# Terminal Sliding Mode Control for Attitude Tracking of Spacecraft Based on Rotation Matrix

Authors: Yong Guo; Shen-Min Song; Xue-Hui Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Two finite-time controllers without unwinding for the attitude tracking control of the spacecraft are investigated based on the rotation matrix, in which a novel modified nonsingular fast terminal sliding manifold is developed to keep tr(R~)≠-1. The first terminal sliding mode controller can compensate external disturbances with known bounds, while the second one can compensate external disturbances with unknown bounds by using an adaptive control method. Since the first terminal sliding mode controller is continuous, it is able to avoid chattering phenomenon. Theoretical analysis shows that both the two controllers can make spacecraft follow a time-varying reference attitude signal in finite time. Numerical simulations also demonstrate that the proposed control schemes are effective.


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## Body

## 1. Introduction

Because of the important applications in many space missions such as in-orbit maintenance and space station installation, the attitude control of the spacecraft has gained extensive interests in recent years. Many researchers have developed various controllers for them based on the unit quaternion [1, 2], dual quaternion [3], and modified Rodrigues parameters (MRP) [4, 5]. However, those representations are unable to represent the set of attitudes both globally and uniquely, so that the controllers usually result in unwinding [6], which means that it will cost extra fuel consumption by the spacecraft traveling a large distance before returning to the desired attitude when the closed-loop system is close to the desired attitude equilibrium.In order to avoid unwinding, many controllers based on the rotation matrix have been developed [7–10]. Weiss et al. [7] derived two controllers for spacecraft attitude tracking problems without unwinding. Sanyal et al. [8] gave a continuous controller for spacecraft attitude tracking of arbitrary continuously differentiable attitude trajectories. Chaturvedi et al. [9] presented a complete analysis for two problems on the stabilization of the inverted 3D pendulum. Because controllers [7–9] adopted the attitude error function e=0.5tr⁡(I-R~) which was not proportional to the rotation angle, the performance of those controllers became worse when the initial error became larger. To deal with this problem, Lee [10] proposed an attitude controller by a new attitude error function for the tracking control system on SO(3). Since the aforementioned controllers provided asymptotic stability [7–9] and exponential stability [10], the systems could converge to the equilibrium only when the time goes to infinity. Thus, the finite-time stabilization implying better performance is attracting more and more attention.Terminal sliding mode (TSM) control is one of finite-time control schemes that provide faster convergence speed. However, TSM control has three disadvantages that are singularity, chattering, and slower convergence speed when the system state is far away from the equilibrium. Therefore, many improved TSM algorithms have been developed. Feng et al. [11] gave a nonsingular TSM controller for a second-order system to eliminate the singularity problem. Yu and Zhihong [12] designed a fast TSM controller combining advantages of the traditional linear hyperplane-based sliding mode control and TSM control to improve the convergence speed. The discontinuous controllers [11, 12] can lead to chattering which may arouse high-frequency unmodeled plant dynamics. In general, the boundary layer approach is used to eliminate the chattering; however, the finite-time stability will be lost in the boundary layer. Yu et al. [13] proposed a continuous finite-time controller that can enjoy benefits of both high precision and chattering attenuation by properly choosing the fractional powers.TSM control has led wide applications in attitude tracking of the spacecraft. A number of research works have been reported on this topic in the past decades. Wu et al. [14] investigated two robust sliding mode controllers based on the quaternion and Lagrange-like model to solve the spacecraft attitude tracking control problem. Lu and Xia [15] investigated the attitude tracking control problem for the rigid spacecraft under input saturation with finite-time stabilization. Zou and Kumar [16] proposed a distributed attitude coordination control scheme using TSM control for a group of spacecraft in the presence of external disturbances. Pukdeboon [17] proposed two second-order sliding mode controllers to solve the attitude tracking control of a spacecraft with external disturbances and inertia uncertain.To the best knowledge of the authors, there are rarely approaches can provide finite-time control without unwinding for a spacecraft except our previous article [18]. In our previous article, we investigated two finite-time controllers for the attitude control of the spacecraft by an adaptive backstepping method. Because the two controllers were designed in the region of attraction, it needs to be further studied.To overcome these drawbacks, we investigate the finite-time control by using TSM control. Compared with the listed literatures, the contributions are summarized as follows. (i) A novel modified nonsingular fast terminal sliding manifold is developed to keeptr⁡(R~)≠-1. (ii) Two controllers without unwinding are proposed for the attitude control of the spacecraft by using TSM control. (iii) Compared with the controllers in [18], the two controllers are almost globally finite-time stabilization.This paper is organized as follows. An attitude dynamic model is established in the following section. In Section3, a state error is given. Then, two controllers are proposed. Furthermore, the corresponding stability proofs are given as well. Numerical simulations are presented in Section 4. The paper is closed with some concluding remarks.


## 2. Spacecraft Attitude Dynamics

The spacecraft is modeled as rigid body based on the rotation matrix. Specifically, the equation of motion of the spacecraft is defined as (1) and (2). Here, R is the rotation matrix that transforms the body frame into the inertial frame resolved in the body frame. ω∈R3×1 is the angular velocity in the body frame. d∈R3×1 and u∈R3×1 are the external disturbance torque and control torque, respectively. J∈R3×3 is the inertia matrix. Consider
(1)R˙=Rω×,(2)Jω˙=-ω×Jω+u+d,(3)ω×=0-ω3ω2ω30-ω1-ω2ω10.R
d and ωd∈R3×1 are the reference attitude and angular velocity in the reference frame, respectively. R~=RdTR and ω~=ω-R~Tωd are the rotation matrix error and angular velocity error in the body frame. Because R~ is a matrix, it cannot be used to design the controller directly. A new attitude error is constructed in [10] which is defined as (4). The map∨ transforms a skew-symmetric matrix to a vector. For example (a×)∨=a and (A∨)×=A, where a∈R3×1 and A is a skew-symmetric matrix. Consider
(4)eR~=121+tr⁡(R~)R~-R~T∨.In combination with (1)–(4), the equation of motion of the spacecraft is given by the following equations:
(5)e˙R~=Eω~,(6)Jω~˙=F+u+d,(7)F=-ω~+R~Tωd×Jω~+R~Tωd+Jω~×R~Tωd-JR~Tω˙d,(8)E=121+tr⁡(R~)(tr⁡(R~)I-R~T+2eR~eR~T).


## 3. Design of the Controller

We employ the idea of the finite-time control to design the robust controllers for the attitude control of the spacecraft by using TSM control. The following lemmas are useful to design the finite-time controllers.Lemma 1 (see [10]).
LetRdTR=exp⁡(x×), and there exists x∈R3 with x≤π. The matrix 2-norm of E is E=1/2. Furthermore, if x≠π, E is an invertible matrix, eR~<1, and -1<tr⁡(R~)≤3.Lemma 2 (see [13]).
Supposing thatα1,α2,…,αn are all positive numbers and 0<ρ<2, then the following inequality holds:
(9)α12+⋯+αn2ρ≤α1ρ+⋯+αnρ2.Lemma 3 (see [13]).
Supposing thatV˙(t)≤-αV(t)-βV(t)γ, ∀t1≥t0, where α>0, β>0, 0<γ<1, and V(t) is a continuous positive definite function, then the system converges to the equilibrium point in finite time:
(10)t1≤t0+1α(1-γ)ln⁡αVt01-γ+ββ.Assumption 4.
d,  ωd , and  ω˙d  are assumed to satisfy d≤dmax⁡ and ωd+ω˙d≤ωdmax⁡, respectively. Here, dmax⁡ and ωdmax⁡ are positive constants.In order to keepeR~<1, we design the fast TSM as (11), where α, β, λ, and η are positive constants, 0<γ<1,  r1=(2-γ)ηγ-1,  r2=(γ-1)ηγ-2,  f(eR~)=[f(eR~,1),f(eR~,2),f(eR~,3)]T, and sig(eR~,i)γ=eR~,iγsign(eR~,i). Consider
(11)S=e˙R~+αeR~+βfeR~+λln⁡11-eR~TeR~eR~,(12)f(eR~,i)=r1eR~,i+r2sign(eR~,i)eR~,i2eR~,i≤η,i=1,2,3sigeR~,iγothers.Based on the TSM control, the control law for the spacecraft is given by the following equation:(13)u=-JE-1E˙ω~+αe˙R~+2λeR~Te˙R~1-eR~TeR~eR~+λln⁡11-eR~TeR~e˙R~hhhhhhhh2λeR~Te˙R~1-eR~TeR~eR~+λln⁡11-eR~TeR~+βf˙eR~+k1S+k2sigSγ-F.Theorem 5.
Consider a spacecraft described by (5)-(6) under Assumption 4 where dmax⁡ is a given positive constant. By applying the proposed control scheme (13), the following results are achieved.(i)
The sliding manifoldS converges to the region S≤Δ in finite time. Here, b and b1 are positive constants satisfying k1-b>0 and b1=fmax⁡2/4b, respectively. fmax⁡ is the maximum value of EJ-1d. Consider
(14)Δ=min⁡b1k1-b1/2,b1k21/1+γ.(ii)
The errorseR~,i and e˙R~,i converge to the regions eR~,i≤ΔeR~ and e˙R~,i≤Δe˙R~ in finite time, respectively. Consider
(15)ΔeR~=max⁡⁡η,min⁡⁡Δα1/2,Δβ1/(1+γ),(16)Δe˙R~=Δ+λln⁡11-3ΔeR~2+αΔeR~+βΔeR~γ.Proof.
We choose the Lyapunov function asV1=(1/2)STS. Applying (13) and (5)–(7), the derivative of V1 can be written as
(17)V˙1=STS˙=STE˙ω~+EJ-1(F+u+d)+αe˙R~+2λeR~Te˙R~1-eR~TeR~eR~+λln⁡11-eR~TeR~e˙R~+βf˙(eR~)≤-k1STS-k2STsigSγ+SEJ-1d≤-(k1-b)STS-k2STsigSγ+b1.
To deal withb1, V˙1 can be rewritten as (18)-(19). We will discuss these situations in Cases 1-2. Consider
(18)V˙1≤-k1-b-b1STSSTS-k2STsigSγ,(19)V˙1≤-k1-bSTS-k2-b1STsigSγSTsigSγ.
Case 1. Assuming η1=k1-b-b1/STS and η2=k2, (18) can be rewritten as (20). If η1>0 and η2>0, the sliding manifold S will converge to the region S≤(b1/(k1-b))1/2 in finite time by using Lemma 3. Consider
(20)V˙1≤-η1STS-η2STS(γ+1)/2.
Case 2. Assuming μ1=k1-b and μ2=k2-b1/STsig(S)γ, (19) can be rewritten as (21). If μ1>0 and μ2>0, the sliding manifold S will converge to the region S≤(b1/k2)1/1+γ in finite time by using Lemma 3. Consider
(21)V˙1≤-μ1STS-μ2STS(γ+1)/2.
Now, (i) has been proved.
The sliding manifoldS converges to the region S≤Δ in finite time, which means that Si≤Δ, i=1,2,3. Therefore, the stability analysis of eR~,i and e˙R~,i is as follows:
Case 1. If eR~,i≤η, we can get (22) by using (11)-(12). Consider
(22)Si=e˙R~,i+αeR~,i+λln⁡11-eR~TeR~eR~,i+βr1eR~,i+r2signeR~,ieR~,i2=ϕi,ϕi≤Δ,(23)e˙R~,i≤Δ+λln⁡11-3η2+αη+βηγ≤Δe˙R~.
From (23), we can get that e˙R~,i converges to e˙R~,i≤Δe˙R~ in finite time.
Case 2. If e˙R~,i>η, we can get (24) by using (11)-(12). Consider
(24)Si=e˙R~,i+αeR~,i+λln⁡11-eR~TeR~eR~,i+βsigeR~,iγ=ϕi.
We choose the Lyapunov function asV2,i=(1/2)eR~,i2. Applying (24), the derivative of V2,i can be written as follows:
(25)V˙2,i=eR~,ie˙R~,i=-eR~,iαeR~,i+λln⁡11-eR~TeR~eR~,ihhhhhhhln⁡11-eR~TeR~+βsigeR~,iγ-ϕi≤-eR~,iαeR~,i+βsigeR~,iγ-ϕi≤-αeR~,i2-βeR~,isigeR~,iγ+ϕi.
To deal withϕi, V˙2,i can be rewritten as follows:
(26)V˙2,i≤-α-ϕieR~,i2eR~,i2-βeR~,isigeR~,iγ,V˙2,i≤-αeR~,i2-β-ϕieR~,isigeR~,iγeR~,isigeR~,iγ.
Ifα-ϕi/eR~,i2>0, eR~,i converges to eR~,i≤(Δ/α)1/2 in finite time and if β-ϕi/eR~,isig(eR~,i)γ>0, eR~,i converges to eR~,i≤(Δ/β)1/(1+γ) in finite time. Consider
(27)e˙R~,i≤ϕi+λln⁡11-eR~TeR~+αeR~,i+βeR~,iγ=Δe˙R~.
Furthermore, from (27) we can get that e˙R~,i converges to e˙R~,i≤Δe˙R~ in finite time.
Now, (ii) has been proved.Remark 6.
From (14)-(15), it is concluded that the controller parameters k1 and k2 determine the accuracy of Si and the controller parameters α and β determine the final accuracy of eR~,i. In order to get small Si and eR~,i, we can select large k1, k2, α, and β.In Theorem5, it is assumed that dmax⁡ is given. However, in applications, dmax⁡ is hard to know. In order to deal with a time-varying unknown bounded disturbance, we design the discontinuous controller (28)-(29), where d^max⁡ is the estimation value of dmax⁡ and d~max⁡=dmax⁡-d^max⁡. Consider
(28)u=-JE-1E˙ω~+αe˙R~+2λeR~Te˙R~1-eR~TeR~eR~+λln⁡11-eR~TeR~e˙R~hhhhhhhh2λeR~Te˙R~1-eR~TeR~+βf˙eR~+k1S+k2signS-d^max⁡sign(J-TETS)-F,(29)d^˙max⁡=STEJ-1.Theorem 7.
Consider a spacecraft described by (5)-(6) under Assumption 4, where dmax⁡ is an unknown positive constant. By applying the proposed control scheme (28)-(29), S and d~max⁡ are all bounded.Proof.
We choose the Lyapunov function asV3=(1/2)STS+(1/2)d~max⁡2. Applying (28)-(29) and (5)–(7), the derivative of V3 can be written as follows:
(30)V˙3=STS˙+d~max⁡d~˙max⁡=STE˙ω~+EJ-1(F+u+d)+2λeR~Te˙R~1-eR~TeR~eR~+αe˙R~+λln⁡11-eR~TeR~e˙R~+βf˙(eR~)-d~max⁡d^˙max⁡≤-k1STS-k2STsignS+STEJ-1d~max⁡-d~max⁡d^˙max⁡≤0.
It can be seen thatV˙3≤0. Thus, it can be concluded that variables S and d~max⁡ are all bounded.Theorem 8.
Consider a spacecraft described by (5)-(6) under Assumption 4, where dmax⁡ is an unknown positive constant. By applying the proposed control scheme (28)-(29), the following results are achieved.(i)
The sliding manifoldS converges to S=0 in finite time.(ii)
The errorseR~,i and e˙R~,i converge to the regions eR~,i≤Δ1eR~ and e˙R~,i≤Δ1e˙R~ in finite time, respectively. Consider
(31)Δ1eR~=η,(32)Δ1e˙R~=λln⁡11-3Δ1eR~2+αΔ1eR~+βΔ1eR~γ.Proof.
We choose the Lyapunov function asV4=(1/2)STS. Applying (28)-(29) and (5)–(7), the derivative of V4 can be written as
(33)V˙4=STS˙=STE˙ω~+EJ-1(F+d)+2λeR~Te˙R~1-eR~TeR~eR~+αe˙R~+λln⁡11-eR~TeR~e˙R~+βf˙(eR~)+STEJ-1u≤-k1STS-(k2-d~max⁡)S.
If k2-d~max⁡>0, then, by using Lemma 3, we can get that the sliding manifold S will converge to S=0 in finite time.
Therefore, the stability analysis ofeR~,i and e˙R~,i is as follows.
Case 1. If eR~,i≤η, we can get (34) by using (11)-(12). Consider
(34)Si=e˙R~,i+αeR~,i+λln⁡11-eR~TeR~eR~,i+βr1eR~,i+r2signeR~,ieR~,i2=0,(35)e˙R~,i≤λln⁡11-3η2+αη+βηγ≤Δ1e˙R~.
From (35), we can get that e˙R~,i converges to e˙R~,i≤Δ1e˙R~ in finite time.
Case 2. If e˙R~,i>η, we can get (36) by using (11)-(12). Consider
(36)Si=e˙R~,i+αeR~,i+λln⁡11-eR~TeR~eR~,i+βsigeR~,iγ=0.
We choose the Lyapunov function asV5,i=(1/2)eR~,i2. Applying (36), the derivative of V5,i can be written as follows:
(37)V˙5,i=eR~,ie˙R~,i=-eR~,iαeR~,i+λln⁡11-eR~TeR~eR~,i+βsigeR~,iγ≤-eR~,iαeR~,i+βsigeR~,iiγ≤-αeR~,i2-βeR~,isigeR~,iγ.
It can be seen thateR~,i converges to eR~,i≤η in finite time. Consider
(38)e˙R~,i≤λln⁡11-eR~TeR~+αeR~,i+βeR~,iγ=Δ1e˙R~.
Furthermore, from (38), we can get that e˙R~,i converges to e˙R~,i≤Δ1e˙R~ in finite time.
Now, (ii) has been proved.
Now, Theorem8 has been proved.Remark 9.
In Theorems7 and 8, d~max⁡ does not converge to the region near zero in finite time. It just guarantees that d~max⁡ is bounded.Remark 10.
Owing to the novel modified nonsingular fast terminal sliding manifold, we can easily ensure thattr⁡(R~)≠-1 from the proofs. Compared with [10, 18], we do not need to consider the region of attraction in the controllers. Because the system initial value can not be tr⁡(R~)≠-1, the two controllers are almost globally finite-time stabilization.


## 4. Simulations

In this section, the simulation is given to illustrate the theoretical results. To validate the effectiveness of the proposed two controllers, numerical simulations are given in the following scenario. The spacecraft tracks a time-varying reference signal.The model parameters and initial value for the spacecraft are defined as follows:(39)J=22.70.2-0.50.223.30.3-0.50.324.5kg·m2,R0=-0.5414-0.7072-0.4546-0.0009-0.54030.8415-0.84070.45600.2919,ω(0)=[0.1,0.1,0.1]rad/s.The initial value and reference angular velocity for the spacecraft are as follows:(40)Rd0=I,ωd=0.1sin⁡t40,-0.1cos⁡⁡t50,-0.1sin⁡t60Trad/s.For the dynamic model described by (5) and (6), it should be noted that system exists external disturbance. The disturbance torque d in (6) is defined as follows:
(41)d=2×10-3sin⁡0.1t,cos⁡⁡0.2t,sin⁡(0.2t)TN·m.We select parameters of the first controller asα=0.01, β=0.15, k1=0.1, k2=0.001,  and γ=0.8. Simulation results of the spacecraft system under the controller (13) are shown in Figures 1–3, from which it can be seen that the maneuver can be completed in less than 70 seconds. Figure 1 gives the attitude curves of the spacecraft. As tr(R~(0))≠-1, the controller (13) is effective. Figure 2 gives the cures of e˙R~. Figure 3 plots the curves of control torque of the system, from which it can be seen that the controller is continuous without chattering. It is easy to find that the controller (13) can obtain better performance when the attitude tracking is performed.Figure 1
The curves ofeR~ under the first controller.Figure 2
The curves ofe˙R~ under the first controller.Figure 3
The control torques of the system under the first controller.To validate the second controller, numerical simulations are given as follows. The parameters of the second controller are selected the same as the first one. In order to avoid chattering, we use saturation function to take the place of sign function. Simulation results of the spacecraft system under the controller (28) are shown in Figures 4, 5, 6, and 7. We can see that the attitude maneuver of the spacecraft can be completed in less than 40 seconds. Compared with Figures 3 and 6, it can be seen that the control torques of the both controllers are similar. Compared with Figures 1 and 4, the second controller has more excellent performance than the first controller.Figure 4
The curves ofeR~ under the second controller.Figure 5
The curves ofe˙R~ under the second controller.Figure 6
The control torques of the system under the second controller.Figure 7
The estimated value of the disturbance torque under the second controller.In order to get similar performance as the second controller, the parameters of the first controller are chosen asα=0.01, β=0.15, k1=0.1, k2=1, and γ=0.8. Figure 8 gives the attitude curves of the spacecraft. Figure 9 gives the curves of control torque of the system. Compared with Figures 6 and 9, the first controller needs larger control torques. The results of simulation show that both controllers can obtain better performance when the attitude tracking is performed and the second controller has more excellent performance than the first controller.Figure 8
The curves ofeR~ under the first controller with k2=1.Figure 9
The control torques of the system under the first controller withk2=1.To further research, the first controller is compared with the controllers(21)–(23) in [18]. We chose the parameters of the first controller as α=0.01, β=0.23, k1=0.35, k2=0.002, and γ=0.8 to get similar performance as the controller in [18]. Figures 10 and 11 give curves of the attitude and control torque of the spacecraft. Figures 12 and 13 give the corresponding curves of the controller in [18]. Compared with Figures 11 and 13, the first controller needs smaller control torques. We can see that both controllers can obtain better performance when the attitude tracking is performed and the first controller outperforms the controller in [18] clearly.Figure 10
The curves ofeR~ under the first controller with k2=0.002.Figure 11
The control torques of the system under the first controller withk2=0.002.Figure 12
The curves ofeR~ under the controller in [18].Figure 13
The control torques of the system under the controller in [18].


## 5. Conclusions

This work develops two robust finite-time controllers without unwinding for the spacecraft. The first proposed robust controller is continuous to avoid the chattering. With the use of adaptive control, the second proposed robust controller does not need the bounds of external disturbances. For both controllers, we can get that the overall closed-loop system is finite-time stability by Lyapunov’s theorems. Simulations have shown that the controllers can make the spacecraft follow a time-varying reference attitude signal without unwinding in finite time.



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