# Nonsingular Integral Sliding Mode Attitude Control for Rigid-Flexible Coupled Spacecraft with High-Inertia Rotating Appendages

Authors: Gaowang Zhang; Xueqin Chen; Ruichen Xi; Huayi Li
Journal: Complexity
 (2021)

---

## Abstract

This study addresses the challenge of attitude tracking control for a rigid-flexible spacecraft with high-inertia rotating appendages. The Lagrange method was used to establish the kinematic and dynamic models of the spacecraft. The translation and rotation of the spacecraft, vibrations of solar panels, and imbalance caused by the rotating appendages, which cause a complex control problem, were considered. To address the complex control problem, a novel, fast nonsingular integral sliding mode control method is proposed to perform the attitude tracking function of spacecraft. A sliding mode control law was established for the high-inertia appendages to maintain an appropriate angular velocity during rotation. Finally, the effectiveness of the proposed attitude control law was verified by numerical simulations for a spacecraft with high-inertia rotating appendages and symmetrical flexible solar panels.


---

## Body

## 1. Introduction

With the increased complexity of modern space missions, modern spacecraft carry many rigid and flexible components, such as manipulators, solar panels, antennas, and cameras. Additionally, many modern spacecraft components possess highly nonlinear characteristics. Owing to the strong nonlinearity of spacecraft systems and the improvements in the spacecraft attitude control and tracking accuracy, attitude control and tracking for spacecraft with multiple appendages have become a very complex problem.Modern spacecraft are predominantly powered by large, lightweight solar panels. To absorb as much solar energy as possible, it is necessary that the solar panels rotate while the spacecraft moves to ensure the panels are oriented perpendicular to the incident light. Notably, during rapid maneuvering, the rotation and vibration of flexible components reduce the stiffness and increase the nonlinearity of the spacecraft system. These components are always coupled with the attitude and orbital motion of the rigid body of spacecraft, thus increasing the complexity of spacecraft attitude control and tracking problem [1, 2]. Li et al. [3] used cooperative tracking protocols to address unknown nonlinear dynamics problems with an adaptive learning law, which solved the leader-following tracking problem of fractional-order multiagent systems. For multiagent systems with unknown disturbances and input saturation, Zhou et al. [4] proposed an event-triggered control method based on disturbance observer to solve the following consistency problem. Li et al. [5] proposed a finite-time fuzzy adaptive control (AC) scheme to address the finite-time adaptive fuzzy control problem for a class of multiinput multioutput (MIMO) nonlinear nonstrict feedback systems. Tong et al. [6] proposed a novel observer-based adaptive fuzzy output-feedback control method to solve the output-feedback backstepping control design problem in uncertain strict-feedback nonlinear systems. To solve the fault observer design problem of Markovian jumped systems, Chen et al. [7] proposed two types of adaptive observer methods that could avoid the sliding surface switching problem. Zhao et al. [8] proposed a control strategy combined with filtered backstepping and adaptive technique to solve the problem of adaptive finite-time attitude control for multiple spacecraft with unknown external disturbances. The solar panel of the Feng Yun-3 (FY-3) spacecraft rotates relative to the spacecraft body to ensure the solar panel generates sufficient energy for the system [9]. In addition, external disturbances, inertial variability, and nonlinearity pose difficulties for the design of fast and accurate attitude control schemes, such as proportional integral derivative (PID) control, active disturbance rejection control (ADRC) [10], AC [11, 12], sliding mode control (SMC) [13], and finite-time control (FTC) [14]. SMC exhibits excellent performance in uncertain nonlinear systems. Thus, various sliding mode variable structure control methods have been proposed for spacecraft attitude tracking, such as integral SMC [15], second-order SMC [16], and terminal SMC [17].As a form of finite-time control, terminal SMC has attracted increasing attention owing to its robustness and anti-interference properties. Terminal SMC has been applied to spacecraft attitude control but with limitations because of singularity and slow convergence. Zhao [18] proposed a finite-time command filter backstepping approach based on a finite-time adaptive fuzzy tracking control scheme for a class of unknown nonlinear systems. In recent years, nonsingular terminal SMC has attracted increasing attention for applications in complex spacecraft attitude control owing to its advantageous characteristics. Lu et al. [19] proposed an effective SMC strategy for a nonlinear spacecraft model with external disturbances and inertial uncertainty that could effectively solve the problem of spacecraft attitude tracking control. Yousefpour et al. [20] proposed an integral terminal SMC method based on a disturbance observer for the tracking control of a hyperchaotic memory resistance oscillator. To solve the precise trajectory tracking control problem of unmanned vehicles, Lv et al. [21] proposed a nonsingular integral SMC law based on a finite-time disturbance observer. In [19–21], effective disturbance observers were designed for controlled systems to achieve high performance in the presence of inertia uncertainties and external disturbances. Na [22] proposed an alternative AC with prescribed performance to solve the output tracking problem of nonlinear systems with unknown dead zones. Chen et al. [23] designed an adaptive finite-time controller based on the fast TSM control principle to overcome the singularity problem of nonlinear systems. Na [22] and Chen et al. [23] designed effective attitude control schemes for systems with problems such as uncertain inertia terms, unknown disturbances, and sudden actuator failure. To solve the attitude tracking problem for a rigid platform spacecraft with rotating flexible appendages, Shi et al. [24] proposed the nonsingular terminal SMC algorithm for the finite-time tracking control of n-order nonlinear dynamical systems. To achieve underactuated attitude stabilization, Yue et al. [25] designed a higher-level SMC part to stabilize the angular velocity combined with a tracking control part to track desired angular velocities. Furthermore, [24, 25] demonstrated effective control laws for nonlinear dynamical systems to achieve finite-time convergence and strong robustness. Alshamali et al. [26] designed an observer-based backstepping controller to solve attitude tracking problems with the angles of satellite dynamics under external disturbances. Wang et al. [27] proposed an adaptive SMC scheme to control the chaotic oscillation of a complex seven-dimensional power system. The proposed scheme avoided the singularity problem and improved the accuracy of attitude control as well. Finally, Peng et al. [28] proposed a fast, nonsingular integral terminal SMC scheme for a class of nonlinear systems by introducing a power integral term. This type of scheme can be used to avoid the singularity problem without constraints and to achieve finite-time convergence of the system state. These control methods exhibit good performance and robustness in many systems but are unsuitable for rigid-flexible coupled spacecraft with high-inertia rotating appendages.In recent years, the number of high-inertia rotating appendages in spacecraft has increased. “high inertia” is mainly in the relative sense, and high-inertia components generally refer to appendages with similar mass as spacecraft or high-inertia matrixes. However, no unified and quantifiable standard is available in the academic field. These components often contain both static and dynamic imbalances, thus producing an unbalanced disturbance torque. Therefore, higher accuracy is required for the attitude control of spacecraft. Miao et al. [29] designed a fast, nonsingular SMC method integrated with an adaptive method, which has been proven to have a higher convergence speed and increased robustness than previous terminal SMC methods. However, high-inertia components and flexible solar panels were not considered in their study. Gui et al. [30] proposed a novel integral TSM controller integrated with an adaptive fault-tolerant control law to solve the attitude tracking problem for a rigid spacecraft; however, a rigid-flexible coupled spacecraft was not considered. This TSM controller exhibited better control performance, including higher convergence speed and estimation accuracy, than the conventional controllers reported in [29, 30]. Xie [31] designed a composite compensation controller to account for the control torques and reduce the impulse characteristics of the disturbance torque during the acceleration of rotating components. Further, the effectiveness of the controller was proven using a semiphysical simulation system; additionally, the coupling effect between the high-inertia components and platform and the characteristics of unbalanced disturbance torque were analyzed. In addition, a PD control algorithm combined with repetitive control was designed. Regarding the unbalanced characteristics of high-inertia rotating appendages, Chai [32, 33] used the Kalman filter method to effectively and accurately estimate unbalanced interference torques caused by appendage rotation.Among the broad array of studies conducted to date, only a few studies address the problem of attitude tracking control of rigid-flexible coupled platforms with high-inertia components. To bridge this divide, this study proposes a novel, fast integral SMC strategy for spacecraft with high-inertia components and flexible solar panels and demonstrates the applicability of this approach to facilitate the attitude tracking of spacecraft systems. The primary contributions of this work are listed below:(1)
A rigid-flexible coupling spacecraft with rotating components is considered, and the mathematical and dynamic models are established(2)
An attitude control law for the high-inertia components of a spacecraft system, integrated with a novel, fast nonsingular integral SMC law for symmetrical solar panels, and a spacecraft platform to enable accurate spacecraft attitude tracking are demonstratedThe remainder of this paper is organized as follows. Section2 introduces the dynamic and kinematic models for spacecraft attitude tracking with high-inertia rotating appendages and symmetrical solar panels. Section 3 introduces the proposed fast nonsingular integral SMC law. Section 4 presents the numerical simulation results to demonstrate the effectiveness of the control algorithm. Finally, Section 5 concludes the study.


## 2. Preliminary Modeling

A rigid spacecraft model with large, rigid rotating inertial components and symmetrical flexible solar panels was considered in this study. The coordinate system included the inertial reference frameOXYZ, orbital reference frame OXoYoZo, spacecraft platform reference frame oxsyszs, large rigid rotating component body-fixed frame oxpypzp, and flexible solar panels body-fixed frame oxayaza (Figure 1)—these were defined as R, Ro, Rs, Rp, andRa, respectively. The y-axes of Rs, Rp, and Ra were parallel. The flexible solar panels were installed on the ±y-axis, and the large rigid rotating components were installed on the x-axis of the platform coordinate system R.Figure 1
Coordinate system of spacecraft with appendages.The rigid-flexible coupling spacecraft model consisted of kinematic and dynamic equations, which are described in the following sections.


### 2.1. Kinematic Model of the Spacecraft System

Because the Euler angle has a singularity, the Euler equation is not suitable for spacecraft with a wide range of attitude motion. To avoid the singularity and complex calculations, the unit quaternions are often used to describe the kinematics of a spacecraft. A unit quaternion is expressed asQs=q0qsTT=q0q1q2q3T, where q0 and qs=q1q2q3Tare the scalar and vector parts of the quaternion, respectively, and q0 and qs satisfy the condition q02+qsTqs=1.The kinematic equation with the quaternion of spacecraft can be given as(1)Q·=12Eqs∗ωb=12−qsTq˜s+q0I3∗ωs,where ωs denotes the angular attitude velocity of the spacecraft relative to the inertial system and q˜s represents the skew symmetric matrix of qssuch that(2)q˜s=0−q3q2q30−q1−q2q10.The desired orbital coordinate system for the spacecraft tracking system is denoted asRo, and Qd is the expected attitude quaternion. Thus, Qd=qd0qdTT, and the kinematic equation is defined as(3)Qd·=12Eqd∗ωd,where ωd is the desired angular velocity from Ro relative to R. In addition, the attitude tracking error quaternion is defined as Qe=qe0qeTT such that(4)Qe=qe0qe=qs0qd0+qdTqsqd0qs+q˜sqd−qs0qd,which describes the error Rs relative to Ro.The principle of two vector attitude determinations was used to determine the desired quaternion of the spacecraft. The desired quaternion could then be used to obtain the attitude transformation matrix of the spacecraft orbitRo relative to the inertial orbit R. According to the known position vector X and velocity vector V of the spacecraft, matrix B is the projection of X and V of the spacecraft in R, and matrix A is the projection of X and V of the spacecraft in Ro. The unique transformation matrix is then defined as(5)CBA=B−1A,where the matrixes B and A can be defined as(6)B=XX,X×VX×V,X×X×VX×X×V,A=00−10−10−100.Thus, the desired quaternion can be obtained as follows:(7)q0=121+CBA1,1+CBA2,2+CBA3,3,q1=14q0CBA2,3−CBA3,2,q2=14q0CBA3,1−CBA1,3,q3=14q0CBA1,2−CBA2,1,where Csd is the coordinate transformation matrix from Ro to Rs, which can be calculated as(8)Csd=qe02−qeTqeI3+2qeqeT−2qe0q˜e.The angular attitude velocity tracking error of the spacecraft is defined as(9)ωe=ωs−Csdωd.The high-inertia component is the rotation around thexs-axis of the spacecraft platform with the desired attitude angular velocity ωpd=ωpd,0,0T. Similarly, the flexible solar panels rotate around the ys-axis of the spacecraft platform with the desired attitude angular velocity ωad=0,ωad,0T. The kinematic equation for the desired attitude of the high-inertia components is Q˙pd=1/2Eqpdωpd and that for the flexible solar panels is Q˙ad=1/2Eqadωad. Thus, the real kinematic equations are Q˙p=1/2Eqpωp and Q˙a=1/2Eqaωa. The quaternion for the attitude tracking error can be obtained using (4), such that the attitude tracking error of the flexible solar panels is(10)Qae=qae0qae=qa0qad0+qadTqaqad0qa+q˜aqad−qa0qad.The angular attitude velocity tracking error of the high-inertia components and flexible solar panels can be, respectively, expressed as(11)ωpe=ωp−ωpd,ωae=ωa−ωad.When the high-inertia components and the flexible solar panels rotate only around thexs-axis and ys-axis, respectively, of the spacecraft platform, the rotational angle of the former is set as θp∈0,2π and that of the latter is set as θa∈0,2π. The rotational angles can be calculated by(12)θp=∫ωpdt,θai=∫ωaidt.The transformation matrixes fromRp and Ra to Rs are then defined as(13)Cps=1000cosθpxsinθpx0−sinθpxcosθpx,Cabi=cosθayi0−sinθayi010sinθayi0cosθayi.


### 2.2. Dynamics of the Spacecraft System

We considered the rotation of the rigid high-inertia components and flexible solar panels on the rigid-flexible coupling spacecraft, as well as the translation and rotation of the spacecraft platform and the modal vibration of the solar panels. The Lagrange method was used to deduce the complex equations and the dynamic model equations of the spacecraft system attitude, as follows:(14a)M+mp+2maX¨=Ps,(14b)Jsω˙s+ω˜sJsωs+Rspωp+Rsaωa+Fsaη¨ai+Rsaω˙ai+Rspω˙p=Ts+Tsdp+d,(14c)Jaiω˙ai+Faiη¨ai+RsaiTω˙s=Taii=−y,y,(14d)η¨ai+2ζaiΩaiη˙ai+Λaiηai+RsaTω˙s+FaiTω˙ai=0i=−y,+y,where M is the mass of the spacecraft platform; mp is the mass of the high-inertia component; mais the mass of a single flexible solar panel; Ps denotes the combined external forces on the spacecraft system; the inertia matrix with respect to Rs of the spacecraft is defined as Js and satisfies the equation Js=Js0+CsaJa0CsaT+CspJpCspT; Ja0 is the inertia matrix of the flexible solar panel relative to the platform when it is not rotating; Jp is the inertia matrix of the high-inertia rotating component with respect to the platform; Ts and Ta denote the torque of the spacecraft platform and the solar panels, respectively; Td is the unknown external disturbance torque; Tsdp is the sum of the dynamic and static unbalanced torques and the bearing torque of high-inertia components under the influence of coupling; ηa denotes the modal coordinates of the solar panel; Ωa is the diagonal matrix of the modal frequency of the solar panel; Λa is the stiffness matrix of the solar panel such that Λa=Ωa2; ξa is the damping ratio of the solar panel. Notably, all these parameters are diagonal matrixes. From (14a)–(14b), Fsa=Fsa_right+Fsa_left and Rsa=Rsa_right+Rsa_left.The connection point between the high-inertia rotation component and the platform was modeled as shown in Figure1. In this paper, rs denotes the vector from the connection point to the origin of the high-inertia rotation component’s Rscoordinate system; rp denotes the vector from the origin of the Rp coordinate system to the connection point; and r denotes the vector from the connection point to the mass element dm on the high-inertia rotation component. When the high-inertia rotation component rotates around the x-axis at the angular velocityωp, the unbalanced torque Tsd can be generated on the satellite platform. The dynamic equations for a high-inertia appendage are as follows:(15)Tsd=−rp×∫pωp×ωp×rdm−r×∫pωp×ωp×rdm=−ωp×Jpωp+mpωpTωprs×rp,(16)Jpω˙p+RspTω˙s=Tp,(17)Tsdp=Tsd−Tp,where Tp is the control torque of the high-inertia component. Because the solar panels only rotate about the y-axis relative to the spacecraft platform, Fsadenotes the coupling coefficient matrix of the rotation of the spacecraft platform with the vibration of the solar panels such that(18)Fsa=fsa1sinθay1cosθay,fsa2sinθay1cosθay,…,fsa6sinθay1cosθay,where fsa is the coefficient matrix of the platform rotation to the solar panel vibration, which is related to the rotating angle θay, Fa is the coupling coefficient matrix between the vibration and rotation of solar panels, Rsa is the coupling coefficient matrix of rotation between the solar panels and the platform and is calculated in a similar manner as equation (18), with rsa as the angle-independent coefficient, and Rsp is the matrix of vibration of the high-inertia component with the rotation of the spacecraft platform. The high-inertia component rotates only around the x-axis relative to the spacecraft platform; thus, Rsp can be described by a variation of (16).Equation (13) shows the dynamic equations of the spacecraft system, including the translation of the platform, as well as the rotation of the high-inertia components and solar panels. Upon substituting (5) and (11) into (14a)-(14d), we obtain(19)Θω˙+Ξ+Δ=T,where(20)ω=ωeωaeT,Θ=Js0−FsaFsaTRsa−FsaFaTRsaT−FaFsaTJa−FaFaTΞ=RsPω˙P+ωs×Js0ωs+Rspωp+Rsaωa+Js0−FsaFsaT−ω˜eCsdωd+Csdω˙d−TsdpRsaT−FaFsaT−ω˜eCsdωd+Csdω˙d,Δ=Δ1Δ2=Fsa2ζaiΩaiη˙ai+Λaiηai+CsaJa0CsaT+CspJpCspTω˙e−ω˜eCsdωd+Csdω˙d−d+ΔFsa,RsaΔFsa,Rsa,T=TsTa.Remark 1.
The moment of inertia of the spacecraft system has the specific formJ=J0+ΔJ, where J0 is the nonsingular nominal constant matrix, and ΔJ is the uncertainty of moment of inertia and is a bounded constant.Remark 2.
ΔFsa,ΔRsa,ΔRsp represent the combined uncertainties on parameters Fsa, Rsa, and Rsp; these uncertainties exist because, although the high-inertia components and solar panels rotate predominantly around one axis of the spacecraft, they exhibit small rotations around the other axes. Thus, the combined uncertainties are bounded.Remark 3.
The disturbanceΔ is bounded, and satisfies Δ≤Δd, where d is a constant.


## 2.1. Kinematic Model of the Spacecraft System

Because the Euler angle has a singularity, the Euler equation is not suitable for spacecraft with a wide range of attitude motion. To avoid the singularity and complex calculations, the unit quaternions are often used to describe the kinematics of a spacecraft. A unit quaternion is expressed asQs=q0qsTT=q0q1q2q3T, where q0 and qs=q1q2q3Tare the scalar and vector parts of the quaternion, respectively, and q0 and qs satisfy the condition q02+qsTqs=1.The kinematic equation with the quaternion of spacecraft can be given as(1)Q·=12Eqs∗ωb=12−qsTq˜s+q0I3∗ωs,where ωs denotes the angular attitude velocity of the spacecraft relative to the inertial system and q˜s represents the skew symmetric matrix of qssuch that(2)q˜s=0−q3q2q30−q1−q2q10.The desired orbital coordinate system for the spacecraft tracking system is denoted asRo, and Qd is the expected attitude quaternion. Thus, Qd=qd0qdTT, and the kinematic equation is defined as(3)Qd·=12Eqd∗ωd,where ωd is the desired angular velocity from Ro relative to R. In addition, the attitude tracking error quaternion is defined as Qe=qe0qeTT such that(4)Qe=qe0qe=qs0qd0+qdTqsqd0qs+q˜sqd−qs0qd,which describes the error Rs relative to Ro.The principle of two vector attitude determinations was used to determine the desired quaternion of the spacecraft. The desired quaternion could then be used to obtain the attitude transformation matrix of the spacecraft orbitRo relative to the inertial orbit R. According to the known position vector X and velocity vector V of the spacecraft, matrix B is the projection of X and V of the spacecraft in R, and matrix A is the projection of X and V of the spacecraft in Ro. The unique transformation matrix is then defined as(5)CBA=B−1A,where the matrixes B and A can be defined as(6)B=XX,X×VX×V,X×X×VX×X×V,A=00−10−10−100.Thus, the desired quaternion can be obtained as follows:(7)q0=121+CBA1,1+CBA2,2+CBA3,3,q1=14q0CBA2,3−CBA3,2,q2=14q0CBA3,1−CBA1,3,q3=14q0CBA1,2−CBA2,1,where Csd is the coordinate transformation matrix from Ro to Rs, which can be calculated as(8)Csd=qe02−qeTqeI3+2qeqeT−2qe0q˜e.The angular attitude velocity tracking error of the spacecraft is defined as(9)ωe=ωs−Csdωd.The high-inertia component is the rotation around thexs-axis of the spacecraft platform with the desired attitude angular velocity ωpd=ωpd,0,0T. Similarly, the flexible solar panels rotate around the ys-axis of the spacecraft platform with the desired attitude angular velocity ωad=0,ωad,0T. The kinematic equation for the desired attitude of the high-inertia components is Q˙pd=1/2Eqpdωpd and that for the flexible solar panels is Q˙ad=1/2Eqadωad. Thus, the real kinematic equations are Q˙p=1/2Eqpωp and Q˙a=1/2Eqaωa. The quaternion for the attitude tracking error can be obtained using (4), such that the attitude tracking error of the flexible solar panels is(10)Qae=qae0qae=qa0qad0+qadTqaqad0qa+q˜aqad−qa0qad.The angular attitude velocity tracking error of the high-inertia components and flexible solar panels can be, respectively, expressed as(11)ωpe=ωp−ωpd,ωae=ωa−ωad.When the high-inertia components and the flexible solar panels rotate only around thexs-axis and ys-axis, respectively, of the spacecraft platform, the rotational angle of the former is set as θp∈0,2π and that of the latter is set as θa∈0,2π. The rotational angles can be calculated by(12)θp=∫ωpdt,θai=∫ωaidt.The transformation matrixes fromRp and Ra to Rs are then defined as(13)Cps=1000cosθpxsinθpx0−sinθpxcosθpx,Cabi=cosθayi0−sinθayi010sinθayi0cosθayi.


## 2.2. Dynamics of the Spacecraft System

We considered the rotation of the rigid high-inertia components and flexible solar panels on the rigid-flexible coupling spacecraft, as well as the translation and rotation of the spacecraft platform and the modal vibration of the solar panels. The Lagrange method was used to deduce the complex equations and the dynamic model equations of the spacecraft system attitude, as follows:(14a)M+mp+2maX¨=Ps,(14b)Jsω˙s+ω˜sJsωs+Rspωp+Rsaωa+Fsaη¨ai+Rsaω˙ai+Rspω˙p=Ts+Tsdp+d,(14c)Jaiω˙ai+Faiη¨ai+RsaiTω˙s=Taii=−y,y,(14d)η¨ai+2ζaiΩaiη˙ai+Λaiηai+RsaTω˙s+FaiTω˙ai=0i=−y,+y,where M is the mass of the spacecraft platform; mp is the mass of the high-inertia component; mais the mass of a single flexible solar panel; Ps denotes the combined external forces on the spacecraft system; the inertia matrix with respect to Rs of the spacecraft is defined as Js and satisfies the equation Js=Js0+CsaJa0CsaT+CspJpCspT; Ja0 is the inertia matrix of the flexible solar panel relative to the platform when it is not rotating; Jp is the inertia matrix of the high-inertia rotating component with respect to the platform; Ts and Ta denote the torque of the spacecraft platform and the solar panels, respectively; Td is the unknown external disturbance torque; Tsdp is the sum of the dynamic and static unbalanced torques and the bearing torque of high-inertia components under the influence of coupling; ηa denotes the modal coordinates of the solar panel; Ωa is the diagonal matrix of the modal frequency of the solar panel; Λa is the stiffness matrix of the solar panel such that Λa=Ωa2; ξa is the damping ratio of the solar panel. Notably, all these parameters are diagonal matrixes. From (14a)–(14b), Fsa=Fsa_right+Fsa_left and Rsa=Rsa_right+Rsa_left.The connection point between the high-inertia rotation component and the platform was modeled as shown in Figure1. In this paper, rs denotes the vector from the connection point to the origin of the high-inertia rotation component’s Rscoordinate system; rp denotes the vector from the origin of the Rp coordinate system to the connection point; and r denotes the vector from the connection point to the mass element dm on the high-inertia rotation component. When the high-inertia rotation component rotates around the x-axis at the angular velocityωp, the unbalanced torque Tsd can be generated on the satellite platform. The dynamic equations for a high-inertia appendage are as follows:(15)Tsd=−rp×∫pωp×ωp×rdm−r×∫pωp×ωp×rdm=−ωp×Jpωp+mpωpTωprs×rp,(16)Jpω˙p+RspTω˙s=Tp,(17)Tsdp=Tsd−Tp,where Tp is the control torque of the high-inertia component. Because the solar panels only rotate about the y-axis relative to the spacecraft platform, Fsadenotes the coupling coefficient matrix of the rotation of the spacecraft platform with the vibration of the solar panels such that(18)Fsa=fsa1sinθay1cosθay,fsa2sinθay1cosθay,…,fsa6sinθay1cosθay,where fsa is the coefficient matrix of the platform rotation to the solar panel vibration, which is related to the rotating angle θay, Fa is the coupling coefficient matrix between the vibration and rotation of solar panels, Rsa is the coupling coefficient matrix of rotation between the solar panels and the platform and is calculated in a similar manner as equation (18), with rsa as the angle-independent coefficient, and Rsp is the matrix of vibration of the high-inertia component with the rotation of the spacecraft platform. The high-inertia component rotates only around the x-axis relative to the spacecraft platform; thus, Rsp can be described by a variation of (16).Equation (13) shows the dynamic equations of the spacecraft system, including the translation of the platform, as well as the rotation of the high-inertia components and solar panels. Upon substituting (5) and (11) into (14a)-(14d), we obtain(19)Θω˙+Ξ+Δ=T,where(20)ω=ωeωaeT,Θ=Js0−FsaFsaTRsa−FsaFaTRsaT−FaFsaTJa−FaFaTΞ=RsPω˙P+ωs×Js0ωs+Rspωp+Rsaωa+Js0−FsaFsaT−ω˜eCsdωd+Csdω˙d−TsdpRsaT−FaFsaT−ω˜eCsdωd+Csdω˙d,Δ=Δ1Δ2=Fsa2ζaiΩaiη˙ai+Λaiηai+CsaJa0CsaT+CspJpCspTω˙e−ω˜eCsdωd+Csdω˙d−d+ΔFsa,RsaΔFsa,Rsa,T=TsTa.Remark 1.
The moment of inertia of the spacecraft system has the specific formJ=J0+ΔJ, where J0 is the nonsingular nominal constant matrix, and ΔJ is the uncertainty of moment of inertia and is a bounded constant.Remark 2.
ΔFsa,ΔRsa,ΔRsp represent the combined uncertainties on parameters Fsa, Rsa, and Rsp; these uncertainties exist because, although the high-inertia components and solar panels rotate predominantly around one axis of the spacecraft, they exhibit small rotations around the other axes. Thus, the combined uncertainties are bounded.Remark 3.
The disturbanceΔ is bounded, and satisfies Δ≤Δd, where d is a constant.


## 3. Sliding Mode Control Law Design

Lemma 1.
The extended Lyapunov of finite-time stability can be expressed in the form of a fast TSM as follows [34, 35]:(21)V·x+λ1Vx+λ2Vγx≥0.If (18) satisfies the conditions λ1>0,λ2>0 and 0<γ<1, the convergence time can be given by(22)T≤1λ11−γlnλ1V1−γx0+λ2λ2.Lemma 2.
For anyxi∈Ri=1,2,…,n and a positive real number γ∈0,1, the following inequality holds [36]:(23)∑i=1nxi21+γ/2≤∑i=1nxi1+γ.Lemma 3.
We consider the systemx˙=fx,u, where f is a continuous function that satisfies f0=0, and a Lyapunov function Vx that satisfies [37](24)V·x+λVαx≤0,where λ>0,0<α<1. Then, the origin is a finite-time stable equilibrium of the systemx˙=fx,u. Let T represent the time required for the system to reach Vx≡0. If x0 is the initial system state, the convergence time satisfies(25)T≤V1−αx0λ1−α.Using the quaternion for the attitude tracking error of the spacecraft platform and solar panel to describe the attitude system state, we can writex=qeqaeT and define(26)x˙=q˙eq˙aeT=12Eω.Thus,x¨ can be given as(27)x¨=14L+12(EΘ)T−Ξ−Δ.Finally, the augmented matrixE can be written as E=Eqe00Eqae and L=qeωeTωeqaeωaeTωae.


### 3.1. Controller Design of the Spacecraft Platform and Solar Panel

#### 3.1.1. Integral Sliding Mode Surface Design

The integral sliding mode surface can be defined by(28)S=kxx.+∫0tC1S1λ1,ρ1,x,ε1+C2S2λ2,ρ2,x.,ε2dt,where C1=diagc11,c12,c13,c14,c15,c16 and C2=diagc21,c22,c23,c24,c25,c26, and(29)γ1=γ11,γ12,γ13,γ14,γ15,γ16T,γ2=γ21,γ22,γ23,γ24,γ25,γ26T,ε1=ε11,ε12,ε13,ε14,ε15,ε16T,ε2=ε21,ε22,ε23,ε24,ε25,ε26T,where c1i>0 and c2i>0 are the sliding surface design matrixes for i = 1,2, …, 6. Furthermore, the vectors S1 and S2 can be defined as(30)S1=S11,S12,S13,S14,S15,S16T,S2=S21,S22,S23,S24,S25,S26T.In (30), S1i and S2i are defined as(31)S1iγ1i,ρ1i,x1i,ε1i=x1iγ1i,ifx1i≤ε1i,ε1iγ1i−ρ1ix1iρ1isignx1i,ifx1i>ε1i,S2iγ2i,ρ2i,x˙2i,ε2i=x˙2iγ2i,ifx˙2i≤ε2i,ε2iγ2i−ρ2ix˙2iρ2isignx˙2i,ifx˙2i>ε2i,where γ1i and γ2i are the positive constant gains, which satisfy 0<γ2i<1 and γ1i=γ2i/2−γ2i. In the same way, ρ1i, ρ2i, ε1i, and ε2i are the positive constant gains, which satisfy ρ1i≥1, ρ2i≥1, ε1i>0, and ε2i>0, respectively.The control torqueT can be expressed as follows:(32)T=2KxEΘ−1−τS−KssgnS−C1S1γ1,ρ1,x,ε1−C2S2γ2,ρ2,x˙,ε2+12EΘ−1L+Ξ+Δ.Remark 4.
From (28), it is observed that the controller parameters γ1i, γ2i, kxi, C1i, and C2i affect the convergences performances of the closed-loop systems. Smaller γ1i and γ2i and larger kxi, C1i, andC2iwill yield a more quick convergence of the attitude tracking errorx(t); xcan be further reduced by increasing the control parameters τ and kS.


#### 3.1.2. Proof

With the spacecraft attitude dynamics model described in (14a)-(14d), the integral sliding mode (see (28)) and the control law (see (32)) were designed. The system state will slide to the equilibrium point in a finite time after reaching the sliding mode surface. Based on Lyapunov theory, it can be proved that(33)V=12STS,because(34)S·=kxx¨+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2=kx14L+12(EΘ)T−Ξ−Δ+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2.DifferentiatingV with respect to time yields (30); from (32), it can be determined that(35)V·=STS·=STkx14L+12(EΘ)T−Ξ−Δ+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2=ST−τS−KssgnS=−τV−ksV1/2.According to Lemma1, the convergence time can be determined as(36)T≤2τlnτV1/2x0+ksks.


### 3.2. Controller Design for High-Inertia Components

The high-inertia components rotate around thex-axis at a certain attitude angular velocity relative to the spacecraft platform. Thus, a control law can be designed for these components.


#### 3.2.1. Sliding Mode Surface Design

The sliding mode function is defined as(37)Sp=KPωPe+Kdsigγω˙Pe,where KP=diagkp1,kp2,kp3, Kd=diagkd1,kd2,kd3, and γ>1 is a positive odd number. When the sliding mode function Sp=0, the attitude angular velocity tracking error of the high-inertia component is equal to 0 (ωPe=0) because(38)ω˙Pe=kpikdiωPe1/γsignkpikdiωPe=kpikdiωPe1/γsignωPe.The control torqueTp can be expressed as(39)Tp=Jpkp−1−τSp−kdsigγω¨pe+RspTω˙s.


#### 3.2.2. Proof

As the high-inertia component dynamics model described in (16), the sliding mode (see (37)) was designed. The system state slides to the equilibrium point in a finite time after reaching the sliding mode surface. Based on Lyapunov theory, it can be proved that(40)V2=12ωPeTJωPe.Differentiation ofV2 with respect to time yields(41)V2=ωPeTJωPe∑13kpikdiωPeTJωPei1/γsignωPei=−∑13kpikdiJωPei2γ+1/2γ≤−minkpikdi2⋅12J∑13ωPeiγ+1/2γ=−minkpikdi⋅2γ+1/2γVγ+1/2γ.Furthermore, lettingξ=−minkpi/kdi·2γ+1/2γ, V˙2 satisfies(42)V˙2+ξV2γ+1/2γ≤0.Finally, the convergence time can be determined according to Lemma3, as follows:(43)Tp≤2γV2γ+1/2γ0ξγ−1.


## 3.1. Controller Design of the Spacecraft Platform and Solar Panel

### 3.1.1. Integral Sliding Mode Surface Design

The integral sliding mode surface can be defined by(28)S=kxx.+∫0tC1S1λ1,ρ1,x,ε1+C2S2λ2,ρ2,x.,ε2dt,where C1=diagc11,c12,c13,c14,c15,c16 and C2=diagc21,c22,c23,c24,c25,c26, and(29)γ1=γ11,γ12,γ13,γ14,γ15,γ16T,γ2=γ21,γ22,γ23,γ24,γ25,γ26T,ε1=ε11,ε12,ε13,ε14,ε15,ε16T,ε2=ε21,ε22,ε23,ε24,ε25,ε26T,where c1i>0 and c2i>0 are the sliding surface design matrixes for i = 1,2, …, 6. Furthermore, the vectors S1 and S2 can be defined as(30)S1=S11,S12,S13,S14,S15,S16T,S2=S21,S22,S23,S24,S25,S26T.In (30), S1i and S2i are defined as(31)S1iγ1i,ρ1i,x1i,ε1i=x1iγ1i,ifx1i≤ε1i,ε1iγ1i−ρ1ix1iρ1isignx1i,ifx1i>ε1i,S2iγ2i,ρ2i,x˙2i,ε2i=x˙2iγ2i,ifx˙2i≤ε2i,ε2iγ2i−ρ2ix˙2iρ2isignx˙2i,ifx˙2i>ε2i,where γ1i and γ2i are the positive constant gains, which satisfy 0<γ2i<1 and γ1i=γ2i/2−γ2i. In the same way, ρ1i, ρ2i, ε1i, and ε2i are the positive constant gains, which satisfy ρ1i≥1, ρ2i≥1, ε1i>0, and ε2i>0, respectively.The control torqueT can be expressed as follows:(32)T=2KxEΘ−1−τS−KssgnS−C1S1γ1,ρ1,x,ε1−C2S2γ2,ρ2,x˙,ε2+12EΘ−1L+Ξ+Δ.Remark 4.
From (28), it is observed that the controller parameters γ1i, γ2i, kxi, C1i, and C2i affect the convergences performances of the closed-loop systems. Smaller γ1i and γ2i and larger kxi, C1i, andC2iwill yield a more quick convergence of the attitude tracking errorx(t); xcan be further reduced by increasing the control parameters τ and kS.


### 3.1.2. Proof

With the spacecraft attitude dynamics model described in (14a)-(14d), the integral sliding mode (see (28)) and the control law (see (32)) were designed. The system state will slide to the equilibrium point in a finite time after reaching the sliding mode surface. Based on Lyapunov theory, it can be proved that(33)V=12STS,because(34)S·=kxx¨+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2=kx14L+12(EΘ)T−Ξ−Δ+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2.DifferentiatingV with respect to time yields (30); from (32), it can be determined that(35)V·=STS·=STkx14L+12(EΘ)T−Ξ−Δ+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2=ST−τS−KssgnS=−τV−ksV1/2.According to Lemma1, the convergence time can be determined as(36)T≤2τlnτV1/2x0+ksks.


## 3.1.1. Integral Sliding Mode Surface Design

The integral sliding mode surface can be defined by(28)S=kxx.+∫0tC1S1λ1,ρ1,x,ε1+C2S2λ2,ρ2,x.,ε2dt,where C1=diagc11,c12,c13,c14,c15,c16 and C2=diagc21,c22,c23,c24,c25,c26, and(29)γ1=γ11,γ12,γ13,γ14,γ15,γ16T,γ2=γ21,γ22,γ23,γ24,γ25,γ26T,ε1=ε11,ε12,ε13,ε14,ε15,ε16T,ε2=ε21,ε22,ε23,ε24,ε25,ε26T,where c1i>0 and c2i>0 are the sliding surface design matrixes for i = 1,2, …, 6. Furthermore, the vectors S1 and S2 can be defined as(30)S1=S11,S12,S13,S14,S15,S16T,S2=S21,S22,S23,S24,S25,S26T.In (30), S1i and S2i are defined as(31)S1iγ1i,ρ1i,x1i,ε1i=x1iγ1i,ifx1i≤ε1i,ε1iγ1i−ρ1ix1iρ1isignx1i,ifx1i>ε1i,S2iγ2i,ρ2i,x˙2i,ε2i=x˙2iγ2i,ifx˙2i≤ε2i,ε2iγ2i−ρ2ix˙2iρ2isignx˙2i,ifx˙2i>ε2i,where γ1i and γ2i are the positive constant gains, which satisfy 0<γ2i<1 and γ1i=γ2i/2−γ2i. In the same way, ρ1i, ρ2i, ε1i, and ε2i are the positive constant gains, which satisfy ρ1i≥1, ρ2i≥1, ε1i>0, and ε2i>0, respectively.The control torqueT can be expressed as follows:(32)T=2KxEΘ−1−τS−KssgnS−C1S1γ1,ρ1,x,ε1−C2S2γ2,ρ2,x˙,ε2+12EΘ−1L+Ξ+Δ.Remark 4.
From (28), it is observed that the controller parameters γ1i, γ2i, kxi, C1i, and C2i affect the convergences performances of the closed-loop systems. Smaller γ1i and γ2i and larger kxi, C1i, andC2iwill yield a more quick convergence of the attitude tracking errorx(t); xcan be further reduced by increasing the control parameters τ and kS.


## 3.1.2. Proof

With the spacecraft attitude dynamics model described in (14a)-(14d), the integral sliding mode (see (28)) and the control law (see (32)) were designed. The system state will slide to the equilibrium point in a finite time after reaching the sliding mode surface. Based on Lyapunov theory, it can be proved that(33)V=12STS,because(34)S·=kxx¨+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2=kx14L+12(EΘ)T−Ξ−Δ+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2.DifferentiatingV with respect to time yields (30); from (32), it can be determined that(35)V·=STS·=STkx14L+12(EΘ)T−Ξ−Δ+C1S1γ1,ρ1,x,ε1+C2S2γ2,ρ2,x˙,ε2=ST−τS−KssgnS=−τV−ksV1/2.According to Lemma1, the convergence time can be determined as(36)T≤2τlnτV1/2x0+ksks.


## 3.2. Controller Design for High-Inertia Components

The high-inertia components rotate around thex-axis at a certain attitude angular velocity relative to the spacecraft platform. Thus, a control law can be designed for these components.


### 3.2.1. Sliding Mode Surface Design

The sliding mode function is defined as(37)Sp=KPωPe+Kdsigγω˙Pe,where KP=diagkp1,kp2,kp3, Kd=diagkd1,kd2,kd3, and γ>1 is a positive odd number. When the sliding mode function Sp=0, the attitude angular velocity tracking error of the high-inertia component is equal to 0 (ωPe=0) because(38)ω˙Pe=kpikdiωPe1/γsignkpikdiωPe=kpikdiωPe1/γsignωPe.The control torqueTp can be expressed as(39)Tp=Jpkp−1−τSp−kdsigγω¨pe+RspTω˙s.


### 3.2.2. Proof

As the high-inertia component dynamics model described in (16), the sliding mode (see (37)) was designed. The system state slides to the equilibrium point in a finite time after reaching the sliding mode surface. Based on Lyapunov theory, it can be proved that(40)V2=12ωPeTJωPe.Differentiation ofV2 with respect to time yields(41)V2=ωPeTJωPe∑13kpikdiωPeTJωPei1/γsignωPei=−∑13kpikdiJωPei2γ+1/2γ≤−minkpikdi2⋅12J∑13ωPeiγ+1/2γ=−minkpikdi⋅2γ+1/2γVγ+1/2γ.Furthermore, lettingξ=−minkpi/kdi·2γ+1/2γ, V˙2 satisfies(42)V˙2+ξV2γ+1/2γ≤0.Finally, the convergence time can be determined according to Lemma3, as follows:(43)Tp≤2γV2γ+1/2γ0ξγ−1.


## 3.2.1. Sliding Mode Surface Design

The sliding mode function is defined as(37)Sp=KPωPe+Kdsigγω˙Pe,where KP=diagkp1,kp2,kp3, Kd=diagkd1,kd2,kd3, and γ>1 is a positive odd number. When the sliding mode function Sp=0, the attitude angular velocity tracking error of the high-inertia component is equal to 0 (ωPe=0) because(38)ω˙Pe=kpikdiωPe1/γsignkpikdiωPe=kpikdiωPe1/γsignωPe.The control torqueTp can be expressed as(39)Tp=Jpkp−1−τSp−kdsigγω¨pe+RspTω˙s.


## 3.2.2. Proof

As the high-inertia component dynamics model described in (16), the sliding mode (see (37)) was designed. The system state slides to the equilibrium point in a finite time after reaching the sliding mode surface. Based on Lyapunov theory, it can be proved that(40)V2=12ωPeTJωPe.Differentiation ofV2 with respect to time yields(41)V2=ωPeTJωPe∑13kpikdiωPeTJωPei1/γsignωPei=−∑13kpikdiJωPei2γ+1/2γ≤−minkpikdi2⋅12J∑13ωPeiγ+1/2γ=−minkpikdi⋅2γ+1/2γVγ+1/2γ.Furthermore, lettingξ=−minkpi/kdi·2γ+1/2γ, V˙2 satisfies(42)V˙2+ξV2γ+1/2γ≤0.Finally, the convergence time can be determined according to Lemma3, as follows:(43)Tp≤2γV2γ+1/2γ0ξγ−1.


## 4. Numerical Simulation Results

This section presents numerical simulations to demonstrate the performance of the proposed nonsingular integral sliding mode controller (NISMC). The results of the proposed method are compared with those of the fast TSM finite-time controller (FSMC) designed by Yang and Yang in [38] and the PD controller. The simulation results of the spacecraft with high-inertia components and flexible solar panels are presented, and the proposed control laws are demonstrated to have superior performance. The fast-sliding mode finite-time controller exhibits higher convergence speed and an accurate attitude tracking and control of the component angle. In this analysis, the desired angular velocity of the flexible solar panels about the y-axis of the spacecraft was set as ωad = 0.1 rad/s and that of the large rigid inertial component about the x-axis of the spacecraft as ωpd = 0.2 rad/s. The expected attitude of the spacecraft platform is the coordinate system Rs that coincides with Ro.Figures2 and 3 present the results of the vector component of the error quaternion and the error of the attitude angular velocity of the platform, respectively. Figure 4 presents the results of the vector component of the error quaternion of the flexible solar panels. Figures 5 and 6 present the results of the error of the attitude angular velocity and the attitude angular velocity of the flexible solar panels relative to the spacecraft platform. Figure 7 shows the attitude angular velocity and error of the angular velocity of the large rigid inertial component that rotates around the x-axis of the spacecraft platform. Figures 3, 5, and 7 indicate that the error of the attitude angular velocity of the spacecraft satisfies the constraint. Therefore, the system can reach the sliding surface in a finite time, and when the system enters the sliding surface, it can maintain sliding mode motion. Therefore, the designed controller can ensure that the system tracks the desired attitude angle and angular velocity.Figure 2
Trajectories of the vector component of the platform error quaternion.Figure 3
Trajectories of the error of attitude angular velocity of platform.Figure 4
Trajectories of vector component of the error quaternion of flexible solar panel.Figure 5
Trajectories of the error of attitude angular velocity of flexible solar panel.Figure 6
Trajectories of they-axis flexible solar panel’s attitude angular velocity.Figure 7
Trajectories of angular velocity and error of angular velocity of the large rigid inertial component.Figures8–11 present the results of the error quaternion and the error of angular velocity of the flexible solar panels and platform. These figures indicate that the proposed method converges faster than the existing methods (FSMC and PD). Table 1 presents the results of the stable error of attitude angular velocity and the stable error quaternion of the flexible solar panels and platform. The table indicates that the stable error of attitude angular velocity of the large rigid inertial component is 3.622 × 10−4.Figure 8
Comparison of platform error quaternions.
(a)(b)(c)(d)Figure 9
Comparison of errors of attitude angular velocity of platform.
(a)(b)(c)Figure 10
Comparison of error quaternions of flexible solar panels.
(a)(b)(c)(d)Figure 11
Comparison of errors of attitude angular velocity of flexible solar panels.
(a)(b)(c)Table 1
Parameter control performance.
Control strategyNISMCFSMCPDStable error of attitude angular velocity of platform1.798 × 10−55.074 × 10−40.0012Stable error of attitude angular velocity of flexible solar panels1.114 × 10−51.243 × 10−40.0018Stable error quaternion of platform3.560 × 10−72.574 × 10−40.00391Stable error quaternion of flexible solar panels1.704 × 10−56.819 × 10−40.00115Figures8 and 9 show that NISMC drives the attitude tracking errors to the origin faster than the existing FSMC and PD algorithms. In specific, NISMC takes ∼60 s for the error quaternion to converge to the equilibrium position, whereas the FSMC and PD require ∼80 s and ∼110 s, respectively. Furthermore, NISMC takes ∼55 s for the error angular velocity to converge to the equilibrium position, whereas the FSMC and PD algorithms require ∼120 s. In the steady-state response region, the final stable error quaternion and the final error of attitude angular velocity of the platform are 3.560 × 10−7 and 1.798 × 10−5, respectively, under the NISMC algorithm.Figures10 and 11 indicate that the NISMC takes ∼70 s for the error quaternion of the flexible solar panels to converge to the equilibrium position, whereas the FSMC and PD require ∼100 s and ∼110 s, respectively. Furthermore, NISMC takes ∼75 s for the error of angular velocity to converge to the equilibrium position, whereas the FSMC and PD algorithms require ∼105 s. The final stable error quaternion of the flexible solar panels is 1.704 × 10−5 and the final error of attitude angular velocity of the platform is 1.114 × 10−5, under the NISMC algorithm.Table1 presents the final stable error quaternion and the final error of attitude angular velocity of the platform under the FSMC and PD algorithms. Clearly, the magnitude of the error of angular velocity and error quaternion is smaller under the NISMC algorithm than under other control laws, indicating that the system has high steady-state accuracy.These results demonstrate the superiority of the proposed NISMC algorithm in terms of faster convergence and higher stable accuracy over other control algorithms.


## 5. Conclusions

A novel integral sliding mode control law was proposed for spacecraft systems with high-inertia components and flexible solar panels. The proposed control scheme exhibited faster convergence and provided precise attitude tracking in terms of external imbalances caused by appendages, rotation and vibration of flexible solar panels, and parametric uncertainties. Using numerical simulations, the control method was shown to allow an accurate tracking of the desired spacecraft attitude and the rotation of the flexible solar panels and high-inertia components around the special axis with a stable angular speed. Therefore, the control scheme offers a valuable contribution to the design of future space missions in both theoretical and practical respects. As future work, high-flexibility appendages, such as flexible antennas and connecting adapters as magnetic bearings, would be considered. Problems such as parameter uncertainty and actuator saturation demand advanced control methods to realize special state-constrained attitude maneuvering without any intrusion into the forbidden zones. Given the increasing complexity of space missions and the resulting changes to spacecraft design, I believe that this study will be of interest to those who work in aerospace engineering and similar fields.



---

*Source: 8812187-2021-02-16.xml*