An Attitude Determination and Sliding Mode Control Method for Agile Whiskbroom Scanning Maneuvers of Microsatellites

Abstract

Microsatellites have significantly impacted space missions by offering advanced technology at a low cost. This study introduces an attitude determination and control algorithm for agile whiskbroom scanning maneuvers in microsatellites to enable wide-swath target detection for low-Earth-orbit microsatellites. First, an angular velocity calculation model for agile whiskbroom scanning is established. A methodology has been developed to calculate the maximum available time for whiskbroom scanning from one side of the sub-satellite point to the other while ensuring the seamless joining of adjacent strips to avoid missing targets. Thereafter, a gyro- and magnetometer-based cubature Kalman filter is put forward for microsatellite attitude estimation. Furthermore, for attitude control, a hybrid manipulation law capable of preventing singularities and escaping singularity surfaces is designed to ensure high-precision torque output from the control moment gyroscopes (CMGs) used as actuators. The benefits of the linear sliding mode and fast terminal sliding mode are integrated, and a non-singular sliding surface is designed, yielding a non-singular fast terminal sliding mode attitude control algorithm for tracking the desired trajectory. This algorithm effectively suppresses chattering and enhances dynamic performance without using a switching term. A semi-physical simulation experiment system is also conducted on the ground to validate the proposed algorithm’s high-precision tracking of the planned whiskbroom scanning path. The experimental results demonstrate an attitude angle control accuracy of 4 × 10⁻² degrees and angular velocity control accuracy of 0.01°/s and thus the effectiveness of the proposed algorithm.

1. Introduction

The rapid development of modern space technology necessitates enhancing the operational abilities of remote-sensing satellites. There is a need not only to process high-resolution satellite payloads but also to detect wide-swath targets to meet the needs of target search. One effective method of enhancing detection swath is to utilize a low-Earth-orbit satellite equipped with a high-speed reciprocating whiskbroom scanning camera that moves perpendicularly to the orbital path. The captured images are subsequently stitched. However, whiskbroom scanning imaging in the perpendicular-to-orbit direction imposes new requirements on satellite attitude control. Frequent and rapid attitude maneuvers are necessary to gather more information and data within the same overpass duration, thus improving operational efficiency. This requires satellites to possess agile attitude maneuver capabilities and the ability to track the planned path with high precision and stability. However, rapid attitude maneuvers may lead to failures of star sensors, which poses challenges for satellite attitude determination.

The methods by which to determine satellite attitude can be categorized into deterministic algorithms and dynamic state estimation methods. Deterministic algorithms can be further classified into reference vector and inertial measurement methods [1,2,3]. However, deterministic algorithms often struggle to handle sensor errors, resulting in inaccurate attitude determination. Dynamic estimation methods [4], such as the Kalman filter algorithm, have been developed to address these challenges. The Kalman filter primarily deals with linear system problems and may not effectively handle the highly nonlinear nature of attitude kinematic equations for the satellite. Therefore, the extended Kalman filter (EKF) has been proposed for satellite attitude estimation [5]. However, the estimation performance of EKF is affected by initial errors, leading to slow convergence rates. The unscented Kalman filter (UKF) was introduced to address these limitations by using specific sample points to map function probability distributions without linearizing the nonlinear functions or computing Jacobian matrices [6]. Consequently, UKF performs significantly better in accurately determining the attitude estimation and tends to be more robust than EKF. The cubature Kalman filter (CKF) proposed by Arasaratnam is a nonlinear Gaussian filtering algorithm closest to Bayesian estimation theory. It offers the advantages of short computation time, high efficiency, precise estimation, real-time performance, and strong robustness [7]. However, the estimation performance CKF is suboptimal in response to state changes, with low tracking accuracy and a tendency to diverge, thereby lacking good adaptability. To address this issue, scholars have proposed numerous improved algorithms. For instance, fading factors combining strong tracking filtering (STF) theory with adaptive square-root CKF was introduced to handle poor measurement issues [8]. Since the position of the fading factors was somewhat arbitrary, this was resolved by incorporating the current statistical model into the enhanced tracking CKF algorithm [9]. Novel methods of repositioning the fading factors were proposed to augment the tracking capabilities for any change in the target state response. Similarly, the covariance matrix for measurement error was constructed using the Huber function to enhance the robustness of the CKF algorithm [10]. Likewise, the Huber-based Bayesian estimation theory was applied to the CKF with measurements processed to enhance the algorithm’s robustness [11]. A robust adaptive cubature Kalman filtering algorithm with a noise estimator was designed that incorporated the Huber function and real-time adaptive adjustment of noise statistical properties based on residual sequences into CKF [12]. To minimize the impact of non-Gaussian noise on the estimation performance, a linear regression model was established using a fixed-point iterative algorithm to solve the Huber minimization problem [13].

The single gimbal control moment gyroscope (SGCMG) is a commonly used actuator for agile spacecraft. It enables the rapid maneuvering of satellite attitude and generates large output torque with low power consumption. However, the SGCMG system suffers from severe singularity issues that act as key impediments to its further advancement. Extensive studies have been conducted domestically and internationally, with a prime focus on creating suitable manipulation laws to prevent singular states. For instance, one study examined the performance of various manipulation laws and developed methods to circumvent singular states [14]. Another study proposed a more robust manipulation law that used Gaussian functions to calculate robust coefficients [15]. This law proved effective in quickly avoiding singularity and completing maneuvers. Additionally, an offline manipulation law was suggested to plan the gimbal angle trajectory offline near singular points yet had to encounter complexity and timing issues [16]. Previous extensive analysis of the SGCMG system verified several manipulation laws using simulations [17].

The design of control laws in attitude control processes is crucial for generating the desired torque and completing attitude maneuvers from an operational perspective. Currently, spacecraft primarily employ proportional–integral–derivative (PID) control and other advanced control methods as the key attitude control techniques. While optimal control methods offer superior dynamic performance and steady-state indicators as compared to the PID control counterpart, they face computational difficulties and lack robustness [18,19]. Sliding mode variable structure control is suitable for nonlinear systems due to its strong robustness and low dependency on system models. However, conventional sliding mode control encounters severe chattering issues, which complicates its application in spacecraft attitude control. Various studies have addressed the chattering issue in spacecraft attitude control. For instance, an attitude control law was designed by introducing a saturation function in the boundary layer [20]. Although chattering was suppressed successfully using this approach, declining control accuracy and increasing convergence time were the main drawbacks that inhibited the usefulness of this method. In this context, the saturation function was replaced with an “arctangent” function that effectively suppressed chattering while improving the control accuracy [21]. Although these approaches managed to suppress chattering, the systems were asymptotically stable, meaning the system state only converged to the equilibrium point over an infinite time. A finite-time control algorithm using homogeneity theory was designed, allowing the system state to converge to the equilibrium point within a finite time [22]. Furthermore, a new high-order sliding mode control law was devised, which effectively avoided control chattering while ensuring high-precision tracking [23]. Terminal sliding mode control was deployed to design control algorithms to compensate for external disturbance torques with parameter adaptive estimation [24]. This strategy effectually quells chattering and external disturbances. However, terminal sliding mode control introduces singularity issues, particularly in attitude tracking control. Therefore, a sliding mode variable structure control based on a novel reaching law was developed to minimize the chattering effect [25]. Terminal sliding mode control and reaching law methods share the same mathematical structure. The two methods differ only in their focus: the former on system maneuvers on the sliding surface and the latter on system maneuvers during the reaching phase, without introducing singularity issues.

This study introduced a gyro- and magnetometer-based cubature Kalman filter for microsatellite attitude estimation and utilized a fast terminal sliding mode control algorithm for designing a non-singular fast terminal sliding mode. The aim is to develop a reliable control method for the attitude control of microsatellites in agile whiskbroom scanning. Moreover, a verification system for the attitude control of microsatellites was constructed. The proposed method demonstrated enhanced accuracy in determining the attitude of microsatellites despite sensors failure and achieved a highly precise, stable tracking of planned paths, thereby meeting the requirements for attitude control of microsatellites in agile whiskbroom scanning.

2. Angular Velocity Calculation Model for Agile Whiskbroom Scanning

2.1. Timing Requirements for Agile Whiskbroom Scanning

When microsatellites are in a low-Earth orbit, the camera has a narrow field of view. Agile whiskbroom scanning maneuvers are necessary to overcome the limitations of the small swath of each imaging for the wide-area detection of specific targets. To complete such maneuvers, the satellite needs to perform large-angle, reciprocating, and rapid attitude maneuvers perpendicular to the orbit on the preset path within a limited time. This will serve to increase the detectable range during the same overpass duration. The time required by the satellite to perform whiskbroom scanning maneuvers has been explained in the coming lines.

Assume the satellite’s orbital altitude is h km, and it needs to detect an area of size $S=m\times n$ (km), where m and n represent the width and length of the area perpendicular to and along the orbit, respectively. During a single overpass, the satellite should perform a complete scan of this area and stitch the images into a wide swath image. To enhance the imaging efficiency of whiskbroom scanning, this paper adopts a double-pass whiskbroom scanning approach. Initially, the satellite maneuvers to a certain angle on one side of the orbit that acts as the starting point for scanning. The satellite then performs large-angle reciprocating maneuvers around the roll axis perpendicular to the orbit. The camera, fixed to the satellite’s yaw axis (z-axis), adjusts its field of view with the satellite’s attitude changes. The whiskbroom scanning maneuvers are illustrated in Figure 1.

However, due to the high orbital velocity of the satellite, the lines connecting the centers of consecutive images are not perpendicular to the orbit but staggered along the orbital direction. As a result, the image strips during forward and backward scans within one whiskbroom scanning cycle do not overlap in the flying direction. This results in incomplete coverage and potential omission of target within the scanned area.

There are two ways by which the scanned image can be stitching thoroughly. One approach is to increase the satellite’s whiskbroom scanning speed to ensure overlap in the along-track direction within one whiskbroom scanning cycle, as shown in Figure 2. Another technique involves adding pitch maneuvers to the roll axis whiskbroom scanning to compensate for the along-track displacement. For investigation purposes, the first method has been adopted in this work.

To achieve all-inclusive image stitching, the images captured at the maximum side-scan points on both sides must overlap with each other. Specifically, the central point of the image captured on the right side of an endpoint must align exactly with the right edge of the image captured on the left side of an endpoint. In other words, the advance distance of the satellite in the along-track direction during a single-pass whiskbroom scanning is only half the swath width of the camera’s single frame. Assuming the satellite camera imaging width is $W\times W$, and the single-pass whiskbroom scanning cycle is $t$. For easier stitching, the adjacent images in the whiskbroom scanning direction have an overlap ratio of $\xi$ in the perpendicular-to-orbit direction. Suppose there are N exposures in a single-pass whiskbroom scanning; this leads to the following expression:

$$(1-\xi )W+(1-2\xi )(N-2)W+\xi W(N-1)=m(\mathrm{km}).$$

(1)

The equation can be simplified to $N=\left[W(1-2\xi )+m\right]/\left[W(1-\xi )\right]$. In the along-track direction, the advance distance per frame should be $\left(W/2\right)/N$. From the satellite’s motion characteristics, the linear velocity and angular velocity at an altitude of $h$ km are $v_{\mathrm{sat}}=\sqrt{G{M}_e/\left(R+h\right)}$ and ${\omega }_{\mathrm{sat}}=\sqrt{G{M}_e/{\left(R+h\right)}^3}$, respectively. As shown in Figure 3, point C represents the satellite’s imaging payload; points A and B represent the maximum detectable range intersecting with the Earth’s surface; O is the Earth’s center; $R$ is the Earth’s radius; $h$ is the satellite’s orbit height; $\alpha$ is half of the maximum side-scanning angle; and $l_{AB}$ is the arc length of $\stackrel{⏜}{AB}$. By geometric relationship, the corresponding central angle is $l_{AB}={\displaystyle \frac{2n}{180°}}\pi R$, yielding $n={\displaystyle \frac{90°l_{AB}}{\pi R}}$.

In $\Delta OAC$, the length AC was determined using the cosine theorem in triangle, $AC=\sqrt{R^2+{\left(R+h\right)}^2-2R\left(R+h\right)\cos \angle AOC}$. The sine theorem was employed to solve for an angle using $\sin \alpha ={\displaystyle \frac{AO\cdot \sin \angle AOC}{AC}}$. Thus, we obtain the required whiskbroom scanning angle for an area of size $S=m\times n$: $2\alpha =2\arcsin \left(AO\sin \angle AOC/AC\right)$.

During agile whiskbroom scanning, the microsatellite’s angular velocity undergoes acceleration, constant speed, and deceleration phases. To ensure imaging consistency during the whiskbroom scanning, the microsatellite must complete one cycle of whiskbroom scanning in the perpendicular-to-orbit direction, with the satellite’s advance distance in the along-track direction not exceeding the swath width of a single-frame image. This means that the advance distance of the satellite during a single-pass whiskbroom scanning should not exceed half the swath width of a single-frame image.

As illustrated in Figure 4, $\stackrel{⏜}{A_1{A}_2}$ corresponds to half the frame swath, $W/2$; thus, ${\displaystyle \frac{\angle A_{1}O{A}_2}{360°}}={\displaystyle \frac{\stackrel{⏜}{A_1{A}_2}}{2\pi R}}$. Consequently, an angle $\angle A_{1}O{A}_2={\displaystyle \frac{90°W}{\pi R}}$ is derived. Thus, the maximum available time to complete a whiskbroom scan from one side of the subsatellite point to the other is as follows:

$$T_{\max }={\displaystyle \frac{\angle A_{1}O{A}_2}{{\omega }_{\mathrm{sat}}}}.$$

(2)

2.2. Path Planning for Rapid Attitude Maneuvering

While analyzing the attitude control of rapid satellite maneuvers, it has been observed that directly delivering axis control commands from the controller to the actuators can result in a strong pulse excitation, thereby causing abrupt changes in angular acceleration and torque output from the actuators. Consequently, severe vibrations can be experienced in flexible components of the satellite, e.g., solar panels. Therefore, it is necessary to plan the changes in angular acceleration rationally during the attitude maneuvering process. The goal is to design a rapid maneuvering process that could enable the transitioning of the satellite from its current attitude to the desired one within a limited time while maintaining high precision and stability during the transition. This approach can reduce the vibration or deformation of flexible components and meet multiple constraints during the satellite maneuver, such as the normal operation of sensors and whether the torque and angular acceleration produced by the actuators are within the normal output range.

There are several methods for planning attitude maneuvering paths, including bang-coast-bang (BCB) path planning, S-curve path planning, sinusoidal path planning, and polynomial path planning. The design process must satisfy the constraints that both the initial and final angular velocities and angular accelerations are zero. During the satellite attitude maneuvering time T, with a change in the attitude angle from ${\phi }_0$ to ${\phi }_d$, the angular velocity varies from ${\dot{\phi }}_0$ to ${\dot{\phi }}_d$, and the angular acceleration correspondingly alters from zero and eventually returns to zero.

In this work, attitude maneuvering paths and variations in angular velocity of the satellite has been planned to be determined by a sinusoidal angular acceleration maneuvering curve. The sinusoidal path uses simple forms and frequencies that are easy to characterize with sinusoidal functions to design angular acceleration. The angular accelerations in the acceleration and deceleration segments are designed as downward-opening and upward-opening half-cycle sinusoidal functions, respectively. Both half-cycle sinusoidal functions have the same amplitude and frequency, while the speed segment has a constant value. Therefore, the sinusoidal path comprises three stages: the sinusoidal acceleration segment, the constant speed segment, and the sinusoidal deceleration segment.

The sinusoidal acceleration segment time is $T_1$; the maximum angular acceleration is $a_{\max }$, with angular acceleration designed as $a_{\max }\sin \left({\omega }_{1}t+{\phi }_1\right)$; the constant speed segment time is $T_2$, with an angular acceleration of zero; and the sinusoidal deceleration segment time is $T_3$, with angular acceleration designated as $-a_{\max }\sin \left({\omega }_{2}t+{\phi }_2\right)$. Then, the planned angular acceleration can be expressed as follows:

$$a(t)=\left\{\begin{array}{l}\begin{array}{ll}{a}_{\max }\sin \left({\displaystyle \frac{\pi }{T_1}}t\right)& \begin{array}{c}\hspace{1em}0\le t<T_1\end{array}\end{array}\\ \begin{array}{ll}0& \begin{array}{c}\hspace{1em}{T}_1\le t<T_1+T_2\end{array}\end{array}\\ \begin{array}{ll}-a_{\max }\sin \left({\displaystyle \frac{\pi }{T_3}}(t-T_1-T_2)\right)& {\begin{array}{c}\hspace{1em}{T}_1+T_2\le t<T_1+T_2+T\end{array}}_3\end{array}\end{array}\right.$$

(3)

The angular velocity can be obtained by integrating the angular acceleration as follows:

$$\dot{\phi }(t)=\left\{\begin{array}{l}\begin{array}{ll}{\displaystyle \frac{a_{\max }{T}_1}{\pi }}\left[1-\cos \left({\displaystyle \frac{\pi }{T_1}}t\right)\right]+{\dot{\phi }}_0& \begin{array}{c}\hspace{1em}0\le t<T_1\end{array}\end{array}\\ \begin{array}{ll}2a_{\max }{\displaystyle \frac{T_1}{\pi }}+{\dot{\phi }}_0& \begin{array}{c}\hspace{1em}\hspace{1em}{T}_1\le t<T_1+T_2\end{array}\end{array}\\ \begin{array}{ll}2a_{\max }{\displaystyle \frac{T_1}{\pi }}+{\dot{\phi }}_0-a_{\max }{\displaystyle \frac{T_3}{\pi }}\left[1-\cos \left({\displaystyle \frac{\pi }{T_3}}(t-T_1-T_2)\right)\right]& {\begin{array}{c}\hspace{1em}{T}_1+T_2\le t<T_1+T_2+T\end{array}}_3\end{array}\end{array}\right.$$

(4)

The following is integrated to obtain the attitude angle change:

$$\phi (t)=\left\{\begin{array}{l}{\displaystyle \frac{a_{\max }{T}_1}{\pi }}\left[t-{\displaystyle \frac{T_1}{\pi }}\sin \left({\displaystyle \frac{\pi }{T_1}}t\right)\right]+{\dot{\phi }}_{0}t+\phi ,0\le t<T_1\\ 2a_{\max }{\displaystyle \frac{T_1}{\pi }}t-a_{\max }{\displaystyle \frac{T_1^2}{\pi }}+{\dot{\phi }}_{0}t+{\phi }_0,T_1\le t<T_1+T_2\\ a_{\max }{\displaystyle \frac{T_1}{\pi }}(2t-T_1)-a_{\max }\left[t-{\displaystyle \frac{T_3}{\pi }}\sin \left({\displaystyle \frac{T_3}{\pi }}(t-T_1-T_2\right)-T_1-T_2\right]+{\dot{\phi }}_{0}t+{\phi }_0,T_1+T_2\le t<T_1+T_2+T_3\end{array}\right.$$

(5)

The following constraints must be met:

$$\left\{\begin{array}{l}2a_{\max }{T}_1\le \left({\dot{\phi }}_{\max }-{\dot{\phi }}_0\right)\pi \\ {\displaystyle \frac{\pi }{2}}{a}_{\max }\left(T_1-T_3\right)={\dot{\phi }}_e-{\dot{\phi }}_0\\ {\displaystyle \frac{T_1}{\pi }}{a}_{\max }\left(T_1+2T_2+2T_3\right)-{\displaystyle \frac{T_3^2}{\pi }}{a}_{\max }+{\dot{\phi }}_{0}t+{\phi }_0={\phi }_e\end{array}\right.$$

(6)

where ${\phi }_0$ and ${\phi }_e$ are the attitude angles at the start and end of the satellite attitude maneuver, respectively; ${\dot{\phi }}_0$ and ${\dot{\phi }}_e$ are the angular velocities at the start and end of the satellite attitude maneuver, respectively; and ${\dot{\phi }}_{\max }$ is the maximum maneuvering angular velocity.

Generally, ${\dot{\phi }}_0={\dot{\phi }}_e=0$, and the satellite accelerates to its maximum angular velocity ${\dot{\phi }}_{\max }$ during agile whiskbroom scanning, so

$$\left\{\begin{array}{l}{T}_1={\displaystyle \frac{{\dot{\phi }}_{\max }\pi }{2a_{\max }}}\\ T_1=T_3\\ T_1+T_2={\displaystyle \frac{\Psi }{{\dot{\phi }}_{\max }}}\end{array}\right.,$$

(7)

where $\Psi$ is the maneuvering angle.

3. “Gyro + Magnetometer”-Based Cubature Kalman Filter for Attitude Estimation

Traditional star sensors cannot adapt to the high-dynamic application scenarios of the agile whiskbroom scanning maneuvers discussed in this paper. In addition, to reduce the cost of microsatellites, we chose a combination of a gyro and a magnetometer. The gyro provides high-precision attitude information in real-time, compensating for the lower attitude determination accuracy of the magnetometer. Meanwhile, the magnetometer does not accumulate errors over time and can be used to correct the gyro’s angular velocity drift.

3.1. Microsatellite Kinematic Model

Microsatellite attitude is typically determined by the orientation of the satellite’s body coordinate system relative to a reference coordinate system. There are various ways to describe this orientation. In this paper, we use quaternions to describe satellite attitude. According to Euler’s finite rotation theorem, any angular displacement of a rigid body around a fixed point can be achieved by rotating it around an axis passing through that point by a certain angle. The transformation between the satellite’s body coordinate system and the reference coordinate system can thus be described by a rotation around an axis passing through the origin of the reference coordinate system. While this axis–angle representation can be cumbersome for calculations, combining the axis and angle information in a certain way yields the attitude quaternion.

The attitude quaternion $\mathit{q}$ is represented as $\mathit{q}={\left({\mathit{q}}_v,q_4\right)}^{\mathrm{T}}={\left(q_1,q_2,q_3,q_4\right)}^{\mathrm{T}}$. Here, ${\mathit{q}}_v={\left(q_1,q_2,q_3\right)}^{\mathrm{T}}=\mathit{e}\sin \left(\Phi /2\right)$, where $\mathit{e}={\left(e_x,e_y,e_z\right)}^{\mathrm{T}}$ and $\Phi$ represent the unit vector of the Euler rotation axis and rotation angle, respectively, and $q_4=\cos \left(\Phi /2\right)$.

Using quaternions to describe satellite attitude avoids singularities and simplifies the calculation process, which only involves matrix multiplications and avoids complex trigonometric operations. Therefore, we adopt a quaternion-based kinematic model for satellite attitude: $\left\{\begin{array}{l}{\dot{\mathit{q}}}_v={\displaystyle \frac{1}{2}}\left({\mathit{q}}_v^{\times }+q_4{\mathit{I}}_3\right)\mathit{\omega }\\ {\dot{q}}_4=-{\displaystyle \frac{1}{2}}{\mathit{q}}_v^{\mathrm{T}}\mathit{\omega }\end{array}\right.$, where ${\mathit{q}}_v^{\times }=\left[\begin{array}{ccc}0& -q_3& q_2\\ q_3& 0& -q_1\\ -q_2& q_1& 0\end{array}\right]$ is the cross-product antisymmetric skew-symmetric matrix of ${\mathit{q}}_v$; $\mathit{\omega }={\left({\omega }_x,{\omega }_y,{\omega }_z\right)}^{\mathrm{T}}$ is the satellite’s angular velocity vector.

3.2. Attitude Measurement Models

The attitude measurement devices selected for this study are a gyro and a magnetometer. The corresponding vector observation model is as follows:

$${\mathit{z}}_k=A(\mathit{q}){\mathit{r}}_k+v_k,$$

(8)

where ${\mathit{v}}_k$ is the measurement noise, which follows a zero-mean normal distribution; ${\mathit{r}}_k$ is the reference vector components in the inertial coordinate system; ${\mathit{z}}_k$ is the reference vector component in the satellite body coordinate system; $A(\mathit{q})=(q_4^2-{‖{\mathit{q}}_v‖}^2){\mathit{I}}_3-2q_4[{{\mathit{q}}_v}^{\times }]+2{\mathit{q}}_v{\mathit{q}}_v^{\mathrm{T}}$ is the attitude rotation matrix; and ${\mathit{I}}_3$ denotes the 3 × 3 identity matrix. The measurement models of the gyro and the magnetometer are given below, respectively.

The gyro’s angular velocity measurement output model is as follows:

$$\left\{\begin{array}{l}{\mathit{\omega }}_g=\mathit{\omega }+{\mathit{\beta }}_g+{\mathit{\eta }}_v\\ {\dot{\mathit{\beta }}}_g={\mathit{\eta }}_u\end{array}\right.,$$

(9)

where ${\mathit{\omega }}_g$ represents the gyro’s measured angular velocity, ${\mathit{\eta }}_v$ and ${\mathit{\eta }}_u$ are unrelated zero-mean Gaussian white noises with standard deviations ${\mathit{\sigma }}_v$ and ${\mathit{\sigma }}_u$, respectively; ${\mathit{\beta }}_g$ is the gyro drift.

The magnetometer’s geomagnetic field vector measurement model is as follows:

$${\mathit{B}}_m={(B_r,B_{\theta },{\mathrm{B}}_{\lambda })}^{\mathrm{T}}=-\nabla \mathit{V},$$

(10)

where $\nabla$ is the gradient operator; and $\mathit{V}$ is the geomagnetic field potential function, which is expressed as follows:

$$\mathit{V}\left(r,\theta ,\lambda \right)=R_e{\displaystyle \sum _{n=1}^{\infty }{\displaystyle \sum _{m=0}^n{\left(\frac{R_e}{r}\right)}^{n+1}\left(g_n^m\cos \left(m\lambda \right)+h_n^m\sin \left(m\lambda \right)\right)P_n^m\left(\cos \theta \right)}},$$

(11)

where $R_e$ is the Earth’s radius (6378.14 km); $r$ is the distance from the Earth’s center; $\lambda$ is the geographic longitude; $\theta ={\displaystyle \frac{\pi }{2}}-\phi$ is the geocentric colatitude; and $\phi$ is the geocentric latitude. $P_n^m\left(\cdot \right)$ denotes the mth-degree quasi-associated Legendre function of nth order; $g_n^m$ and $h_n^m$ are Gaussian coefficients; and the IGRF model provides corresponding Gaussian coefficient tables that can be accessed via the internet.

Then, the components of ${\mathit{B}}_m$ in the Earth-fixed coordinate system are as follows:

$$\left\{\begin{array}{l}{B}_r=-{\displaystyle \frac{\partial \mathit{V}}{\partial r}}={\displaystyle \sum _{n=1}^k{\left(\frac{R_e}{r}\right)}^{n+2}\left(n+1\right){\displaystyle \sum _{m=0}^n\left(g_n^m\cos \left(m\lambda \right)+h_n^m\sin \left(m\lambda \right)\right)P_n^m\left(\cos \theta \right)}}\\ B_{\theta }=-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \mathit{V}}{\partial \theta }}=-{\displaystyle \sum _{n=1}^k{\left(\frac{R_e}{r}\right)}^{n+2}{\displaystyle \sum _{m=0}^n\left(g_n^m\cos \left(m\lambda \right)+h_n^m\sin \left(m\lambda \right)\right)\frac{\partial P_n^m\left(\cos \theta \right)}{\partial \theta }}}\\ B_{\lambda }=-{\displaystyle \frac{1}{r\sin \theta }}{\displaystyle \frac{\partial \mathit{V}}{\partial \lambda }}=-{\displaystyle \frac{1}{\sin \theta }}{\displaystyle \sum _{n=1}^k{\left(\frac{R_e}{r}\right)}^{n+2}{\displaystyle \sum _{m=0}^n\left(-g_n^m\cos \left(m\lambda \right)+h_n^m\sin \left(m\lambda \right)\right)P_n^m\left(\cos \theta \right)}}\end{array}\right..$$

(12)

3.3. Cubature Kalman Filter Based Mirosatellite Attitude Estimation

Since quaternions need to satisfy the constraint ${\mathit{q}}^{\mathrm{T}}\mathit{q}=1$, normalization is required during the attitude update process, which may lead to singularity issues. This makes it impossible to determine the predicted mean, potentially resulting in the singularity of the state prediction covariance. Therefore, the modified Rodrigues parameters (MRPs) are selected. MRPs only become singular when the attitude angle is $\pm 2\pi$, and they do not require normalization. Thus, a combination of quaternions and MRPs is used for attitude description, involving the conversion between these two attitude representations, as shown below:

$$\delta \mathit{p}=f_c{\displaystyle \frac{\delta {\mathit{q}}_v}{a_c+\delta q_4}},$$

(13)

$$\left\{\begin{array}{l}\delta q_4={\displaystyle \frac{-a_c{‖\delta \mathit{p}‖}^2+f_c\sqrt{f_c^2+(1-a_c^2){‖\delta \mathit{p}‖}^2}}{f_c^2+{‖\delta \mathit{p}‖}^2}}\\ \delta {\mathit{q}}_v=f_c^{-1}(a_c+\delta q_4)\delta \mathit{p}\end{array}\right.,$$

(14)

where p represents the MRPs; and $\delta \mathit{p}$ represents the error Rodrigues parameters. Conversely, $\delta \mathit{q}=\left(\delta {\mathit{q}}_v,\delta q_4\right)$ represents the error quaternion. In addition, $a_c$ represents a parameter; and $f_c$ is a proportional adjustment factor. In general, $a_c=1$ and $f_c=4$; thus, Equations (13) and (14) can be expressed in the following simplified forms:

$$\delta \mathit{p}={\displaystyle \frac{4\delta {\mathit{q}}_v}{1+\delta q_4}},$$

(15)

$$\left\{\begin{array}{l}\delta q_4={\displaystyle \frac{16-{‖\delta \mathit{p}‖}^2}{16+{‖\delta \mathit{p}‖}^2}}\\ \delta {\mathit{q}}_v={\displaystyle \frac{1}{4}}(1+\delta q_4)\delta \mathit{p}\end{array}\right.,$$

(16)

where Equation (15) describes the transformation from error quaternion to error Rodrigues parameters; while Equation (16) depicts the conversion from error Rodrigues parameters to error quaternion.

This paper presents the completion of attitude estimation for a microsatellite using the cubature Kalman filter algorithm. For more detailed information on the cubature Kalman filter, please refer to the literature [7].

The state of attitude estimation in this paper is defined as $\mathit{x}={\left(\delta {\mathit{p}}^{\mathrm{T}},{\mathit{\beta }}_g^{\mathrm{T}}\right)}^{\mathrm{T}}$, where $\delta \mathit{p}$ and ${\mathit{\beta }}_g$ represents the error Rodrigues parameters and gyro drift mentioned above, respectively. It can be seen that the system dimension is 6. In this section, numerous subscripts are present. To enhance the clarity of variable expressions, we utilize semicolons to delimit the columns of a vector. For instance, ${\left(\delta {\mathit{p}}^{\mathrm{T}},{\mathit{\beta }}_g^{\mathrm{T}}\right)}^{\mathrm{T}}$ is denoted as $\left(\delta \mathit{p};{\mathit{\beta }}_g\right)$.

Here, we will provide a concise description of the variables.

The symbol $\widehat{\cdot }$ above the variable denotes the estimated value of the variable.

The superscript “−” indicates the prior estimate; whereas the superscript “+” signifies the posterior estimate.

The superscript “(i)” denotes the ith cubature point.

The subscript “k” denotes the time step.

The following steps outline the process for conducting microsatellite attitude estimation.

Step 1 Filter initialization.

(1)

${\mathit{x}}_0=\left(\delta {\mathit{p}}_0;{\hat{\mathit{\beta }}}_{g,0}\right)$ is the initial state for microsatellite attitude estimation, where $\delta {\mathit{p}}_0={\left(0,0,0\right)}^T$, and ${\widehat{\mathit{\beta }}}_{g,0}$ represents the initial estimate of the gyroscope drift.

(2)

${\mathit{P}}_0=\mathbf{E}\left[\left({\mathit{x}}_0-{\hat{\mathit{x}}}_0\right){\left({\mathit{x}}_0-{\hat{\mathit{x}}}_0\right)}^{\mathrm{T}}\right]$ is the initial attitude estimation error covariance matrix.

(3)

Let ${\widehat{\mathit{q}}}_0$ represent the initial estimated attitude quaternion.

Let k = 1, 2, …, and repeat the following steps.

Step 2 Time Prediction.

(1)

Calculate the cubature points ${\hat{\mathit{x}}}_{k-1}^{\left(i\right)}=\left(\delta {\widehat{\mathit{p}}}_{k-1}^{\left(i\right)};{\widehat{\mathit{\beta }}}_{k-1}^{\left(i\right)}\right)=\sqrt{{\mathit{P}}_{k-1}^{+}}{\mathit{\xi }}_i+{\hat{\mathit{x}}}_{k-1}^{+}$, $i=1,2,\cdots ,12$, where ${\hat{\mathit{x}}}_{k-1}^{+}=\left(\delta {\hat{\mathit{p}}}_{k-1}^{+};{\hat{\mathit{\beta }}}_{g,k-1}^{+}\right)$ denotes the posterior estimation of the state at time k − 1; ${\mathit{P}}_{k-1}^{+}$ represents the posterior error covariance matrix at time k − 1; ${\mathit{\xi }}_i$ denotes the ith column of the matrix $\sqrt{6}\times \left[{\mathit{I}}_6,-{\mathit{I}}_6\right]$; and ${\mathit{I}}_6$ represents the 6 × 6 identity matrix.

(2)

Convert the error MRPs cubature points to the error quaternions cubature points according to Formula (16) as $\delta q_{4,k-1}^{\left(i\right)}={\displaystyle \frac{16-{‖\delta {\widehat{\mathit{p}}}_{k-1}^{\left(i\right)}‖}^2}{16+{‖\delta {\widehat{\mathit{p}}}_{k-1}^{\left(i\right)}‖}^2}}$, $\delta {\mathit{q}}_{v,k-1}^{\left(i\right)}={\displaystyle \frac{1}{4}}\left(1+\delta q_{4,k-1}^{\left(i\right)}\right)\delta {\widehat{\mathit{p}}}_{k-1}^{\left(i\right)}$, and we have $\delta {\mathit{q}}_{k-1}^{\left(i\right)}=\left(\delta {\mathit{q}}_{v,k-1}^{\left(i\right)};\delta q_{4,k-1}^{\left(i\right)}\right)$.

(3)

Compute the quaternion cubature point set ${\hat{\mathit{q}}}_{k-1}^{\left(i\right)}=\delta {\mathit{q}}_{k-1}^{\left(i\right)}\otimes {\hat{\mathit{q}}}_{k-1}^{+}$, where $\otimes$ denotes the quaternion multiplication.

(4)

Calculate the corresponding estimated angular velocity ${\hat{\mathit{\omega }}}_{k-1}^{\left(i\right)}={\mathit{\omega }}_{g,k-1}-{\hat{\mathit{\beta }}}_{g,k-1}^{\left(i\right)}$.

(5)

Use the discrete form of the attitude kinematic equation to compute the updated quaternion cubature points ${\hat{\mathit{q}}}_{\mathit{pre},k}^{\left(i\right)}=\mathit{T}\left({\hat{\mathit{\omega }}}_{k-1}^{\left(i\right)}\right){\hat{\mathit{q}}}_{k-1}^{\left(i\right)}$, where

$$\mathit{T}\left({\hat{\mathit{\omega }}}_{k-1}^{\left(i\right)}\right)=\left[\begin{array}{cc}\cos ({\displaystyle \frac{1}{2}}‖{\hat{\mathit{\omega }}}_{k-1}^{\left(i\right)}‖\Delta t){\mathit{I}}_3-\left[{\mathit{L}}_{k-1}^{\times }\right]& {\mathit{L}}_{k-1}\\ -{\mathit{L}}_{k-1}^T& \cos ({\displaystyle \frac{1}{2}}‖{\hat{\mathit{\omega }}}_{k-1}^{\left(i\right)}‖\Delta t){\mathit{I}}_3\end{array}\right], {\mathit{L}}_{k-1}={\displaystyle \frac{\sin (\frac{1}{2}‖{\hat{\mathit{\omega }}}_{k-1}‖\Delta t){\hat{\mathit{\omega }}}_{k-1}}{‖{\hat{\mathit{\omega }}}_{k-1}‖}}.$$

(6)

Since the attitude error is selected as the state variable, the attitude error quaternions need to be calculated as $\delta {\mathit{q}}_{pre,k}^{\left(i\right)}={\widehat{\mathit{q}}}_{pre,k}^{\left(i\right)}\otimes {\left({\widehat{\mathit{q}}}_{k-1}^{+}\right)}^{-1}$.

(7)

Convert the attitude error quaternions to attitude error MRPs $\delta {\widehat{p}}_{pre,k}^{\left(i\right)}={\displaystyle \frac{4\delta q_{v,pre,k}^{\left(i\right)}}{1+\delta q_{4,pre,k}^{\left(i\right)}}}$, while the gyro drift ${\hat{\mathit{\beta }}}_{g,k}^{\left(i\right)}={\hat{\mathit{\beta }}}_{g,k-1}^{\left(i\right)}$, and then we have ${\hat{\mathit{x}}}_{pre,k}^{\left(i\right)}=\left(\delta {\mathit{p}}_{pre,k}^{\left(i\right)};{\hat{\mathit{\beta }}}_k^{\left(i\right)}\right)$.

(8)

Calculate the predicted state ${\widehat{\mathit{x}}}_k^{-}=\left(\delta {\widehat{\mathit{p}}}_k^{-};{\widehat{\mathit{\beta }}}_k^{-}\right)={\displaystyle \frac{1}{12}}{\displaystyle \sum _{i=1}^{12}{\widehat{\mathit{x}}}_{pre,k}^{\left(i\right)}}$ and the predicted state error covariance matrix ${\mathit{P}}_k^{-}={\displaystyle \frac{1}{12}}{\displaystyle \sum _{i=1}^{12}\left({\hat{\mathit{x}}}_{pre,k}^{\left(i\right)}-{\hat{\mathit{x}}}_k^{-}\right){\left({\hat{\mathit{x}}}_{pre,k}^{\left(i\right)}-{\hat{\mathit{x}}}_k^{-}\right)}^{\mathrm{T}}}+{\mathit{Q}}_{k-1}$, where

$${\mathit{Q}}_{k-1}=\left[\begin{array}{cc}\left({\sigma }_v^2\Delta t+{\displaystyle \frac{1}{3}}{\sigma }_u^2\Delta t^3\right){\mathit{I}}_3& -\left({\displaystyle \frac{1}{2}}{\sigma }_u^2\Delta t^2\right){\mathit{I}}_3\\ -\left({\displaystyle \frac{1}{2}}{\sigma }_u^2\Delta t^2\right){\mathit{I}}_3& \left({\sigma }_u^2\Delta t\right){\mathit{I}}_3\end{array}\right].$$

(9)

Calculate the attitude error quaternion $\delta q_{4,k}^{-}={\displaystyle \frac{16-{‖\delta {\hat{\mathit{p}}}_k^{-}‖}^2}{16+{‖\delta {\hat{\mathit{p}}}_k^{-}‖}^2}}$ and $\delta {\mathit{q}}_{v,k}^{-}={\displaystyle \frac{1}{4}}\left(1+\delta q_{4,k}^{-}\right)\delta {\hat{\mathit{p}}}_k^{-}$, and we have $\delta {\mathit{q}}_k^{-}=\left(\delta {\mathit{q}}_{v,k}^{-};\delta q_{4,k}^{-}\right)$.

(10)

Calculate the estimated attitude quaternion ${\hat{\mathit{q}}}_k^{-}=\delta {\mathit{q}}_k^{-}\otimes {\hat{\mathit{q}}}_{k-1}^{+}$.

Step 3 Measurement Update.

(1)

Calculate the cubature points ${\hat{\mathit{x}}}_k^{\left(i\right)}=\left(\delta {\widehat{\mathit{p}}}_k^{\left(i\right)};{\widehat{\mathit{\beta }}}_k^{\left(i\right)}\right)=\sqrt{{\mathit{P}}_k^{-}}{\mathit{\xi }}_i+{\hat{\mathit{x}}}_k^{-}$.

(2)

Convert the error MRPs cubature points to the error quaternion cubature points according to Formula (16) as $\delta q_{4,k}^{\left(i\right)}={\displaystyle \frac{16-{‖\delta {\widehat{\mathit{p}}}_k^{\left(i\right)}‖}^2}{16+{‖\delta {\widehat{\mathit{p}}}_k^{\left(i\right)}‖}^2}}$ and $\delta {\mathit{q}}_{v,k}^{\left(i\right)}={\displaystyle \frac{1}{4}}\left(1+\delta q_{4,k}^{\left(i\right)}\right)\delta {\widehat{\mathit{p}}}_k^{\left(i\right)}$, and we have $\delta {\mathit{q}}_k^{\left(i\right)}=\left(\delta {\mathit{q}}_{v,k}^{\left(i\right)};\delta q_{4,k}^{\left(i\right)}\right)$.

(3)

Compute the quaternion cubature point set: ${\hat{\mathit{q}}}_k^{\left(i\right)}=\delta {\mathit{q}}_k^{\left(i\right)}\otimes {\hat{\mathit{q}}}_k^{-}$.

(4)

Obtain the cubature points through the propagation of the nonlinear measurement function as ${\mathit{Z}}_k^{\left(i\right)}={\left[\mathit{A}\left({\hat{\mathit{q}}}_k^{\left(i\right)}\right){\mathit{r}}_1,\mathit{A}\left({\hat{\mathit{q}}}_k^{\left(i\right)}\right){\mathit{r}}_2,\cdots ,\mathit{A}\left({\hat{\mathit{q}}}_k^{\left(i\right)}\right){\mathit{r}}_m\right]}_k^{\mathrm{T}}$.

(5)

Calculate the measurement prediction, measurement covariance matrix, and cross-covariance matrix as ${\widehat{\mathit{z}}}_k={\displaystyle \frac{1}{12}}{\displaystyle \sum _{i=1}^{12}{\mathit{Z}}_k^{\left(i\right)}}$, ${\mathit{P}}_{z,k}={\displaystyle \frac{1}{12}}{\displaystyle \sum _{i=1}^{12}\left({\mathit{Z}}_k^{\left(i\right)}-{\hat{\mathit{z}}}_k\right){\left({\mathit{Z}}_k^{\left(i\right)}-{\hat{\mathit{z}}}_k\right)}^{\mathrm{T}}}+{\mathit{R}}_k$, and ${\mathit{P}}_{xz,k}={\displaystyle \frac{1}{12}}{\displaystyle \sum _{i=1}^{12}\left({\hat{\mathit{x}}}_k^{\left(i\right)}-{\hat{\mathit{x}}}_k^{-}\right)\left({\mathit{Z}}_k^{\left(i\right)}-{\hat{\mathit{z}}}_k\right)}$, respectively.

(6)

Calculate the Kalman gain ${\mathit{K}}_k={\mathit{P}}_{xz,k}{\mathit{P}}_{z,k}^{-1}$.

(7)

Calculate the posterior state estimation, ${\hat{\mathit{x}}}_k^{+}=\left(\delta {\hat{\mathit{p}}}_k^{+};{\hat{\mathit{\beta }}}_{g,k}^{+}\right)={\hat{\mathit{x}}}_k^{-}+{\mathit{K}}_k({\mathit{z}}_k-{\hat{\mathit{z}}}_k)$, and the posterior state error covariance ${\mathit{P}}_k^{+}={\mathit{P}}_k^{-}-{\mathit{K}}_k{P}_{z,k}{\mathit{K}}_k^{\mathrm{T}}$.

(8)

Convert the error MRPs to the error quaternion as $\delta q_{4,k}^{+}={\displaystyle \frac{16-{‖\delta {\hat{\mathit{p}}}_k^{+}‖}^2}{16+{‖\delta {\hat{\mathit{p}}}_k^{+}‖}^2}}$ and $\delta {\mathit{q}}_{v,k}^{+}={\displaystyle \frac{1}{4}}\left(1+\delta q_{4,k}^{+}\right)\delta {\hat{\mathit{p}}}_k^{+}$, and we have $\delta {\mathit{q}}_k^{+}=\left(\delta {\mathit{q}}_{v,k}^{+};\delta q_{4,k}^{+}\right)$.

(9)

Calculate the posterior estimated attitude quaternion ${\hat{\mathit{q}}}_k^{+}=\delta {\mathit{q}}_k^{+}\otimes {\hat{\mathit{q}}}_k^{-}$.

(10)

Reset the first three elements of ${\hat{\mathit{x}}}_k^{+}$ to zero.

4. Non-Singular Fast Terminal Sliding Mode Control for Agile Whiskbroom Scanning Based on CMGs

4.1. Satellite Attitude Dynamics Model

In this paper, the microsatellite is treated as a rigid body with its center of mass as the reference point. The satellite’s dynamics model can be described as follows:

$$\mathit{J}{\dot{\mathit{\omega }}}_{\mathrm{bi}}+{\mathit{\omega }}_{\mathrm{bi}}\times \left(\mathit{J}{\mathit{\omega }}_{\mathrm{bi}}+\mathit{C}\mathit{h}\right)=-\mathit{C}\dot{\mathit{h}}+{\mathit{T}}_f,$$

(17)

where $\mathit{J}$ is the satellite’s moment of inertia; ${\mathit{\omega }}_{\mathrm{bi}}$ is the angular velocity vector of the satellite’s body coordinate system relative to the inertial coordinate system, expressed in the body coordinate system components; $\mathit{h}$ is the sum of the angular momentum of CMGs; $\mathit{C}$ is the installation matrix; and ${\mathit{T}}_f$ is the external torque acting on the satellite’s center of mass.

The angular velocity error is given by ${\mathit{\omega }}_e={\mathit{\omega }}_{bi}-{\mathit{C}}_e{\mathit{\omega }}_r$, where ${\mathit{C}}_e=(q_{e0}^2-{\mathit{q}}_{ev}^T{\mathit{q}}_{ev})I_3+2{\mathit{q}}_{ev}{\mathit{q}}_{ev}^T-2q_{e0}{\mathit{q}}_{ev}^{\times }$ is the transformation matrix from the target coordinate system to the body coordinate system, and ${\mathit{\omega }}_r$ is the desired angular velocity. Substituting this into Equation (17) yields the dynamics equation based on angular velocity error:

$$\mathit{J}{\dot{\mathit{\omega }}}_e=-({\mathit{\omega }}_e-{\mathit{C}}_e{\mathit{\omega }}_r)\times [\mathit{J}({\mathit{\omega }}_e-{\mathit{C}}_e{\mathit{\omega }}_r)+\mathit{C}\mathit{h}]+\mathit{J}({\mathit{\omega }}_e\times {\mathit{C}}_e{\mathit{\omega }}_r-{\mathit{C}}_e{\dot{\mathit{\omega }}}_r)-\mathit{C}\dot{\mathit{h}}+{\mathit{T}}_f.$$

(18)

4.2. CMG Configuration and Manipulation Law Design

(1)

CMG Configuration Design

Actuators capable of providing very large control torques are required for measuring the angular velocity as the satellite in this study has been assumed to undergo rapid attitude maneuvers. Therefore, SGCMGs have been selected as the actuators. To achieve three-axis attitude control of the satellite, a pyramid configuration has been chosen whose structural schematic is shown in Figure 5:

The pyramid configuration consists of four SGCMGs installed symmetrically in pairs, forming a pyramid shape with minimal redundancy. The SGCMGs are installed at an angle $\beta$ to the horizontal plane of the pyramid base, resulting in a frame axis angle of $90°-\beta$ to the plane. The angular momentum constraint is distributed across the four sides, rotating around the frame axes perpendicular to the pyramid faces. With an installation inclination of $\beta =54.73°$ and a consistent angular momentum of each SGCMG along its spin axis, the angular momentum envelope of the SGCMGs approaches a sphere. This results in a symmetric and highly linear angular momentum envelope, making the satellite’s attitude maneuvering path smoother and uninterrupted, thereby minimizing the challenges associated with the manipulation law design.

The four frame axes in the pyramid configuration can be expressed in the satellite’s body coordinate system as follows:

$$\left\{\begin{array}{l}{\mathbf{g}}^1=i\sin \beta +k\cos \beta \\ {\mathbf{g}}^2=j\sin \beta +k\cos \beta \\ {\mathbf{g}}^3=-i\sin \beta +k\cos \beta \\ {\mathbf{g}}^4=-j\sin \beta +k\cos \beta \end{array}\right.,$$

(19)

where i, j, and k are the base vectors of the three axes, xb, yb, and zb, in the satellite’s body coordinate system, respectively.

The total angular momentum of the CMGs in the pyramid configuration can be expressed in the satellite’s body coordinate system as follows:

$$\begin{array}{ll}h& =h_{x}i+h_{y}k+h_{z}j={\displaystyle \sum _{i=1}^4{h}_i}({\delta }_i)\\ & =h_1({\delta }_1)+h_2({\delta }_2)+h_3({\delta }_3)+h_4({\delta }_4)\\ & =\left[\begin{array}{l}-\cos \beta \sin {\delta }_1\\ \cos {\delta }_1\\ \sin \beta \sin {\delta }_1\end{array}\right]h_0+\left[\begin{array}{l}-\cos {\delta }_2\\ -\cos \beta \sin {\delta }_2\\ \sin \beta \sin {\delta }_2\end{array}\right]h_0+\left[\begin{array}{l}\cos \beta \sin {\delta }_3\\ \cos {\delta }_3\\ \sin \beta \sin {\delta }_3\end{array}\right]h_0+\left[\begin{array}{l}\cos {\delta }_4\\ \cos \beta \sin {\delta }_4\\ \sin \beta \sin {\delta }_4\end{array}\right]h_0\\ & =h_0\left[\begin{array}{c}-\cos \beta \sin {\delta }_1-\cos {\delta }_2+\cos \beta \sin {\delta }_3+\cos {\delta }_4\\ \cos {\delta }_1-\cos \beta \sin {\delta }_2+\cos {\delta }_3\cos \beta \sin {\delta }_4\\ \sin \beta \sin {\delta }_1+\sin \beta \sin {\delta }_2+\sin \beta \sin {\delta }_3+\sin \beta \sin {\delta }_4\end{array}\right]\end{array}$$

(20)

where ${\delta }_i$ is the frame angle; $h_x$, $h_y$, and $h_z$ are the components of the total angular momentum of the SGCMGs in the satellite’s body coordinate system; and $h_0$ is the angular momentum of a single SGCMG.

Differentiating the total angular momentum:

$$\dot{h}=h_0({\mathit{A}}_1{\dot{\delta }}_1+{\mathit{A}}_2{\dot{\delta }}_2+{\mathit{A}}_3{\dot{\delta }}_3+{\mathit{A}}_4{\dot{\delta }}_4)=h_0\mathit{A}\dot{\delta },$$

(21)

where $\mathit{A}$ is the Jacobian matrix, expressed as follows:

$$\begin{array}{ll}\mathit{A}& =\left[\begin{array}{cccc}{\mathit{A}}_1& {\mathit{A}}_2& {\mathit{A}}_3& {\mathit{A}}_4\end{array}\right]\\ & =\left[\begin{array}{cccc}-\cos \beta \cos {\delta }_1& \sin {\delta }_2& \cos \beta \cos {\delta }_3& -\sin {\delta }_4\\ -\sin {\delta }_1& -\cos \beta \cos {\delta }_2& \sin {\delta }_3& \cos \beta \cos {\delta }_4\\ \sin \beta \cos {\delta }_1& \sin \beta \cos {\delta }_2& \sin \beta \cos {\delta }_3& \sin \beta \cos {\delta }_4\end{array}\right]\end{array}.$$

(22)

If the desired control torque $u$ is known, the desired rate of change of the SGCMG angular momentum can be expressed as follows:

$$\dot{h}=-u-{\omega }^{\times }h.$$

(23)

(2)

Manipulation Law Design

The manipulation law can obtain frame angle commands based on output directives of the control torque by the controller. The law assists in reasonably distributing the frame angular velocity $\dot{\delta }$ of each CMG unit within hardware capability limits. In this work, a hybrid manipulation law has been adopted, which is explained as follows.

Initially, the following equation is used to determine whether zero motion exists:

$${\lambda }^{\mathrm{T}}{N}^{\mathrm{T}}PN\lambda ={\lambda }^{\mathrm{T}}Q\lambda =\mathbf{0}.$$

(24)

The hybrid manipulation law differs from other manipulation laws in that it requires a positive definiteness assessment of the mapping matrix $Q$ in Equation (24) to determine the type of singularity surface. If zero motion is present, it is subsequently added to avoid singularities on the hyperbolic singularity surface. If zero motion does not exist, torque errors are directed near the elliptical singularity surface to prevent singularity from appearing on the surface.

The hybrid manipulation law can be expressed as

$$\dot{\delta }={\displaystyle \frac{1}{h_0}}{\mathit{A}}^{\mathrm{SDA},\mathsf{\alpha }}\dot{h}+\rho \left[{\mathit{I}}_4-{\mathit{A}}^{\mathrm{T}}{{\displaystyle (}\mathit{A}{\mathit{A}}^{\mathrm{T}}{\displaystyle )}}^{-1}\mathit{A}\right]d,$$

(25)

where the null vector $d$ is in the direction of the gradient of $f=-\det (\mathit{A}{\mathit{A}}^T)=-m^2$ and maximizes the distance from the singularity.

$${\mathit{A}}^{SDA,\alpha }=V\left[\begin{array}{ccc}{\displaystyle \frac{1}{{\sigma }_1}}& 0& 0\\ 0& {\displaystyle \frac{1}{{\sigma }_2}}& 0\\ 0& 0& {\displaystyle \frac{{\sigma }_3}{{\sigma }_3^2+\alpha }}\\ 0& 0& 0\end{array}\right]U^T,$$

(26)

$$\alpha ={\alpha }_0{e}^{-a\overline{\alpha }-{\mu }_{1}m}$$

(27)

$$\rho ={\rho }_0{e}^{-b\overline{\rho }-{\mu }_{2}m}$$

(28)

where $a$, $b$, ${\alpha }_0$, ${\rho }_0$, ${\mu }_1$, and ${\mu }_2$ are positive scalar constants.

$$m=\sqrt{\det (\mathit{A}{\mathit{A}}^{\mathrm{T}})}$$

(29)

$$\overline{\alpha }=\left|Q_0-\det (Q)\right|$$

(30)

$$\overline{\rho }={\displaystyle \frac{1}{\overline{\alpha }}}={\displaystyle \frac{1}{\left|Q_0-\det (Q)\right|}}$$

(31)

The meanings of matrices Q, P, N, U, V, and m are the same as defined as in reference [26]. Q is the singularity definition matrix; P is the projection matrix for the singularity metric; N is null-space basis for the Jacobian; U is the left unitary matrix found from the singular value decomposition of Jacobian; and V is the left unitary matrix found from the singular value decomposition of Jacobian. The next step is to determine the type of singularity. If the singularity is located near the elliptical singularity surface, the parameter $\alpha$ in the hybrid manipulation law can be increased to steer the torque error away from the elliptical singularity surface. Similarly, if it is situated near the hyperbolic singularity surface, the parameter $\alpha$ in the hybrid manipulation law can be decreased while simultaneously increasing $\rho$ and adding zero motion to avoid the hyperbolic singularity surface.

4.3. Design of Non-Singular Fast Terminal Sliding Mode Controller

Conventional sliding mode control typically selects a linear sliding hyperplane, ensuring that once the system attains the sliding mode, the tracking error asymptotically converges to zero. The convergence rate can be adjusted by choosing appropriate sliding surface parameters. However, theoretically, the system state tracking error cannot converge to zero within a finite time. Terminal sliding mode control achieves state tracking by designing a dynamic nonlinear sliding surface that enables complete tracking of the desired attitude within a finite time while maintaining the stability of the sliding mode control. Fast terminal sliding mode control (FTSMC) transcends the characteristic of the traditional sliding mode control whereby the state asymptotically converges under the condition of a linear sliding surface. FTSMC ensures error convergence to zero and, unlike conventional linear sliding mode control, does not include a switching term. Therefore, it effectively suppresses chattering and provides superior dynamic performance. However, in traditional FTSMC, the convergence rate of the nonlinear sliding surface is rather slower than that of the linear sliding surface when the system state is near equilibrium, and singularities are present. To address these issues, linear sliding mode and fast terminal sliding mode can be combined, along with the design of a non-singular sliding surface, to obtain non-singular fast terminal sliding mode control (NFTSMC). NFTSMC ensures the satellite’s attitude stability and high-precision tracking of the preset trajectory, thus enabling rapid error convergence and avoiding singularities.

The sliding surface of NFTSMC can be designed as

$$s=q_{ev}+\alpha q_{ev}^{{\displaystyle \frac{g}{h}}}+\beta {\omega }_e^{{\displaystyle \frac{m}{n}}},$$

(32)

where α > 0; β > 0; $1<{\displaystyle \frac{m}{n}}<2$; ${\displaystyle \frac{m}{n}}<{\displaystyle \frac{g}{h}}$.

Thus, the dynamic equation for the satellite rapid whiskbroom scanning attitude tracking error is

$$\left\{\begin{array}{ll}{\dot{q}}_{ev}& ={\displaystyle \frac{1}{2}}\left(q_{e0}{I}_{3\times 3}+q_{ev}^{\times }\right){\omega }_e\\ {\dot{\omega }}_e& =f+g\cdot u+d(t)\\ & =J^{-1}\left[-{\omega }_b^{\times }\left(J{\omega }_b+h\right)\right]+{\omega }_e^{\times }{R}_d^b(q_e){\omega }_d-R_d^b(q_e){\dot{\omega }}_d+J^{-1}u+J^{-1}{d}_0\end{array}\right..$$

(33)

The NFTSMC sliding mode control law can be designed as

$$u=u_{eq}+u_s,$$

(34)

where $u_{eq}$ is the equivalent control term; and $u_s$ is the switching control term.

The sliding mode reaching law can be expressed as follows:

$$\begin{array}{ll}\dot{s}& ={\dot{q}}_{ev}+\alpha {\displaystyle \frac{g}{h}}{q_{ev}}^{{\displaystyle \frac{g}{h}}-1}{\dot{q}}_{ev}+\beta {\displaystyle \frac{m}{n}}{{\omega }_e}^{{\displaystyle \frac{m}{n}}-1}{\dot{\omega }}_e\\ & =\left(I_{3\times 3}+\alpha {\displaystyle \frac{g}{h}}{q_{ev}}^{{\displaystyle \frac{g}{h}}-1}\right){\dot{q}}_{ev}+\beta {\displaystyle \frac{m}{n}}{{\omega }_e}^{{\displaystyle \frac{m}{n}}-1}{\dot{\omega }}_e\\ & =\left(I_{3\times 3}+\alpha {\displaystyle \frac{g}{h}}{q_{ev}}^{{\displaystyle \frac{g}{h}}-1}\right){\dot{q}}_{ev}+\beta {\displaystyle \frac{m}{n}}{{\omega }_e}^{{\displaystyle \frac{m}{n}}-1}\left[-J^{-1}{\omega }_b^{\times }\left(J{\omega }_b+H_{CMG}\right)\right.\\ & \left.+{\omega }_e^{\times }{R}_d^b(q_e){\omega }_d-R_d^b(q_e){\dot{\omega }}_d+J^{-1}u+J^{-1}{d}_0\right]\end{array}.$$

(35)

When $\dot{s}=0$, the equivalent control term is as follows:

$$u_{eq}=-J\left[{\displaystyle \frac{n}{2\beta m}}A(q_e){\omega }_e^{2-{\displaystyle \frac{m}{n}}}+{\displaystyle \frac{\alpha gn}{hm}}{q}_{ev}^{{\displaystyle \frac{g}{h}}-1}{\omega }_e^{2-{\displaystyle \frac{m}{n}}}\right]+{\omega }_b^{\times }\left(J{\omega }_b+H_{CMG}\right)-J\left[{\omega }_e^{2-{\displaystyle \frac{m}{n}}}{R}_d^b(q_e){\omega }_d-R_d^b(q_e){\dot{\omega }}_d\right].$$

(36)

The switching control term $u_s$ is given by

$$u_s=-K\mathrm{sgn}(s)-\eta s,$$

(37)

where $K=diag(K_1,K_2,K_3)$.

Therefore, combining Equations (34), (36) and (37), the control law can be obtained as follows:

$$\begin{array}{l}u=u_{eq}+u_s=-J\left[{\displaystyle \frac{n}{2\beta m}}A(q_e){\omega }_e^{2-{\displaystyle \frac{m}{n}}}+{\displaystyle \frac{\alpha gn}{hm}}{q}_{ev}^{{\displaystyle \frac{g}{h}}-1}{\omega }_e^{2-{\displaystyle \frac{m}{n}}}\right]\\ +{\omega }_b^{\times }\left(J{\omega }_b+H_{CMG}\right)-J\left[{\omega }_e^{\times }{R}_d^b(q_e){\omega }_d-R_d^b(q_e){\dot{\omega }}_d\right]-K\mathrm{sgn}(s)-\eta s\end{array}.$$

(38)

To prove the stability of this control algorithm, the Lyapunov function has been used:

$$V={\displaystyle \frac{n}{2\beta m}}{s}^{\mathrm{T}}J{\omega }_e^{1-{\displaystyle \frac{m}{n}}}s.$$

(39)

Thus,

$$\begin{array}{ll}\dot{V}& ={\displaystyle \frac{n}{\beta m}}{s}^{\mathrm{T}}J{\omega }_e^{1-{\displaystyle \frac{m}{n}}}\dot{s}\\ & ={\displaystyle \frac{n}{\beta m}}{s}^{\mathrm{T}}J{\omega }_e^{1-{\displaystyle \frac{m}{n}}}({\dot{q}}_{ev}+\alpha {\displaystyle \frac{g}{h}}{q_{ev}}^{{\displaystyle \frac{g}{h}}-1}{\dot{q}}_{ev}+\beta {\displaystyle \frac{m}{n}}{{\omega }_e}^{{\displaystyle \frac{m}{n}}-1}{\dot{\omega }}_e)\\ & ={\displaystyle \frac{n}{\beta m}}{s}^{\mathrm{T}}J{{\omega }_e}^{1-{\displaystyle \frac{m}{n}}}\left(I_{3\times 3}+\alpha {\displaystyle \frac{g}{h}}{q_{ev}}^{{\displaystyle \frac{g}{h}}-1}\right){\dot{q}}_{ev}\\ & +s^{\mathrm{T}}\left[-J^{-1}{\omega }_b^{\times }\left(J{\omega }_b+H_{CMG}\right)\right.\left.+{\omega }_e^{\times }{R}_d^b(q_e){\omega }_d-R_d^b(q_e){\dot{\omega }}_d+J^{-1}u+J^{-1}{d}_0\right]\end{array}$$

(40)

Substituting Equation (35) into (40) yields:

$$\begin{array}{ll}\dot{V}& ={\displaystyle \frac{n}{\beta m}}{s}^{\mathrm{T}}J{{\omega }_e}^{1-{\displaystyle \frac{m}{n}}}\left(I_{3\times 3}\left.+\alpha {\displaystyle \frac{g}{h}}{q_{ev}}^{{\displaystyle \frac{g}{h}}-1}\right)\right.{\dot{q}}_{ev}+s^{\mathrm{T}}J\left[-J^{-1}{\omega }_b^{\times }\left(J{\omega }_b+H_{CMG}\right)\right.+{\omega }_e^{\times }{R}_d^b(q_e){\omega }_d\\ & -R_d^b(q_e){\dot{\omega }}_d-J^{-1}J\left({\displaystyle \frac{n}{2\beta m}}A(q_e){\omega }_e^{2-{\displaystyle \frac{m}{n}}}\left.+{\displaystyle \frac{\alpha gn}{hm}}{q}_{ev}^{{\displaystyle \frac{g}{h}}-1}{\omega }_e^{2-{\displaystyle \frac{m}{n}}}\right)\right.+J^{-1}{\omega }_b^{\times }\left(J{\omega }_b+H_{CMG}\right)\\ & -J^{-1}J\left({\mathbf{\omega }}_e^{\times }{R}_d^b(q_e){\omega }_d-R_d^b(q_e){\dot{\omega }}_d\right)\left.\left.-J^{-1}K\mathrm{sgn}(s)-\eta s\right)+J^{-1}{d}_0\right]\\ & ={\displaystyle \frac{n}{\beta m}}{s}^{\mathrm{T}}J\left\{{\displaystyle \frac{\alpha g}{h}}{{\omega }_e}^{1-{\displaystyle \frac{m}{n}}}{q_{ev}}^{{\displaystyle \frac{g}{h}}-1}\left[A(q_e)-\beta I_{3\times 3}\right]{\omega }_e+J^{-1}\left[-K\mathrm{sgn}(s)-\eta s+d_0\right]\right\}\end{array}$$

(41)

Given ${\displaystyle \frac{\alpha g}{h}}{{\omega }_e}^{1-{\displaystyle \frac{m}{n}}}{q_{ev}}^{{\displaystyle \frac{g}{h}}-1}\left(A(q_e)-\beta I_{3\times 3}\right){\omega }_e\le 0$, it follows that

$$\dot{V}\le s^{\mathrm{T}}\left[-K\mathrm{sgn}(s)-\eta s\right]=-K{s}^{\mathrm{T}}‖s‖-\eta s^{\mathrm{T}}s\le 0.$$

(42)

Therefore, based on the standard Lyapunov function theory, the system is asymptotically stable. As a result, $q_e$ and ${\omega }_e$ can reach the sliding surface and converge to the origin within a finite time.

5. Simulation Experiment and Analysis

5.1. Microsatellite Attitude Agile Whiskbroom Scanning Simulation System

To validate the effectiveness of the proposed methods, a microsatellite attitude agile whiskbroom scanning verification system was designed, as shown in Figure 6, Figure 7, Figure 8 and Figure 9.

This system adopts a virtual–real combination approach. It is primarily composed of a microsatellite electrical model, a real-time simulator, a space magnetic field simulator, and testing software. The microsatellite electrical model is consistent with the actual on-orbit satellite. It can physically simulate the working states of various devices during an on-orbit operation of the satellite. The attitude estimation algorithm and attitude control algorithm designed in this study can be integrated into the comprehensive electronic system of the microsatellite for validation. Satellite attitude dynamics, orbital dynamics, space environment gyroscopes, and CMG can be virtually modelled via the real-time simulator. The space magnetic field simulator can model the space magnetic field, providing input for the magnetometer, while the testing software can monitor and control the state of the satellite.

5.2. Simulation Experiment Results and Analysis

Table 1. Parameters for the whiskbroom scanning mission scenario.

Parameter Name	Parameter Value
Imaging area $S=m\times n$	600 km × 1000 km
Orbital altitude $h$	900 km
Single-frame image swath width $W\times W$	100 km × 100 km
Satellite moment of inertia ${\mathbf{J}}_{sat}$	$\left[\begin{array}{ccc}4.5& 0& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]\mathrm{kg}\cdot {\mathrm{m}}^2$

tabulates the input parameters utilized for the whiskbroom scanning mission scenario.

By substituting these parameters into the calculations, the whiskbroom angle was determined to be $\pm 38°$ with $a=7.5°/{\mathrm{s}}^2$. As per the planned angular acceleration formula, the single-pass whiskbroom cycle was measured to be $t=7.28 \mathrm{s}$, including the acceleration segment time $T_1$ = 1.2671 s, the deceleration segment time $T_3$ = 1.2671 s, and the constant speed segment time $T_2=t-T_1-T_3=5.0139 \mathrm{s}$. Thus, the maximum angular velocity was found to be ${\omega }_{\max }=6.05°/\mathrm{s}$. Traditional reaction wheels cannot provide such a large angular velocity; hence, it has been established that utilizing CMGs is beneficial in achieving the high torque and the high angular velocity output.

The following controller parameters have been used (

Table 2. Controller parameters.

Parameter Name	Parameter Value
entry 1{g, h, m, n}	{9, 3, 9, 5}
{$\alpha$, $\rho$}	{0.5, 4}
$\eta$	40
$\gamma$	7
$K$	[1.5 1.5 15]
$e$	[0.02 0.02 0.02]

).

The CMGs with the single frame that were developed by a specific institute were used. Their basic configurations are listed in

Table 3. Model and parameters of the selected SGCMGs.

Maximum Output Torque (Nm)	Rotor Speed (r/min)	Weight (kg)	Dimensions (mm × mm × mm)
5	6000	6	301.4 × 194.6 × 181.6

.

The external environmental disturbance torque was configured as follows:

$${\mathit{T}}_f=\left[\begin{array}{c}3\sin \left({\displaystyle \frac{\pi }{14.56}}t\right)-6\\ 3\sin \left({\displaystyle \frac{\pi }{14.56}}t\right)-5\\ 3\sin \left({\displaystyle \frac{\pi }{14.56}}t\right)+2\end{array}\right]\times {10}^{-5}\left(\mathrm{N}\cdot \mathrm{m}\right).$$

The slew period is 29.12 s, so the frequency of disturbance torque is set to ${\displaystyle \frac{\pi }{14.56}}$. Aerodynamic moment is the main factor underpinning the external environmental disturbance torque for low-Earth-orbit satellites, and its magnitude is about 10⁻⁵ (N·m), so the assumed magnitude has been set to 10⁻⁵ (N·m).

The designed rapid attitude maneuvering path planning method was employed to predict the satellite whiskbroom path. The desired Euler angles and angular velocity are depicted in Figure 10 and Figure 11.

The designed non-singular fast terminal sliding mode controller was applied to track the satellite whiskbroom process. The simulation results are shown in Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17.

Figure 12 and Figure 13 illustrate the errors in the Euler angles and angular velocity of the NFTSMC controller. The maximum attitude angle errors were observed to be about $0.04°$ on the roll axis. The error accuracy generally fell within the $4\times {10}^{-2}$ magnitude. The maximum attitude angle errors were observed to be about $0.01°/\mathrm{s}$ on the x axis. The reason for this is that the satellite only performs agile whiskbroom scanning maneuvers in the rolling axis direction, and the motion in the other two axis directions is almost zero.

Figure 14 illustrates control torques of NFTSMC. The maximum torques were observed to be about $1.5 \mathrm{Nm}$ on the x axis, which is within the maximum output torque range. From Figure 15, Figure 16 and Figure 17, it can be seen that the CMGs are working well, and there is no singularity. The angles of the CMGs’ rotation are almost the same. Because the satellite performs agile whiskbroom scanning maneuvers in the direction of the rolling axis, the control torque is also along this direction. Four rotor frames rotating in the same way can provide torque along this direction.

6. Conclusions

This study addressed the challenge of controlling the attitude of high-speed reciprocating whiskbroom for microsatellites in low-Earth orbit. This study proposed a non-singular fast terminal sliding mode control method to enable agile whiskbroom scanning in the direction perpendicular to the orbit. The proposed method successfully achieved high-precision tracking control of the planned path, which was validated through the construction of a microsatellite attitude agile whiskbroom scanning verification system. The simulation study concluded that the optimized controller designed in this work achieved high-precision tracking control of the planned whiskbroom path. It was demonstrated that the error accuracy was within the $4\times {10}^{-2}$ magnitude, and angular velocity control accuracy was reached ($0.01°/\mathrm{s}$). These results demonstrate the effectiveness of the proposed algorithm and its potential to lead to the development of an optimized controller for improved space exploration.