# Finite Time Output Feedback Attitude Tracking Control for Rigid Body Based on Extended State Observer

Authors: Meng Duan; Yingmin Jia
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the attitude tracking control problem of output feedback is investigated. A finite time extended state observer (FTESO) is designed through the homogeneous Lyapunov method to estimate the virtual angular velocity and total disturbances. Based on these estimated states, a finite time attitude tracking controller is developed. The numerical simulations are given to illustrate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

Recently, the issue of attitude control of rigid body has received more and more attention due to its wide practical applications such as spacecraft, robotics, and maglev planar motor [1–3]. Also, the complex nonlinearity of the attitude dynamics raises potential value in theory. A great deal of control methods have been proposed to deal with the attitude control problems.In [3], two time-varying terminal sliding mode attitude control algorithms were designed to deal with the system uncertainties and external disturbances. In [4], an adaptive regulation technique was proposed to reject the disturbances whose frequency is unknown. In [5], another adaptive attitude tracking control was implemented for the stability of a rigid spacecraft with time-varying inertia components. In [6], a backstepping method was used to design the robust attitude control for the agile satellite.Various attitude control algorithms above need the information of full states including attitudes and angular velocities for controller design. However, in actual engineering applications, due to energy cost, weight, and other factors, an angular velocity sensor cannot be installed in the mechanical systems, or the angular velocity sensors fail to work. Thus, the design of output feedback attitude control in which just attitude measurements are available is required.Many great research studies have been proposed to deal with the problem of output feedback attitude control. These methods can be divided into two strategies: the passivity-based controller and the observer-based controller. In [7], the passivity of attitude dynamics represented by quaternion was first proved and a passivity-based attitude control was proposed to guarantee the stability of rigid body attitude. This work was further researched in [8]; the passivity of attitude dynamics represented by modified Rodrigues parameters was proven. In [9], an auxiliary dynamical system was derived to compensate the unknown angular velocities. In [10], a passivity filter which adopted attitude Euler angles was introduced; compared to other passivity-based controllers, the proposed output feedback controller can deal with external disturbances.The examples of observer-based attitude controller can be found in [11–14]. In [11], a finite time observer was proposed through geometric homogeneity and Lyapunov methods; a virtual angular velocity is introduced in this method. In [12], a continuous finite time angular velocity observer was designed, but the settling time could not be given. In [13], an observer was designed by adding a power integrator, thus the settling time can be estimated and the disturbances could be rejected by this method. In [14], a continuous terminal sliding mode observer was designed.However, the u-modeled system dynamics and external disturbances are also the difficulties in the attitude control design. In [15–17], the unknown dynamics and external disturbances are addressed by simple structure estimators for nonlinear systems. Extended state observer (ESO) is another method to compensate these problems. The main characteristic of ESO is that all disturbances and uncertainties can be regarded as total disturbances. The linear ESO (LESO) was introduced in [18] and the structure of ESO became more concise and simple, thus ESO is preferred by engineering applications whose plants are too complex [19–21].Inspired by [22, 23], a FTESO is introduced for the attitude control without angular velocity in this paper. The main contributions of this article are listed as follows:Compared with other observer-based controllers, the presented control scheme can estimate angular velocity and disturbances of system simultaneously. The disturbances of the attitude system can be compensated by the outputs of proposed FTESO when the attitude controller is designed.A finite time attitude controller which combined with FTESO is designed; it is an improvement of [11]. Due to the introduction of ESO, the proposed control framework has higher control accuracy and faster response.This article is organized as follows. In Section2, system dynamical models, some definitions, and lemmas are shown; the proposed FTESO-based controller is shown in Section 3; simulations are given in Section 4 and Section 5 shows the conclusions and future work.


## 2. System Models and Preliminaries

### 2.1. Equations of Attitude Kinematics and Dynamics

In this section, attitude system equations of a rigid body, which include attitude kinematics and dynamics, are presented.The modified Rodrigues parameters (MRPs) are introduced to express the attitude of the rigid body; the equations of attitude kinematics are given as follows:(1)σ˙=Tσω,where σ=σ1,σ2,σ3T∈R3 is the MRPs which denotes the rotation from the body frame to the inertial frame and ω=ω1,ω2,ω3T∈R3 is the angular velocity of the rigid body expressed in the body frame. The matrix Tσ is defined as follows:(2)Tσ=141−σTσI3×3+2σ×+2σσT,where the operator σ× represents the skew symmetric matrix of vector σ and has the following form:(3)σ×=0−σ3σ2σ30−σ1−σ2σ10.The attitude dynamics of rigid body is modeled as:(4)Jω˙=−ω×Jω+τ+d,where τ=τ1,τ2,τ3T∈R3 denotes the control torque, d=d1,d2,d3T∈R3 denotes the external disturbances applied to the rigid body, and J∈R3×3 denotes the inertial matrix of the rigid body which is symmetric.Denote thatx1=σ, x2=σ˙, x2 is the first time derivative of the attitude MRPs and is introduced as the virtual angular velocity, then equations (1) and (3) can be transformed to the following equations:(5)x˙1=x2,x˙2=fx1,x2+gx1τ+d∗t,where fx1,x2=T˙Px2−TJ−1Px2×JPx2 is defined as system function, P=T−1x, gx1=TJ−1, and d∗t=gx1d are the external disturbances in system equation (4).


### 2.2. Definitions and Lemmas

For simplicity, some definitions about vectorx=x1,x2,…,xnT∈Rn are given:(6)xα=x1α,x2α,…,xnαT,signx=signx1,signx2,…,signxnT,diagx=diagx1,x2,…,xn,where sign⋅ represents the signum function and diag⋅ represents the diagonal matrix.Lemma 1 (see [24]).
For anyx∈R,y∈R,a>0,b>0, xayb≤cxa+b/a+b+aya+b/a+b.Lemma 2 (see [25]).
For anyxi∈R,i=1,2,…,n, r∈0,1, ∑i=1nxir≤∑i=1nxir≤n1−r∑i=1nxir.Lemma 3 (see [22]).
Consider the system like equation (4); assume that a Lyapunov function Vx is defined on a neighbourhood of U⊂Rn of the origin, and(7)V˙x+l1Vxm+l2Vxn<0,x∈N\0,where m≥1,n∈0,1andl1>0,l2>0. Then, the system is locally finite time stable, and the Vx which starts from U can arrive at Vx≡0 in finite time T:(8)T≤1l21−nV01−nF1,1−nm−n,1+1−nm−n,−l1l2V0m−n,where V0 is the starting value of Vx, and F⋅ denotes the Gaussian hypergeometric function.


## 2.1. Equations of Attitude Kinematics and Dynamics

In this section, attitude system equations of a rigid body, which include attitude kinematics and dynamics, are presented.The modified Rodrigues parameters (MRPs) are introduced to express the attitude of the rigid body; the equations of attitude kinematics are given as follows:(1)σ˙=Tσω,where σ=σ1,σ2,σ3T∈R3 is the MRPs which denotes the rotation from the body frame to the inertial frame and ω=ω1,ω2,ω3T∈R3 is the angular velocity of the rigid body expressed in the body frame. The matrix Tσ is defined as follows:(2)Tσ=141−σTσI3×3+2σ×+2σσT,where the operator σ× represents the skew symmetric matrix of vector σ and has the following form:(3)σ×=0−σ3σ2σ30−σ1−σ2σ10.The attitude dynamics of rigid body is modeled as:(4)Jω˙=−ω×Jω+τ+d,where τ=τ1,τ2,τ3T∈R3 denotes the control torque, d=d1,d2,d3T∈R3 denotes the external disturbances applied to the rigid body, and J∈R3×3 denotes the inertial matrix of the rigid body which is symmetric.Denote thatx1=σ, x2=σ˙, x2 is the first time derivative of the attitude MRPs and is introduced as the virtual angular velocity, then equations (1) and (3) can be transformed to the following equations:(5)x˙1=x2,x˙2=fx1,x2+gx1τ+d∗t,where fx1,x2=T˙Px2−TJ−1Px2×JPx2 is defined as system function, P=T−1x, gx1=TJ−1, and d∗t=gx1d are the external disturbances in system equation (4).


## 2.2. Definitions and Lemmas

For simplicity, some definitions about vectorx=x1,x2,…,xnT∈Rn are given:(6)xα=x1α,x2α,…,xnαT,signx=signx1,signx2,…,signxnT,diagx=diagx1,x2,…,xn,where sign⋅ represents the signum function and diag⋅ represents the diagonal matrix.Lemma 1 (see [24]).
For anyx∈R,y∈R,a>0,b>0, xayb≤cxa+b/a+b+aya+b/a+b.Lemma 2 (see [25]).
For anyxi∈R,i=1,2,…,n, r∈0,1, ∑i=1nxir≤∑i=1nxir≤n1−r∑i=1nxir.Lemma 3 (see [22]).
Consider the system like equation (4); assume that a Lyapunov function Vx is defined on a neighbourhood of U⊂Rn of the origin, and(7)V˙x+l1Vxm+l2Vxn<0,x∈N\0,where m≥1,n∈0,1andl1>0,l2>0. Then, the system is locally finite time stable, and the Vx which starts from U can arrive at Vx≡0 in finite time T:(8)T≤1l21−nV01−nF1,1−nm−n,1+1−nm−n,−l1l2V0m−n,where V0 is the starting value of Vx, and F⋅ denotes the Gaussian hypergeometric function.


## 3. Main Results

### 3.1. Finite Time Extended State Observer Design

In this section, only the attitude MRPsσ is the available information; an ESO which ensures finite time convergence is presented to estimate the virtual angular velocity x2 and the external disturbances d∗t.Designate the external disturbancesd∗t as an extended state x3; the system in equation (4) can be transformed to a new third-order dynamics equation which is shown in the following formulas:(9)x˙1=x2,x˙2=x3+fx1,x2+gx1τ,x˙3=wt,where wt=d˙∗t, and the following assumption is given.Assumption 1.
The external disturbancesd∗t are unknown but continuously differential and bounded; its first time derivative wt is also unknown and bounded; the following inequalities hold:(10)d∗t≤d∗¯,wt≤w¯,where d¯∗ and w¯ represent the upper bound of d∗t and wt, respectively.
Letz1,z2, and z3 represent the estimates of x1,x2, and x3 in equation (7), respectively, then the FTESO is designed as follows:(11)z˙1=z2+ρϵ1α1signϵ1+ρϵ1β1signϵ1+θ1signϵ1,z˙2=z3+fx1,z2+gx1τ+ρ2ϵ1α2signϵ1+ρ2ϵ1β2signϵ1+θ2signϵ1,z˙3=ρ3ϵ1α3signϵ1+ρ3ϵ1β3signϵ1+θ3signϵ1,where ϵ1=x1−z1 represents the estimation error of attitude MRPs x1, 0<ρ<+∞, 0.5<α1<1, α2=2α1−1, α3=3α1−2,β1=1/α1, and β2=1/β1+β1−1,β2=2/β1+β1−2, and θ1, θ2, and θ3 are the parameters to deal with the external disturbances.
Letϵ2=x2−z2, ϵ3=x3−z3, and the error dynamical equations of the proposed FTESO is expressed as:(12)ϵ.1=ϵ2−ρϵ1α1signϵ1−ρϵ1β1signϵ1−θ1signϵ1,ϵ.2=ϵ3+fx1,x2−fx1,z2−ρ2ϵ1α2signϵ1−ρ2ϵ1β2signϵ1−θ2signϵ1,ϵ.3=ωt−ρ3ϵ1α3signϵ1−ρ3ϵ1β3signϵ1−θ3signϵ1.
The following compact set is constructed for any constantΔ:(13)Ω1=x1,x2,z1,z2x1≤Δ,x2≤Δ,z1≤Δ,z2≤Δ.
This compact set can be easily obtained by designing a proper desired attitude reference trajectory. If the setΩ1 is established, there will exist a positive constant λ such that fx1,x2−fx1,z2≤λϵ2.
The stability properties of the extended state observer is presented in the following theorem.Theorem 1.
Consider the error dynamical system given by equation (10), where 0.5<α1<1, α2=2α1−1, α3=3α1−2,β1=1/α1, β2=1/β1+β1−1,andβ2=2/β1+β1−2; if the compact set Ω1 holds, ρ, θ1, θ2, and θ3 are sufficiently large, then the estimation errors would converge to a residual region in finite time.Proof.
Consider the error dynamics in equation (10); ignore the terms ρϵ1β1signϵ1signϵ1−θ1signϵ1, −ρ2ϵ1β2signϵ1−θ2signϵ1, and wt−ρ3ϵ1β3signϵ1−θ3signϵ1, so equation (10) can be reduced to the following form:(14)ϵ.1=ϵ2−ρϵ1α1signϵ1,ϵ.2=ϵ3+fx1,x2−fx1,z2−ρ2ϵ1α2signϵ1,ϵ.3=−ρ3ϵ1α3signϵ1.
It can be acquired that equation (12) is homogeneous of degree α1−1<0 with respect to the weight 1,α1,2α1−1.
The Lyapunov function is proposed as follows:(15)Vαζ1,ζ2,ζ3=ζTPζ.where ζ=ζ1,ζ2,ζ3T=ϵ11/φsignϵ1,ϵ21/φα1signϵ2,ϵ31/φ2α1−1signϵ3T, 0.5<φ=α1<1,P denotes a positive definite symmetric matrix. Make fα represent the vector field of system equation (12); it can be obtained that Vαζ and LfαVαζ are homogeneous of degrees 2/φ and 2/φ+α1−1 with the same weights in system equation (12), respectively.
According to [26, 27], the inequality is obtained as follows:(16)LfαVαζ≤μ1α,θVaζγ1,where μ1=−maxy:Vαy=1LfaVαy, γ1=1+α1φ/2−α1φ/2≤1, and limα1⟶1μ1≥ρ/λmaxP.
Same as the previous analysis, ignore the termsϵ2−ρϵ1α1signϵ1−θ1signϵ1, ϵ3+fx1,x2−fx1,z2−ρ2ϵ1α2signϵ1−θ2signϵ1, and ωt−ρ3ϵ1α3signϵ1−θ3signϵ1. The other homogeneous system can be got from equation (10):(17)ϵ.1=−ρϵ1β1signϵ1,ϵ.2=−ρ2ϵ1β2signϵ1,ϵ.3=−ρ3ϵ1β3signϵ1.
Similarly, the system of equation (15) is homogeneous of degree β1−1>0 with respect to the weight 1,α1,2α1−1.
Define the same Lyapunov function as equation (13) as follows:(18)Vβζ1,ζ2,ζ3=ζTPζ,where ζ=ζ1,ζ2,ζ3T=ϵ11/φsignϵ1,ϵ21/φα1signϵ2,ϵ31/φ2α1−1signϵ3T, 0.5<φ=α1<1, P denotes a positive definite symmetric matrix. Make fβ represent the vector field of system equation (15); it can be obtained that Vβζ and LfβVβζ are homogeneous of degrees 2/φ and 2/φ+β1−1 with the same weights in system equation (15), respectively.
Once again the following inequality is derived according to [26, 27](19)LfβVβζ≤μ2β,θVβζγ2,where μ2=−maxy:Vβy=1LfβVβy, γ2=1+φ/2β1−φ/2≥1, and limα1⟶1μ2≥ρ/λmaxP.
Considering the error dynamics in equation (10), the following Lyapunov function is chosen as(20)Vζ1,ζ2,ζ3=ζTPζ,where ζ=ζ1,ζ2,ζ3T=ϵ11/φsignϵ1,ϵ21/φα1signϵ2,ϵ31/φ2α1−1signϵ3T, 0.5<φ=α1<1, P denotes a positive definite symmetric matrix same as in equations (13) and (16), then take first time derivative of V:(21)V˙ζ=LfaVζ+LfβVζ+2ζTP01φα1diagϵ21/φα−1fx1,x2−fx1,z20+2ζTPdiagϵ11/φα−1−θ1φsignϵ1diagϵ21/φα1−1−θ2φα1signϵ2diagϵ31/φ2α1−1wt−θ1φsignϵ3.
Substituting equations (14) and (17) into equation (19), the following formula can be got:(22)V˙ζ≤−μ1Vαζγ1−μ2Vβζγ2+2ζTP01φα1diagϵ21/φα1−1fx1,x2−fx1,z20+2ζTPdiagϵ11/φα1−1−θ1φsignϵ1diagϵ21/φα1−1−θ2φα1signϵ2diagϵ31/2φα1−1wt−θ3φsignϵ3≤−μ1Vαζγ1−μ2Vβζγ2+2λλmaxPζ∑i=13ε2i1/φα1−1∑i=13ε2iφα1+2θ1λmaxPζ∑i=13ϵ1i1/φ−1φ+2θ2λmaxPζ∑i=13ϵ2i1/φα1−1φα1+2θ3+w¯λmaxPζ∑i=13ϵ2i1/φ2α1−1−1φ2α1−1,where λmaxP refers to the maximum eigenvalue of matrix P.
The following inequality is derived according to Lemma 1:(23)ϵ2j1/φα1−1∑i=13ϵ2i≤31/φα1ϵ2j1/φα+φα1∑i=13ϵ2i1/φα1,wherej=1,2,3, and the following result is got:(24)∑i=13ϵ2i1/φα1−1∑i=13ϵ2i≤3∑i=13ϵ2i1/φα1≤33ζ2≤33ζ.
The following inequality is derived according to Lemma2:(25)∑i=13ϵ2i1/φα−1=∑i=13ϵ2i1/φα1−φα≤3φα1∑i=13ζ2i1−φα1≤3φα13ζ21−φα1≤31+φα1/2ζ1−φα1.
The following inequalities can be derived in the same way according to Lemma2:(26)∑i=13ϵ1i1/φ−1≤31/φ/2ζ1−φ,(27)∑i=13ϵ1i1/φ2α1−1−1≤31/φ2α1−1/2ζ1−φ2α1−1.
So, substituting equations (22)–(25) into the inequality equation (20), the following inequality can be got:(28)V˙ζ≤−μ1Vαζγ1−μ2Vβζγ2=+62λλmaxPζ2φα1+2×31+φα1/2θ1λmaxPζ2−φφ+2×31+φ/2θ2λmaxPζ2−φα1φα1+2×31+φ2α1−1/2θ3+w¯λmaxPζ2−φ2α1−1φ2α1−1≤−μ1Vγ1−μ2Vγ2+μ3V+μ4V1−φ−2+μ5V1−φα1/2+μ6V1−φ2α1−1,where(29)μ3=62λλmaxPφλminP,μ4=2×31+φα1/2θ1λmaxPφλminP1−φ/2,μ5=2×31+φ/2θ2λmaxPφλminP1−φ/2,μ6=2×31+φ/2α1/2θ3+w¯λmaxPφλminP1−φα1/2.
Two cases are discussed in further analysis.Case 1. V≥1.Consider the parameters in equation (26):(30)0<1−φ2<1<γ2,0<1−φα12<1<γ2,0<1−φ2α1−12<1<γ2.So, simplify equation (26) to the following formula:(31)V˙≤−μ1Vγ1−μ7Vγ2.whereμ7=μ2−μ3−μ4−μ5−μ6; select the parameter ρ sufficiently large to ensure μ7>0; according to Lemma 3, the Lyapunov function V will converge to Vζ≡1 in finite time t1.(32)t1<1μ11−γ1V01−γ1F1,1−γ1γ2−γ1,1+1−γ1γ2−γ1,−μ7μ1V0γ2−γ1,whereV0 denotes the initial value of V.Case 2.V<1. In this case, equation (26) can be simplified to(33)V˙≤−μ1Vγ1−μ2Vγ2+μ3Vγ1+μ4+μ5+μ6V1−φ/2≤−μ11−ρ0Vγ1+μ3Vγ1+μ4+μ5+μ6V1−q/2−μ1ρ0Vγ1−μ2Vγ2≤−μ11−ρ0−μ3Vγ1−1+q/2−μ4+μ5+μ6V1−q/2−μ1ρ0Vγ1−μ2Vγ2,where the parameterρ0 is selected to ensure that 0<ρ0<1.
From inequality equation (30), when μ11−ρ0−μ3Vγ1−1+φ/2−μ4+μ5+μ6≥0, the Lyapunov function V will converge to the residual region Ω2 in finite time T1=t1+t2:(34)Ω2=VV<μ4+μ5+μ6μ11−ρ0−μ32/2γ1−2+φ,(35)t2<1μ1ρ01−γ1Vt11−γ1×F1,1−γ1γ2−γ1,1+1−γ1γ2−γ1,−μ2μ1ρ0Vt1γ2−γ1,where Vt1 represents the value of V at time t1. The estimation error can be written as(36)Ω3=ζ1λminPμ4+μ5+μ6μ11−ρ0−μ31/2γ1−2+φ.
In next step, the parametersθ1,θ2, and θ3 will be analyzed to ensure ϵ1,ϵ2,ϵ3 converge to zeros within finite time.
Define the following Lyapunov function:(37)Vϵ1ϵ1=12ϵ1Tϵ1.
Take time derivative ofVϵ1:(38)V˙ϵ1=ϵ1Tϵ.1=ϵ1Tϵ2−ρ∑i=13ϵ1iα1+1−ρ∑i=13ϵ1iβ1+1−θ1∑i=13ϵ1≤−ρϵ1α1+1/2−ρϵ1β1+1/2−θ1−ϵ2ϵ1.
According to equation (33), ϵ2 will converge to the bounded region Ω3, and then selecting the parameter θ1>Ω3, the inequality equation (35) can be simplified to(39)V˙ϵ1≤−2α1+1/2Vϵ1α1+1/2−2β1+1/2ρVϵ1β1+1/2.
So,ϵ1 can converge to zeros within time T2=T1+t3:(40)t3<12α1−1/21−α1×F1,1−α1β1−α1,1+1−α1β1−α1,−2β1−α1/2Ve1T1β1−α1,where Vϵ1T1 is the value of Vϵ1 at time t1.
So,ϵ1=ϵ.1=0 is got; according to [28], discontinuous terms include that signϵ1 can be considered as equivalent input; the error dynamical system equation (10) can be reduced to(41)0=ϵ2−θ1signϵ1,ϵ.2=ϵ3+λϵ2−θ2signϵ1,ϵ.3=ωt−θ3signϵ1.
It can be derived thatsignϵ2=signϵ1. Define the Lyapunov function Vϵ2ϵ2=1/2ϵ2Tϵ2 and Vϵ3ϵ3=1/2ϵ3Tϵ3. Analyze these two Lyapunov functions like we did before; it can be easily obtained that, let θ2>Ω3 and θ1>w¯, the estimation errors ϵ2 and ϵ3 will converge to zeros within finite time T3=T2+t4 and T4=T3+t5, respectively [11]:(42)t4<21/2θ2−Ω3Vϵ2T21/2,t5<21/2θ3−w¯Vϵ3T31/2,where Vϵ2T2 denotes the value of Vϵ2 at time T2andVϵ3T3 denotes the value of Vϵ3 at time T3.
At this point, the proof is complete.


### 3.2. Finite Time Attitude Controller Design Based on FTESO

In this section, inspired by Zou [11], a finite time controller is introduced to stabilize the rigid body to the desired attitude. Because only the attitude measurement is available, the estimation of virtual angular velocity z2 and external disturbances z3 are utilized in the design procedure.Defineξ=η1/v−ηd1/v and e=x1−σd, where η=z2−σ˙d, ηd=−k1eν, 1/2<ν=ν1/ν2<1, and ν1andν2 are positive odd integers.The proposed finite time attitude controller is shown as follow:(43)τ=g−1−k3signξ−k2ξ2ν−1−z3−fx1,z2+q¨d−ρ2ϵ1α2signϵ1−ρ2ϵ1β2signϵ1−θ2signϵ1.Theorem 2.
Consider the system expressed in equation (4); if finite time control torque in equation (40) combined with FTESO in equation (9) is employed, the desired attitude can be tracked within finite time.Proof.
Define the Lyapunov function:(44)Vf=V+Vc,where(45)Vc=V0+∑i=13Vi,(46)V0=12eTe.
Vi is defined as(47)Vi=12−ν21−vk11+1/v∫ηdiηis1/v−ηdi1/v2−vds.
Taking time derivative of Lyapunov functionVf, the result can be got as follows:(48)V˙f=V˙+V˙c=V˙+V˙0+∑i=13V˙i≤−μ1Vγ1−μ2Vγ2+μ3V+μ4V1−φ/2+μ5V1−φα1/2+μ6V1−φ2α1−1/2+V˙0+∑i=13V˙i.
In this section, just the termsV˙0 and ∑i=13V˙i need to be derived. After tedious derivation, it could be obtained that(49)V˙0≤−k1eTeν+∑i=1321−ν1+νei1+ν+νξi1+v+eTϵ2,(50)V˙i≤1k1ϵ2iξi+k1+21−ν1+νξi1+νk11+ν+νei1+ν1+ν+12−ν21−νk11+1/vξi2−ντ¯i,where τ¯i=∂ηi/∂t.
Substitute the control torque equation (40) into equation (47). The V˙i can be derived as follows:(51)V˙i≤1k1ϵ2iξi+k1+21−ν1+νξi1+νk11+ν+νei1+ν1+ν+−k22−ν21−νk11+1/vξi1+ν+−k32−ν21−νk11+1/vξi2−ν.
Combining equation (48) with equations (42) and (46),(52)V˙c≤−k1−21−ν+ν1+ν∑i=13ei1+ν+eTζ2+∑i=13ζ2iζik1−k22−ν21−νk11+1/v−k1+21−ν1+νk11+ν×∑i=13ξi1+ν+−k32−ν21−νk11+1/v∑i=13ξi2−ν.
Choosek3>0 and let(53)k1≥21−ν+ν1+ν+k4k2≥2−ν21−νk11+1/vk1+21−ν1+νk11+ν+k4,where k4 is the positive constant. The following inequality is obtained:(54)V˙c≤−k3∑i=13ei1+ν−k3∑i=13ξi1+ν+eTϵ2+∑i=13ϵ2iξik1≤c1∑i=13ei2+∑i=13ξi2,where c1=max1/2,1/2−vk11+1/v.
Notice that(55)Vi≤12−ν21−νk11+1/vηi−ηdiξi2−ν≤12−νk11+1/vξi2.
Hence, the following inequality is obtained:(56)Vc=V0+∑i=13Vi≤c2∑i=13ϵi2+∑i=13ξi2,where c2=max1/2,1/2−vk11+1/v.
According to Lemma2:(57)V˙c≤−c3V1+ν/2+eTϵ2+∑i=13ϵ2iξik1,where c3=k3/c21+ν/2.
From equations (45) and (54),(58)V˙f=V˙+V˙c=V˙+V˙0+∑i=13V˙i≤−μ1Vγ1−μ2Vγ2+μ3V+μ4V1−φ/2+μ5V1−φα1/2+μ6V1−φ2α1−1/2−c2V1+ν/2+eTϵ2+∑i=13ϵ2iξik1.
According to Theorem1, ϵt would converge to zeros in finite time, so equation (55) becomes(59)V˙f=V˙c≤−c2Vc1+ν/2.
Because0<1+ν/2<1, the tracking errors e converge to zeros in finite time [11].
The proof is completed.Remark 1.
In practical situations, due to the absence of certain parameters, the system functionfx1,x2 can not be calculated accurately. To deal with this problem, the system function fx1,x2 and external disturbances d∗t can be regarded as total disturbances dtotalt. The control algorithm proposed in this article can be written as(60)z˙1=z2+ρϵ1α1signϵ1+ρϵ1β1signϵ1+θ1signϵ1,z˙2=z3+gx1τ+ρ2ϵ1α2signϵ1+ρ2ϵ1β2signϵ1+θ2signϵ1,z˙3=ρ3ϵ1α3signϵ1+ρ3ϵ1β3signϵ1+θ3signϵ1,τ=g−1−k3signξ−k2ξ2ν−1−z3+q¨d−ρ2ϵ1α2signϵ1−ρ2ϵ1β2signϵ1−θ2signϵ1.Remark 2.
In order to eliminate the chattering issue caused by discontinuous terms in the observer and controller, the signum function can be replaced by the following function:(61)sgmfe1=211+exp−γoe1−12,e1≤δ,signe1,e1>δ,where γo and δ are constants which can be designed.


## 3.1. Finite Time Extended State Observer Design

In this section, only the attitude MRPsσ is the available information; an ESO which ensures finite time convergence is presented to estimate the virtual angular velocity x2 and the external disturbances d∗t.Designate the external disturbancesd∗t as an extended state x3; the system in equation (4) can be transformed to a new third-order dynamics equation which is shown in the following formulas:(9)x˙1=x2,x˙2=x3+fx1,x2+gx1τ,x˙3=wt,where wt=d˙∗t, and the following assumption is given.Assumption 1.
The external disturbancesd∗t are unknown but continuously differential and bounded; its first time derivative wt is also unknown and bounded; the following inequalities hold:(10)d∗t≤d∗¯,wt≤w¯,where d¯∗ and w¯ represent the upper bound of d∗t and wt, respectively.
Letz1,z2, and z3 represent the estimates of x1,x2, and x3 in equation (7), respectively, then the FTESO is designed as follows:(11)z˙1=z2+ρϵ1α1signϵ1+ρϵ1β1signϵ1+θ1signϵ1,z˙2=z3+fx1,z2+gx1τ+ρ2ϵ1α2signϵ1+ρ2ϵ1β2signϵ1+θ2signϵ1,z˙3=ρ3ϵ1α3signϵ1+ρ3ϵ1β3signϵ1+θ3signϵ1,where ϵ1=x1−z1 represents the estimation error of attitude MRPs x1, 0<ρ<+∞, 0.5<α1<1, α2=2α1−1, α3=3α1−2,β1=1/α1, and β2=1/β1+β1−1,β2=2/β1+β1−2, and θ1, θ2, and θ3 are the parameters to deal with the external disturbances.
Letϵ2=x2−z2, ϵ3=x3−z3, and the error dynamical equations of the proposed FTESO is expressed as:(12)ϵ.1=ϵ2−ρϵ1α1signϵ1−ρϵ1β1signϵ1−θ1signϵ1,ϵ.2=ϵ3+fx1,x2−fx1,z2−ρ2ϵ1α2signϵ1−ρ2ϵ1β2signϵ1−θ2signϵ1,ϵ.3=ωt−ρ3ϵ1α3signϵ1−ρ3ϵ1β3signϵ1−θ3signϵ1.
The following compact set is constructed for any constantΔ:(13)Ω1=x1,x2,z1,z2x1≤Δ,x2≤Δ,z1≤Δ,z2≤Δ.
This compact set can be easily obtained by designing a proper desired attitude reference trajectory. If the setΩ1 is established, there will exist a positive constant λ such that fx1,x2−fx1,z2≤λϵ2.
The stability properties of the extended state observer is presented in the following theorem.Theorem 1.
Consider the error dynamical system given by equation (10), where 0.5<α1<1, α2=2α1−1, α3=3α1−2,β1=1/α1, β2=1/β1+β1−1,andβ2=2/β1+β1−2; if the compact set Ω1 holds, ρ, θ1, θ2, and θ3 are sufficiently large, then the estimation errors would converge to a residual region in finite time.Proof.
Consider the error dynamics in equation (10); ignore the terms ρϵ1β1signϵ1signϵ1−θ1signϵ1, −ρ2ϵ1β2signϵ1−θ2signϵ1, and wt−ρ3ϵ1β3signϵ1−θ3signϵ1, so equation (10) can be reduced to the following form:(14)ϵ.1=ϵ2−ρϵ1α1signϵ1,ϵ.2=ϵ3+fx1,x2−fx1,z2−ρ2ϵ1α2signϵ1,ϵ.3=−ρ3ϵ1α3signϵ1.
It can be acquired that equation (12) is homogeneous of degree α1−1<0 with respect to the weight 1,α1,2α1−1.
The Lyapunov function is proposed as follows:(15)Vαζ1,ζ2,ζ3=ζTPζ.where ζ=ζ1,ζ2,ζ3T=ϵ11/φsignϵ1,ϵ21/φα1signϵ2,ϵ31/φ2α1−1signϵ3T, 0.5<φ=α1<1,P denotes a positive definite symmetric matrix. Make fα represent the vector field of system equation (12); it can be obtained that Vαζ and LfαVαζ are homogeneous of degrees 2/φ and 2/φ+α1−1 with the same weights in system equation (12), respectively.
According to [26, 27], the inequality is obtained as follows:(16)LfαVαζ≤μ1α,θVaζγ1,where μ1=−maxy:Vαy=1LfaVαy, γ1=1+α1φ/2−α1φ/2≤1, and limα1⟶1μ1≥ρ/λmaxP.
Same as the previous analysis, ignore the termsϵ2−ρϵ1α1signϵ1−θ1signϵ1, ϵ3+fx1,x2−fx1,z2−ρ2ϵ1α2signϵ1−θ2signϵ1, and ωt−ρ3ϵ1α3signϵ1−θ3signϵ1. The other homogeneous system can be got from equation (10):(17)ϵ.1=−ρϵ1β1signϵ1,ϵ.2=−ρ2ϵ1β2signϵ1,ϵ.3=−ρ3ϵ1β3signϵ1.
Similarly, the system of equation (15) is homogeneous of degree β1−1>0 with respect to the weight 1,α1,2α1−1.
Define the same Lyapunov function as equation (13) as follows:(18)Vβζ1,ζ2,ζ3=ζTPζ,where ζ=ζ1,ζ2,ζ3T=ϵ11/φsignϵ1,ϵ21/φα1signϵ2,ϵ31/φ2α1−1signϵ3T, 0.5<φ=α1<1, P denotes a positive definite symmetric matrix. Make fβ represent the vector field of system equation (15); it can be obtained that Vβζ and LfβVβζ are homogeneous of degrees 2/φ and 2/φ+β1−1 with the same weights in system equation (15), respectively.
Once again the following inequality is derived according to [26, 27](19)LfβVβζ≤μ2β,θVβζγ2,where μ2=−maxy:Vβy=1LfβVβy, γ2=1+φ/2β1−φ/2≥1, and limα1⟶1μ2≥ρ/λmaxP.
Considering the error dynamics in equation (10), the following Lyapunov function is chosen as(20)Vζ1,ζ2,ζ3=ζTPζ,where ζ=ζ1,ζ2,ζ3T=ϵ11/φsignϵ1,ϵ21/φα1signϵ2,ϵ31/φ2α1−1signϵ3T, 0.5<φ=α1<1, P denotes a positive definite symmetric matrix same as in equations (13) and (16), then take first time derivative of V:(21)V˙ζ=LfaVζ+LfβVζ+2ζTP01φα1diagϵ21/φα−1fx1,x2−fx1,z20+2ζTPdiagϵ11/φα−1−θ1φsignϵ1diagϵ21/φα1−1−θ2φα1signϵ2diagϵ31/φ2α1−1wt−θ1φsignϵ3.
Substituting equations (14) and (17) into equation (19), the following formula can be got:(22)V˙ζ≤−μ1Vαζγ1−μ2Vβζγ2+2ζTP01φα1diagϵ21/φα1−1fx1,x2−fx1,z20+2ζTPdiagϵ11/φα1−1−θ1φsignϵ1diagϵ21/φα1−1−θ2φα1signϵ2diagϵ31/2φα1−1wt−θ3φsignϵ3≤−μ1Vαζγ1−μ2Vβζγ2+2λλmaxPζ∑i=13ε2i1/φα1−1∑i=13ε2iφα1+2θ1λmaxPζ∑i=13ϵ1i1/φ−1φ+2θ2λmaxPζ∑i=13ϵ2i1/φα1−1φα1+2θ3+w¯λmaxPζ∑i=13ϵ2i1/φ2α1−1−1φ2α1−1,where λmaxP refers to the maximum eigenvalue of matrix P.
The following inequality is derived according to Lemma 1:(23)ϵ2j1/φα1−1∑i=13ϵ2i≤31/φα1ϵ2j1/φα+φα1∑i=13ϵ2i1/φα1,wherej=1,2,3, and the following result is got:(24)∑i=13ϵ2i1/φα1−1∑i=13ϵ2i≤3∑i=13ϵ2i1/φα1≤33ζ2≤33ζ.
The following inequality is derived according to Lemma2:(25)∑i=13ϵ2i1/φα−1=∑i=13ϵ2i1/φα1−φα≤3φα1∑i=13ζ2i1−φα1≤3φα13ζ21−φα1≤31+φα1/2ζ1−φα1.
The following inequalities can be derived in the same way according to Lemma2:(26)∑i=13ϵ1i1/φ−1≤31/φ/2ζ1−φ,(27)∑i=13ϵ1i1/φ2α1−1−1≤31/φ2α1−1/2ζ1−φ2α1−1.
So, substituting equations (22)–(25) into the inequality equation (20), the following inequality can be got:(28)V˙ζ≤−μ1Vαζγ1−μ2Vβζγ2=+62λλmaxPζ2φα1+2×31+φα1/2θ1λmaxPζ2−φφ+2×31+φ/2θ2λmaxPζ2−φα1φα1+2×31+φ2α1−1/2θ3+w¯λmaxPζ2−φ2α1−1φ2α1−1≤−μ1Vγ1−μ2Vγ2+μ3V+μ4V1−φ−2+μ5V1−φα1/2+μ6V1−φ2α1−1,where(29)μ3=62λλmaxPφλminP,μ4=2×31+φα1/2θ1λmaxPφλminP1−φ/2,μ5=2×31+φ/2θ2λmaxPφλminP1−φ/2,μ6=2×31+φ/2α1/2θ3+w¯λmaxPφλminP1−φα1/2.
Two cases are discussed in further analysis.Case 1. V≥1.Consider the parameters in equation (26):(30)0<1−φ2<1<γ2,0<1−φα12<1<γ2,0<1−φ2α1−12<1<γ2.So, simplify equation (26) to the following formula:(31)V˙≤−μ1Vγ1−μ7Vγ2.whereμ7=μ2−μ3−μ4−μ5−μ6; select the parameter ρ sufficiently large to ensure μ7>0; according to Lemma 3, the Lyapunov function V will converge to Vζ≡1 in finite time t1.(32)t1<1μ11−γ1V01−γ1F1,1−γ1γ2−γ1,1+1−γ1γ2−γ1,−μ7μ1V0γ2−γ1,whereV0 denotes the initial value of V.Case 2.V<1. In this case, equation (26) can be simplified to(33)V˙≤−μ1Vγ1−μ2Vγ2+μ3Vγ1+μ4+μ5+μ6V1−φ/2≤−μ11−ρ0Vγ1+μ3Vγ1+μ4+μ5+μ6V1−q/2−μ1ρ0Vγ1−μ2Vγ2≤−μ11−ρ0−μ3Vγ1−1+q/2−μ4+μ5+μ6V1−q/2−μ1ρ0Vγ1−μ2Vγ2,where the parameterρ0 is selected to ensure that 0<ρ0<1.
From inequality equation (30), when μ11−ρ0−μ3Vγ1−1+φ/2−μ4+μ5+μ6≥0, the Lyapunov function V will converge to the residual region Ω2 in finite time T1=t1+t2:(34)Ω2=VV<μ4+μ5+μ6μ11−ρ0−μ32/2γ1−2+φ,(35)t2<1μ1ρ01−γ1Vt11−γ1×F1,1−γ1γ2−γ1,1+1−γ1γ2−γ1,−μ2μ1ρ0Vt1γ2−γ1,where Vt1 represents the value of V at time t1. The estimation error can be written as(36)Ω3=ζ1λminPμ4+μ5+μ6μ11−ρ0−μ31/2γ1−2+φ.
In next step, the parametersθ1,θ2, and θ3 will be analyzed to ensure ϵ1,ϵ2,ϵ3 converge to zeros within finite time.
Define the following Lyapunov function:(37)Vϵ1ϵ1=12ϵ1Tϵ1.
Take time derivative ofVϵ1:(38)V˙ϵ1=ϵ1Tϵ.1=ϵ1Tϵ2−ρ∑i=13ϵ1iα1+1−ρ∑i=13ϵ1iβ1+1−θ1∑i=13ϵ1≤−ρϵ1α1+1/2−ρϵ1β1+1/2−θ1−ϵ2ϵ1.
According to equation (33), ϵ2 will converge to the bounded region Ω3, and then selecting the parameter θ1>Ω3, the inequality equation (35) can be simplified to(39)V˙ϵ1≤−2α1+1/2Vϵ1α1+1/2−2β1+1/2ρVϵ1β1+1/2.
So,ϵ1 can converge to zeros within time T2=T1+t3:(40)t3<12α1−1/21−α1×F1,1−α1β1−α1,1+1−α1β1−α1,−2β1−α1/2Ve1T1β1−α1,where Vϵ1T1 is the value of Vϵ1 at time t1.
So,ϵ1=ϵ.1=0 is got; according to [28], discontinuous terms include that signϵ1 can be considered as equivalent input; the error dynamical system equation (10) can be reduced to(41)0=ϵ2−θ1signϵ1,ϵ.2=ϵ3+λϵ2−θ2signϵ1,ϵ.3=ωt−θ3signϵ1.
It can be derived thatsignϵ2=signϵ1. Define the Lyapunov function Vϵ2ϵ2=1/2ϵ2Tϵ2 and Vϵ3ϵ3=1/2ϵ3Tϵ3. Analyze these two Lyapunov functions like we did before; it can be easily obtained that, let θ2>Ω3 and θ1>w¯, the estimation errors ϵ2 and ϵ3 will converge to zeros within finite time T3=T2+t4 and T4=T3+t5, respectively [11]:(42)t4<21/2θ2−Ω3Vϵ2T21/2,t5<21/2θ3−w¯Vϵ3T31/2,where Vϵ2T2 denotes the value of Vϵ2 at time T2andVϵ3T3 denotes the value of Vϵ3 at time T3.
At this point, the proof is complete.


## 3.2. Finite Time Attitude Controller Design Based on FTESO

In this section, inspired by Zou [11], a finite time controller is introduced to stabilize the rigid body to the desired attitude. Because only the attitude measurement is available, the estimation of virtual angular velocity z2 and external disturbances z3 are utilized in the design procedure.Defineξ=η1/v−ηd1/v and e=x1−σd, where η=z2−σ˙d, ηd=−k1eν, 1/2<ν=ν1/ν2<1, and ν1andν2 are positive odd integers.The proposed finite time attitude controller is shown as follow:(43)τ=g−1−k3signξ−k2ξ2ν−1−z3−fx1,z2+q¨d−ρ2ϵ1α2signϵ1−ρ2ϵ1β2signϵ1−θ2signϵ1.Theorem 2.
Consider the system expressed in equation (4); if finite time control torque in equation (40) combined with FTESO in equation (9) is employed, the desired attitude can be tracked within finite time.Proof.
Define the Lyapunov function:(44)Vf=V+Vc,where(45)Vc=V0+∑i=13Vi,(46)V0=12eTe.
Vi is defined as(47)Vi=12−ν21−vk11+1/v∫ηdiηis1/v−ηdi1/v2−vds.
Taking time derivative of Lyapunov functionVf, the result can be got as follows:(48)V˙f=V˙+V˙c=V˙+V˙0+∑i=13V˙i≤−μ1Vγ1−μ2Vγ2+μ3V+μ4V1−φ/2+μ5V1−φα1/2+μ6V1−φ2α1−1/2+V˙0+∑i=13V˙i.
In this section, just the termsV˙0 and ∑i=13V˙i need to be derived. After tedious derivation, it could be obtained that(49)V˙0≤−k1eTeν+∑i=1321−ν1+νei1+ν+νξi1+v+eTϵ2,(50)V˙i≤1k1ϵ2iξi+k1+21−ν1+νξi1+νk11+ν+νei1+ν1+ν+12−ν21−νk11+1/vξi2−ντ¯i,where τ¯i=∂ηi/∂t.
Substitute the control torque equation (40) into equation (47). The V˙i can be derived as follows:(51)V˙i≤1k1ϵ2iξi+k1+21−ν1+νξi1+νk11+ν+νei1+ν1+ν+−k22−ν21−νk11+1/vξi1+ν+−k32−ν21−νk11+1/vξi2−ν.
Combining equation (48) with equations (42) and (46),(52)V˙c≤−k1−21−ν+ν1+ν∑i=13ei1+ν+eTζ2+∑i=13ζ2iζik1−k22−ν21−νk11+1/v−k1+21−ν1+νk11+ν×∑i=13ξi1+ν+−k32−ν21−νk11+1/v∑i=13ξi2−ν.
Choosek3>0 and let(53)k1≥21−ν+ν1+ν+k4k2≥2−ν21−νk11+1/vk1+21−ν1+νk11+ν+k4,where k4 is the positive constant. The following inequality is obtained:(54)V˙c≤−k3∑i=13ei1+ν−k3∑i=13ξi1+ν+eTϵ2+∑i=13ϵ2iξik1≤c1∑i=13ei2+∑i=13ξi2,where c1=max1/2,1/2−vk11+1/v.
Notice that(55)Vi≤12−ν21−νk11+1/vηi−ηdiξi2−ν≤12−νk11+1/vξi2.
Hence, the following inequality is obtained:(56)Vc=V0+∑i=13Vi≤c2∑i=13ϵi2+∑i=13ξi2,where c2=max1/2,1/2−vk11+1/v.
According to Lemma2:(57)V˙c≤−c3V1+ν/2+eTϵ2+∑i=13ϵ2iξik1,where c3=k3/c21+ν/2.
From equations (45) and (54),(58)V˙f=V˙+V˙c=V˙+V˙0+∑i=13V˙i≤−μ1Vγ1−μ2Vγ2+μ3V+μ4V1−φ/2+μ5V1−φα1/2+μ6V1−φ2α1−1/2−c2V1+ν/2+eTϵ2+∑i=13ϵ2iξik1.
According to Theorem1, ϵt would converge to zeros in finite time, so equation (55) becomes(59)V˙f=V˙c≤−c2Vc1+ν/2.
Because0<1+ν/2<1, the tracking errors e converge to zeros in finite time [11].
The proof is completed.Remark 1.
In practical situations, due to the absence of certain parameters, the system functionfx1,x2 can not be calculated accurately. To deal with this problem, the system function fx1,x2 and external disturbances d∗t can be regarded as total disturbances dtotalt. The control algorithm proposed in this article can be written as(60)z˙1=z2+ρϵ1α1signϵ1+ρϵ1β1signϵ1+θ1signϵ1,z˙2=z3+gx1τ+ρ2ϵ1α2signϵ1+ρ2ϵ1β2signϵ1+θ2signϵ1,z˙3=ρ3ϵ1α3signϵ1+ρ3ϵ1β3signϵ1+θ3signϵ1,τ=g−1−k3signξ−k2ξ2ν−1−z3+q¨d−ρ2ϵ1α2signϵ1−ρ2ϵ1β2signϵ1−θ2signϵ1.Remark 2.
In order to eliminate the chattering issue caused by discontinuous terms in the observer and controller, the signum function can be replaced by the following function:(61)sgmfe1=211+exp−γoe1−12,e1≤δ,signe1,e1>δ,where γo and δ are constants which can be designed.


## 4. Simulation Results

In this section, several numerical simulations are implemented to show the effectiveness of the FTESO-based control scheme proposed in this paper; meanwhile, the control scheme with general LESO and the control scheme without ESO in reference [11] are introduced as comparisons.The initial states are designed asx10=0,0,0T, x20=0,0,0T, z10=0,0,0T,z20=0,0,0T,z30=0,0,0T. The desired attitude trajectory is designed as σd=0.5cos0.2t,sin0.2t,3T, and the external disturbances d∗=0.2sin0.5t,cos1/3t,sin1/4t. The parameters of the proposed ESO and controller are ρ=5, α1=0.8, θ1=1, θ2=5, θ3=1,ν=7/9, k1=1.1, k2=1.5, and k3=0.01. The inertia matrix of rigid body is chosen as(62)J=201.20.91.2171.40.91.415kgm2.The upper bound of control torque is selected asτi≤50Nm, where i=1,2,3.First, the proposed scheme includes that equations (9) and (40) are simulated. In this simulation, the system function fx1,x2 is assumed to be known; the proposed FTESO only estimates the external disturbances d∗ and the virtual angular velocity x2.The results of attitude tracking errors and external disturbances estimate for the FTESO-based control scheme are shown in Figures1 and 2. According to Figure 1, the proposed control scheme can converge the tracking errors to zeros within about 4 seconds, and the extended state z3 can also estimate the total disturbances d∗t in finite time.Figure 1
Tracking errors for FTESO-based controller with external disturbances.Figure 2
External disturbances estimate for FTESO. (a)z31. (b) z32. (c) z33.
(a)(b)(c)Then, consider the situation that the system functionfx1,x2 is unknown, which is regarded as total disturbances dtotalt together with external disturbances d∗t. The simulation results are shown in Figures 3 and 4. Compared with above results, the total disturbances dtotalt is much larger than d∗t in Figure 2, but it can be fast estimated by FTESO. Because the estimated total disturbances can be compensated in controller, even the system function fx1,x2 is unknown; the proposed control scheme can also converge the tracking errors to zeros within finite time. The FTESO guarantees that the two simulations in Figures 1 and 3 have the similar results.Figure 3
Tracking errors for the FTESO-based controller with total disturbances.Figure 4
Total disturbances estimate for FTESO. (a)z31. (b) z32. (c) z33.
(a)(b)(c)For comparison, the introduced linear LESO is designed as follows:(63)z˙1=z2+ρ1ε1,z˙2=z3+fx1,z2+gx1τ+ρ12ε1,z˙3=ρ13ε1.The corresponding controller is similar to the one designed in this article:(64)τ=g−1−k3signξ−k2ξ2ν−1−z3−fx1,z2+q¨d−ρ12ε1.The parameters are designed asρ1=28, ν=7/9, k1=1.1, k2=1.5, and k3=0.01.The other control scheme in [11] is expressed as(65)z˙1=z2+θc1ε1α1signε1,z˙2=z3+fx1,z2+gx1τ+θ2c2ε1α2signε1,τ=g−1−k2ξ2v−1−fx1,z2+q¨d−θ2c2ε1α2signε1,where the parameters are selected to be same as in [11]: θ=3, c1=1, c2=0.2,ν=7/9, k1=1.1, k2=1.5, and k3=0.01.The comparison results between these three attitude control schemes are shown in Figure5. As shown in Figure 5, the proposed control scheme in this article and the control algorithm based on LESO in equations (59) and (60) can attain faster convergence than the controller in equation (61), and the proposed control scheme gets the best disturbances rejection.Figure 5
Tracking errorse under different control schemes.
(a)(b)


## 5. Conclusion

The finite time attitude tracking control for rigid body is studied in this article; a FTESO is designed through the homogeneous Lyapunov method. The angular velocity and total disturbances can be estimated by the proposed observer. Based on this, a finite time attitude controller is introduced. Numerical simulations show that the proposed control scheme can achieve fast convergence and good disturbances rejection. Further work will be focused on reducing power cost and dealing with the input saturation problem.



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