https://orcid.org/0009-0005-3310-7933
Figures
Abstract
In this paper, the finite-time attitude tracking control problem for spacecraft based on the backstepping method is addressed. Firstly, a finite-time controller is designed, which can provide robustness for external disturbance by employing an improved adaptation law. Secondly, a novel finite-time controller with input saturation is proposed by introducing the hyperbolic function and auxiliary system to guarantee the control torques below a predetermined value. The above two controllers are continuous, thus, they are chattering-free. Finally, simulation results are presented to illustrate the effectiveness of the control strategies.
Citation: Ren Y, Xing A (2025) Finite-time attitude tracking control for spacecraft based on backstepping method with input saturation. PLoS One 20(6): e0326150. https://doi.org/10.1371/journal.pone.0326150
Editor: Gang Wang, University of Shanghai for Science and Technology, CHINA
Received: November 8, 2024; Accepted: May 27, 2025; Published: June 18, 2025
Copyright: © 2025 Ren, Xing. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: “All relevant data are within the paper and its Supporting Information files.”
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
In recent years, the attitude control problem of spacecraft has gained extensive attention. This has been motivated by the benefits gained via its space applications, such as space rendezvous and docking, spacecraft formation flying, deep space exploration, etc. Recently, various nonlinear control methods have been proposed for solving the attitude tracking control problem of spacecraft, such as adaptive sliding film attitude control [1], asymptotic tracking control [2], hybrid attitude saturation and fault-tolerant control [3], Quaternion-Based Hybrid Control [4], robust control [5,6], arbitrary perturbation control based on Tan-type BLFs [7], adaptive control based on filtered concurrent learning [8], and decentralized coordinated [9,10] control for spacecraft formation flying, etc. In reference [11], for the input amplitude/rate constraints of unmanned aerial systems, the algorithm is constructed using a saturation function and a damping term, relying on Barbalat’s Lemma and graph theory to achieve group synergy, but the control objective is only asymptotically stable, and does not achieve the finite-time convergence characteristic. The above attitude control laws are asymptotically stable, which means tracking errors of closed-loop systems converge to equilibrium as time goes to infinity. Compared with asymptotic control methods, the finite-time control method can provide faster convergence and higher control precision. Thus, it is applied in spacecraft attitude tracking control widely.
Up to now, the existing finite-time control strategies mainly contain two categories: the homogeneous theory approach and the Lyapunov-based approach. In reference [12], the homogeneous theory is used to propose a finite-time control algorithm for spacecraft formation flying. References [13] and [14] propose robust and fault-tolerant finite-time control methods for spacecraft attitude control by using sliding mode control. In reference [15], finite-time control methods based on fast terminal sliding mode control are derived for spacecraft attitude synchronization and tracking. Reference [16] developed a finite-time output feedback attitude control method by adding a power integrator technique. The rigid spacecraft system is a standard cascade system and the backstepping method is a great cascade design tool, so the backstepping method [17,18] has been applied in finite-time attitude tracking control of rigid spacecraft successfully. Although the above methods solve the saturation problem, they face limitations. For example, reference [19] focuses on robust and consistent control of second-order uncertain multi-intelligent body systems, which is based on the fixed-time stability theory and Lyapunov function to deal with the speed and input constraints, but its controllers introduce discontinuities due to the sgn() function and the segmented saturation function, which may lead to jittering problems in the real system.
It is worthwhile to mention that to guarantee a fast convergence rate, the finite-time control output always reaches a large magnitude, especially within the initial phase of system response. Thus, many researchers have made effort in the study of finite-time control methods with actuator saturation. The homogeneous theory and saturated function in reference [20] are used to design a bounded proportional-derivative-type finite-time attitude control law. In order to deal with external disturbances, the finite-time observer [21] is adopted to develop a finite-time controller via output feedback with the adaptive method to ensure bounded input. The fast terminal sliding mode and Chebyshev neural network [22]are employed to derive a bounded finite-time distributed cooperative attitude control law for multiple spacecrafts. So far, although many finite-time controllers with actuator saturation have been proposed, there are few results on bounded finite-time controllers based on the backstepping method. References [23–26] use the backstepping technique to design controllers for spacecraft with actuator saturation, but the first three proposed control laws can not achieve finite-time stability and the last one is not continuous. References [11] and [19] further illustrate trade-offs between asymptotic stability and robustness under input/velocity constraints, emphasizing the need for continuous, chattering-free finite-time controllers.
Inspired by the realities stated above, this paper designs two finite-time attitude tracking controllers for the spacecraft by using the backstepping method. The main contributions of the results in this paper are: (i) The first controller can inhibit external disturbances requiring no information about them without chattering.(ii) The second controller can overcome the input saturation problem effectively in the presence of external disturbances with known bounds.
This paper is organized as follows. Sect 2 states the attitude tracking dynamics model of the spacecraft. In Sect 3, the design procedure of two continuous finite-time controllers is given in detail, and the stability of the closed-loop system is proved. A simulation example is then shown in Sect 4. Sect 5 concludes this paper.
2 Posture control model and problem description
2.1 Spacecraft attitude control model
The attitude, kinematics, and dynamics of a rigid spacecraft are modeled as
where is the rotation matrix that transforms the body frame to the inertial frame.
is the spacecraft’s angular velocity.
and
are the control torque and external disturbance torque, respectively. They are expressed in the body frame.
is the spacecraft’s inertial matrix.
is the skew-symmetric matrix that transforms a vector in
to a
skew-symmetric matrix.
To address the attitude tracking problem, a time-varying target attitude trajectory is described by , the attitude kinematics is written as
where is the target angular velocity.
The error rotation matrix is defined as
The angular velocity error resolved in the body frame is introduced as
As the error rotation matrix can’t be used to design the controller, basically, we use a new attitude error demonstrated [27]
where the map denotes the inverse of the cross-product operation.
Then, based on the equation , the attitude error kinematics and dynamics of the spacecraft are derived as follows.
Note that (Eqs 8) and (11) are singular, when the case occurs. So in order to solve this problem, the attitude error
and E should be defined in the sublevel
.
2.2 Problem description
In this paper, a tracking spacecraft tracks a target spacecraft with a time-varying motion state, wherein the tracking spacecraft and the target spacecraft measure their own attitude and angular velocity through their respective measuring elements, and the attitude and angular velocity of the target spacecraft are transmitted to the tracking spacecraft in real time through wireless communication. The tracking spacecraft uses its own state information and the state information of the target spacecraft to calculate the relative information of the rendezvous and docking.
The finite-time attitude tracking control problem of spacecraft studied in this paper can be described as: under the condition that the two spacecrafts of attitude tracking each measure their own state and the tracking spacecraft can obtain the state of the target spacecraft in real time, according to the attitude control model, the attitude tracking controller and the input saturation controller are designed using the finite-time control algorithm, the hyperbolic tangent function, and the auxiliary system. The relevant lemmas used in designing the controller are given below.
Lemma 1 [28]. Suppose are all positive numbers and
, and then the following inequality holds
Lemma 2 [29]. V(x) is a continuous positive definite function. An extended Lyapunov description of finite-time stability can be given as
where ,
,
, and the converging time can be given as
Lemma 3 [15]. The inertia matrix J is symmetric and positive, which is also bounded as , where
,
and
are the maximum eigenvalue and minimum eigenvalue of J respectively.
Definition 1 [30]. For the following system , where x is the system state, u is the control input. x(t0) = x(0), if there exist
and
, makes the inequality
hold for
, then the system is practically finite-time stable.
Lemma 4 [31]. For the above system, if there exists a continuous positive definite function V(x), for real numbers ,
,
and an open neighbourhood base
containing the origin, a Lyapunov condition of finite-time stability can be given as
, then the system is practically finite-time stable.
Lemma 5 [32]. V(x) is a continuous positive definite function, for any real number c>0 and an open neighbourhood base containing the origin, a Lyapunov condition of finite-time stability can be given as
, and if
, then the system is globally finite-time stable.
3 Design of the finite-time controller
3.1 Attitude tracking control with external disturbances
In this section, we assume that the external disturbances are bounded,
are all unknown constants,
and
are bounded. Consider the rigid spacecraft system (8) and (9), the variables x1 and x2 are introduced as follows.
In the light of (Eq 8), a virtual controller that makes the attitude error always satisfy
is proposed as
where ,
,
,
,
,
,
,
.
Proposition 1. Consider (Eq 5), when and
are bounded, x1,i(i = 1,2,3) will converge to
in finite time with the virtual controller (16).
Proof. Consider the following Lyapunov function.
Its time derivative is , substituting (Eq 16) into it yields
Based on (Eq 17), using Lemma 1 obtains
When
When
According to Lemma 2, it can be concluded that x1,i will converge to in finite time.
Based on the virtual controller (16), a continuous controller is designed as
where ,
,
is the estimation of
,
,
is the estimation error which is defined as
,
,
,
,
.
The following Lyapunov function is chosen
Its time derivative along the trajectory of the dynamical system (9) is
Using Lemma 1 and Lemma 3, and substituting (Eq 22) and (Eq 23) into (Eq 25) obtain
where ,
. As
is bounded, based on the boundedness theorem,
,
and
are all uniformly ultimately bounded. In order to achieve the finite-time convergence, a finite-time controller is proposed as follows.
where ,
.
Theorem 1. Consider the spacecraft system described by (8) and (9), the following conclusions can be satisfied with the control law (27).
(i) and
are practically finite-time stable.
(ii) is practically finite-time stable.
Proof. (i) Consider the Lyapunov function , based on (Eq 26), its time derivative along the trajectory of the dynamical system (9) can be written as
When ,
When ,
.
When ,
has it been in the convergence region
where .
As is positive and bounded, according to the Lemma 4,
and
are all practically finite-time stable.
(ii) Based on (Eqs 15) and (16), the angular velocity error can be written as follows
Furthermore, considering the finite-time convergence of and
, it can be concluded that
is also practically finite-time stable.
Now the proof has been completed.
Remark 1. The control law (27) is continuous and chattering-free. in (Eq 23) can not be strictly positive for all the time, which can avoid
growing without bound and reduce the control torque in a steady stage.
3.2 Attitude tracking control with external disturbances and input saturation
It should be noted that Sect 3.1 does not consider the input saturation problem, which can affect the control effect and stability of the system. In order to solve this problem, a novel finite-time controller with input saturation is proposed by introducing the hyperbolic function and auxiliary system in this section. We assume that the external disturbances d are bounded, ,
is a known positive constant,
and
are bounded. The finite-time controller with input saturation is proposed as
where ,
,
,
,
,
,
,
,
.
Theorem 2. Consider the spacecraft system described by (8) and (9), the following conclusions can be satisfied with the control laws (31)–(34).
(i) When ,
and
will converge to equilibrium
and
in finite time, when
, and k4 satisfying
, then
and
will converge to equilibrium
and
in finite time.
(ii) In finite time, will converge to
.
Proof. (i) Consider the Lyapunov function , its time derivative along the trajectory of the dynamical system (9) can be written as
When ,
When ,
,
Using Lemma 5 obtains that V3 will converge to the region V3 = 0 in finite time. Therefore, it can be concluded that and
will converge to equilibrium
and
in finite time.
Based on (Eqs 15) and (16), considering the finite-time convergence of and
, it can be concluded that
will converge to the region
in finite time.
Now the proof has been completed.
Remark 2. When , (Eq 33) will appear singular. To solve this problem, the term
is substituted by the following term
4 Simulation results
In this section, the proposed control schemes are applied to spacecraft attitude tracking control.
The inertia matrix and initial conditions of the spacecraft are:
.
The desired attitude and angular velocity for the spacecraft are: ,
.
External disturbances are: .
Parameters of the controller (Eq 27) are selected as: , k2 = 0.4,
,
,
,
,
,
,
,
.
Figs 1–5 depict the performance of the controller (Eq 27). In the figures, i represents the i th element of the corresponding vector. It follows from Figs 1–3 that the attitude tracking maneuver can be achieved in 15 seconds. The control input is shown in Fig 4, where chattering is avoided because the controller can maintain continuity when dealing with external disturbances. The estimated parameters are depicted in Fig 5.
In order to verify that the finite-time controller has faster convergence speed and higher steady-state accuracy, this paper will compare it with the finite-time controller based on the boundary layer theorem. In order to ensure the continuity of the controller, based on the boundary layer theorem, the controller is first designed as shown in (Eq 39).
where ,
and
are small positive numbers, and
,
and
are expressed by (Eq 40).
In order to solve the problem that the estimated value of the traditional adaptive update law may increase without limit, an improved adaptive update law is designed as shown in (Eq 41).
where ,
. According to the boundedness theorem, in order to ensure the finite time convergence of the system, the finite time controller is finally designed as (Eq 42).
where ,
. The specific control parameters are: k1 = 0.1, k2 = 0.2,
,
,
,
,
,
,
,
. The simulation results are shown in Figs 6–10.
It can be seen from Figs 6–8 that in the presence of external disturbances, the attitude error, angular velocity error, and achieve high-precision convergence within 20 seconds. In addition, it can be seen from the partial enlarged views of Figs 7 and 8 that the simulation curve in the steady-state stage is smooth and there is no jitter phenomenon. Comparing Fig 1 and Fig 2 with Fig 6 and Fig 7, it can be seen that the finite time controller designed in this paper has a faster convergence speed and higher steady-state accuracy. Fig 9 is a simulation curve of the control torque of the tracking spacecraft, and Fig 10 is a simulation curve of the upper limit estimate of the external disturbance torque.
Parameters of the controller (Eq 31) are set as ,
,
,
,
,
,
,
,
,
,
,
,
,
. Figs 11–14 depict the performance of the controller (Eq 31). It follows from Figs 11–14 that the attitude tracking maneuver can be achieved in 20 seconds under the condition that the control input satisfying
.
Comparing Figs 1–4 and Figs 11–14, there is no remarkable change except that the controller (Eq 27) can provide a little faster convergence than the controller (Eq 31), so it concludes that the controller (Eq 31) can resolve the problem of actuator saturation well. The control input is shown in Fig 14, where chattering is avoided and the control input satisfies the physical limit of the actuators.
5 Conclusion
In this paper, the finite-time control to achieve spacecraft attitude tracking based on the backstepping method is presented. The external disturbances, chattering, and input saturation are considered in the controller’s design. The proposed controllers can prevent chattering phenomena and achieve finite-time tracking. In addition, the second controller has been demonstrated to have superior performance while still holding stability in the presence of input saturation. A simulation example is shown to support the above analysis and demonstrate excellent dynamic tracking performance using the proposed finite-time controllers. The results show that the presented controllers can achieve spacecraft attitude tracking accurately in finite time.
References
- 1. Huo B, Du M, Yan Z. Adaptive sliding mode attitude tracking control for rigid spacecraft considering the unwinding problem. Mathematics. 2023;11(20):4372.
- 2. Lee D. Nonlinear disturbance observer-based robust control for spacecraft formation flying. Aerosp Sci Technol. 2018;76:82–90.
- 3. Ma J, Wang Z, Wang C. Hybrid attitude saturation and fault-tolerant control for rigid spacecraft without unwinding. Mathematics. 2023;11(15):3431.
- 4. Mayhew CG, Sanfelice RG, Teel AR. Quaternion-based hybrid control for robust global attitude tracking. IEEE Trans Autom Control. 2011;56(11):2555–66.
- 5. Soken HE, Hajiyev C. Pico satellite attitude estimation via Robust Unscented Kalman Filter in the presence of measurement faults. ISA Trans. 2010;49(3):249–56. pmid:20452589
- 6. Akman T, Ben-Asher JZ, Holzapfel F. Approximation of closed-loop sensitivities in robust trajectory optimization under parametric uncertainty. Aerospace. 2024;11(8):640.
- 7. Xuan-Mung N, Golestani M, Hong SK. Tan-type BLF-based attitude tracking control design for rigid spacecraft with arbitrary disturbances. Mathematics. 2022;10(23):4548.
- 8. Long J, Guo Y, Liu Z. Filtering-based concurrent learning adaptive attitude tracking control of rigid spacecraft with inertia parameter identification. Int J Robust Nonlinear Control. 2023;33(8).
- 9. Min H, Wang S, Sun F. Decentralized adaptive attitude synchronization of spacecraft formation. Syst Control Lett. 2012;61(1):238–46.
- 10. Zou A-M, Kumar KD. Distributed Attitude Coordination Control for Spacecraft Formation Flying. IEEE Trans Aerosp Electron Syst. 2012;48(2):1329–46.
- 11. Wang G, Zuo Z, Li P, Shen Y. Controllers for multiagent systems with input amplitude and rate constraints and their application to quadrotor rendezvous. IEEE Trans Autom Sci Eng. 2025;22:6354–64.
- 12. Zou A-M, de Ruiter AHJ, Kumar KD. Distributed finite-time velocity-free attitude coordination control for spacecraft formations. Automatica. 2016;67:46–53.
- 13. Gao J, Cai Y. Fixed-time control for spacecraft attitude tracking based on quaternion. Acta Astronaut. 2015;115:303–13.
- 14. Ran D, Chen X, de Ruiter A. Adaptive extended-state observer-based fault tolerant attitude control for spacecraft with reaction wheels. Acta Astronaut. 2018;145:501–14.
- 15. Lu K, Xia Y, Fu M. Controller design for rigid spacecraft attitude tracking with actuator saturation. Inf Sci. 2013;220:343–66.
- 16. Jiang B, Li C, Ma G. Finite-time output feedback attitude control for spacecraft using “adding a power integrator” technique. Aerosp Sci Technol. 2017;66:342–54.
- 17. Hu Q, Niu G. Attitude output feedback control for rigid spacecraft with finite-time convergence. ISA Trans. 2017;70:173–86. pmid:28789773
- 18. Huo J, Meng T, Song R. Adaptive prediction backstepping attitude control for liquid-filled micro-satellite with flexible appendages. Acta Astronaut. 2018;152:557–66.
- 19. Wang G, Zuo Z, Wang C. Robust consensus control of second-order uncertain multiagent systems with velocity and input constraints. Automatica. 2023;157:111226.
- 20. Gui H, Jin L, Xu S. Simple finite-time attitude stabilization laws for rigid spacecraft with bounded inputs. Aerosp Sci Technol. 2015;42:176–86.
- 21. Zhang J, Hu Q, Wang D. Bounded finite-time attitude tracking control for rigid spacecraft via output feedback. Aerosp Sci Technol. 2017;64:75–84.
- 22. Lu P, Gan C, Liu X. Finite-time distributed cooperative attitude control for multiple spacecraft with actuator saturation. IET Control Theory Appl. 2014;8(18):2186–98.
- 23. Xia K, Huo W. Robust adaptive backstepping neural networks control for spacecraft rendezvous and docking with input saturation. ISA Trans. 2016;62:249–57. pmid:26892402
- 24. Lu K, Xia Y. Adaptive attitude tracking control for rigid spacecraft with finite-time convergence. Automatica. 2013;49(12):3591–9.
- 25. Jia Z, Li T, Lu K. Adaptive backstepping control for longitudinal dynamics of hypersonic vehicle subject to actuator saturation and disturbance. Math Probl Eng. 2019;2019(1):1019621.
- 26. Guo Y, Guo J, Song S. Backstepping control for attitude tracking of the spacecraft under input saturation. Acta Astronaut. 2017;138:318–25.
- 27. Lee T. Exponential stability of an attitude tracking control system on SO (3) for large-angle rotational maneuvers. Syst Control Lett. 2012;61(1):231–7.
- 28. Du H, Li S, Qian C. Finite-time attitude tracking control of spacecraft with application to attitude synchronization. IEEE Trans Autom Control. 2011;56(11):2711–7.
- 29. Yu S, Yu X, Shirinzadeh B. Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica. 2005;41(11).
- 30. Zhu Z, Xia Y, Fu M. Attitude stabilization of rigid spacecraft with finite-time convergence. Int J Robust Nonlinear Control. 2011;21(6):686–702.
- 31. Hu Q, Jiang B, Friswell MI. Robust saturated finite time output feedback attitude stabilization for rigid spacecraft. J Guid Control Dyn. 2014;37(6).
- 32. Huo M, Huo X, Karimi HR. Finite-time control for attitude tracking maneuver of rigid satellite. Abstr Appl Anal. 2014;2014(1):302982.