## 1 Introduction  
Aircraft spin, as a highly nonlinear and dangerous flight condition, is characterized by a supercritical angle of attack, a large sideslip angle, and a significant yaw rate. Recovering from aircraft spin to avoid ground collisions is a challenging task that has attracted considerable attention [1; 2; 3; 4]. The spin characteristics of an aircraft can be researched through rotary balance tests, vertical spin tunnels, bifurcation analysis, radio-controlled flight tests, and real aircraft flight tests. Recovering from aircraft spin is difficult and depends on its spin characteristics, including entry speed, intensity, and recovery time. The combination of a steep descent, delayed recognition of the spin mode, and inadequate recovery actions creates a situation where it becomes almost improbable for the pilot to achieve a successful recovery before colliding with the ground [5; 6]. Hence, research on control technology for automatic recovery from aircraft spin is of utmost importance. The first critical issue in recovering from an aircraft spin is predicting and identifying the spin state. Bifurcation analysis techniques have been widely used to investigate the spin solutions of the open-loop aircraft dynamics model. The open-loop bifurcation results have been used to provide the reference inputs for the spin recovery controller [7; 8; 9]. Therefore, to address the spin recovery control problem within a reasonable time, several control techniques have been investigated. Ref. [5] introduced a spin automatic recovery flight control system, minimizing altitude loss. The study highlighted the importance of initiating recovery within the first turn and quantified the relationship between altitude loss and recovery delay. In Ref. [10], a reinforcement learning intelligent control approach has been applied for fully developed spin recovery that demonstrated time optimality. In Ref. [11], spin recovery was formulated as a trajectory optimization problem, with time and altitude loss serving as cost functions to be minimized and addressed through the utilization of the direct multiple shooting method. Ref. [12] delved into the application of nonlinear dynamic inversion (NDI) techniques for resolving the spin recovery problem. However, controllers based on NDI have been criticized in [13] for their lack of robustness and inability to handle control saturation limits. Sliding mode control (SMC) has attracted significant interest due to its simplicity, efficiency, remarkable adaptability to external disturbances, and insensitivity to fluctuations in system parameters [14]. Therefore, in Ref. [15], linear sliding mode (LSM) control was used to spin recovery, which was achieved at low trim state without thrust vectoring. Refs. [16; 17] improved SMC controller design by incorporating the TSM-type reaching law, enabling direct spin recovery through aerodynamic forces, and thrust utilization. Moreover, through repetitive altitude simulations, the altitude loss allowance to prevent ground collisions was achieved [18]. In Ref. [19], an investigation was conducted to examine the influence of open-closed loop thrust control on spin recovery, employing SMC. The findings of the study revealed that the implementation of a closed-loop thrust controller effectively mitigated altitude loss during the spin recovery process, showcasing its efficacy in enhancing control precision and system performance. Furthermore, in Ref. [20], the thrust vectoring control command was strategically designed as a function of the angle of attack in both pitch and yaw directions, and its optimal integration with the aircraft's primary control system was explored. Ref. [17] investigated the impact of fin positioning on SMC spin recovery, finding that optimal fin placement notably reduced altitude loss. In addition, Ref. [21] discusses the problem of spin recovery in aircraft and proposes a robust SMC control solution to address the issue of improper feedback data.