# High Precision Control of Rotating Payload Satellite considering Static and Dynamic Unbalanced Disturbance

Authors: Yatao Zhao; Cheng Wei; Qingsheng Wei; Yang Zhao
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

This paper is devoted to suppressing the lumped disturbance, which is mainly composed of static and dynamic unbalanced disturbance, to ensure the imaging quality of the rotating payload satellite system with a five-degree-of-freedom active magnetic bearing. The dynamic model of lumped disturbance in imaging phase is established to design the balancing error index of payload unbalance, and the effect of bearing mechanical characteristics is analyzed. A novel fixed-time extended state observer is proposed to estimate unknown lumped disturbance and uncertainty. On this basis, a novel quaternion-based fixed-time nonsingular terminal sliding mode controller is presented to achieve high precision, high stability, and chattering-free attitude control. The complete proof on the faster convergence performance of the presented sliding mode surface compared to the existing sliding mode surfaces and the fixed-time convergence of the presented controller is provided. Numerical simulation results are carried out to verify the lumped disturbance modeling accuracy and the effectiveness of the proposed controller.


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## Body

## 1. Introduction

In earth observation missions, remote sensing images acquired by optical observation payloads are widely used in military and civil fields because of their wide coverage and rich spectral bands. The imaging coverage capability is an important index of optical remote sensing satellites, and its demand is increasing to obtain more information per unit time. However, the relationship between high resolution and wide field of view is contradictory. In the traditional push-broom imaging mode, in order to increase the field of view width on the basis of ensuring the imaging resolution, the common methods are inner field of view stitching [1] and external field of view stitching [2, 3]. However, these methods are difficult to implement due to the limitations of production process, development cost, and satellite carrying capacity, and the improvement of imaging width is limited.To further increase the imaging width, a novel spin-scan imaging mode is proposed in [4], where the optical payload is installed on the axial direction of the satellite platform, and the optical axis is perpendicular to the advancing direction of the satellite. When the payload rotates relative to the satellite platform, a larger stripe-size area can be obtained. In addition, since the advantages of longer life, no wear, no lubrication, and higher rotational speed [5, 6], the five degree-of-freedom active magnetic bearing (5-DOF AMB) is utilized as a connecting element between the payload and the platform instead of conventional mechanical bearing to improve the rotational accuracy of payload. The specific structure of the rotating payload satellite (denoted as RPS) can be found in [7]. Note that the mass and inertia of the payload considered in the RPS system are comparable to the satellite platform.In practice, the operational phases of the RPS system are complex. The satellite platform has two phases: attitude maneuvering phase and earth orientation phase, the rotating payload has three phases: spin-up phase, constant-speed rotation phase, and despinning phase. When the satellite platform maintains earth orientation and the payload keeps rotating at a constant speed, the payload enters the remote sensing imaging mode, denoted as the imaging phase, with the demand of high attitude accuracy and stability, which is the concern of this paper.However, the large mass payload is unbalanced due to production and assembly errors and cannot be completely compensated by the existing balancing technology [8]. The payload unbalance consists of static unbalance and dynamic unbalance, where the former is related to the centroid deviation relative to the rotation axis, and the latter is related to the inertia product. When the large mass payload rotates, static unbalanced disturbance, dynamic unbalanced disturbance, and inertia difference disturbance caused by the radial principal inertia difference of the payload and axial angular momentum disturbance are transmitted to the satellite platform through the AMB. The abovementioned multiple disturbances are collectively referred to as lumped disturbance, which stimulate the nutation motion of the RPS system and seriously affect the control accuracy of payload in the imaging phase. In addition, the dynamic unbalance and static unbalance of the payload are randomly valued within the error boundary and change in orbit, indicating that the lumped disturbance is unknown in practice, which makes the suppression of the lumped disturbance complicated.To suppress the disturbance effect, the maximum torque of the actuator needs to be greater than the boundary of the lumped disturbance with a certain margin. However, due to the complex composition of lumped disturbance and the mechanical characteristics of the AMB, modeling the lumped disturbance in the imaging phase is a challenging problem. Furthermore, plenty of related research works have focused on controller design with disturbance, uncertainty, and nonlinearity, including the model predictive control [9], the robust H-infinity control [10, 11], and the active disturbance rejection control [12, 13]. Moreover, the sliding mode control (SMC) has been widely used in satellite attitude control due to its high accuracy and strong robustness against disturbances [14–16]. However, the above sliding mode controllers drive the system states to the equilibrium point asymptotically in infinite time, which cannot meet the requirement of rapid stability. Therefore, the research on finite-time sliding mode control for satellite attitude using terminal sliding mode (TSM) surface [17] has been carried out, which can make the system states converge to the equilibrium with a finite settling time and better resistance to disturbances. Recent researches on finite-time sliding mode control have focused on designing sliding mode surface with faster convergence rate, such as fast terminal sliding mode (FTSM) surface [18–20], and solving inherent singular problem, such as nonsingular TSM surface [21, 22] and nonsingular FTSM surface [23–25]. However, finite-time stability theory cannot guarantee that the system states converge within a bounded time independent of the initial values, and the convergence time cannot be estimated when the initial values are unknown. Fixed-time sliding mode control, with faster convergence speed and higher control accuracy, ensures that the convergence time is bounded regardless of the initial values and has been widely studied in [26–29], where the convergence speed of the utilized sliding mode surfaces gradually increase and the singularity problem is avoided.The requirement of disturbance’s boundary and the chattering phenomenon are the two main drawbacks of SMC that need to be improved. When the disturbance boundary is unknown, the switching gain needs to be selected sufficiently large to ensure the stability and adaptability of the controller, resulting in large energy consumption [30] and aggravating the chattering phenomenon. Although the maximum torque of the actuator can be used as the boundary in this paper, it is still conservative. In order to obtain the true boundary of disturbance to attenuate the chattering, an adaptive law can be adopted to adjust the switching gain, denoted as adaptive SMC, and uses the continuous function [17, 20, 28, 29] or boundary layer method [24, 31] to replace the discontinuous term in the controller, which is an effective way to improve the SMC’s drawbacks. But in the references [18, 24, 25, 31, 32] of adaptive SMC, the system states are only practically finite-time or fixed-time stable. In addition, utilizing disturbance observer algorithm to estimate and compensate disturbance to further improve control accuracy is an alternative solution [19, 30, 33, 34], where the switching gain needs to be greater than zero instead of disturbance’s boundary, or directly omitted, achieving chattering-free control while ensuring the system states are finite-time or fixed-time stable.According to the above discussion, there are two research objectives in this paper. First, establish the dynamic model of lumped disturbance in the imaging phase, and then, the disturbance analysis is studied. Second, in order to improve the attitude accuracy and stability of the payload in the presence of unknown lumped disturbance and uncertainty, a novel quaternion-based fixed-time nonsingular fast terminal sliding mode control (QFNFTSMC) is proposed as the controller for the satellite platform. In the proposed controller, a novel quaternion-based fixed-time nonsingular fast terminal sliding mode surface (QFNFTSMS) and a novel fixed-time extended state observer (FESO) are utilized. To the best knowledge of the authors, there are few studies on the dynamic model and attitude controller of the RPS system with 5-DOF AMB. The main contributions of this paper can be summarized as follows:(1)
The dynamic model of the lumped disturbance is established, and the disturbance analysis including the balancing error index, actuator selection, and the effect of AMB mechanical characteristics is carried out(2)
A proof of the fast convergence performance of the novel QFNFTSMS compared to other sliding mode surfaces is given, and a complete proof of fixed-time convergence of the states on the QFNFTSMS is proposed(3)
Based on FESO algorithm, a novel QFNFTSMC is proposed to achieve high precision, high stability, and chattering-free attitude control of the RPS system in the presence of unknown lumped disturbance and uncertaintyThe rest of this paper is organized as follows. In Section2, the dynamic model of the RPS system and the dynamic model of lumped disturbance are proposed, followed by a brief disturbance analysis. The quaternion-based fixed-time nonsingular fast terminal sliding mode control with sliding mode surface and extended state observer is presented in Section 3. Simulations and results are given in Section 4. Lastly, the conclusion obtained in this paper is summarized in Section 5.


## 2. Modeling of RPS System

As shown in Figure1, the RPS system consists of platform subsystem, payload subsystem, and AMB subsystem. The platform subsystem consists of a satellite platform, solar panels, and a wheel control system, and the payload subsystem consists of a large mass rotating payload and a single-axis momentum wheel used to control the axial rotation speed of the payload relative to the satellite platform.Figure 1
Structure of the RPS system.The AMB subsystem is used to connect the platform subsystem and the payload subsystem. As shown in Figure2, the AMB consists of a thrust active magnetic bearing (TAMB), a shaft, a touchdown bearing (TB), eddy-current position sensors, and two identical radial active magnetic bearings (RAMBs), where the one near the satellite platform is noted as the left RAMB, and the one near the rotating payload is noted as the right RAMB. The shaft with a thrust disk is fixed on the rotating payload to form the AMB rotor; RAMBs and TAMB are fixed on the satellite platform to form the AMB stator, where the RAMB is configured with two magnetic pole pairs to provide radial displacement stiffness and angular stiffness; the TAMB is configured with a pair of annular magnetic poles to provide axial displacement stiffness. When the rotor moves relative to the stator, the position sensors detect the air gap changes of each magnetic pole pair; then, the RAMB generates radial electromagnetic force and the TAMB generates axial electromagnetic force to constrain the actual geometric center position of AMB to track the resting geometric center position, ensuring that the 5-DOF constraint of the payload relative to the platform except for axial rotation holds. As an alternative, the TB works when the AMB fails, avoiding structural damage and mission failure.Figure 2
Structure of the AMB subsystem.The coordinate frames and position vectors used in this section are shown in Figure3.Figure 3
Coordinate frames and vectors used for modeling.Whereo is the inertial coordinate frame, b is the platform’s body-fixed coordinate frame, and p is the payload’s body-fixed coordinate frame. wii=x,y,z is the momentum wheel i body-fixed coordinate frame (Figure 3 takes the momentum wheel x as an example), wt is the body-fixed coordinate frame of momentum wheel t installed in the payload, and akk=1,2 is the solar panel k body-fixed coordinate frame (Figure 3 takes the solar panel 1 as an example).WhereXc, Xpf, and Xpl are the position vectors of the RPS system’s centroid, the platform subsystem’s centroid, and the payload subsystem’s centroid, respectively. rs is the position vector from the platform subsystem’s centroid to the coordinate origin of b. rak is the position vector from the platform subsystem’s centroid to the solar panel k installation position. ρk and ρck are the position vectors from the installation position to a certain mass point dm and the centroid of solar panel k, respectively. rP is the position vector from the RPS system’s centroid to the coordinate origin of p. rk is the position vector from the coordinate origin of p to the payload subsystem’s centroid. rl is the position vector from the coordinate origin of b to the coordinate origin of p.Notation: for a given vector projectionr=r1,r2,r3T∈R3, we use r~∈R3×3 to denote the cross-product antisymmetric matrix operation. r is the Euclidean norm, diagr∈R3×3 is the diagonal matrix with r as the diagonal element, ṙ∈R3 is the time derivative, and r¨ is the second time derivative. sgnr=sgnr1,sgnr2,sgnr3T∈R3 and sigar=r1asgnr1,r2asgnr2,r3asgnr3T∈R3, where a∈ℝ and sgn. denote the sign function.


### 2.1. Dynamic Model of RPS

The dynamic model of the platform subsystem is given as follows.(1)MpfX¨pf+∑k=1,2PakDakθ¨ak+∑k=1,2γtkη¨k=−AbFmb,Jpfω̇pf+J̇pfωpf+∑i=x,y,zRwiDwiθ¨wi+∑k=1,2RakDakθ¨ak+γskη¨k+ω~pfJpfωpf+∑i=x,y,zRwiDwiθ̇wi+∑k=1,2RakDakθ̇ak+γskη̇k=τb,JWiθ¨wi+ω̇pfTRwiDwi=τwii=x,y,z,JAkθ¨ak+X¨pfTPakDak+ω̇pfTRakDak+DakTγakη¨k=τakk=1,2,η¨k+γtkTX¨pf+γskTω̇pf+γakTDakθ¨ak+Ωk2ηk+2ζkΩkη̇k=0k=1,2.And the dynamic model of the payload subsystem is given as follows.(2)MplX¨pl=AbFmb,Jplω̇pl+RwtDwtθ¨wt+ω~plJplωpl+RwtDwtθ̇wt=τp,JWtθ¨wt+ω̇plTRwtDwt=τwt,where Xpf and Xpl are the projections of Xpf and Xpl in o, respectively. Mpf and Jpf are the mass and inertia matrix of the platform subsystem, while Mpl and Jpl are the definitions of the payload subsystem. ωpf and ωpl denote the angular velocity vector projection of the satellite platform and the rotating payload under their body-fixed coordinate frame, respectively. θwi, Rwi, and Jwi denote the rotation angle relative to the satellite platform, rotation coupling matrix with the satellite platform attitude, and inertia matrix of the momentum wheel i, respectively. In addition, JWi is the i-axis principal inertia of Jwi, and Dwi is the projection matrix of the relative angular velocity vector of momentum wheel i under its body-fixed reference frame, such as Dwx=1,0,0T. Similarly, θwt, Rwt, Jwt, JWt, and Dwt are the definitions of momentum wheel t. Furthermore, θak, Rak, Jak, JAk, and Dak are the definitions of the solar panel k, and Pak is the translation coupling matrix with the satellite platform. Solar panel vibration is described by mode coordinate ηk and mode frequency matrix Ωk. γak, γtk, and γsk denote the coupling matrix between the solar panel vibration and θak, Xpf, and ωpf, respectively. τwi and τwt denote the driving torques of the momentum wheel i and t, respectively. Ab is the attitude matrix of b relative to o. To give a more thorough description of Equation (1), the expressions of the coupling matrixes mentioned above are as follows.
(3)Pak=AbAkMkρ~ckT,γtk=AbAk∫kΦkdm,Rak=r~akAkMkρ~ckT+AkJak,γak=∫kρ~kΦkdm,γsk=r~akAk∫kΦkdm+Ak∫kρ~kΦkdm,where rak is the projection of rak in b, ρk and ρck are the projections of ρk and ρck in ak, Ak is the attitude matrix of ak relative to b, and Mk and Φk are the mass and mode shape function of solar panel k, respectively.DefiningFmb and τmb as the resultant force projection and torque projection of AMB acting on payload in b, respectively, which can be expressed as follows.
(4)Fmb=Kr5b+Cṙ5b,τmb=Kaϑ+Caϑ̇,where K∈R3×3, Ka∈R3×3, C∈R3×3, and Ca∈R3×3 are the bearing mechanical characteristics parameters, specifically, K and Ka are the displacement stiffness matrix and angular stiffness matrix of the AMB, respectively, and C and Ca are the displacement damping matrix and angular damping matrix of the AMB, respectively. r5b is the projection of the vector from the AMB’s resting geometric center position to the AMB’s actual geometric center position in b, i.e., the position deviation of the AMB. ϑ=φ,θ,ψT is the Euler angle of the payload relative to the satellite platform, i.e., the angular deviation of the AMB. Based on Equation (4), τb and τp can be expressed as follows.
(5)τb=−r~fbKr5b+Cṙ5b−Kaϑ−Caϑ̇,(6)τp=r~gpAbpTKr5b+Cṙ5b+AbpTKaϑ+Caϑ̇,where rfb is the projection of the vector from platform subsystem’s centroid to the AMB’s actual geometric center position in b, rgp is the projection of the vector from payload subsystem’s centroid to the AMB’s actual geometric center position in p, and Abp is the attitude matrix of p relative to b. Combining with Equations (1)–(6), the refined dynamic model of the RPS system has been established, and more details about Equations (1)–(6) can be found in [7].


### 2.2. Dynamic Model of Lumped Disturbance

Note that the models established in the previous subsection are applicable to any phase. In the imaging phase, the inertial force inFmb and the inertial torque in τmb are small, i.e., the lumped torque τb transmitted from the AMB to the satellite platform, denoted as the lumped disturbance d, is mainly composed of dynamic unbalanced disturbance and static unbalanced disturbance. In this section, the dynamic model of the lumped disturbance d is proposed. Before modeling, the following assumptions are made to simplify the analysis.Assumption 1.
The radial angle deviation and displacement deviation of AMB can be omitted, i.e., the following equations approximately.(7)ϑ=0,0,ψT,ϑ̇=0,0,ψ̇T,r5b=ṙ5b=03×1.Assumption 2.
The system states are stable,ω̇pf=ϑ¨=03×1 hold. The orbital angular velocity is denoted as ω0, and the desired rotational speed of payload is ψ̇d. Assuming that the state tracking errors are small, i.e., the following equations approximately.
(8)ωpf=ω0,0,0T,ψ̇=ψ̇d.As shown in Figure3, the centroid of momentum wheel t and the coordinate origin of p are coincident. Define rk to be the projection of rk in p, rj=rjx,rjy,rjzT to be the projection of the vector from coordinate origin of p to payload’s centroid in p (i.e., static unbalance) and Mp to be the payload’s mass, then rk=Mprj/Mpl holds. On this basis, the following equation can be obtained.
(9)MplXpl=MplXp+Aprk=MplXc+AbrP+ApMpMplrj=MpXc+AbrP+Aprj+MwtXc+AbrP,where Xc is the projection of Xc in o, Xp is the position vector projection of the coordinate origin of p in o, Ap is the attitude matrix of p relative to o, Mwt is the mass of momentum wheel t, and Mpl=Mp+Mwt holds. rP=rPx,rPy,rPzT is the projection of rP in b, which can be expressed as follows.
(10)rP=σrs+rl−εAbprj,where rs=rsx,rsy,rszT is the projection of rs in b, rl is the projection of rl in b, and σ=Mpf+Mpl−1Mpf and ε=Mpf+Mpl−1Mp hold.Combining Equations (2) and (9), one has
(11)Fmb=MplAbTX¨c+ω~pfω~pfrP+ω̇~pfrP+2ω~pfṙP+r¨P+MpAbpω~plω~plrj+Abpω̇~plrj.When Assumption1 holds, it follows that
(12)Abp=cosψ−sinψ0sinψcosψ0001,ωlf=ϑ̇,ṙP=−εAbpω~lfrj,r¨P=−εAbpω~lfω~lfrj−εAbpω̇~lfrj,where ωlf is the projection of the angular velocity of the payload relative to the satellite platform in p. Furthermore, when Assumption 2 holds, it follows that
(13)ωpl=ω0cosψd−ω0sinψdψ̇d,ω̇pl=−ψ̇dω0sinψdψ̇dω0cosψd0,ṙP=−εψ̇d−rjycosψd−rjxsinψd−rjysinψd+rjxcosψd0,r¨P=−εψ̇d2−rjxcosψd+rjysinψd−rjxsinψd−rjycosψd0.Bringing Equations (12) and (13) into Equation (11) and notingFmb=Fmb,x,Fmb,y,Fmb,zT, one has
(14)Fmb,x=Mplε−Mpψ̇d2rjxcosψd−rjysinψd,Fmb,y=Mplε−Mpψ̇d2rjxsinψd+rjycosψd−Mplω02rPy,Fmb,z=2Mp−2Mplεω0ψ̇drjxcosψd−rjysinψd−Mprjz+rPzω02−ω02rPzMwt.For the convenience of subsequent modeling,Fmb is divided into Fmb1 and Fmb2, where Fmb1=Fmb1,x,Fmb1,y,Fmb1,zT is the resultant force on the payload, and Fmb2=Fmb2,x,Fmb2,y,Fmb2,zT is the resultant force on the momentum wheel t, one has
(15)Fmb1,x=Mpε−1ψ̇d2rjxcosψd−rjysinψd,Fmb1,y=Mpε−1ψ̇d2rjxsinψd+rjycosψd−Mpω02rPy,Fmb1,z=2Mp1−εω0ψ̇drjxcosψd−rjysinψd−Mprjz+rPzω02,(16)Fmb2,x=Mwtεψ̇d2rjxcosψd−rjysinψd,Fmb2,y=Mwtεψ̇d2rjxsinψd+rjycosψd−Mwtω02rPy,Fmb2,z=−2Mwtεω0ψ̇drjxcosψd−rjysinψd−Mwtω02rPz.DividingrP into rP1 and rP2=rP2,x,rP2,y,rP2,zT, where rP1 is the projection of the vector from RPS system’s centroid to AMB’s actual geometric center position in b, and rP2=rP−rP1 holds. Then, based on Equations (2) and (5), the following equation can be obtained.
(17)τmb=AbpJplω̇pl+AbpRwtDwtθ¨wt+Abpω~plJplωpl+RwtDwtθ̇wt+r~P2+MpMplAbpr~jAbpTFmb.The inertia matrix of the rotating payload is denoted asJp and consists of the principal inertia Ix, Iy, and Iz and the inertia product Ixy, Ixz, and Iyz (i.e., dynamic unbalance). The following equation holds.
(18)Jpl=Jp+MpMplMwtr~jr~jT+Jwt.Bringing Equation (18) into Equation (17), when Assumption 1 and Assumption 2 hold, τmb=τmb,x,τmb,y,τmb,zTcan be given as
(19)τmb,x=−Ix−Iyψ̇dω0sin2ψd+2Ixyψ̇dω0cos2ψd+ψ̇d2Ixzsinψd+Iyzcosψd−rP2,z+rjzFmb1,y+rjycosψd+rjxsinψdFmb1,z−rP2,zFmb2,y,τmb,y=ω0ψ̇dIx−Iycos2ψd+2ψ̇dω0Ixysin2ψd+ψ̇d2Iyzsinψd−Ixzcosψd−ω0Izψ̇d+Jwtzψ̇d+θ̇wt+rP2,z+rjzFmb1,x+rjysinψd−rjxcosψdFmb1,z+rP2,zFmb2,x,τmb,z=12Ix−Iyω02sin2ψd−Ixyω02cos2ψd−rjycosψd+rjxsinψdFmb1,x+−rjysinψd+rjxcosψdFmb1,y.According to the definitions ofrfb, rs, rl, and rP2, it is clear that rfb=rs+rl−rp2 holds. Furthermore, based on Equations (5), (15), (16), and (19), the lumped disturbance d=dx,dy,dzT transmitted by the AMB, consisting of inertia difference disturbance d1, axial angular momentum disturbance d2, dynamic unbalanced disturbance d3, and static unbalanced disturbance d4, can be modeled as
(20)d1=Ix−Iyψ̇dω0sin2ψd−ω0ψ̇dIx−Iycos2ψd−Ix−Iyω02sin2ψd2,d2=0ω0Izψ̇d+ω0Jwtzψ̇d+θ̇wt0,d3=−2Ixyψ̇dω0cos2ψd−ψ̇d2Ixzsinψd+Iyzcosψd−2ψ̇dω0Ixysin2ψd−ψ̇d2Iyzsinψd−IxzcosψdIxyω02cos2ψd,(21)d4x=rsz+rlz+rjzFmb1,y−rsy+rjycosψd+rjxsinψdFmb1,z+rsz+rlzFmb2,y−rsyFmb2,z,d4y=−rsz+rlz+rjzFmb1,x+rsx+rjxcosψd−rjysinψdFmb1,z−rsz+rlzFmb2,x+rsxFmb2,z,d4z=rsy+rjycosψd+rjxsinψdFmb1,x−rsx−rjysinψd+rjxcosψdFmb1,y+rsyFmb2,x−rsxFmb2,y.


### 2.3. Disturbance Analysis

Equations (20) and (21) provide mechanical constraints on the payload mass parameters such as the radial principal inertia, dynamic unbalance, and static unbalance. Based on these two equations, the feasibility of the current balancing error index can be judged. To illustrate this point, the maximum torque of the satellite platform momentum wheel is 1 Nm, ω0=0.0011rad/s, ψ̇d=16deg/s, and the compensation accuracy of axial angular momentum disturbance is 10%, three balancing errors are considered: 1kg⋅m2 and 1mm, 3.9kg⋅m2 and 5mm, and 6.9kg⋅m2 and 7mm. The real dynamic unbalance and static unbalance are taken randomly within the respective error boundary, where the cases that make the lumped disturbance take their respective maximum values, denoted as cases 1-3, as shown in Table 1.Table 1
Payload inertia parameter values for different cases.
Principal inertiakg⋅m2Dynamic unbalancekg⋅m2Static unbalancemmCase 1Ix=114.4Ixz=1x=1, y=0, z=−9.6Iy=311.8Ixy=Iyz=0Iz=252.9Case 2Ix=214.4Ixz=3.9x=5, y=0, z=−9.6Iy=211.8Ixy=Iyz=0Iz=252.9Case 3Ix=214.4Ixz=6.9x=7, y=0, z=−9.6Iy=211.8Ixy=Iyz=0Iz=252.9The theoretical results ofd1-d4 for case 1 are summarized in Table 2. From case 1, it can be concluded that the inertia difference of payload does not lead to excessive disturbance even if it increases to 197.4 kgm2, and the axial angular momentum disturbance d2 is small constant. Therefore, when the dynamic unbalance and static unbalance increase, the lumped disturbance is mainly composed of d3 and d4.Table 2
Theoretical results of case 1.
d1d2d3d4Amplitude (Nm)0.0610.0080.0780.099Frequency (Hz)0.088/0.0440.044The theoretical results of the lumped disturbance in the three cases are shown in Figure4. In case 2, the lumped disturbance increases significantly when the unbalance increases. Furthermore, there is a risk that the lumped disturbance may exceed the maximum torque of the momentum wheel, as shown in case 3, and the control law cannot adjust the excess part, which will inevitably affect the attitude accuracy and stability of the satellite system. Therefore, more attention should be paid to the payload balancing error index. Under the condition of the maximum torque of 1 Nm, case 2 has a certain margin, which is a reasonable scheme. Similarly, using Equations (20) and (21), the actuator can be selected according to the balancing error index, e.g., in case 3, a momentum wheel with a maximum torque of 1.5 Nm needs to be selected instead of 1 Nm.Figure 4
Theoretical results of the lumped disturbance.It should be noted that Equations (20) and (21) are established on the basis that Assumptions 1 and 2 hold. In practice, when the stiffness and damping of AMB is low, the radial angle deviation and displacement deviation of AMB cannot be omitted, i.e., Assumption 1 does not hold, which is manifested as the lumped disturbance d will be amplified in transfer process. Similarly, when the control accuracy of the satellite platform is low, i.e., Assumption 2 does not hold, the lumped disturbance d will also be amplified because of the inertial part. However, even if the two assumptions are violated, Equations (20) and (21) have sufficient accuracy, and the related simulation results will be given in Section 4.


## 2.1. Dynamic Model of RPS

The dynamic model of the platform subsystem is given as follows.(1)MpfX¨pf+∑k=1,2PakDakθ¨ak+∑k=1,2γtkη¨k=−AbFmb,Jpfω̇pf+J̇pfωpf+∑i=x,y,zRwiDwiθ¨wi+∑k=1,2RakDakθ¨ak+γskη¨k+ω~pfJpfωpf+∑i=x,y,zRwiDwiθ̇wi+∑k=1,2RakDakθ̇ak+γskη̇k=τb,JWiθ¨wi+ω̇pfTRwiDwi=τwii=x,y,z,JAkθ¨ak+X¨pfTPakDak+ω̇pfTRakDak+DakTγakη¨k=τakk=1,2,η¨k+γtkTX¨pf+γskTω̇pf+γakTDakθ¨ak+Ωk2ηk+2ζkΩkη̇k=0k=1,2.And the dynamic model of the payload subsystem is given as follows.(2)MplX¨pl=AbFmb,Jplω̇pl+RwtDwtθ¨wt+ω~plJplωpl+RwtDwtθ̇wt=τp,JWtθ¨wt+ω̇plTRwtDwt=τwt,where Xpf and Xpl are the projections of Xpf and Xpl in o, respectively. Mpf and Jpf are the mass and inertia matrix of the platform subsystem, while Mpl and Jpl are the definitions of the payload subsystem. ωpf and ωpl denote the angular velocity vector projection of the satellite platform and the rotating payload under their body-fixed coordinate frame, respectively. θwi, Rwi, and Jwi denote the rotation angle relative to the satellite platform, rotation coupling matrix with the satellite platform attitude, and inertia matrix of the momentum wheel i, respectively. In addition, JWi is the i-axis principal inertia of Jwi, and Dwi is the projection matrix of the relative angular velocity vector of momentum wheel i under its body-fixed reference frame, such as Dwx=1,0,0T. Similarly, θwt, Rwt, Jwt, JWt, and Dwt are the definitions of momentum wheel t. Furthermore, θak, Rak, Jak, JAk, and Dak are the definitions of the solar panel k, and Pak is the translation coupling matrix with the satellite platform. Solar panel vibration is described by mode coordinate ηk and mode frequency matrix Ωk. γak, γtk, and γsk denote the coupling matrix between the solar panel vibration and θak, Xpf, and ωpf, respectively. τwi and τwt denote the driving torques of the momentum wheel i and t, respectively. Ab is the attitude matrix of b relative to o. To give a more thorough description of Equation (1), the expressions of the coupling matrixes mentioned above are as follows.
(3)Pak=AbAkMkρ~ckT,γtk=AbAk∫kΦkdm,Rak=r~akAkMkρ~ckT+AkJak,γak=∫kρ~kΦkdm,γsk=r~akAk∫kΦkdm+Ak∫kρ~kΦkdm,where rak is the projection of rak in b, ρk and ρck are the projections of ρk and ρck in ak, Ak is the attitude matrix of ak relative to b, and Mk and Φk are the mass and mode shape function of solar panel k, respectively.DefiningFmb and τmb as the resultant force projection and torque projection of AMB acting on payload in b, respectively, which can be expressed as follows.
(4)Fmb=Kr5b+Cṙ5b,τmb=Kaϑ+Caϑ̇,where K∈R3×3, Ka∈R3×3, C∈R3×3, and Ca∈R3×3 are the bearing mechanical characteristics parameters, specifically, K and Ka are the displacement stiffness matrix and angular stiffness matrix of the AMB, respectively, and C and Ca are the displacement damping matrix and angular damping matrix of the AMB, respectively. r5b is the projection of the vector from the AMB’s resting geometric center position to the AMB’s actual geometric center position in b, i.e., the position deviation of the AMB. ϑ=φ,θ,ψT is the Euler angle of the payload relative to the satellite platform, i.e., the angular deviation of the AMB. Based on Equation (4), τb and τp can be expressed as follows.
(5)τb=−r~fbKr5b+Cṙ5b−Kaϑ−Caϑ̇,(6)τp=r~gpAbpTKr5b+Cṙ5b+AbpTKaϑ+Caϑ̇,where rfb is the projection of the vector from platform subsystem’s centroid to the AMB’s actual geometric center position in b, rgp is the projection of the vector from payload subsystem’s centroid to the AMB’s actual geometric center position in p, and Abp is the attitude matrix of p relative to b. Combining with Equations (1)–(6), the refined dynamic model of the RPS system has been established, and more details about Equations (1)–(6) can be found in [7].


## 2.2. Dynamic Model of Lumped Disturbance

Note that the models established in the previous subsection are applicable to any phase. In the imaging phase, the inertial force inFmb and the inertial torque in τmb are small, i.e., the lumped torque τb transmitted from the AMB to the satellite platform, denoted as the lumped disturbance d, is mainly composed of dynamic unbalanced disturbance and static unbalanced disturbance. In this section, the dynamic model of the lumped disturbance d is proposed. Before modeling, the following assumptions are made to simplify the analysis.Assumption 1.
The radial angle deviation and displacement deviation of AMB can be omitted, i.e., the following equations approximately.(7)ϑ=0,0,ψT,ϑ̇=0,0,ψ̇T,r5b=ṙ5b=03×1.Assumption 2.
The system states are stable,ω̇pf=ϑ¨=03×1 hold. The orbital angular velocity is denoted as ω0, and the desired rotational speed of payload is ψ̇d. Assuming that the state tracking errors are small, i.e., the following equations approximately.
(8)ωpf=ω0,0,0T,ψ̇=ψ̇d.As shown in Figure3, the centroid of momentum wheel t and the coordinate origin of p are coincident. Define rk to be the projection of rk in p, rj=rjx,rjy,rjzT to be the projection of the vector from coordinate origin of p to payload’s centroid in p (i.e., static unbalance) and Mp to be the payload’s mass, then rk=Mprj/Mpl holds. On this basis, the following equation can be obtained.
(9)MplXpl=MplXp+Aprk=MplXc+AbrP+ApMpMplrj=MpXc+AbrP+Aprj+MwtXc+AbrP,where Xc is the projection of Xc in o, Xp is the position vector projection of the coordinate origin of p in o, Ap is the attitude matrix of p relative to o, Mwt is the mass of momentum wheel t, and Mpl=Mp+Mwt holds. rP=rPx,rPy,rPzT is the projection of rP in b, which can be expressed as follows.
(10)rP=σrs+rl−εAbprj,where rs=rsx,rsy,rszT is the projection of rs in b, rl is the projection of rl in b, and σ=Mpf+Mpl−1Mpf and ε=Mpf+Mpl−1Mp hold.Combining Equations (2) and (9), one has
(11)Fmb=MplAbTX¨c+ω~pfω~pfrP+ω̇~pfrP+2ω~pfṙP+r¨P+MpAbpω~plω~plrj+Abpω̇~plrj.When Assumption1 holds, it follows that
(12)Abp=cosψ−sinψ0sinψcosψ0001,ωlf=ϑ̇,ṙP=−εAbpω~lfrj,r¨P=−εAbpω~lfω~lfrj−εAbpω̇~lfrj,where ωlf is the projection of the angular velocity of the payload relative to the satellite platform in p. Furthermore, when Assumption 2 holds, it follows that
(13)ωpl=ω0cosψd−ω0sinψdψ̇d,ω̇pl=−ψ̇dω0sinψdψ̇dω0cosψd0,ṙP=−εψ̇d−rjycosψd−rjxsinψd−rjysinψd+rjxcosψd0,r¨P=−εψ̇d2−rjxcosψd+rjysinψd−rjxsinψd−rjycosψd0.Bringing Equations (12) and (13) into Equation (11) and notingFmb=Fmb,x,Fmb,y,Fmb,zT, one has
(14)Fmb,x=Mplε−Mpψ̇d2rjxcosψd−rjysinψd,Fmb,y=Mplε−Mpψ̇d2rjxsinψd+rjycosψd−Mplω02rPy,Fmb,z=2Mp−2Mplεω0ψ̇drjxcosψd−rjysinψd−Mprjz+rPzω02−ω02rPzMwt.For the convenience of subsequent modeling,Fmb is divided into Fmb1 and Fmb2, where Fmb1=Fmb1,x,Fmb1,y,Fmb1,zT is the resultant force on the payload, and Fmb2=Fmb2,x,Fmb2,y,Fmb2,zT is the resultant force on the momentum wheel t, one has
(15)Fmb1,x=Mpε−1ψ̇d2rjxcosψd−rjysinψd,Fmb1,y=Mpε−1ψ̇d2rjxsinψd+rjycosψd−Mpω02rPy,Fmb1,z=2Mp1−εω0ψ̇drjxcosψd−rjysinψd−Mprjz+rPzω02,(16)Fmb2,x=Mwtεψ̇d2rjxcosψd−rjysinψd,Fmb2,y=Mwtεψ̇d2rjxsinψd+rjycosψd−Mwtω02rPy,Fmb2,z=−2Mwtεω0ψ̇drjxcosψd−rjysinψd−Mwtω02rPz.DividingrP into rP1 and rP2=rP2,x,rP2,y,rP2,zT, where rP1 is the projection of the vector from RPS system’s centroid to AMB’s actual geometric center position in b, and rP2=rP−rP1 holds. Then, based on Equations (2) and (5), the following equation can be obtained.
(17)τmb=AbpJplω̇pl+AbpRwtDwtθ¨wt+Abpω~plJplωpl+RwtDwtθ̇wt+r~P2+MpMplAbpr~jAbpTFmb.The inertia matrix of the rotating payload is denoted asJp and consists of the principal inertia Ix, Iy, and Iz and the inertia product Ixy, Ixz, and Iyz (i.e., dynamic unbalance). The following equation holds.
(18)Jpl=Jp+MpMplMwtr~jr~jT+Jwt.Bringing Equation (18) into Equation (17), when Assumption 1 and Assumption 2 hold, τmb=τmb,x,τmb,y,τmb,zTcan be given as
(19)τmb,x=−Ix−Iyψ̇dω0sin2ψd+2Ixyψ̇dω0cos2ψd+ψ̇d2Ixzsinψd+Iyzcosψd−rP2,z+rjzFmb1,y+rjycosψd+rjxsinψdFmb1,z−rP2,zFmb2,y,τmb,y=ω0ψ̇dIx−Iycos2ψd+2ψ̇dω0Ixysin2ψd+ψ̇d2Iyzsinψd−Ixzcosψd−ω0Izψ̇d+Jwtzψ̇d+θ̇wt+rP2,z+rjzFmb1,x+rjysinψd−rjxcosψdFmb1,z+rP2,zFmb2,x,τmb,z=12Ix−Iyω02sin2ψd−Ixyω02cos2ψd−rjycosψd+rjxsinψdFmb1,x+−rjysinψd+rjxcosψdFmb1,y.According to the definitions ofrfb, rs, rl, and rP2, it is clear that rfb=rs+rl−rp2 holds. Furthermore, based on Equations (5), (15), (16), and (19), the lumped disturbance d=dx,dy,dzT transmitted by the AMB, consisting of inertia difference disturbance d1, axial angular momentum disturbance d2, dynamic unbalanced disturbance d3, and static unbalanced disturbance d4, can be modeled as
(20)d1=Ix−Iyψ̇dω0sin2ψd−ω0ψ̇dIx−Iycos2ψd−Ix−Iyω02sin2ψd2,d2=0ω0Izψ̇d+ω0Jwtzψ̇d+θ̇wt0,d3=−2Ixyψ̇dω0cos2ψd−ψ̇d2Ixzsinψd+Iyzcosψd−2ψ̇dω0Ixysin2ψd−ψ̇d2Iyzsinψd−IxzcosψdIxyω02cos2ψd,(21)d4x=rsz+rlz+rjzFmb1,y−rsy+rjycosψd+rjxsinψdFmb1,z+rsz+rlzFmb2,y−rsyFmb2,z,d4y=−rsz+rlz+rjzFmb1,x+rsx+rjxcosψd−rjysinψdFmb1,z−rsz+rlzFmb2,x+rsxFmb2,z,d4z=rsy+rjycosψd+rjxsinψdFmb1,x−rsx−rjysinψd+rjxcosψdFmb1,y+rsyFmb2,x−rsxFmb2,y.


## 2.3. Disturbance Analysis

Equations (20) and (21) provide mechanical constraints on the payload mass parameters such as the radial principal inertia, dynamic unbalance, and static unbalance. Based on these two equations, the feasibility of the current balancing error index can be judged. To illustrate this point, the maximum torque of the satellite platform momentum wheel is 1 Nm, ω0=0.0011rad/s, ψ̇d=16deg/s, and the compensation accuracy of axial angular momentum disturbance is 10%, three balancing errors are considered: 1kg⋅m2 and 1mm, 3.9kg⋅m2 and 5mm, and 6.9kg⋅m2 and 7mm. The real dynamic unbalance and static unbalance are taken randomly within the respective error boundary, where the cases that make the lumped disturbance take their respective maximum values, denoted as cases 1-3, as shown in Table 1.Table 1
Payload inertia parameter values for different cases.
Principal inertiakg⋅m2Dynamic unbalancekg⋅m2Static unbalancemmCase 1Ix=114.4Ixz=1x=1, y=0, z=−9.6Iy=311.8Ixy=Iyz=0Iz=252.9Case 2Ix=214.4Ixz=3.9x=5, y=0, z=−9.6Iy=211.8Ixy=Iyz=0Iz=252.9Case 3Ix=214.4Ixz=6.9x=7, y=0, z=−9.6Iy=211.8Ixy=Iyz=0Iz=252.9The theoretical results ofd1-d4 for case 1 are summarized in Table 2. From case 1, it can be concluded that the inertia difference of payload does not lead to excessive disturbance even if it increases to 197.4 kgm2, and the axial angular momentum disturbance d2 is small constant. Therefore, when the dynamic unbalance and static unbalance increase, the lumped disturbance is mainly composed of d3 and d4.Table 2
Theoretical results of case 1.
d1d2d3d4Amplitude (Nm)0.0610.0080.0780.099Frequency (Hz)0.088/0.0440.044The theoretical results of the lumped disturbance in the three cases are shown in Figure4. In case 2, the lumped disturbance increases significantly when the unbalance increases. Furthermore, there is a risk that the lumped disturbance may exceed the maximum torque of the momentum wheel, as shown in case 3, and the control law cannot adjust the excess part, which will inevitably affect the attitude accuracy and stability of the satellite system. Therefore, more attention should be paid to the payload balancing error index. Under the condition of the maximum torque of 1 Nm, case 2 has a certain margin, which is a reasonable scheme. Similarly, using Equations (20) and (21), the actuator can be selected according to the balancing error index, e.g., in case 3, a momentum wheel with a maximum torque of 1.5 Nm needs to be selected instead of 1 Nm.Figure 4
Theoretical results of the lumped disturbance.It should be noted that Equations (20) and (21) are established on the basis that Assumptions 1 and 2 hold. In practice, when the stiffness and damping of AMB is low, the radial angle deviation and displacement deviation of AMB cannot be omitted, i.e., Assumption 1 does not hold, which is manifested as the lumped disturbance d will be amplified in transfer process. Similarly, when the control accuracy of the satellite platform is low, i.e., Assumption 2 does not hold, the lumped disturbance d will also be amplified because of the inertial part. However, even if the two assumptions are violated, Equations (20) and (21) have sufficient accuracy, and the related simulation results will be given in Section 4.


## 3. Controller Design

### 3.1. Control System Model

To simplify the analysis, the solar panel is considered as a rigid body, and its rotation relative to the satellite platform is ignored. Then, Equation (1) can be simplified as
(22)MpfX¨pf=−AbFmb,Jpfω̇pf+∑i=x,y,zRwiDwiθ¨wi+ω~pfJpfωpf+∑i=x,y,zRwiDwiθ̇wi=τb,JWiθ¨wi+ω̇pfTRwiDwi=τwii=x,y,z.Furthermore, simplifying the driving equations of the momentum wheelsi, and assuming that the flywheels are all installed in the forward direction. Note that u=−τwx,τwy,τwzT is the control torque, one has
(23)Jpfnω̇pf+ω~pfJpfnωpf+ω~pf∑i=x,y,zRwiDwiθ̇wi=u+d′,where Jpfn denotes the nominal value of Jpf, ΔJpf=Jpf−Jpfn denotes the parameter uncertainty, and d′=d−ΔJpfω̇pf−ω~pfΔJpfωpf holds. Define the quaternion of the satellite platform as qpf=qpf,0,qpf,bTT, where qpf,0 and qpf,b∈R3 are the scalar and vector components of the quaternion qpf, respectively, satisfying qpf,02+qpf,bTqpf,b=1. In addition, if the desired quaternion of the satellite platform is qd=qd0,qdbTT, the error quaternion qe=qe0,qebTT is obtained as follows:
(24)qe0=qdbTqpf,b+qd0qpf,0,qeb=qd0qpf,b−q~dbqpf,b−qpf,0qdb.Furthermore, defining the attitude error matrix asAqe and the desired angular velocity as ωd, then, the error angular velocity ωe can be expressed as ωe=ωpf−ωr, where ωr=AqeTωd. Therefore, Equation (23) can be expressed as
(25)ω̇e=−ω̇r−Jpfn−1ω~pfJpfnωpf−Jpfn−1ω~pf∑i=x,y,zRwiDwiθ̇wi+Jpfn−1u+Jpfn−1d′.And the error quaternion attitude kinematics equations are(26)q̇e0=−12ωeTqeb,q̇eb=Qqeωe,where Qqe=q~eb+qe0I3/2 holds. On the basis of Q̇qeωe=−qebωeTωe/4 [21] holds, combining Equations (25) and (26), one has
(27)q¨eb=h+Bu+D,where B=QqeJpfn−1, D=Bd′, and h satisfies
(28)h=−14qebωeTωe+Qqe−ω̇r−Jpfn−1ω~pfJpfnωpf−Jpfn−1ω~pf∑i=x,y,zRwiDwiθ̇wi.Up to now, the dynamic model for controller design has been established. In this paper, the control objective is to design the controlleru, such that the states qpf and ωpf can follow the desired states qd and ωd in a fixed time in spite of unknown lumped disturbance and uncertainty, i.e., there exist a constant Tmax>0 such that qeb=03×1 and ωe=03×1 hold for all t>Tmax.


### 3.2. Definition and Lemmas

Before the controller design, some useful definitions and theorems need to be provided.Consider the following nonlinear system:(29)ẋ=fxt,x0=x0,f0=0,where x∈Rn and f:Rn⟶Rn are a nonlinear function.Definition 3 (see [35]).
The system (29) is called finite-time convergent to the origin for an initial condition x0, if it is Lyapunov stable and there exists a settling time function Tx0 such that the system state xt,x0∈Rn is equal to zero, for all t≥Tx0.Definition 4 (see [36]).
The system (29) is called fixed-time convergent to the origin, if the origin is globally finite-time stable and the settling time function Tx0 is bounded by positive constant Tmax>0, i.e., the system state xt,x0∈Rn is equal to zero, for all t≥Tmax, starting from any initial condition x0.Lemma 5 (see [29]).
For system (29), suppose there exists a Lyapunov function Vx, scalars α1, β1, p, q, k∈R+, pk<1, and qk>1, such that V̇x≤−α1Vxp+β1Vxqk. Then, the system is fixed-time stable. Furthermore, the upper bound of the convergence time is given as follows.
(30)T≤1α1k1−pk+1β1kqk−1.Lemma 6 (see [27]).
Forxi∈ℝ, i=1,2,⋯,n, 0<χ1≤1, and χ2>1, the following inequalities hold:
(31)∑i=1nxiχ1≤∑i=1nxiχ1,∑i=1nxiχ2≤nχ2−1∑i=1nxiχ2.Assumption 7.
The unknown disturbanceD is bounded and continuously differentiable with respect to time, such that satisfy D≤Dm and Ḋ≤L, where Dm and L are a positive constant.Assumption 8.
System statesqeb and q̇eb in dynamic model (27) can be both measured.


### 3.3. Fixed-Time Extended State Observer

To estimate the unknown disturbanceD in dynamic model (27), define x1=q̇eb, and the extended state x2=D, the fixed-time extended state observer equation is designed as follows:
(32)Ż1=Z2−λ1e1e11/2−λ2e1e1ζ−1+h+Bu,Ż2=−λ3e1e1,where Z1∈R3, Z2∈R3, λ3>4L, λ1>2λ3, λ2>0, and ζ>1. Then, the observer error e1=Z1−x1 and e2=Z2−x2 can be expressed as
(33)ė1=−λ1e1e11/2−λ2e1e1ζ−1+e2,ė2=−λ3e1e1−Ḋ.According to the theorem proposed in [36], both states e1t and e2t converge to the origin uniformly in fixed time To, where To satisfies
(34)To≤1λ2ζ−1ηζ−1+23η1/2λ1+KM×1+Mm1−2λ3/λ1+3Km,where M=λ3+L, m=λ3−L, η=31/4λ1/λ21/ζ+0.5, and K≥e20. In this paper, since the initial value of Z2 is taken as zero, so K can take Dm according to Assumption 7. More information on the derivation of the upper bound for To can be found in [36].Remark 9.
It should be noted that the proposed FESO (32) is modified according to [36]. Compared with the fixed-time observers in [33, 34], observer (32) is much simpler for designing the observer parameters. Taking the observer in [33] as an example, a total of six parameters need to be designed in order to achieve unknown disturbance estimation, so this observer is much more difficult to be applied to space missions. However, the FESO (32) has only λ1, λ2, λ3, and ζ to be designed, and it is clear that the proposed FESO achieves a more concise structure and fewer parameters to be designed. Therefore, in space missions, it is more convenient for the proposed FESO (32) to estimate the disturbance D within a fixed time.


### 3.4. Design of Controller

#### 3.4.1. Fixed-Time Sliding Mode Surface

In this section, a novel QFNFTSMS is proposed:(35)s=Kqebqeb+sigr2q̇eb,where Kqeb is a diagonal matrix, and the diagonal elements kqebii=1,2,3 are denoted as
(36)kqebi=aqebip−1/kr2+bqebig−1/kr2kr2,where a>0, b>0, k>1, 2>r2>1, 1<gk≤2k−1, and 1/r2<pk<1.Theorem 10.
Whens=0 satisfies, qeb and q̇eb can converge to the origin uniformly within fixed time Ts in spite of the initial conditions. Meanwhile, qe0 and ωe will converge to 1 (or -1) and 03×1, respectively: if qe00>0, qe0 will converge to 1, if qe00<0, qe0 will converge to -1. Where the upper bound of convergence time Ts satisfies
(37)Ts≤21−pk/2ak31−k1−pk+21−gk/2bk31−kgk−1.Proof.
Ifs=0 holds, that is, q̇eb=−sig1/r2Kqebqeb. Furthermore, we can obtain
(38)q̇ebi=−sig1/r2kqebiqebi=−sig1/r2aqebip+bqebigkr2signqebi=−aqebip+bqebigksignqebi=−sigkasigpqebi+bsiggqebi.Therefore,q̇eb=−sigkasigpqeb+bsiggqeb holds, and qeb=03×1 is the equilibrium point. Let the positive Lyapunov function be V1=0.5qebTqeb, then, the derivative of V1 is obtained as follows.
(39)V̇1=−qebTsigkasigpqeb+bsiggqeb=−∑i=13qebiasigpqebi+bsiggqebiksignqebi=−∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2k.By Lemma6, one has
(40)V̇1≤−31−k∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2k=−31−ka∑i=13qebi2p+1/k/2+b∑i=13qebi2g+1/k/2k≤−31−k/k2p+1/k/2aVp+1/k/2+31−k/k2p+1/k/2bVg+1/k/2k.Then based on Lemma5, qeb can converge to the origin uniformly in bounded convergence time T3 and T3 satisfies Equation (37). Moreover, q̇eb will converge to the origin, and ωe will converge to the origin because of q̇eb=Qqeωe. However, whether qe0 will converge to 1 or -1 cannot be judged from V1 and needs further discussion.Note that whens=0 satisfies, the following equation can be obtained based on Equation (26).
(41)q̇e0=qebTsigkasigpqeb+bsiggqebqe0.Ifqe00>0, then q̇e0>0 holds at t>0, so qe0t>0 holds for all t>0; if qe00<0, q̇e0<0 holds at t>0, soqe0t<0 holds for all t>0. Furthermore, when qe00>0, define V2=qebTqeb+1−qe02, V3=qebTqeb+1+qe02, the following inequalities hold.
(42)V̇2=−2∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2kqe0<0,V̇3=2∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2kqe0>0.Equation (42) indicates that qe0=1 is the equilibrium point and qe0=−1 is the nonequilibrium point. Similarly, when qe00<0, V̇2>0, and V3<0 hold, i.e., qe0=−1is the equilibrium point, and qe0=1 is the nonequilibrium point. This completes the proof of Theorem 10.Furthermore, the proposed QFNFTSMS has a faster convergence performance than other sliding mode surfaces, such as the nonsingular TSM surface [21] and the sliding mode surfaces used in [26–28], as listed below.
(43)s1=qeb+σ1sign1q̇eb,s2=qeb+σ2sign2qeb+σ3sign3q̇eb,s3=q̇eb+σ4sign4qeb+σ5sign5qeb,where σi>0,i=1,2,⋯,5, 1<ni<2,i=1,3, n2>n3, n4>1, and 0<n5<1 hold. In order to make the sliding mode surfaces have the same coefficients and exponents, the sliding mode surface parameters are selected as follows.
(44)σ1=σ3,n1=n3,σ4=1/n3,n4=n2n3,σ5=σ3−1/n3,pk=n5=n3−1,a=σ51/k,b=σ41/k,gk=n4.Based ons1, we can derive that
(45)q̇ebi=−qebiσ11/n1sgnqebi.Definey=qebi, dt=−σ11/n1y−1/n1dy holds. Therefore, the convergence time of s1 can be solved as follows.
(46)Ts1=∫0y01σ1y−1/n1dy,where y0=qebi0. Similarly, the convergence time of the others can be solved as follows.
(47)Ts2=∫0y011/σ3y+σ2/σ3yn21/n3dy,Ts3=∫0y01σ4yn4+σ5yn5dy,Ts=∫0y01ayp+bygkdy.Combining Equation (44) and Lemma 6, one has
(48)1σ3y+σ2σ3yn21/n3=1σ1y+σ2σ1yn21/n1>1σ1y1/n1,1σ3y+σ2σ3yn21/n3≤1σ31/n3y1/n3+σ2σ31/n3yn2/n3=σ4yn4+σ5yn5=akypk+bkygk≤ayp+bygk.Which impliesTs≤Ts3≤Ts2<Ts1. With the parameters chosen as σ1=σ2=1, n1=1.5, n2=2, and k=2, the convergence time comparison for different sliding mode surfaces is shown in Figure 5, indicating that the proposed QFNFTSMS has a faster convergence performance.Remark 11.
It should be noted that the proposed QFNFTSMS is modified with respect to [29]. In addition to the faster convergence performance proved above, the proposed QFNFTSMS can avoid the singularity without any extra structure. Furthermore, compared with Euler angles used for attitude representation in [34], quaternion has the advantage of avoiding singularity in the kinematic equations. However, the sliding mode surface designed by qeb and q̇eb in [21, 31] and the conclusion that qe0=−1 is nonequilibrium point in [17] are both misquoted, whose premise is that the sliding mode surface needs to be designed by qeb and ωe. In this paper, Theorem 10 provides a complete proof, whose premise is that qe00 should not be equal to zero.Figure 5
Convergence time comparison.


#### 3.4.2. Fixed-Time Controller

In this section, a novel quaternion-based fixed-time nonsingular fast terminal sliding mode control law is designed as follows.(49)u=−1r2B−1K¯qeb+Kqebsig2−r2q̇eb−B−1h−1r2B−1diagκq̇ebr2−1diagq̇eb1−r2×sigk1αsigp1s+βsigg1s−B−1ρsigns−B−1Z2,where k1>1, 0<p1k1<1, 1<g1k1<2k1−1, α>0, β>0, and ρ>0. κ is a nonlinear function defined as follows.
(50)κ=sinπx22τ2x<τ,1x≥τ,where τ is a sufficiently small positive number. K¯qeb is a diagonal matrix, and the diagonal elements k¯qebii=1,2,3 are denoted as
(51)k¯qebi=kr2aqebip−1/kr2+bqebig−1/kr2kr2−1×ap−1kr2qebip−1/kr2+bg−1kr2qebig−1/kr2.Theorem 12.
For dynamic model (27), with the proposed FESO (32), the proposed QFNFTSMS (35), and the proposed QFNFTSMC (49), qeb and q̇eb can converge to the origin uniformly within fixed time T in spite of the initial conditions, and the upper bound of convergence time T satisfies.
(52)T≤To+Tc+Ts,where Tc satisfies
(53)Tc≤231−k1αk12p1k1+1/21−p1k1+231−k1βk12g1k1+1/2g1k1−1+2τ1/r2−1ρ.Proof.
Let the positive Lyapunov function beV4=sTs/2, then, the derivative of V4 is obtained as follows.
(54)V̇4=sTK¯qeb+Kqebq̇eb+sTr2diagq̇ebr2−1h+Bu+D.
Substituting Equation (49) into Equation (54), we can get
(55)V̇4=−sTdiagκq̇ebr2−1sigk1αsigp1s+βsigg1s+sTr2diagq̇ebr2−1−ρsgns−Z2+D.
BecauseZ2⟶D within a fixed time To, Equation (55) can be simplified as follows.
(56)V̇4<−sTdiagκq̇ebr2−1sigk1αsigp1s+βsigg1s≤−31−k1αminκi1/k1∑i=13si2p1k1+1/2k1+βminκi1/k1∑i=13si2g1k1+1/2k1k1≤−31−k1/k1αminκi1/k12p1k1+1/2k1Vp1k1+1/2k1+31−k1/k1βminκi1/k12g1k1+1/2k1Vg1k1+1/2k1k1≤−αmminκi1/k1Vp1k1+1/2k1+βmminκi1/k1Vg1k1+1/2k1k1,where αm and βm are defined as
(57)αm=31−k1/k1α2p1k1+1/2k1,βm=31−k1/k1β2g1k1+1/2k1.
The state space is divided into two areas as(58)δ1=qebi,q̇ebiq̇ebir2−1≥τ,∀i=1,2,3,δ2=qebi,q̇ebiq̇ebir2−1<τ,∃i=1,2,3.When the system statesqeb and q̇eb are in the region δ1, q̇ebi≥τ1/r2−1,∀i=1,2,3 and minκi1/k1=1 satisfies, so V̇≤−αmVp1k1+1/2k1+βmVg1k1+1/2k1k1, therefore, system states qeb and q̇eb will be driven on the sliding mode surface s=0 or into the region δ2 in fixed convergence time Tc1 based on Lemma 5. Where Tc1 satisfies
(59)Tc1≤231−k1αk12p1k1+1/21−p1k1+231−k1βk12g1k1+1/2g1k1−1.If the system statesqeband q̇eb are in the region δ2, i.e., 0≤q̇ebi<τ1/r2−1,∃i=1,2,3 holds. When q̇ebi≠0,∀i=1,2,3, it can also be verified that s=0 is still an attractor. When q̇ebi=0,∃i=1,2,3, combined with Equation (27) and Equation (49), we can get q¨ebi=−ρsgnsi, if si>0, then q¨ebi=−ρ<0, if si<0, then q¨ebi=ρ>0. Therefore, q̇ebi=0 is not an attractor except for the origin. Furthermore, assuming that q¨ebi=−ρsgnsi holds approximately when q̇ebi<τ1/r2−1, the system states will pass through the region δ2 monotonically in fixed time Tc2, where Tc2≤2τ1/r2−1/ρ. So, system states qeb and q̇eb will be driven on the sliding mode surface s=0 within fixed time Tc=Tc1+Tc2. Then, combined with the observer convergence time To and the sliding mode surface convergence time Ts, Theorem 12 is proved.Remark 13.
It should be noted that the control law (49) is modified with respect to [29, 34]. Because the unknown disturbance can be estimated by the fixed-time observer, the proposed control law does not require the disturbance boundary in advance. Compared with the controller proposed in [34], the disturbance effect on the satellite system can be offset by feeding back Z2 to improve control accuracy and the stability condition for ρ≥0, which greatly reduces the parameter selection range to attenuate the chattering. In this paper, in order to enable the system state to pass through the region δ2, ρ is selected as a small constant greater than zero, and then a saturation function is used to achieve chattering-free attitude control.


## 3.1. Control System Model

To simplify the analysis, the solar panel is considered as a rigid body, and its rotation relative to the satellite platform is ignored. Then, Equation (1) can be simplified as
(22)MpfX¨pf=−AbFmb,Jpfω̇pf+∑i=x,y,zRwiDwiθ¨wi+ω~pfJpfωpf+∑i=x,y,zRwiDwiθ̇wi=τb,JWiθ¨wi+ω̇pfTRwiDwi=τwii=x,y,z.Furthermore, simplifying the driving equations of the momentum wheelsi, and assuming that the flywheels are all installed in the forward direction. Note that u=−τwx,τwy,τwzT is the control torque, one has
(23)Jpfnω̇pf+ω~pfJpfnωpf+ω~pf∑i=x,y,zRwiDwiθ̇wi=u+d′,where Jpfn denotes the nominal value of Jpf, ΔJpf=Jpf−Jpfn denotes the parameter uncertainty, and d′=d−ΔJpfω̇pf−ω~pfΔJpfωpf holds. Define the quaternion of the satellite platform as qpf=qpf,0,qpf,bTT, where qpf,0 and qpf,b∈R3 are the scalar and vector components of the quaternion qpf, respectively, satisfying qpf,02+qpf,bTqpf,b=1. In addition, if the desired quaternion of the satellite platform is qd=qd0,qdbTT, the error quaternion qe=qe0,qebTT is obtained as follows:
(24)qe0=qdbTqpf,b+qd0qpf,0,qeb=qd0qpf,b−q~dbqpf,b−qpf,0qdb.Furthermore, defining the attitude error matrix asAqe and the desired angular velocity as ωd, then, the error angular velocity ωe can be expressed as ωe=ωpf−ωr, where ωr=AqeTωd. Therefore, Equation (23) can be expressed as
(25)ω̇e=−ω̇r−Jpfn−1ω~pfJpfnωpf−Jpfn−1ω~pf∑i=x,y,zRwiDwiθ̇wi+Jpfn−1u+Jpfn−1d′.And the error quaternion attitude kinematics equations are(26)q̇e0=−12ωeTqeb,q̇eb=Qqeωe,where Qqe=q~eb+qe0I3/2 holds. On the basis of Q̇qeωe=−qebωeTωe/4 [21] holds, combining Equations (25) and (26), one has
(27)q¨eb=h+Bu+D,where B=QqeJpfn−1, D=Bd′, and h satisfies
(28)h=−14qebωeTωe+Qqe−ω̇r−Jpfn−1ω~pfJpfnωpf−Jpfn−1ω~pf∑i=x,y,zRwiDwiθ̇wi.Up to now, the dynamic model for controller design has been established. In this paper, the control objective is to design the controlleru, such that the states qpf and ωpf can follow the desired states qd and ωd in a fixed time in spite of unknown lumped disturbance and uncertainty, i.e., there exist a constant Tmax>0 such that qeb=03×1 and ωe=03×1 hold for all t>Tmax.


## 3.2. Definition and Lemmas

Before the controller design, some useful definitions and theorems need to be provided.Consider the following nonlinear system:(29)ẋ=fxt,x0=x0,f0=0,where x∈Rn and f:Rn⟶Rn are a nonlinear function.Definition 3 (see [35]).
The system (29) is called finite-time convergent to the origin for an initial condition x0, if it is Lyapunov stable and there exists a settling time function Tx0 such that the system state xt,x0∈Rn is equal to zero, for all t≥Tx0.Definition 4 (see [36]).
The system (29) is called fixed-time convergent to the origin, if the origin is globally finite-time stable and the settling time function Tx0 is bounded by positive constant Tmax>0, i.e., the system state xt,x0∈Rn is equal to zero, for all t≥Tmax, starting from any initial condition x0.Lemma 5 (see [29]).
For system (29), suppose there exists a Lyapunov function Vx, scalars α1, β1, p, q, k∈R+, pk<1, and qk>1, such that V̇x≤−α1Vxp+β1Vxqk. Then, the system is fixed-time stable. Furthermore, the upper bound of the convergence time is given as follows.
(30)T≤1α1k1−pk+1β1kqk−1.Lemma 6 (see [27]).
Forxi∈ℝ, i=1,2,⋯,n, 0<χ1≤1, and χ2>1, the following inequalities hold:
(31)∑i=1nxiχ1≤∑i=1nxiχ1,∑i=1nxiχ2≤nχ2−1∑i=1nxiχ2.Assumption 7.
The unknown disturbanceD is bounded and continuously differentiable with respect to time, such that satisfy D≤Dm and Ḋ≤L, where Dm and L are a positive constant.Assumption 8.
System statesqeb and q̇eb in dynamic model (27) can be both measured.


## 3.3. Fixed-Time Extended State Observer

To estimate the unknown disturbanceD in dynamic model (27), define x1=q̇eb, and the extended state x2=D, the fixed-time extended state observer equation is designed as follows:
(32)Ż1=Z2−λ1e1e11/2−λ2e1e1ζ−1+h+Bu,Ż2=−λ3e1e1,where Z1∈R3, Z2∈R3, λ3>4L, λ1>2λ3, λ2>0, and ζ>1. Then, the observer error e1=Z1−x1 and e2=Z2−x2 can be expressed as
(33)ė1=−λ1e1e11/2−λ2e1e1ζ−1+e2,ė2=−λ3e1e1−Ḋ.According to the theorem proposed in [36], both states e1t and e2t converge to the origin uniformly in fixed time To, where To satisfies
(34)To≤1λ2ζ−1ηζ−1+23η1/2λ1+KM×1+Mm1−2λ3/λ1+3Km,where M=λ3+L, m=λ3−L, η=31/4λ1/λ21/ζ+0.5, and K≥e20. In this paper, since the initial value of Z2 is taken as zero, so K can take Dm according to Assumption 7. More information on the derivation of the upper bound for To can be found in [36].Remark 9.
It should be noted that the proposed FESO (32) is modified according to [36]. Compared with the fixed-time observers in [33, 34], observer (32) is much simpler for designing the observer parameters. Taking the observer in [33] as an example, a total of six parameters need to be designed in order to achieve unknown disturbance estimation, so this observer is much more difficult to be applied to space missions. However, the FESO (32) has only λ1, λ2, λ3, and ζ to be designed, and it is clear that the proposed FESO achieves a more concise structure and fewer parameters to be designed. Therefore, in space missions, it is more convenient for the proposed FESO (32) to estimate the disturbance D within a fixed time.


## 3.4. Design of Controller

### 3.4.1. Fixed-Time Sliding Mode Surface

In this section, a novel QFNFTSMS is proposed:(35)s=Kqebqeb+sigr2q̇eb,where Kqeb is a diagonal matrix, and the diagonal elements kqebii=1,2,3 are denoted as
(36)kqebi=aqebip−1/kr2+bqebig−1/kr2kr2,where a>0, b>0, k>1, 2>r2>1, 1<gk≤2k−1, and 1/r2<pk<1.Theorem 10.
Whens=0 satisfies, qeb and q̇eb can converge to the origin uniformly within fixed time Ts in spite of the initial conditions. Meanwhile, qe0 and ωe will converge to 1 (or -1) and 03×1, respectively: if qe00>0, qe0 will converge to 1, if qe00<0, qe0 will converge to -1. Where the upper bound of convergence time Ts satisfies
(37)Ts≤21−pk/2ak31−k1−pk+21−gk/2bk31−kgk−1.Proof.
Ifs=0 holds, that is, q̇eb=−sig1/r2Kqebqeb. Furthermore, we can obtain
(38)q̇ebi=−sig1/r2kqebiqebi=−sig1/r2aqebip+bqebigkr2signqebi=−aqebip+bqebigksignqebi=−sigkasigpqebi+bsiggqebi.Therefore,q̇eb=−sigkasigpqeb+bsiggqeb holds, and qeb=03×1 is the equilibrium point. Let the positive Lyapunov function be V1=0.5qebTqeb, then, the derivative of V1 is obtained as follows.
(39)V̇1=−qebTsigkasigpqeb+bsiggqeb=−∑i=13qebiasigpqebi+bsiggqebiksignqebi=−∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2k.By Lemma6, one has
(40)V̇1≤−31−k∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2k=−31−ka∑i=13qebi2p+1/k/2+b∑i=13qebi2g+1/k/2k≤−31−k/k2p+1/k/2aVp+1/k/2+31−k/k2p+1/k/2bVg+1/k/2k.Then based on Lemma5, qeb can converge to the origin uniformly in bounded convergence time T3 and T3 satisfies Equation (37). Moreover, q̇eb will converge to the origin, and ωe will converge to the origin because of q̇eb=Qqeωe. However, whether qe0 will converge to 1 or -1 cannot be judged from V1 and needs further discussion.Note that whens=0 satisfies, the following equation can be obtained based on Equation (26).
(41)q̇e0=qebTsigkasigpqeb+bsiggqebqe0.Ifqe00>0, then q̇e0>0 holds at t>0, so qe0t>0 holds for all t>0; if qe00<0, q̇e0<0 holds at t>0, soqe0t<0 holds for all t>0. Furthermore, when qe00>0, define V2=qebTqeb+1−qe02, V3=qebTqeb+1+qe02, the following inequalities hold.
(42)V̇2=−2∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2kqe0<0,V̇3=2∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2kqe0>0.Equation (42) indicates that qe0=1 is the equilibrium point and qe0=−1 is the nonequilibrium point. Similarly, when qe00<0, V̇2>0, and V3<0 hold, i.e., qe0=−1is the equilibrium point, and qe0=1 is the nonequilibrium point. This completes the proof of Theorem 10.Furthermore, the proposed QFNFTSMS has a faster convergence performance than other sliding mode surfaces, such as the nonsingular TSM surface [21] and the sliding mode surfaces used in [26–28], as listed below.
(43)s1=qeb+σ1sign1q̇eb,s2=qeb+σ2sign2qeb+σ3sign3q̇eb,s3=q̇eb+σ4sign4qeb+σ5sign5qeb,where σi>0,i=1,2,⋯,5, 1<ni<2,i=1,3, n2>n3, n4>1, and 0<n5<1 hold. In order to make the sliding mode surfaces have the same coefficients and exponents, the sliding mode surface parameters are selected as follows.
(44)σ1=σ3,n1=n3,σ4=1/n3,n4=n2n3,σ5=σ3−1/n3,pk=n5=n3−1,a=σ51/k,b=σ41/k,gk=n4.Based ons1, we can derive that
(45)q̇ebi=−qebiσ11/n1sgnqebi.Definey=qebi, dt=−σ11/n1y−1/n1dy holds. Therefore, the convergence time of s1 can be solved as follows.
(46)Ts1=∫0y01σ1y−1/n1dy,where y0=qebi0. Similarly, the convergence time of the others can be solved as follows.
(47)Ts2=∫0y011/σ3y+σ2/σ3yn21/n3dy,Ts3=∫0y01σ4yn4+σ5yn5dy,Ts=∫0y01ayp+bygkdy.Combining Equation (44) and Lemma 6, one has
(48)1σ3y+σ2σ3yn21/n3=1σ1y+σ2σ1yn21/n1>1σ1y1/n1,1σ3y+σ2σ3yn21/n3≤1σ31/n3y1/n3+σ2σ31/n3yn2/n3=σ4yn4+σ5yn5=akypk+bkygk≤ayp+bygk.Which impliesTs≤Ts3≤Ts2<Ts1. With the parameters chosen as σ1=σ2=1, n1=1.5, n2=2, and k=2, the convergence time comparison for different sliding mode surfaces is shown in Figure 5, indicating that the proposed QFNFTSMS has a faster convergence performance.Remark 11.
It should be noted that the proposed QFNFTSMS is modified with respect to [29]. In addition to the faster convergence performance proved above, the proposed QFNFTSMS can avoid the singularity without any extra structure. Furthermore, compared with Euler angles used for attitude representation in [34], quaternion has the advantage of avoiding singularity in the kinematic equations. However, the sliding mode surface designed by qeb and q̇eb in [21, 31] and the conclusion that qe0=−1 is nonequilibrium point in [17] are both misquoted, whose premise is that the sliding mode surface needs to be designed by qeb and ωe. In this paper, Theorem 10 provides a complete proof, whose premise is that qe00 should not be equal to zero.Figure 5
Convergence time comparison.


### 3.4.2. Fixed-Time Controller

In this section, a novel quaternion-based fixed-time nonsingular fast terminal sliding mode control law is designed as follows.(49)u=−1r2B−1K¯qeb+Kqebsig2−r2q̇eb−B−1h−1r2B−1diagκq̇ebr2−1diagq̇eb1−r2×sigk1αsigp1s+βsigg1s−B−1ρsigns−B−1Z2,where k1>1, 0<p1k1<1, 1<g1k1<2k1−1, α>0, β>0, and ρ>0. κ is a nonlinear function defined as follows.
(50)κ=sinπx22τ2x<τ,1x≥τ,where τ is a sufficiently small positive number. K¯qeb is a diagonal matrix, and the diagonal elements k¯qebii=1,2,3 are denoted as
(51)k¯qebi=kr2aqebip−1/kr2+bqebig−1/kr2kr2−1×ap−1kr2qebip−1/kr2+bg−1kr2qebig−1/kr2.Theorem 12.
For dynamic model (27), with the proposed FESO (32), the proposed QFNFTSMS (35), and the proposed QFNFTSMC (49), qeb and q̇eb can converge to the origin uniformly within fixed time T in spite of the initial conditions, and the upper bound of convergence time T satisfies.
(52)T≤To+Tc+Ts,where Tc satisfies
(53)Tc≤231−k1αk12p1k1+1/21−p1k1+231−k1βk12g1k1+1/2g1k1−1+2τ1/r2−1ρ.Proof.
Let the positive Lyapunov function beV4=sTs/2, then, the derivative of V4 is obtained as follows.
(54)V̇4=sTK¯qeb+Kqebq̇eb+sTr2diagq̇ebr2−1h+Bu+D.
Substituting Equation (49) into Equation (54), we can get
(55)V̇4=−sTdiagκq̇ebr2−1sigk1αsigp1s+βsigg1s+sTr2diagq̇ebr2−1−ρsgns−Z2+D.
BecauseZ2⟶D within a fixed time To, Equation (55) can be simplified as follows.
(56)V̇4<−sTdiagκq̇ebr2−1sigk1αsigp1s+βsigg1s≤−31−k1αminκi1/k1∑i=13si2p1k1+1/2k1+βminκi1/k1∑i=13si2g1k1+1/2k1k1≤−31−k1/k1αminκi1/k12p1k1+1/2k1Vp1k1+1/2k1+31−k1/k1βminκi1/k12g1k1+1/2k1Vg1k1+1/2k1k1≤−αmminκi1/k1Vp1k1+1/2k1+βmminκi1/k1Vg1k1+1/2k1k1,where αm and βm are defined as
(57)αm=31−k1/k1α2p1k1+1/2k1,βm=31−k1/k1β2g1k1+1/2k1.
The state space is divided into two areas as(58)δ1=qebi,q̇ebiq̇ebir2−1≥τ,∀i=1,2,3,δ2=qebi,q̇ebiq̇ebir2−1<τ,∃i=1,2,3.When the system statesqeb and q̇eb are in the region δ1, q̇ebi≥τ1/r2−1,∀i=1,2,3 and minκi1/k1=1 satisfies, so V̇≤−αmVp1k1+1/2k1+βmVg1k1+1/2k1k1, therefore, system states qeb and q̇eb will be driven on the sliding mode surface s=0 or into the region δ2 in fixed convergence time Tc1 based on Lemma 5. Where Tc1 satisfies
(59)Tc1≤231−k1αk12p1k1+1/21−p1k1+231−k1βk12g1k1+1/2g1k1−1.If the system statesqeband q̇eb are in the region δ2, i.e., 0≤q̇ebi<τ1/r2−1,∃i=1,2,3 holds. When q̇ebi≠0,∀i=1,2,3, it can also be verified that s=0 is still an attractor. When q̇ebi=0,∃i=1,2,3, combined with Equation (27) and Equation (49), we can get q¨ebi=−ρsgnsi, if si>0, then q¨ebi=−ρ<0, if si<0, then q¨ebi=ρ>0. Therefore, q̇ebi=0 is not an attractor except for the origin. Furthermore, assuming that q¨ebi=−ρsgnsi holds approximately when q̇ebi<τ1/r2−1, the system states will pass through the region δ2 monotonically in fixed time Tc2, where Tc2≤2τ1/r2−1/ρ. So, system states qeb and q̇eb will be driven on the sliding mode surface s=0 within fixed time Tc=Tc1+Tc2. Then, combined with the observer convergence time To and the sliding mode surface convergence time Ts, Theorem 12 is proved.Remark 13.
It should be noted that the control law (49) is modified with respect to [29, 34]. Because the unknown disturbance can be estimated by the fixed-time observer, the proposed control law does not require the disturbance boundary in advance. Compared with the controller proposed in [34], the disturbance effect on the satellite system can be offset by feeding back Z2 to improve control accuracy and the stability condition for ρ≥0, which greatly reduces the parameter selection range to attenuate the chattering. In this paper, in order to enable the system state to pass through the region δ2, ρ is selected as a small constant greater than zero, and then a saturation function is used to achieve chattering-free attitude control.


## 3.4.1. Fixed-Time Sliding Mode Surface

In this section, a novel QFNFTSMS is proposed:(35)s=Kqebqeb+sigr2q̇eb,where Kqeb is a diagonal matrix, and the diagonal elements kqebii=1,2,3 are denoted as
(36)kqebi=aqebip−1/kr2+bqebig−1/kr2kr2,where a>0, b>0, k>1, 2>r2>1, 1<gk≤2k−1, and 1/r2<pk<1.Theorem 10.
Whens=0 satisfies, qeb and q̇eb can converge to the origin uniformly within fixed time Ts in spite of the initial conditions. Meanwhile, qe0 and ωe will converge to 1 (or -1) and 03×1, respectively: if qe00>0, qe0 will converge to 1, if qe00<0, qe0 will converge to -1. Where the upper bound of convergence time Ts satisfies
(37)Ts≤21−pk/2ak31−k1−pk+21−gk/2bk31−kgk−1.Proof.
Ifs=0 holds, that is, q̇eb=−sig1/r2Kqebqeb. Furthermore, we can obtain
(38)q̇ebi=−sig1/r2kqebiqebi=−sig1/r2aqebip+bqebigkr2signqebi=−aqebip+bqebigksignqebi=−sigkasigpqebi+bsiggqebi.Therefore,q̇eb=−sigkasigpqeb+bsiggqeb holds, and qeb=03×1 is the equilibrium point. Let the positive Lyapunov function be V1=0.5qebTqeb, then, the derivative of V1 is obtained as follows.
(39)V̇1=−qebTsigkasigpqeb+bsiggqeb=−∑i=13qebiasigpqebi+bsiggqebiksignqebi=−∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2k.By Lemma6, one has
(40)V̇1≤−31−k∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2k=−31−ka∑i=13qebi2p+1/k/2+b∑i=13qebi2g+1/k/2k≤−31−k/k2p+1/k/2aVp+1/k/2+31−k/k2p+1/k/2bVg+1/k/2k.Then based on Lemma5, qeb can converge to the origin uniformly in bounded convergence time T3 and T3 satisfies Equation (37). Moreover, q̇eb will converge to the origin, and ωe will converge to the origin because of q̇eb=Qqeωe. However, whether qe0 will converge to 1 or -1 cannot be judged from V1 and needs further discussion.Note that whens=0 satisfies, the following equation can be obtained based on Equation (26).
(41)q̇e0=qebTsigkasigpqeb+bsiggqebqe0.Ifqe00>0, then q̇e0>0 holds at t>0, so qe0t>0 holds for all t>0; if qe00<0, q̇e0<0 holds at t>0, soqe0t<0 holds for all t>0. Furthermore, when qe00>0, define V2=qebTqeb+1−qe02, V3=qebTqeb+1+qe02, the following inequalities hold.
(42)V̇2=−2∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2kqe0<0,V̇3=2∑i=13aqebi2p+1/k/2+bqebi2g+1/k/2kqe0>0.Equation (42) indicates that qe0=1 is the equilibrium point and qe0=−1 is the nonequilibrium point. Similarly, when qe00<0, V̇2>0, and V3<0 hold, i.e., qe0=−1is the equilibrium point, and qe0=1 is the nonequilibrium point. This completes the proof of Theorem 10.Furthermore, the proposed QFNFTSMS has a faster convergence performance than other sliding mode surfaces, such as the nonsingular TSM surface [21] and the sliding mode surfaces used in [26–28], as listed below.
(43)s1=qeb+σ1sign1q̇eb,s2=qeb+σ2sign2qeb+σ3sign3q̇eb,s3=q̇eb+σ4sign4qeb+σ5sign5qeb,where σi>0,i=1,2,⋯,5, 1<ni<2,i=1,3, n2>n3, n4>1, and 0<n5<1 hold. In order to make the sliding mode surfaces have the same coefficients and exponents, the sliding mode surface parameters are selected as follows.
(44)σ1=σ3,n1=n3,σ4=1/n3,n4=n2n3,σ5=σ3−1/n3,pk=n5=n3−1,a=σ51/k,b=σ41/k,gk=n4.Based ons1, we can derive that
(45)q̇ebi=−qebiσ11/n1sgnqebi.Definey=qebi, dt=−σ11/n1y−1/n1dy holds. Therefore, the convergence time of s1 can be solved as follows.
(46)Ts1=∫0y01σ1y−1/n1dy,where y0=qebi0. Similarly, the convergence time of the others can be solved as follows.
(47)Ts2=∫0y011/σ3y+σ2/σ3yn21/n3dy,Ts3=∫0y01σ4yn4+σ5yn5dy,Ts=∫0y01ayp+bygkdy.Combining Equation (44) and Lemma 6, one has
(48)1σ3y+σ2σ3yn21/n3=1σ1y+σ2σ1yn21/n1>1σ1y1/n1,1σ3y+σ2σ3yn21/n3≤1σ31/n3y1/n3+σ2σ31/n3yn2/n3=σ4yn4+σ5yn5=akypk+bkygk≤ayp+bygk.Which impliesTs≤Ts3≤Ts2<Ts1. With the parameters chosen as σ1=σ2=1, n1=1.5, n2=2, and k=2, the convergence time comparison for different sliding mode surfaces is shown in Figure 5, indicating that the proposed QFNFTSMS has a faster convergence performance.Remark 11.
It should be noted that the proposed QFNFTSMS is modified with respect to [29]. In addition to the faster convergence performance proved above, the proposed QFNFTSMS can avoid the singularity without any extra structure. Furthermore, compared with Euler angles used for attitude representation in [34], quaternion has the advantage of avoiding singularity in the kinematic equations. However, the sliding mode surface designed by qeb and q̇eb in [21, 31] and the conclusion that qe0=−1 is nonequilibrium point in [17] are both misquoted, whose premise is that the sliding mode surface needs to be designed by qeb and ωe. In this paper, Theorem 10 provides a complete proof, whose premise is that qe00 should not be equal to zero.Figure 5
Convergence time comparison.


## 3.4.2. Fixed-Time Controller

In this section, a novel quaternion-based fixed-time nonsingular fast terminal sliding mode control law is designed as follows.(49)u=−1r2B−1K¯qeb+Kqebsig2−r2q̇eb−B−1h−1r2B−1diagκq̇ebr2−1diagq̇eb1−r2×sigk1αsigp1s+βsigg1s−B−1ρsigns−B−1Z2,where k1>1, 0<p1k1<1, 1<g1k1<2k1−1, α>0, β>0, and ρ>0. κ is a nonlinear function defined as follows.
(50)κ=sinπx22τ2x<τ,1x≥τ,where τ is a sufficiently small positive number. K¯qeb is a diagonal matrix, and the diagonal elements k¯qebii=1,2,3 are denoted as
(51)k¯qebi=kr2aqebip−1/kr2+bqebig−1/kr2kr2−1×ap−1kr2qebip−1/kr2+bg−1kr2qebig−1/kr2.Theorem 12.
For dynamic model (27), with the proposed FESO (32), the proposed QFNFTSMS (35), and the proposed QFNFTSMC (49), qeb and q̇eb can converge to the origin uniformly within fixed time T in spite of the initial conditions, and the upper bound of convergence time T satisfies.
(52)T≤To+Tc+Ts,where Tc satisfies
(53)Tc≤231−k1αk12p1k1+1/21−p1k1+231−k1βk12g1k1+1/2g1k1−1+2τ1/r2−1ρ.Proof.
Let the positive Lyapunov function beV4=sTs/2, then, the derivative of V4 is obtained as follows.
(54)V̇4=sTK¯qeb+Kqebq̇eb+sTr2diagq̇ebr2−1h+Bu+D.
Substituting Equation (49) into Equation (54), we can get
(55)V̇4=−sTdiagκq̇ebr2−1sigk1αsigp1s+βsigg1s+sTr2diagq̇ebr2−1−ρsgns−Z2+D.
BecauseZ2⟶D within a fixed time To, Equation (55) can be simplified as follows.
(56)V̇4<−sTdiagκq̇ebr2−1sigk1αsigp1s+βsigg1s≤−31−k1αminκi1/k1∑i=13si2p1k1+1/2k1+βminκi1/k1∑i=13si2g1k1+1/2k1k1≤−31−k1/k1αminκi1/k12p1k1+1/2k1Vp1k1+1/2k1+31−k1/k1βminκi1/k12g1k1+1/2k1Vg1k1+1/2k1k1≤−αmminκi1/k1Vp1k1+1/2k1+βmminκi1/k1Vg1k1+1/2k1k1,where αm and βm are defined as
(57)αm=31−k1/k1α2p1k1+1/2k1,βm=31−k1/k1β2g1k1+1/2k1.
The state space is divided into two areas as(58)δ1=qebi,q̇ebiq̇ebir2−1≥τ,∀i=1,2,3,δ2=qebi,q̇ebiq̇ebir2−1<τ,∃i=1,2,3.When the system statesqeb and q̇eb are in the region δ1, q̇ebi≥τ1/r2−1,∀i=1,2,3 and minκi1/k1=1 satisfies, so V̇≤−αmVp1k1+1/2k1+βmVg1k1+1/2k1k1, therefore, system states qeb and q̇eb will be driven on the sliding mode surface s=0 or into the region δ2 in fixed convergence time Tc1 based on Lemma 5. Where Tc1 satisfies
(59)Tc1≤231−k1αk12p1k1+1/21−p1k1+231−k1βk12g1k1+1/2g1k1−1.If the system statesqeband q̇eb are in the region δ2, i.e., 0≤q̇ebi<τ1/r2−1,∃i=1,2,3 holds. When q̇ebi≠0,∀i=1,2,3, it can also be verified that s=0 is still an attractor. When q̇ebi=0,∃i=1,2,3, combined with Equation (27) and Equation (49), we can get q¨ebi=−ρsgnsi, if si>0, then q¨ebi=−ρ<0, if si<0, then q¨ebi=ρ>0. Therefore, q̇ebi=0 is not an attractor except for the origin. Furthermore, assuming that q¨ebi=−ρsgnsi holds approximately when q̇ebi<τ1/r2−1, the system states will pass through the region δ2 monotonically in fixed time Tc2, where Tc2≤2τ1/r2−1/ρ. So, system states qeb and q̇eb will be driven on the sliding mode surface s=0 within fixed time Tc=Tc1+Tc2. Then, combined with the observer convergence time To and the sliding mode surface convergence time Ts, Theorem 12 is proved.Remark 13.
It should be noted that the control law (49) is modified with respect to [29, 34]. Because the unknown disturbance can be estimated by the fixed-time observer, the proposed control law does not require the disturbance boundary in advance. Compared with the controller proposed in [34], the disturbance effect on the satellite system can be offset by feeding back Z2 to improve control accuracy and the stability condition for ρ≥0, which greatly reduces the parameter selection range to attenuate the chattering. In this paper, in order to enable the system state to pass through the region δ2, ρ is selected as a small constant greater than zero, and then a saturation function is used to achieve chattering-free attitude control.


## 4. Simulation Results

### 4.1. Disturbance Effect

In this section, the lumped disturbance simulation results when Assumption1 and Assumption 2 do not hold are given, then the AMB mechanical characteristics are determined, and finally, the effects of lumped disturbance and uncertainty on the RPS system are discussed. The RPS system parameters are shown in Table 3.Table 3
Parameters of the RPS system.
Mass of platform subsystemMpf=1368kgInertia of platform subsystemJpf=566.7−3.7−3.7−3.73658.9−3.7−3.7−3.73705.6kg⋅m2Axial inertia of momentum wheelii=x,y,zJWi=0.052kg⋅m2Mass of payload subsystemMpl=838kgInertia of payload subsystemJpl=214.50−3.90211.90−3.90253.3kg⋅m2Axial inertia of momentum wheeltJWt=0.362kg⋅m2Static unbalancerj=50−9.6TmmDynamic unbalanceIxz=3.9kg⋅m2To simulate the imaging phase, initial values of quaternion of the platform subsystem and the payload subsystem are both [1,0,0,0]T, and initial values of angular velocity of the platform subsystem and the payload subsystem are [0.001,0,0]Trad/s and [0.001,0,0.28]Trad/s, respectively. The desired angular velocity of the platform subsystem is ω0=0.0011 rad/s, and the expected attitude quaternion of the platform subsystem is cosω0t/2,sinω0t/2,0,0T. The control method used in this section refers to the PD control method in [37]. The adjusting time and damping ratio of the control system is 10 s and 1, respectively. The uncertainty of inertia parameter is 10%, i.e., Jpfn=0.9Jpf. The maximum output of the satellite platform momentum wheel is 1 Nm. In general, the parameter settings in this section are consistent with case 2 in Section 2.In addition, the stiffness and damping parameters of AMB are considered in two cases. Case A: angle stiffness 2450.5 Nm/rad, angle damping 40546.3 Ns/rad. Case B: angle stiffness 2450.5 Nm/rad, angle damping 810.9 Ns/rad. The relative angular deviation of AMB under the two mechanical characteristics is shown in Figure6, where the relative angular deviation of case B increases from 0.002 deg to 0.011 deg compared to case A, so Assumption 1 is not accurate in case B. Furthermore, influenced by the disturbance d′, the satellite platform control accuracy is low in both cases, so Assumption 2 is not valid in both cases.Figure 6
Radial angle deviation of AMB.The lumped disturbance transmitted by AMB is shown as Figure7, where the maximum value of case A increases from 0.8007 Nm of theoretical results to 0.8418 Nm, an increase of 5.1%, and the maximum value of case B further increases to 0.8609 Nm, an increase of 7.5%. Therefore, the mechanical characteristics of low stiffness and low damping, i.e., case B, will amplify the lumped disturbance to a certain extent, but the theoretical results still have more than 90% accuracy.Figure 7
Simulation results of the lumped disturbance.More factors need to be considered in the selection of mechanical characteristics of AMB. The relative angle deviation of low stiffness and low damping is large, as shown in case B of Figure6, which seriously affects the attitude accuracy and stability of the payload. Meanwhile, increasing stiffness is not conducive to isolating the medium and high-frequency vibration of satellite platform caused by flywheel eccentricity. Therefore, the mechanical characteristics of AMB with low stiffness and high damping, i.e., case A, are finally determined.Then, the simulation results of the RPS system attitude under the lumped disturbance and uncertainty are shown in Figure8. As shown in Figures 8(a), 8(c), and 8(h), the lumped disturbance and uncertainty result in nutation motion of the satellite platform, and the nutation motion tends to be stable in 10 s, in which angle control accuracy is 0.011 deg, and angular velocity control accuracy is 0.0032 deg/s. From Figure 8(d), it can be seen that the control torque tends to be stable in 10 s with an amplitude of 0.85 Nm.Figure 8
RPS system attitude results under the lumped disturbance and uncertainty.
(a)
Satellite platform attitude errors(b)
Satellite platform angular velocity errors(c)
Satellite platform nutation trajectory(d)
Control torques(e)
Air gap changes of left RAMB(f)
Air gap changes of right RAMB(g)
Payload nutation trajectory(h)
Nutation angles(i)
Payload angular velocity errorsAs shown in case A of Figures6, 8(e), 8(f), and 8(h), the nutation of the payload relative to the platform also exists, wherein the air gap changing ranges of AMB are stable within 6e-6 m. In addition, the change of the air gaps of the left RAMB and right RAMB are symmetric with respect to the AMB’s geometric center. Finally, as shown in Figures 8(h) and 8(i), the attitude accuracy and angular velocity accuracy of the payload under the lumped disturbance and uncertainty are 0.014 deg and 0.002 deg/s, respectively, which affect the accuracy of the payload in the imaging phase.


### 4.2. Fixed-Time Control

In this section, in order to verify the effectiveness of the proposed controller to reject the lumped disturbance and uncertainty effect in the imaging phase, numerical simulation results are presented. The sliding mode surface parameters are selected as follows:g=0.7, p=0.4, k=2, r2=5/3, and a=b=1.5. The controller parameters are selected as p1=1/3, g1=3/4, k1=1.85, α=β=1, τ=0.005, and ρ=0.0001. The fixed-time observer parameters are selected as follows:L=0.001, ζ=1.5, λ1=0.101, λ2=0.001, λ3=0.005, and Z10=Z20=03×1. And the other parameters refer to Table 3, then, the simulation results of the RPS system attitude under the proposed controller (49) are shown in Figure 9.Figure 9
RPS system attitude results under the proposed controller.
(a)
Satellite platform attitude errors(b)
Satellite platform angular velocity errors(c)
Satellite platform nutation trajectory(d)
Control torques(e)
Sliding mode surface(f)
Estimation errors of observer(g)
Payload nutation trajectory(h)
Nutation angles(i)
Payload angular velocity errorsAs shown in Figures9(a)–9(c) and 9(h), nutation motion of the satellite platform is suppressed by the proposed fixed-time controller. Compared with Figures 8(a)–8(c), the satellite platform has a higher attitude accuracy and avoids chattering. From Figure 9(d), it can be seen that the control torque tends to be stable in 0.8 Nm without chattering phenomenon. In addition, the sliding mode surface converges to zero within a bounded time, as shown in Figure 9(e). Meanwhile, as shown in Figure 9(f), the estimation error of the fixed-time observer also converges to the neighborhood of zero, which is caused by using the boundary layer method to deal with the chattering problem. This indicates that the disturbance d′ is well estimated by the observer. Then, the nutation of the payload relative to the platform and the air gap changes of AMB under the proposed controller is the same as the PD control and can refer to the previous section. Finally, as shown in Figures 9(h) and 9(i), based on the high precision and high stability attitude control of the platform, the payload nutation is consistent with the nutation motion of the payload relative to the platform. Among them, the nutation angle is stable within 0.003 deg, and the angular velocity accuracy converges to within 0.0001 deg/s. The above simulation results show that the proposed controller has excellent suppression effect on the lumped disturbance and uncertainty, which improves the attitude accuracy and attitude stability of the payload in the imaging phase.


## 4.1. Disturbance Effect

In this section, the lumped disturbance simulation results when Assumption1 and Assumption 2 do not hold are given, then the AMB mechanical characteristics are determined, and finally, the effects of lumped disturbance and uncertainty on the RPS system are discussed. The RPS system parameters are shown in Table 3.Table 3
Parameters of the RPS system.
Mass of platform subsystemMpf=1368kgInertia of platform subsystemJpf=566.7−3.7−3.7−3.73658.9−3.7−3.7−3.73705.6kg⋅m2Axial inertia of momentum wheelii=x,y,zJWi=0.052kg⋅m2Mass of payload subsystemMpl=838kgInertia of payload subsystemJpl=214.50−3.90211.90−3.90253.3kg⋅m2Axial inertia of momentum wheeltJWt=0.362kg⋅m2Static unbalancerj=50−9.6TmmDynamic unbalanceIxz=3.9kg⋅m2To simulate the imaging phase, initial values of quaternion of the platform subsystem and the payload subsystem are both [1,0,0,0]T, and initial values of angular velocity of the platform subsystem and the payload subsystem are [0.001,0,0]Trad/s and [0.001,0,0.28]Trad/s, respectively. The desired angular velocity of the platform subsystem is ω0=0.0011 rad/s, and the expected attitude quaternion of the platform subsystem is cosω0t/2,sinω0t/2,0,0T. The control method used in this section refers to the PD control method in [37]. The adjusting time and damping ratio of the control system is 10 s and 1, respectively. The uncertainty of inertia parameter is 10%, i.e., Jpfn=0.9Jpf. The maximum output of the satellite platform momentum wheel is 1 Nm. In general, the parameter settings in this section are consistent with case 2 in Section 2.In addition, the stiffness and damping parameters of AMB are considered in two cases. Case A: angle stiffness 2450.5 Nm/rad, angle damping 40546.3 Ns/rad. Case B: angle stiffness 2450.5 Nm/rad, angle damping 810.9 Ns/rad. The relative angular deviation of AMB under the two mechanical characteristics is shown in Figure6, where the relative angular deviation of case B increases from 0.002 deg to 0.011 deg compared to case A, so Assumption 1 is not accurate in case B. Furthermore, influenced by the disturbance d′, the satellite platform control accuracy is low in both cases, so Assumption 2 is not valid in both cases.Figure 6
Radial angle deviation of AMB.The lumped disturbance transmitted by AMB is shown as Figure7, where the maximum value of case A increases from 0.8007 Nm of theoretical results to 0.8418 Nm, an increase of 5.1%, and the maximum value of case B further increases to 0.8609 Nm, an increase of 7.5%. Therefore, the mechanical characteristics of low stiffness and low damping, i.e., case B, will amplify the lumped disturbance to a certain extent, but the theoretical results still have more than 90% accuracy.Figure 7
Simulation results of the lumped disturbance.More factors need to be considered in the selection of mechanical characteristics of AMB. The relative angle deviation of low stiffness and low damping is large, as shown in case B of Figure6, which seriously affects the attitude accuracy and stability of the payload. Meanwhile, increasing stiffness is not conducive to isolating the medium and high-frequency vibration of satellite platform caused by flywheel eccentricity. Therefore, the mechanical characteristics of AMB with low stiffness and high damping, i.e., case A, are finally determined.Then, the simulation results of the RPS system attitude under the lumped disturbance and uncertainty are shown in Figure8. As shown in Figures 8(a), 8(c), and 8(h), the lumped disturbance and uncertainty result in nutation motion of the satellite platform, and the nutation motion tends to be stable in 10 s, in which angle control accuracy is 0.011 deg, and angular velocity control accuracy is 0.0032 deg/s. From Figure 8(d), it can be seen that the control torque tends to be stable in 10 s with an amplitude of 0.85 Nm.Figure 8
RPS system attitude results under the lumped disturbance and uncertainty.
(a)
Satellite platform attitude errors(b)
Satellite platform angular velocity errors(c)
Satellite platform nutation trajectory(d)
Control torques(e)
Air gap changes of left RAMB(f)
Air gap changes of right RAMB(g)
Payload nutation trajectory(h)
Nutation angles(i)
Payload angular velocity errorsAs shown in case A of Figures6, 8(e), 8(f), and 8(h), the nutation of the payload relative to the platform also exists, wherein the air gap changing ranges of AMB are stable within 6e-6 m. In addition, the change of the air gaps of the left RAMB and right RAMB are symmetric with respect to the AMB’s geometric center. Finally, as shown in Figures 8(h) and 8(i), the attitude accuracy and angular velocity accuracy of the payload under the lumped disturbance and uncertainty are 0.014 deg and 0.002 deg/s, respectively, which affect the accuracy of the payload in the imaging phase.


## 4.2. Fixed-Time Control

In this section, in order to verify the effectiveness of the proposed controller to reject the lumped disturbance and uncertainty effect in the imaging phase, numerical simulation results are presented. The sliding mode surface parameters are selected as follows:g=0.7, p=0.4, k=2, r2=5/3, and a=b=1.5. The controller parameters are selected as p1=1/3, g1=3/4, k1=1.85, α=β=1, τ=0.005, and ρ=0.0001. The fixed-time observer parameters are selected as follows:L=0.001, ζ=1.5, λ1=0.101, λ2=0.001, λ3=0.005, and Z10=Z20=03×1. And the other parameters refer to Table 3, then, the simulation results of the RPS system attitude under the proposed controller (49) are shown in Figure 9.Figure 9
RPS system attitude results under the proposed controller.
(a)
Satellite platform attitude errors(b)
Satellite platform angular velocity errors(c)
Satellite platform nutation trajectory(d)
Control torques(e)
Sliding mode surface(f)
Estimation errors of observer(g)
Payload nutation trajectory(h)
Nutation angles(i)
Payload angular velocity errorsAs shown in Figures9(a)–9(c) and 9(h), nutation motion of the satellite platform is suppressed by the proposed fixed-time controller. Compared with Figures 8(a)–8(c), the satellite platform has a higher attitude accuracy and avoids chattering. From Figure 9(d), it can be seen that the control torque tends to be stable in 0.8 Nm without chattering phenomenon. In addition, the sliding mode surface converges to zero within a bounded time, as shown in Figure 9(e). Meanwhile, as shown in Figure 9(f), the estimation error of the fixed-time observer also converges to the neighborhood of zero, which is caused by using the boundary layer method to deal with the chattering problem. This indicates that the disturbance d′ is well estimated by the observer. Then, the nutation of the payload relative to the platform and the air gap changes of AMB under the proposed controller is the same as the PD control and can refer to the previous section. Finally, as shown in Figures 9(h) and 9(i), based on the high precision and high stability attitude control of the platform, the payload nutation is consistent with the nutation motion of the payload relative to the platform. Among them, the nutation angle is stable within 0.003 deg, and the angular velocity accuracy converges to within 0.0001 deg/s. The above simulation results show that the proposed controller has excellent suppression effect on the lumped disturbance and uncertainty, which improves the attitude accuracy and attitude stability of the payload in the imaging phase.


## 5. Conclusions

In this paper, dynamic model and attitude controller have been studied for the RPS system with a 5-DOF AMB and an unbalanced rotating payload. The dynamic model of the lumped disturbance is established, and the balancing error index of payload unbalance is designed with the maximum torque of actuator as constraint. Subsequently, a disturbance analysis including the effect of AMB mechanical characteristics is carried out, and the simulation results show that the proposed model has more than 90% accuracy in practice, ensuring the effectiveness of the balancing error index. On this basis, a novel QFNFTSMC integrating FESO is proposed in the presence of unknown lumped disturbance and uncertainty. Compared with the previous work, the proposed controller has faster convergence speed, and the disturbance effect is compensated to improve control accuracy and reduce switching gain significantly. Furthermore, the effectiveness of the proposed controller is demonstrated through simulation. Simulation results show that the attitude accuracy and angular velocity accuracy of payload increase from 0.014 deg to 0.003 deg and 0.002 deg/s to 0.0001 deg/s, respectively, which achieves high precision, high stability, and chattering-free attitude control and ensures the imaging quality of the RPS system in the imaging phase.



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*Source: 8281092-2022-06-18.xml*