observer, Luenberger observer, Actuator saturation, State estimation.

Attitude control of spacecraft is a fundamental part of guidance, navigation, and control systems, directly affecting pointing accuracy, communication performance, and mission effectiveness. Large-angle attitude manoeuvres pose a serious control challenge because of the nonlinear nature of spacecraft rotational dynamics and attitude kinematics [1–4].

Therefore, developing control strategies with accurate tracking, acceptable control effort, and satisfactory performance under practical constraints remains an important research topic. Several mathematical representations describe spacecraft orientation, such as Euler angles, rotation matrices, unit quaternions, and modified Rodrigues parameters [1], [2]. Euler angles are intuitive, but their kinematic equations can have singularities. For large angle manoeuvres, unit quaternions are commonly used to avoid these singularities. However, the unit-norm constraint has to be maintained [3], [4]. Linear control techniques, especially the Linear Quadratic Regulator (LQR), offer systematic ways to design feedback controllers by minimising a quadratic performance index which includes state errors and control effort. LQR-based methods have been explored for spacecraft attitude tracking and optimal manoeuvring, with applications involving flexible spacecraft and online system identification [5–7].

However, controllers based on linearised models may degrade when the spacecraft operates far from the nominal equilibrium or performs large-angle manoeuvres. Feedback linearisation is a well-known nonlinear control technique that uses a state-dependent control transformation to compensate for modelled nonlinearities, so that linear control techniques can be applied to the resulting dynamics under suitable conditions [8], [9]. Singh and Yim [10] considered feedback linearisation for satellite attitude control using solar-radiation-pressure-generated torque. Bang et al. [11] considered feedback-linearization-based attitude control for rigid spacecraft. Nonlinear spacecraft manoeuvring and prescribed attitude-error dynamics have been further studied in [12].

Thus, combining feedback linearisation with LQR provides a potentially efficient framework that unifies nonlinear compensation with a systematic approach to optimal control design. However, its performance depends on model accuracy and the validity of the feedback transformation.Another major challenge is the availability of the state variables required for the feedback control. Spacecraft use gyroscopes to measure angular velocities, and attitude sensors like star trackers and Sun sensors provide orientation information [2]. Gyroscope measurements are subject to noise, bias, and drift, so integrating angular velocities alone does not generally give a permanently accurate estimate of absolute attitude.

Hence, reliable state estimation must consider measurement availability, initial conditions, and system observability. State observers use available measurements and a mathematical model to reconstruct unmeasured variables. For linear systems, the classical approach is the Luenberger observer, where convergence depends on the system's observability properties and the choice of observer gain [15]. If the remaining states are available from measurements, a reduced-order observer estimates only the unmeasured states, reducing computational requirements.

Crassidis and Markley [13] studied unscented filtering for attitude estimation for spacecraft applications using gyroscopic propagation and attitude measurements, and Crassidis et al. [14] reviewed nonlinear attitude estimation techniques and their applications. These studies emphasise the importance of sensor modelling, estimation accuracy, and having enough information for attitude reconstruction.

In addition, actuator saturation is a practical limitation, since the controller may request torque beyond the actuator's physical capacity. Such limitations may increase tracking errors, slow down transient response, and impair closed-loop stability. Bang et al. [16] studied large-angle attitude control with actuator saturation. Lu et al. [17] studied spacecraft attitude tracking with saturation, inertia uncertainties, and external disturbances. Actuator faults and constrained control inputs have also been investigated by adaptive and sliding-mode approaches [18]. These results highlight the need to test spacecraft controllers under realistic actuator constraints.Other approaches have developed methods to improve robustness to uncertainty and disturbances. For flexible spacecraft attitude manoeuvring under external disturbances, inertia uncertainty, and input constraints, Bai et al. [19] proposed robust nonlinear (H_\infty) output-feedback control.

Moreover, large-angle manoeuvres with guaranteed-cost control based on sum-of-squares optimisation and linear parameter-varying representations have been considered [20]. Further studies have been carried out on rigid-body attitude control, adaptive manoeuvring and robust control of flexible spacecraft [21–23]. Further, nonlinear attitude recovery and stability were studied by Tafazoli and Khorasani [24]. Kothari et al. [25] developed a unified anti-windup framework to address the effects of actuator constraints in feedback systems. Although these methods offer valuable advantages, their implementation may require greater computational resources or more sophisticated controller-design procedures.Despite these advances, nonlinear dynamics, incomplete state measurements, and actuator saturation still significantly affect spacecraft attitude tracking. More specifically, observer estimation errors can affect the performance of a controller designed using the full state vector, and actuator saturation can cause the actual control input to differ from the desired control command.

Hence, this paper examines a nonlinear spacecraft attitude-control scheme based on feedback linearisation and LQR with a reduced-order Luenberger observer. The simulation model considers actuator torque saturation. The proposed approach is validated using TURKSAT-1B spacecraft parameters. The main contributions are: 1) the combination of feedback linearisation with LQR for large-angle attitude tracking, 2) the introduction of reduced-order state estimation under given measurement and observability conditions; and, 3) the evaluation of tracking performance, control effort, observer behaviour, and actuator saturation by numerical simulations.

This section briefly reviews the kinematic and dynamic equations of motion for a three-axis-stabilized spacecraft.

The attitude dynamic equations are obtained from Euler's moment equations as follows:

τ = ḣ I = ḣ B + ω × h. (1)

In this equation, the subscript "I" indicates a derivative in the inertial frame, while the subscript "B" indicates a derivative in the rotating body frame. Hence, the attitude dynamics in principal axes can be written as:

Ixω̇ x + (Iz − Iy)ωyωz = τx, Iyω̇ y + (Ix − Iz)ωzωx = τy, (2) Izω̇ z + (Iy − Ix)ωxωy = τz,

Where U = [u1 u2 u3]T = [τx τy τz]Tare the control torques acting on the satellite IxIy, and Iz are the principal moments of inertia.

The Euler parameter θ used for attitude representation of the body-fixed frame with respect to an inertial frame is given by:

θ̇ = Rω, 1 0 − sinθ2 R−1 = [0 cos θ1 sin θ1 cosθ2 ]. (4) 0 − sinθ1 cos θ1 cosθ2

For small Euler angles, the angular velocities of the body frame are described by the following equation:

ωx ≈ ϕ̇ − ω0ψ, ωy ≈ θ̇ − ω0, (5) ωz ≈ ψ̇ + ω0ϕ.

Therefore, the nonlinear equations of motion for spacecraft can be written as:

ϕ̈ = f1(X) + g1(X)u1, {θ̈ = f2(X) + g2(X)u2, (6) ψ̈ = f3(X) + g3(X)u3,

where:

(Iz − Iy)ωyωz f1(X) = ω0ψ̇ − , Ix (Ix − Iz)ωzωx f2(X) = − , (7) Iy (Iy − Ix)ωxωy f3(X) = −ω0ϕ̇ − .

{ Iz

And

g1(X) = , Ix

g2(X) = , (8) Iy

g3(X) = , { Iz

X = [ϕ θ ψ ϕ̇ θ̇ ψ̇ ]T. (9)

The feedback linearization technique is based on a linearized input-output relationship. It consists of two controller loops: An inner loop (nonlinear controller) to eliminate nonlinear terms in the model dynamics and an outer loop (linear controller) to control the linearized model (see Fig. 1).

By considering the spacecraft dynamic equations Eqs. (6)-(8), also by assuming the output Y such as:

y1 ϕ y = [y2] = [θ] . (10) y3 ψ

The successive differentiation process is done on the output (attitude angle) until the control signal appears:

ÿ = [θ̈ ] = [f(X)] + [g(X)] [u]. (11) ÿi = νi, i = 1,2,3. (12)

The control law can be obtained from the following equation:

[ui] = [νi − fi][gi]−1, i = 1,2,3. (13)

Considering saturation, we choose a larger control-weight matrix, as shown in Fig. 2 [1].

To assess the performance of the proposed nonlinear feedback linearisation controller combined with the LQR method and reduced-order observer, numerical simulations are conducted. Gyroscopes measure the spacecraft's angular velocities, and the observer estimates the Euler angles. The reference spacecraft is chosen as the satellite TURKSAT-1B [8], and the moment of inertia matrix is given by

I = diag( 3770,730,4020) kg, m2.

The initial attitude and angular velocities are specified according to the simulation conditions, while the desired Euler angles are set to

ϕin 3 [θin ] = [4] Deg.

ψin 5 ωix 0 [ωiy] = [0] Rad/sec.

ωiz 0

ϕf 10 [θf ] = [20] Deg.

ψf 30

Fig. 4 compares the actual and estimated Euler angles with their respective reference values. All attitude components move toward their desired values, indicating good tracking performance during the large-angle manoeuvre. The estimated roll and yaw angles show moderate transient overshoots, whereas the pitch angle responds faster with relatively small estimation error. The differences between the real and estimated attitudes decrease gradually, showing convergence of the observer estimates.

Fig. 5 shows the angular velocity responses for the spacecraft's three axes. The angular velocities decrease to zero with a damped transient behaviour after an initial increase during the manoeuvre. This demonstrates the proposed controller's effectiveness in attenuating rotational motion as the spacecraft approaches the target

The control torque components with real states and observer-estimated states are shown in Fig. 6. The biggest transient torque is along the third axis, and it is about 17 N·m. In contrast, the other components are smaller. The differences between the two control strategies are significant in the first transient, but the torque responses tend to zero as the system is stabilised.

The simulation results verify the effectiveness of the proposed control strategy for attitude tracking and observer-based state estimation. Combining feedback linearization with the LQR approach provides satisfactory transient performance, and the reduced-order observer reconstructs the unmeasured attitude states from the available measurements. The satisfaction of actuator saturation constraints should be checked in accordance with the specified torque limits.

This paper investigates a nonlinear spacecraft attitude control strategy based on feedback linearisation, LQR, and a reduced-order Luenberger observer. We performed numerical simulations using the dynamic parameters of the TURKSAT-1B spacecraft to test the proposed framework. According to the results obtained, we can make the following conclusions:

I. The combined feedback linearisation and LQR controller achieved satisfactory attitude tracking during the simulated large-angle manoeuvre, with all three Euler angles tending toward their stated reference values. II. The reduced-order observer could reconstruct the unmeasured attitude states from the available angular velocity measurements. The differences between the actual and estimated attitudes decreased over time, showing convergence of the estimated states.

III. The angular velocities showed damped transient responses and gradually approached zero, indicating that the rotational motion was effectively suppressed when the spacecraft reached the desired orientation.

IV. Following the initial transient, the control torques reduced to zero, with the maximum reported torque being approximately 17 N·m about the third axis. The differences between the control inputs calculated from the actual and estimated states decreased as the system settled down.

The overall results demonstrate the promise of integrating nonlinear feedback linearisation, optimal control, and reduced-order state estimation for spacecraft attitude tracking under partial state measurements. However, further investigations are needed to quantify performance under actuator saturation, measurement noise, external disturbances, and model uncertainties, and to establish robustness under more realistic operating conditions.

The authors declare that they have no competing interests.

The data supporting the findings of this study are available within the article.

This research did not receive any external funding.