 qdv T qe0 = qdv qv + q0 qdv

ωe = ω − Cωd ,

Here, the control objective is to design a robust control law that ensure the finite time attitude tracking for the rigid spacecraft i.e.   lim (q − qd ) = 0 t→tf (6)  lim (ω − ωd ) = 0,

(3)

where qev = [qe1 qe2 qe3 ]T and qe0 are the vector and scalar part of the error quaternion, respectively, qdv = [qd1 qd2 qd3 ]T ∈ R3 , qd0 ∈ R, and ωd = 264

2.3 Assumptions For the control design purpose, the following assumptions are made. Assumption 1. In the spacecraft mission, the quaternion q and the body angular velocity vector ω are measurable and available throughout the space mission for attitude control design. Assumption 2. The desired attitude frame angular velocity ωd and its first time derivative ω˙d are bounded, and the bounds are known. Assumption 3. Inertia matrix J is having two components, the nominal component J0 and the bounded uncertain component ∆J. Assumption 4. Disturbance torque ud is bounded , and the bound limit is known in advance. 2.4 Useful lemma for finite time stability Lemma 1. (Yu et al., 2005) For a continuous system x˙ = f (x), f (0) = 0, x ∈ Rn , suppose there exists a continuous positive definite function V : Rn → R, a real number a > 0 and α ∈ (0, 1) and an open neighborhood U0 ⊆ Rn of the origin such that an extended Lyapunov description is defined by V˙ (x) + aV (x) + bV α (x) ≤ 0. (7) Then the origin is a finite time stable equilibrium. If U0 = Rn , then the origin is a globally finite time stable

2014 ACODS March 13-15, 2014. Kanpur, India

equilibrium. Further, depending on the initial state x(0) = x0 , the origin can be achieved in finite time t1 given by 1 aV 1−α (x0 ) + b t≤ ln (8) a(1 − α) b

3.2 Control law design

3. FINITE TIME CONTROL DESIGN

To guarantee that the attitude trajectory of the closed loop system (4)-(5) reach onto the sliding surface in finite time , and consequently to converge to the equilibrium in finite time, the proposed control law is

The above-mentioned control objective will be accomplished by the following steps.

u(t) = −k1 η − k2 sig l η + T

(17)

3.1 Sliding surface design

where

To ensure the faster convergence to the equilibria in finite time, the singularity free sliding surface is defined by the following structure

k1 = diag(k11 , k12 , k13 ) with k1i(i=1,2,3) > 0, k2 = diag(k21 , k22 , k23 ) with k2i(i=1,2,3) > 0, l ∈ (0, 1), and

η = sig ρ (ωe ) + c sig ρ (qev ) + d (qev ),

(9)

where, η = [η1 η2 η3 ]T ∈ R3 , c = diag(c1 , c2 , c3 ) with ci=1,2,3 > 0, d = diag(d1 , d2 , d3 ) with di=1,2,3 > 0 , ρ ∈ (1, 2), and for any vector χ = [χ1 χ2 χ3 ]T the following notation is adopted sig ρ (χ) = [|χ1 |ρ sign(χ1 ), |χ2 |ρ sign(χ2 ), |χ3 |ρ sign(χ3 )]T . Finite time stability in sliding phase Theorem 1. Once the attitude trajectory of closed loop system (4)-(5) falls on the sliding surface (9), the attitude states will converge to the equilibrium in finite time. Proof. On the sliding surface, η = 0 is satisfied, therefore sig ρ (ωe ) + c sig ρ (qev ) + d qev = 0 1 ρ

1 ρ

1 ρ

ωe ≤ −c qev − d sig qev Define the Lyapunov function T qev . V1 = qev

Remark 1. Evidently, in (18), two terms are having with fractional power (ρ − 1). Since ρ ∈ (1, 2), the fractional power will be nonnegative, and so, the control structure would be free from any singularity problem, caused either by the initial condition or by the zero crossings . Remark 2. Most importantly, in contrary to (Erdong and Zhaowei, 2008) and the control law (5) in (Lu and Xia, 2013), in (17), the control law is not based on pure sign term; and hence the control law will significantly alleviate chattering problem.

(10) (11)

(12)

The first time derivative of (12) is given by T V˙ 1 = 2qev q˙ev T × = qev (qe0 I + qev )ωe . (13) × Applying the inequality (10), and the fact ||(qe0 I +qev )|| ≤ 1 , the above expression modifies to T V˙ 1 ≤ qev (−c1/ρ (qev ) − d1/ρ sig 1/ρ (qev ))

T = (ωe + Cωd )× J0 (ωe + Cωd ) − J0 (ωe× Cωd − C ω˙d ) ) 2−ρ ( J0 × (18) ωe − (ρ2 c diag|qev |ρ−1 + d) qe0 I + qev 2ρ

(14)

(ρ+1)/2ρ

≤ −c1/ρ V1 − d1/ρ V1

Finite time stability in reaching phase Theorem 2. With the control (17), the attitude trajectory of (4)-(5) will be driven to the neighborhood of η = 0 in finite time. Proof. Define the Lyapunov function 1 V2 = η T Jη, 2 that satisfies 1 1 λmin (J)||η||2 ≤ λmax (J)||η||2 2 2

≤ −λ1 V1 − where λ1 and λ2 are the minimum eigen values of c1/ρ and d1/ρ , respectively. Therefore, (ρ+1)/2ρ ≤0 (15) V˙ 1 + λ1 V1 + λ2 V 1

The above equation (15) satisfies the finite time stability criteria (7), and hence, once the attitude trajectory falls on to the sliding surface, then the attitude states will be able to converge to the equilibrium in finite time (ρ−1)/2ρ

aV (qev (t1 )) + λ2 1 ln 1 , (16) λ1 (ρ − 1)/2ρ λ2 where t1 is the time to cross the reaching phase and to enter in to band η ≤ ±∆. This completes the proof.  t2 ≤

265

(20)

The first time derivative of (19) V˙2 = η T J η˙

(ρ+1)/2ρ λ2 V1

(19)

(21)

Now, evaluating the term J η˙ ( J η˙ = ρ diag(|ωe |ρ−1 ) − (ωe + Cωd )× J0 (ωe (22) ) +Cωd ) + J0 (ωe× Cωd − C ω˙d ) + u )( ) J0 ( × ρ c diag(|qev |ρ−1 ) + d qe0 I + qev ωe + 2 +Lu (∆J, ωe , qe ). where, Lu (∆J, ωe , qe ) represents the lumped uncertainty, and its mathematical expression is

2014 ACODS March 13-15, 2014. Kanpur, India

( Lu = ρ1 diag(|ωe |ρ−1 ) (ωe + Cωd )× ∆J(ωe + Cωd )(23) ∆J (ρ α diag|qev |ρ−1 2ρ ) ) 2−ρ ( × ωe + ud , +β) qe0 I + qev +∆J(ωe× Cωd − C ω˙d ) +

problem (Erdong and Zhaowei, 2008) , and results are illustrated in this section. In addition, the simulation results have been compared with the responses of the existing controllers TSM (11) in (Erdong and Zhaowei, 2008) and NTSM (5) in (Lu and Xia, 2013).

with maximum bound Lumax is known a priori. Further, by using (17) and (18), (22) reduced to ( J η˙ = ρ diag(|ωe |ρ1 −1 ) − k1 η − k2 sig l η) ) +Lu (∆J, ωe , qe )

(24)

Therefore, (21) could be rewritten as ( V˙2 = η T ρ diag(|ωe |ρ−1 ) − k1 η − k2 sig l η ) +Lu (∆J, ωe , qe )

(25)

Now, using the fact that ωe = 0 is not an attractor in the reaching phase (Ding et al., 2009), for the pure condition i.e. Lu (∆J, ωe , qe ) = 0 , we have ( ) ( ) l+1 2 l+1 2λ3 2 ˙ V2 + λ4 V2 2 ≤ 0, (26) V2 + λmax (J) λmax (J) where λ3 and λ4 are the minimum eigen values of ρ diag(|ωe |ρ−1 )k1 and ρ diag(|ωe |ρ−1 )k2 , respectively. Again the structure of (26) conforms the structure of faster finite time stability (Yu et al., 2005); and hence the attitude trajectory will reach the sliding surface in finite time t1 with the faster convergence rate.

The example spacecraft nominal mass inertia matrix is J0 = diag(22.7, 23.3, 24.5) kg.m2 , and the allowed mass inertia uncertainty and the total external disturbances are given by ∆J = diag(2.7, 2.3, 2.5) kg.m2 , ud (t) = [−3 sin(0.5t), 4 sin(0.5t), 6 sin(0.5t)]T × 10−2 N-m, respectively. The simulation cases and the selected controller parameters for the simulations are summarized in Table 1 and Table 2, respectively. In the Case 1, the desired angular velocity is zero, and the desired quaternion is not time varying; hence, the problem is of attitude stabilization. Whereas in the Case 2, the desired angular velocity is varying with the time; hence, the problem is of attitude tracking. Table 1. Simulation cases Cases

1

Initial conditions q(0) = [0.8832, 0.3, − 0.2, − 0.3]T (Lu et al., 2013) qd (0) = [0.693, 0.6157, 0.2652, − 0.2652]T (Lu et al., 2013) ω(0) = [0, 0, 0]T rad/sec. ωd = [0, 0, 0]T rad/sec.

However, for the more realistic case, L (∆J, ωe , qe ) ̸= 0.

q(0) = [0.8832, 0.3, − 0.2, − 0.3]T qd (0) = [1, 0, 0, 0]T (Lu et al., 2013) ω(0) = [0.12, − 0.15, 0.11]T rad/sec. (Erdong and Zhaowei, 2008)   sin(πt/100)

Therefore, on revisiting the (25), we found two structures

ωd =  sin(2πt/100)  × 0.05 rad/sec.

u

( ) V˙2 = −η T Ωk1 η − η T Ω k2 − Lu diag −1 (sig l η) sig l η (27)

2

 

 

sin(3πt/100) (Lu et al., 2013)

Table 2. Controller parameters

and ( ) V˙2 = −η T Ωk2 sig l η − η T Ω k1 − Lu diag −1 (η) η,

(28)

where Ω = ρ diag(|ωe |ρ−1 ). In (27) and (28), by ensuring the positive definiteness of k2 − Lu diag −1 (sig l η) and k1 − Lu diag −1 (η), faster finite time convergence to the neighborhood of η = 0 is guaranteed. Where, the neighborhood region will satisfy the condition η ≤ ∆ = min

{( Lu

This completes the proof.

max

k2

)1/l ( Lu )} max , k1

(29)



4. SIMULATIONS To substantiate the effectiveness of the proposed NFTSM control law (17), simulations are performed on example 266

Controller parameters ρ l c d k1 k2

Values 1.1 0.47 diag(0.42, 0.42, 0.42) diag(1.9, 1.9, 1.9) diag(7, 7, 7) diag(0.5, 0.5, 0.5)

4.1 Controller performance under simulation Case1 The NFTSM controller (17) attitude stabilization performance under the effect of inertia uncertainty and the external disturbances is depicted in Fig.(1-2) . Fig. (1) shows the time evolution of the error quaternion and the body frame quaternion . The angular velocity tracking error, sliding surface, and the control input patterns are shown in Fig.(2). The results reveal that the controller is robust enough to negate the effects of inertia uncertainty and the external disturbances; and ensures also that attitude trajectories reach to the equilibrium with faster convergence speed . The bottom frame of Fig. (2) depicts that

2014 ACODS March 13-15, 2014. Kanpur, India

the control input is continuous in nature. Additionally, through the zoomed frames, it is depicted that without any chattering, the high precision accuracy in the range e-7 has been attained in the steady state. 1.5 1

−7

x 10

qe

2

qe0

0

0.5

−2 20

40

In conclusion, the NFTSM control law (17) performance, is more valuable in terms of total convergence time, steady state accuracy, and the applied control input magnitude and nature.

qe1

60

q

0

e2

q

e3

−0.5

0

20

40

60

80

to the sliding surface |η| < 2e-2 little earlier than with the NFTSM control law (17). Even though; the total convergence time in the case of NFTSM control law, is approximately half of NTSM , and one sixth of TSM . Actually, this illustration, confirms that the sliding surface (9) in NFTSM control law (17) offers the faster convergence speed than the sliding surface in (Erdong and Zhaowei, 2008) and (Lu and Xia, 2013); which is already been proved in (13).

100

1

1.5 1

q

0.1

q

0

qe

1

q2

0

20

40

60

80

0

0.5

−1 40

60

80

0

−0.5

100

time (sec)

−2

40

60

80

100

0

0

20

40

60

80

−0.05

100

0.1

Fig. 1. Error quaternion, absolute quaternion: Case1

−1

0.05

0

q3 −0.5

1

1

q 0

−4

x 10

−5

x 10

ω−ωd

0.5

0.15

0

20

0

20

40

60

80 100

0.5 −4

x 10 1

0.05

0 −1 20

40

60

80

0

u

0

η

Further, for the comparative analysis, simulations for the controllers TSM (11) in (Erdong and Zhaowei, 2008) and NTSM (5) in (Lu and Xia, 2013) are performed for the simulation Case 1, and the results are illustrated in Figs. (3-4). Additionally also; numerical results for the reaching time, the total convergence time, and the steady state accuracy are mentioned in Table 3. The TSM controller and the NTSM controller simulations are conducted with ′ α = 0.66, ci = 0.1, ρ = 0.3, and τ = 2I3 , βi = |Gi=1,2,3 | + 0.007, ki=1,2,3 = 1, respectively.

−0.05 −0.1

0

20

40 60 time (sec)

80

−0.5

100

40 60 80 100 time (sec)

Fig. 3. Control law (11)(Erdong and Zhaowei, 2008) response: Case1

0.3 ω1−ωd1

ω2−ωd2

ω3−ωd3

1.5

0.1 0

−0.2 0.5

1

−7

4 2 0 −2 −4

−0.1

x 10

20

0

20

40

−6

1

40

60

qe

60

100

η2

η3

−0.5

η

1

0

20

10

0

20

40

40

60

80

80 u1

5

60

ω−ωd

x 10

20

−1

0

40

60

60

80

0.5

100

−6

x 10

x 10

1 0

0.5

0

−1

−2 20

40

60

80

0

20

0

40

60

80

100

u2

−0.5

u3

0

20

40 60 time (sec)

80

100

−0.5

0

20

40 60 time (sec)

80

100

50

−3

x 10

0.5

5

−5

0

0

−5

−0.5 20

−10

0

20

40

time (sec)

60

40

60

80

u

u [N−m]

40

−4

2

−7

−0.5

20

0

0 4 2 0 −2 −4

x 10

0 −1

80 η1

0.5

η

ω−ωd

0.2

80

0

40

60

80

100

−50

Fig. 2. Angular velocity, sliding vector, control input: Case1 From Table 3, it is noticeable that the attitude trajectory with the controllers TSM or with NTSM, reaches 267

0

20

40

60

80

100

time (sec)

Fig. 4. Control law (5)(Lu and Xia, 2013) response: Case1

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4.2 Controller performance under simulation Case2

q

q q0, qd0

In this subsection, the tracking performance of NFTSM control law (17) is illustrated; and in similar to the stabilization, the tracking performance is compared with the existing controllers .

d

1 0.5 0

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

Further, similar to the discussion made in the previous subsection; it is obvious from Table 3, that with the NFTSM control law, the reaching phase is slight slower than the NTSM controller (5), but the total convergence time is approximately half. In summary, the proposed control law is superior in terms of robustness, convergence speed and chattering elimination. 1.5

qe

q2, qd2 q3, qd3

−0.5

time (sec)

Fig. 6. Quaternion tracking pattern: Case2

0.2

−5

x 10 2 0 −2 20

0

40

ω1−ωd1 −0.2

0

10

20

30

40

80

ω2−ωd2

50

1

60

60

70

ω3−ωd3 80

90

100

−5

5

x 10

0

−5 20

30

40

50

0 η1 −1

0

10

20

30

40

10

50

60

η2 70

80

η3 90

100

0.05 0 −0.05 −0.1 20

40

0 u1 0

20

40

60

u2 80

u3 100

q

−4

x 10

e0

5 0

qe1

−5 20

40

60

80

Fig. 7. Angular velocity, sliding vector, control input: Case2

qe2

0

qe3 0

20

40

60

80

100

5. CONCLUSION

0.4

In this paper, finite time attitude tracking of a rigid spacecraft has been discussed. The control law has been designed using a novel NFTSM with a faster reaching law. The designed control ensures the finite time stability with faster convergence speed. In addition, the controller robustness has been tested under the subjection of external disturbances and inertia uncertainty. The simulation performances under different initial conditions have been reported, which reveal the improved steady state accuracy, better robustness, faster convergence speed, chattering less non-singular control. The finite stability has been proved using the Lyapunov stability theorem.

0.2 q−qd

0

time (sec)

0.5

q0−qd0 0

q1−qd1 q −q

−0.2

2

d2

q −q −0.4

0.5

−10

1

−0.5

0 −1

ω−ωd

For the comparison view, controllers TSM (11) in (Erdong and Zhaowei, 2008) and NTSM (5) in (Lu and Xia, 2013) simulation results for the initial condition Case2 have been shown in Figs.(8-9). Fig. (8) demonstrates that the controller NTSM (5) took 14.24 sec. to steer the attitude states in to the region (|qei |, |ωei |) < ±2e-3, but the controller TSM (11) couldn’t succeed to drive the attitude states with this much accuracy.

1

η

Next , to give more clarity, the tracking pattern of quaternion has also been shown in Fig. (6), which displays that the desired quaternion has been achieved quickly . The angular velocity tracking error, sliding surface, and the control input time evolution are illustrated in Fig. (7). Through these illustration, it is noticed that the control input is free from chattering , and in finite time t = 6.15 sec. the attitude states have been driven into the steady state region (|qei |, |ωei |) < (±5.5e-4, ±3.03e-5)(see zoomed frame).

0.2 0

u [N−m]

The NFTSM control law (17) tracking performance is demonstrated in Figs. (5-7). Fig.(5) illustrates the time evolution of the error quaternion and the quaternion tracking error; which displays that the faster convergence speed has been maintained also in this case .

q1, qd1

0.4

3

0

20

40

60

80

d3

100

time(sec)

Fig. 5. Error quaternion, tracking quaternion: Case2 268

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Table 3. Comparative performance summary Control law

Steady state value η qe

Reaching time (sec.) (|η| < ±2e-2) I ±4.13e-7 ±2.15e-7 ±4.12e-7 5.51 NFTSM (17) II ±2.66e-4 ±5.50e-4 ±3.03e-5 4.60 I ±1.3e-4 ±1.0e-5 ±1.5e-4 3.18 TSM (11)(Erdong and Zhaowei, 2008) II ±2.44e-4 ±8.12e-2 ±4.42e-3 12.17 I ±8.02e-7 ±8.89e-7 ±1.51e-4 2.86 NTSM (5)(Lu and Xia, 2013) II ±4.71e-3 ±4.72e-3∗∗ ±5.6e-4 3.12 *The attitude trajectory can reaches only to near ±e-2 ** Quaternion reach to 2e-3, but finally settles in the region ±4.72e-3 case

1

Ding, S. and Li, S. (2009) ‘Stabilization of the attitude of a rigid spacecraft with external disturbances using finite-time control techniques’, Aerospace Science and Technology, volume 13, pp.256–265.

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Erdong, J. and Zhaowei, S. (2008) ‘Robust controllers design with finite time convergence for rigid spacecraft attitude tracking control’, Aerospace Science and Technology, volume 12, pp.324–330.

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Feng, Y., Yu, X.H and Man, Z. (2002) ‘Non-singular terminal sliding mode control of rigid manipulator’, Automatica, volume 38, no. 12 pp.2159–2167.

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Guoqiang, Z.,Hu. M. (2012) ’Finite-time control for electromagnetic satellite formations’, Acta Astrnautica, volume. 74, pp.120–130.

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Hung, J.Y., Gao, W. and Hung, J.C. (1993) ‘Variable structure control: A survey’, IEEE. Trans. on Industrial Electronics, volume. 40, pp.1–12.

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Fig. 8. Control law (5)(Lu and Xia, 2013) response: Case2 1

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Lu, Kunfeng, and Yuanqing Xia. (2013) ‘Finite-time attitude stabilization for rigid spacecraft’, International Journal of Robust and Nonlinear Control , doi: 10.1002/rnc.3071.

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Lu, Kunfeng, Yuanqing Xia and Fu. M. (2013) ‘Controller design for rigid spacecraft attitude tracking with actuator saturation’, Information Sciences , volume 220, pp.343– 366.

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Li, S., Wang, Z. and Fei, S. (2011) ‘Comments on paper: Robust controllers design with finite time convergence for rigid spacecraft attitude tracking control’, Aerospace Science and Technology, volume 15, no. 3 pp.193–195.

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Hu, Q., Huo, X., Xiao,B. and Zhang, Z. (2012) ‘Robust finitetime control for spacecraft attitude stabilization under actuator fault’, The Institution of Mechanical Engineers, Part I: Journal of Systems and Control, volume. 226, no. 3 pp.416–428. Hu, Q., Huo, X., Xiao,B.(2012) ‘Reaction wheel fault tolerant control for spacecraft attitude stabilization with finitetime convergence’, International Journal of Robust and Nonlinear Control, volume. 23, no. 15 pp.1737–1752.

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Total convergence time (sec.) ((|qei |, |ωei |) < ±2e-3) 7.07 6.15 41.98 ∗ 17.79 14.24

minal sliding mode’, Advances in Space Research, Vol. 51, no. 12 pp.2374–2393.

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Fig. 9. Control law (11)(Erdong and Zhaowei, 2008) response: Case2 REFERENCES Cao, L., Chen, X. Q. and Sheng, T. (2013) ‘Fault tolerant small satellite attitude control using adaptive non-singular ter269

Liang, H., Sun, Z. Wang, J. (2012) ‘Robust decentralized attitude control of spacecraft formations under time-varying topologies, model uncertainties and disturbances’, Acta Astronautica, volume 81, no. 2 pp.445–455. Man, Z.and Yu, X.H. (1997) ‘Terminal sliding mode control of MIMO linear systems’, IEEE. Trans. on Circuits and Systems, volume 44, no. 11, pp.1065–1070.

2014 ACODS March 13-15, 2014. Kanpur, India

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