## 1 Introduction  
Developments of the rechargeable battery source, brushless DC motor, and various related techniques lead to increased research unmanned aerial vehicles (UAVs). The multi-rotor UAVs exhibit significant advantages in terms of hovering capability and maneuverability involving side and backward movement, vertical take-off and landing in 3D space, and lower maintenance cost than other small aerial vehicles [1, 2, 3, 4]. Thus, UAVs were initially used for military applications. However, recently, its applications were extended to agriculture, aerial photography, inspection, surveillance, rescue, transport monitoring, and exploration of inaccessible areas. Most multi-rotor UAVs adopted a four-rotor mounted quadrotor structure, and they exhibit limited power for heavy cargo delivery. Currently, UAVs mounted with more rotors, such as hexacopter and octocopter UAVs, when compared to four rotors, were developed to enforce lifting and higher capabilities in terms of flying time [5, 6, 7, 8, 9, 10]. When considering the efficiency of UAVs, the hexacopter is more feasible than the octocopter because an increase in the number of rotors leads to an increase in size and cost. However, safe and reliable flight of multi-rotor UAVs should be secured. This is because severe damages can result if the UAV falls or crashes, and the increased speed rotation of a multi-rotor can lead to more severe accidents. Thus, to guarantee safety and reliability in the flight of multi-rotor UAVs, an effective and robust controller is required to overcome the following challenges: (1) under-actuated control scheme for six degrees of freedom (DOF) with four control inputs; (2) strongly coupled nonlinear dynamics between translational and rotational subsystems and kinematic coupling terms; and (3) uncertainties including external disturbances, such as wind and rain, in outdoor environment and unknown dynamics due to varying load. Several control methods have been utilized to handle the above issues. Linear control schemes, including proportional-integral-derivative (PID) [7,9,11], linear quadratic regulator (LQR) [12,13], predictive control [14], and \(H_{\infty}\) control [15] have been initially applied for the fundamental flight task of the multi-rotor UAVs. However, these methods undergo their application limits due to performance degradation when the UAV leaves away its designed trimming points or accepts excessive perturbations. To handle these limits, recursive backstepping control (BSC) [6,16,17] is generally utilized as a baseline controller because BSC is appropriate for the aforementioned coupled higher order nonlinear systems and especially for multi-rotor UAVs [18, 19, 20, 21, 22, 23]. The procedures for stabilizing system states are recursively repeated in the BSC design by utilizing virtual controllers, and the final controller is obtained in the last recursive step. Thus, a controller for any nonlinear system can be systematically designed without utilizing any linearization, which neglects useful nonlinearities. Conversely, despite the popularity and advantage of the BSC scheme, there exist several disadvantages in the traditional BSC scheme. First, the virtual stabilizing controls contain the repeated differentiation terms with respect to virtual controls designed during the pre-recursive steps for high-order nonlinear systems. Hence, this leads to an increase in complexity. The tracking differentiator methods [24, 25, 26, 27, 28, 29, 30] were recently investigated as a potential solution. Furthermore, the second-order finite-time command filter [31, 32, 33] was considered to obtain improved properties to guarantee finite-time convergence and avoid increases in complexity, given that the traditional BSC schemes were designed in the point of infinite-time convergence. However, the traditional tracking differentiators exhibit the disadvantage of slow convergence time, though the recently developed tracking differentiators in [27, 28, 29] show more improved performance than those of the previous ones.