#### 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.

#### 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 uncertainty

The rest of this paper is organized as follows. In Section 2, 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 Figure 1, 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.

The AMB subsystem is used to connect the platform subsystem and the payload subsystem. As shown in Figure 2, 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.

The coordinate frames and position vectors used in this section are shown in Figure 3.

Where is the inertial coordinate frame, is the platform’s body-fixed coordinate frame, and is the payload’s body-fixed coordinate frame. is the momentum wheel body-fixed coordinate frame (Figure 3 takes the momentum wheel as an example), is the body-fixed coordinate frame of momentum wheel installed in the payload, and is the solar panel body-fixed coordinate frame (Figure 3 takes the solar panel 1 as an example).

Where , , and are the position vectors of the RPS system’s centroid, the platform subsystem’s centroid, and the payload subsystem’s centroid, respectively. is the position vector from the platform subsystem’s centroid to the coordinate origin of . is the position vector from the platform subsystem’s centroid to the solar panel installation position. and are the position vectors from the installation position to a certain mass point and the centroid of solar panel , respectively. is the position vector from the RPS system’s centroid to the coordinate origin of . is the position vector from the coordinate origin of to the payload subsystem’s centroid. is the position vector from the coordinate origin of to the coordinate origin of .

Notation: for a given vector projection , we use to denote the cross-product antisymmetric matrix operation. is the Euclidean norm, is the diagonal matrix with as the diagonal element, is the time derivative, and is the second time derivative. and , where and denote the sign function.

##### 2.1. Dynamic Model of RPS

The dynamic model of the platform subsystem is given as follows.

And the dynamic model of the payload subsystem is given as follows. where and are the projections of and in , respectively. and are the mass and inertia matrix of the platform subsystem, while and are the definitions of the payload subsystem. and denote the angular velocity vector projection of the satellite platform and the rotating payload under their body-fixed coordinate frame, respectively. , , and denote the rotation angle relative to the satellite platform, rotation coupling matrix with the satellite platform attitude, and inertia matrix of the momentum wheel , respectively. In addition, is the -axis principal inertia of , and is the projection matrix of the relative angular velocity vector of momentum wheel under its body-fixed reference frame, such as . Similarly, , , , , and are the definitions of momentum wheel . Furthermore, , , , , and are the definitions of the solar panel , and is the translation coupling matrix with the satellite platform. Solar panel vibration is described by mode coordinate and mode frequency matrix . , , and denote the coupling matrix between the solar panel vibration and , , and , respectively. and denote the driving torques of the momentum wheel and , respectively. is the attitude matrix of relative to . To give a more thorough description of Equation (1), the expressions of the coupling matrixes mentioned above are as follows. where is the projection of in , and are the projections of and in , is the attitude matrix of relative to , and and are the mass and mode shape function of solar panel , respectively.

Defining and as the resultant force projection and torque projection of AMB acting on payload in , respectively, which can be expressed as follows. where , , , and are the bearing mechanical characteristics parameters, specifically, and are the displacement stiffness matrix and angular stiffness matrix of the AMB, respectively, and and are the displacement damping matrix and angular damping matrix of the AMB, respectively. is the projection of the vector from the AMB’s resting geometric center position to the AMB’s actual geometric center position in , i.e., the position deviation of the AMB. is the Euler angle of the payload relative to the satellite platform, i.e., the angular deviation of the AMB. Based on Equation (4), and can be expressed as follows. where is the projection of the vector from platform subsystem’s centroid to the AMB’s actual geometric center position in , is the projection of the vector from payload subsystem’s centroid to the AMB’s actual geometric center position in , and is the attitude matrix of relative to . 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 in and the inertial torque in are small, i.e., the lumped torque transmitted from the AMB to the satellite platform, denoted as the lumped disturbance , is mainly composed of dynamic unbalanced disturbance and static unbalanced disturbance. In this section, the dynamic model of the lumped disturbance 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.

*Assumption 2. *The system states are stable, hold. The orbital angular velocity is denoted as , and the desired rotational speed of payload is . Assuming that the state tracking errors are small, i.e., the following equations approximately.

As shown in Figure 3, the centroid of momentum wheel and the coordinate origin of are coincident. Define to be the projection of in , to be the projection of the vector from coordinate origin of to payload’s centroid in (i.e., static unbalance) and to be the payload’s mass, then holds. On this basis, the following equation can be obtained. where is the projection of in , is the position vector projection of the coordinate origin of in , is the attitude matrix of relative to , is the mass of momentum wheel , and holds. is the projection of in , which can be expressed as follows. where is the projection of in , is the projection of in , and and hold.

Combining Equations (2) and (9), one has

When Assumption 1 holds, it follows that where is the projection of the angular velocity of the payload relative to the satellite platform in . Furthermore, when Assumption 2 holds, it follows that

Bringing Equations (12) and (13) into Equation (11) and noting, one has

For the convenience of subsequent modeling, is divided into and , where is the resultant force on the payload, and is the resultant force on the momentum wheel , one has

Dividing into and , where is the projection of the vector from RPS system’s centroid to AMB’s actual geometric center position in , and holds. Then, based on Equations (2) and (5), the following equation can be obtained.

The inertia matrix of the rotating payload is denoted as and consists of the principal inertia , , and and the inertia product , , and (i.e., dynamic unbalance). The following equation holds.

Bringing Equation (18) into Equation (17), when Assumption 1 and Assumption 2 hold, can be given as

According to the definitions of , , , and , it is clear that holds. Furthermore, based on Equations (5), (15), (16), and (19), the lumped disturbance transmitted by the AMB, consisting of inertia difference disturbance , axial angular momentum disturbance , dynamic unbalanced disturbance , and static unbalanced disturbance , can be modeled as

##### 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, , , and the compensation accuracy of axial angular momentum disturbance is 10%, three balancing errors are considered: and , and , and and . 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.

The theoretical results of - 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 kgm^{2}, and the axial angular momentum disturbance is small constant. Therefore, when the dynamic unbalance and static unbalance increase, the lumped disturbance is mainly composed of and .

The theoretical results of the lumped disturbance in the three cases are shown in Figure 4. 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.

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 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 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

Furthermore, simplifying the driving equations of the momentum wheels , and assuming that the flywheels are all installed in the forward direction. Note that is the control torque, one has where denotes the nominal value of , denotes the parameter uncertainty, and holds. Define the quaternion of the satellite platform as , where and are the scalar and vector components of the quaternion , respectively, satisfying . In addition, if the desired quaternion of the satellite platform is , the error quaternion is obtained as follows:

Furthermore, defining the attitude error matrix as and the desired angular velocity as , then, the error angular velocity can be expressed as , where . Therefore, Equation (23) can be expressed as

And the error quaternion attitude kinematics equations are where holds. On the basis of [21] holds, combining Equations (25) and (26), one has where , , and satisfies

Up to now, the dynamic model for controller design has been established. In this paper, the control objective is to design the controller , such that the states and can follow the desired states and in a fixed time in spite of unknown lumped disturbance and uncertainty, i.e., there exist a constant such that and hold for all .

##### 3.2. Definition and Lemmas

Before the controller design, some useful definitions and theorems need to be provided.

Consider the following nonlinear system: where and are a nonlinear function.

*Definition 3 (see [35]). *The system (29) is called finite-time convergent to the origin for an initial condition , if it is Lyapunov stable and there exists a settling time function such that the system state is equal to zero, for all .

*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 is bounded by positive constant , i.e., the system state is equal to zero, for all , starting from any initial condition .

Lemma 5 (see [29]). *For system (29), suppose there exists a Lyapunov function , scalars , , , , , , and , such that . Then, the system is fixed-time stable. Furthermore, the upper bound of the convergence time is given as follows.
*

Lemma 6 (see [27]). *For , , , and , the following inequalities hold:
*

*Assumption 7. *The unknown disturbance is bounded and continuously differentiable with respect to time, such that satisfy and , where and are a positive constant.

*Assumption 8. *System states and in dynamic model (27) can be both measured.

##### 3.3. Fixed-Time Extended State Observer

To estimate the unknown disturbance in dynamic model (27), define , and the extended state , the fixed-time extended state observer equation is designed as follows: where , , , , , and . Then, the observer error and can be expressed as

According to the theorem proposed in [36], both states and converge to the origin uniformly in fixed time , where satisfies where , , , and . In this paper, since the initial value of is taken as zero, so can take according to Assumption 7. More information on the derivation of the upper bound for 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 , , , and 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 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: where is a diagonal matrix, and the diagonal elements are denoted as where , , , , , and .

Theorem 10. *When satisfies, and can converge to the origin uniformly within fixed time in spite of the initial conditions. Meanwhile, and will converge to 1 (or -1) and , respectively: if , will converge to 1, if , will converge to -1. Where the upper bound of convergence time satisfies
*

*Proof. *If holds, that is, . Furthermore, we can obtain

Therefore, holds, and is the equilibrium point. Let the positive Lyapunov function be , then, the derivative of is obtained as follows.

By Lemma 6, one has

Then based on Lemma 5, can converge to the origin uniformly in bounded convergence time and satisfies Equation (37). Moreover, will converge to the origin, and will converge to the origin because of . However, whether will converge to 1 or -1 cannot be judged from and needs further discussion.

Note that when satisfies, the following equation can be obtained based on Equation (26).

If , then holds at , so holds for all ; if , holds at , so holds for all . Furthermore, when , define , , the following inequalities hold.

Equation (42) indicates that is the equilibrium point and is the nonequilibrium point. Similarly, when , , and hold, i.e., is the equilibrium point, and 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. where , , , , and hold. In order to make the sliding mode surfaces have the same coefficients and exponents, the sliding mode surface parameters are selected as follows.

Based on , we can derive that

Define , holds. Therefore, the convergence time of can be solved as follows. where . Similarly, the convergence time of the others can be solved as follows.

Combining Equation (44) and Lemma 6, one has

Which implies . With the parameters chosen as , , , and , 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 and in [21, 31] and the conclusion that is nonequilibrium point in [17] are both misquoted, whose premise is that the sliding mode surface needs to be designed by and . In this paper, Theorem 10 provides a complete proof, whose premise is that should not be equal to zero.

###### 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. where , , , , , and . is a nonlinear function defined as follows. where is a sufficiently small positive number. is a diagonal matrix, and the diagonal elements are denoted as

Theorem 12. *For dynamic model (27), with the proposed FESO (32), the proposed QFNFTSMS (35), and the proposed QFNFTSMC (49), and can converge to the origin uniformly within fixed time in spite of the initial conditions, and the upper bound of convergence time satisfies.
where satisfies
*

*Proof. *Let the positive Lyapunov function be , then, the derivative of is obtained as follows.
Substituting Equation (49) into Equation (54), we can get
Because within a fixed time , Equation (55) can be simplified as follows.