Cascaded ADRC Framework for Robust Control of Coaxial UAVs with Uncertainties and Disturbances

Highlights

What are the main findings?

• We proposed a cascade control scheme, which is composed of a classical ADRC and a geometric ESO-based controller, dealing with disturbance and couplings in both outer and inner loops of coaxial UAV dynamics.
• We validated the effectiveness and performance of this ADRC-C controller with numerical simulations.

What are the implications of the main findings?

• This study offers an efficient and robust control scheme for coaxial UAVs, which can effectively handle external disturbances and internal uncertainties across different time scales and dynamic levels.
• This study presents the design of a modular coaxial UAV which offers distinct advantages for industrial applications.

Abstract

Coaxial contra-rotor unmanned aerial vehicles (UAVs) are attractive for their compact structure and aerodynamic efficiency, making them suitable for long-endurance and heavy-payload operations. However, the coaxial configuration introduces strong rotor coupling, phase lag, and additional disturbances, which pose significant challenges for stable control. To overcome these issues, we propose a cascaded Active Disturbance Rejection Control (ADRC-C) framework, in which a two-level control structure is adopted. The outer loop employs a classical ADRC controller to estimate and compensate for the lumped external forces, providing the compensated attitude command to the inner loop. The inner loop, in turn, adopts an $SO\left(3\right)$-based Extended State Observer (ESO) to handle high-frequency torque disturbances through real-time estimation and compensation. The proposed approach is validated through numerical simulations. Results confirm that the cascaded ADRC consistently outperforms conventional PID control in tracking accuracy, transient response, and disturbance rejection, demonstrating strong robustness for demanding coaxial UAV missions.

1. Introduction

Unmanned aerial vehicles (UAVs) are being extensively employed in surveillance, inspection, and logistics, largely attributable to their flexibility, ease of deployment, and cost-effectiveness [1,2]. Among various UAV architectures, coaxial contra-rotor designs are particularly notable for enhanced thrust efficiency and compact structural layouts, making them particularly suitable for long-duration and payload-demanding tasks [3,4]. Nevertheless, compared to conventional quadrotor configurations, this design also introduces significant challenges in flight dynamics and control. One major issue lies in the incomplete cancellation of gyroscopic precession between the two contra-rotors [5]. The residual precession would induce uncertain phase lag in the angular velocity response and complicate attitude stabilization, particularly during aggressive maneuvers. Another typical problem is the aerodynamic coupling between the rotors. The induced flows from the upper and lower rotors interact in a highly nonlinear and time-varying manner, producing aerodynamic forces and moments which are hardly predictable with good precision [6,7,8]. Moreover, other issues, including structural vibration coupling, aeroelastic blade deformation, and vortex ring effects, also contribute to modeling uncertainty and degrade the performance of classical cascaded PID controllers, as well as model predictive control (MPC) schemes, in coaxial UAV applications [9].

To address these challenges and enhance the robustness of coaxial UAV control systems, a variety of advanced control strategies have been investigated. While these methods indeed provide improved performance under high-disturbance conditions, they also exhibit distinct limitations, such as a strong dependence on accurate system models, high computational requirements, and implementation issues like control chattering [10,11]. In comparison, Active Disturbance Rejection Control (ADRC) presents a promising alternative, which offers a favorable trade-off among computational efficiency, model dependency, and control performance [12,13,14]. As the core component of ADRC, the Extended State Observer (ESO) simultaneously estimates both external disturbances and internal model mismatches by treating them as a generalized disturbance, thereby enabling real-time compensation without the need for an accurate system model [15]. However, most existing studies on ADRC-based control of coaxial UAVs adopt a single-loop configuration, which is insufficient to handle the multi-timescale dynamics inherent to such systems. In particular, the fast inner-loop attitude dynamics is primarily influenced by actuator delays and rotor coupling, whereas the slower outer-loop position dynamics is largely affected by external disturbances [16].

In order to better accommodate the dynamic characteristics of coaxial UAVs, we propose a cascaded ADRC framework. By embedding ESO-based disturbance estimation into both the inner and outer control loops, the proposed framework establishes a two-layer active compensation structure that simultaneously handles fast dynamic disturbances induced by actuator coupling and phase lag, as well as slow dynamic disturbances caused by wind and turbulence. This cascaded design not only preserves the modularity of conventional cascaded control but also enhances disturbance rejection and simplifies controller tuning compared with single-loop ADRC or PID-based schemes. In addition, a new coaxial UAV platform has been specifically designed for long-term industrial applications and serves as an experimental testbed to validate the performance of the proposed cascaded ADRC framework under realistic operating conditions.

The main contributions of this paper are summarized as follows:

• Coaxial UAV structural design: We design a new coaxial UAV platform which emphasizes structural compactness, modularization, endurance potential, and feasibility for practical industrial deployment, which is shown in Figure 1.
• Cascaded ADRC framework: We develop a cascaded ADRC scheme that integrates ESO-based disturbance estimation into both inner- and outer-loop controllers. The proposed method explicitly addresses phase delay, rotor coupling, and external disturbances, while offering improved robustness and tuning simplicity compared with single-loop ADRC or PID control.
• Validation of performance: We have validated the proposed framework with both numerical and high-fidelity HIL simulations, in which the results demonstrate superior disturbance rejection, stable trajectory tracking, and enhanced robustness compared with conventional PID controllers.

2. Related Works

Research on disturbance-resistant control of unmanned aerial vehicles (UAVs) has long been an important topic in the field of flight control. A variety of control strategies have been developed to improve the reliability and robustness of flight control systems in the presence of significant disturbances and modeling uncertainties. Representative approaches such as SMC [17], ADRC [18], AMPC [19], and incremental nonlinear dynamic inversion (INDI) [20] have been extensively investigated and successfully applied to fixed-wing and multirotor UAVs. However, despite these advances, their adaptation to coaxial contra-rotor configurations remains relatively unexplored due to the complex aerodynamic coupling and multi-time-scale dynamics, which are intrinsic to such systems [21].

Compared to traditional quadrotor configurations, coaxial contra-rotor UAVs present significantly more complex dynamic characteristics due to strong aerodynamic interference and mechanical coupling between the rotors [22]. The mutual induced flows between the upper and lower rotors result in nonlinear aerodynamic interactions and cross-axis torque effects, which contribute to residual gyroscopic precession, phase lag, and multi-axis dynamic coupling [23,24,25]. These phenomena pose substantial challenges to control linearization and system decoupling, especially under aggressive maneuvers [26]. In addition, structural vibrations and sensor noise commonly observed in practical implementations can further impair attitude estimation accuracy and reduce the effective control bandwidth of coaxial UAVs [27,28].

In response to these challenges, researchers have been extensively investigating different control schemes. Some early efforts primarily focused on fuzzy sliding-mode decoupling control to enhance longitudinal stability and to mitigate rotor coupling effects under strong aerodynamic interference [29]. Subsequently, ADRC was introduced to this field, which is able to efficiently enhance system robustness and disturbance-rejection capability [30]. MPC and model-tracking inversion approaches were also employed for flight control of coaxial UAVs, which improve attitude control accuracy but require extensive computational resources and precise aerodynamic modeling [26,31]. More recently, a hybrid control framework combining ADRC and backstepping was proposed, which is specifically designed to improve dynamic adaptability and disturbance-rejection performance in high-mass-ratio coaxial systems with large inertial asymmetry [32]. However, despite the progress achieved in the aforementioned studies, the coexistence of aerodynamic coupling, phase lag, and multi-scale dynamics remains a major obstacle to realizing robust real-time control in coaxial UAV systems [33,34]. This motivates the development of our cascaded ADRC framework, which provides hierarchical disturbance rejection across multiple dynamic loops and achieves robust yet cost-efficient control performance.

3. Configuration and Modeling of the System

3.1. System Configuration

Although coaxial UAVs share the common architecture of two counter-rotating rotors coaxially aligned, their specific structural implementations differ distinctively in terms of trade-offs between mechanical simplicity, aerodynamic performance, and system integration [35]. Since our primary objective in this study is to develop a UAV suited for deployment in practical industrial scenarios, a simple, compact, and modular configuration is eventually adopted.

The overall structure consists of two mechanically decoupled modules: a cabin module which contains the avionics system and a propulsion module which integrates all the actuators. These two modules are connected through a custom standardized electrical interface as shown in Figure 2, which enables unified power and signal transmission and allows the propulsion unit to be easily replaced as a whole to accommodate different payload requirements. Figure 3 illustrates a compact and practical propulsion scheme adopted in this work, where the propulsion assembly is built upon a fixed central shaft. Each motor is directly connected to its corresponding propeller base, thereby eliminating the need for complex gear mechanisms. Notably, only the lower rotor is equipped with a cyclic pitch control mechanism implemented via a two-degree-of-freedom (DoF) swashplate manipulated by two servos, which enables cyclic pitch modulation of the blades. Both rotors have employed rigid hubs without flapping hinges, which reduces structural complexity and further lowers manufacturing cost.

This design choice offers several noticeable advantages for practical applications. First, the adoption of a swappable propulsion module significantly enhances system maintainability, allowing for rapid replacement or upgrading of the propulsion unit without requiring extensive disassembly. Second, the propulsion module features a structurally simple yet highly integrated configuration, which effectively minimizes mechanical complexity and reduces the manufacturing and tuning costs of the entire system. Third, the minimized number of movable components further enhances the overall reliability of the system, which not only extends its maintenance interval, but also contributes to a reduction in the total structural weight.

Despite its engineering merits, the proposed structural design inevitably involves trade-offs between mechanical simplicity and aerodynamic performance. As previously discussed, coaxial rotor UAVs inherently pose substantial challenges in dynamic modeling and control due to strong aerodynamic coupling, phase lag, and cross-axis interactions. By adopting direct-drive motors, rigid rotor hubs, and a single swashplate configuration, the proposed design intentionally eliminates mechanical complexity while concurrently accentuates control challenges. Specifically, the rigid rotor structure restricts the natural damping of aerodynamic disturbances, while the single-swashplate mechanism limits the independent control authority of the upper rotor. Therefore, the system exhibits stronger dynamic coupling and reduced capability for fine-tuned attitude correction. These limitations become particularly evident under strong wind disturbances or during aggressive attitude maneuvers, as conventional control schemes often fail to suppress the coupling dynamics effectively, resulting in the highly dangerous oscillatory behavior. This highlights the necessity for an advanced control framework capable of achieving multi-timescale disturbance rejection and dynamic decoupling, as addressed by the proposed cascaded ADRC strategy.

3.2. Simplified Dynamic Model

Accurate modeling of a coaxial UAV is intrinsically challenging due to several indeterminable factors, including mutual aerodynamic interference between the upper and lower rotors, unsteady induced inflow, blade flexibility, etc. However, since the proposed control approach is capable of handling both external disturbances and model uncertainties, a certain degree of simplification can be reasonably introduced in the modeling stage.

Although the proposed coaxial UAV design employs a helicopter-like cyclic pitch control mechanism for the lower rotor, its rigid hub design prevents the rotor-disk plane from tilting with respect to the fuselage. In other words, it can reasonably be assumed that the total lift generated by the rotor blades always acts along the Z-axis of the body frame. Therefore, given the definitions of constant variables as shown in

Table 1. Definitions of constant variables.

Variables	Definitions
m	Mass of the vehicle
g	Gravitational acceleration
$\mathit{J}=\mathrm{diag}(J_x,J_y,J_z)$	Inertia matrix of the vehicle (principal moments)
${\mathit{e}}_{\mathbf{3}}={\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T$	Downward unit vector in world frame
${\kappa }_{Tu},{\kappa }_{Tl}$	Lift coefficient for upper and lower rotors
$k_{\Delta }$	Reaction torque coefficient of both rotors (on average)
$k_x,k_y$	Torque coefficient for cyclic modulation
$C_v=\mathrm{diag}\left(C_{dv}\right)$	Translational damping matrix
$C_{\omega }=\mathrm{diag}\left(C_{dw}\right)$	Rotational damping matrix

, the simplified model of this coaxial UAV is nearly identical to that of a classical quadrotor at the kinematic level, which can be expressed as

$$\begin{array}{cc}\hfill \dot{\mathit{p}}& =\mathit{v},\hfill \\ \hfill m\dot{\mathit{v}}& =mg{\mathit{e}}_{\mathbf{3}}-\mathit{T}R{\mathit{e}}_{\mathbf{3}}+{\mathit{f}}_d,\hfill \\ \hfill \mathit{T}& =\left[\begin{array}{ccc}0& 0& T_u+T_l\end{array}\right],\mathit{p},\mathit{v}\in {\mathbb{R}}^3,R\in SO\left(3\right),\hfill \end{array}$$

(1)

where $\mathit{p}$ and $\mathit{v}$ denote the position and velocity of the UAV in world frame $\left\{\mathcal{W}\right\}$, respectively. $\mathit{T}$ denotes the total lift force in theory, which consists of lift of the upper rotor $T_u$ and lift of the lower rotor $T_l$. The rotation matrix R represents the rotation matrix describing the orientation of the body frame $\left\{\mathcal{B}\right\}$ relative to $\left\{\mathcal{W}\right\}$. It should be noted that all external forces affecting the translational motion of the vehicle—including wind disturbances, aerodynamic drag and payload-induced perturbations are collectively represented by a single lumped external disturbance force ${\mathit{f}}_{\mathit{d}}$ expressed in $\left\{\mathcal{W}\right\}$.

Similarly, in the rotational dynamics, all torque disturbances acting on the vehicle, which are possibly caused by aerodynamic asymmetries, gyroscopic effects, or unmodeled couplings, can be collectively represented by one equivalent disturbance torque ${\mathit{\tau }}_d$ in $\left\{\mathcal{B}\right\}$. Thus, the rotational motion of the UAV can be approximated by the following dynamic model on $SO\left(3\right)$:

$$\begin{array}{cc}\hfill \dot{R}& =R{\left[\mathrm{\Omega }\right]}_{\times },\hfill \\ \hfill \mathit{J}\dot{\mathbf{\Omega }}& ={\mathit{\tau }}_c-\mathbf{\Omega }\times \left(\mathit{J}\mathbf{\Omega }\right)+{\mathit{\tau }}_d,\hfill \end{array}$$

(2)

where $\mathit{a}\times \mathit{b}={\left[\mathit{a}\right]}_{\times }\mathit{b}$. $\mathbf{\Omega }={\left[\begin{array}{ccc}{\omega }_p& {\omega }_q& {\omega }_r\end{array}\right]}^T\in {\mathbb{R}}^3$ is the angular velocity expressed in $\left\{\mathcal{B}\right\}$, and ${\left[\mathbf{\Omega }\right]}_{\times }$ is the corresponding skew-symmetric matrix, defined as

$${\left[\mathbf{\Omega }\right]}_{\times }=\left[\begin{array}{ccc}0& -{\mathrm{\Omega }}_z& {\mathrm{\Omega }}_y\\ {\mathrm{\Omega }}_z& 0& -{\mathrm{\Omega }}_x\\ -{\mathrm{\Omega }}_y& {\mathrm{\Omega }}_x& 0\end{array}\right].$$

The control torque ${\mathit{\tau }}_c$ represents the theoretical averaged torque generated by the cyclic pitch modulation of the lower rotor blades and the differential reaction torques between the upper and lower rotors. Any unexpected variations or dynamic delays in this control torque are also incorporated into the disturbance term ${\mathit{\tau }}_d$.

By combining the outer-loop translational model (1) and the inner-loop rotational model (2), a highly simplified yet physically consistent representation of the system can be obtained. However, since $T_u$, $T_l$ and ${\mathit{\tau }}_c$ are not directly actuated variables, additional mapping from the actual inputs to these variables is required. To enable direct control allocation, the actuation model is expressed in terms of the rotational speeds of upper and lower rotor $n_u,n_l$ (in rpm) and the swashplate cyclic pitch commands ${\delta }_{\mathrm{long}},{\delta }_{\mathrm{lat}}$, which correspond to longitudinal and lateral cyclic, respectively.

With the assistance of the disturbance term, some variables in the lift model can be treated as constants, while the resulting model discrepancies are compensated by the observer. Therefore, the lift model can be highly simplified as follows:

$$T_u={\kappa }_{Tu}{n}_u^2,T_l={\kappa }_{Tl}{n}_l^2,$$

(3)

where ${\kappa }_{Tu}$ and ${\kappa }_{Tl}$ (N/${\mathrm{rpm}}^2$) are the thrust coefficients for the upper and lower rotors, respectively.

As for the control torques, they are determined by the swashplate deflection angles and the rotational speed difference between the upper and lower rotors, which can be denoted as:

$${\mathit{\tau }}_c=\left[\begin{array}{c}{\tau }_x\\ {\tau }_y\\ {\tau }_z\end{array}\right]=\left[\begin{array}{ccc}{k}_x& 0& 0\\ 0& k_y& 0\\ 0& 0& k_{\Delta }\end{array}\right]\left[\begin{array}{c}{\delta }_{\mathrm{long}}\\ {\delta }_{\mathrm{lat}}\\ n_u^2-n_l^2\end{array}\right],$$

(4)

where $k_x,k_y$ (N · m/rad) are the roll and pitch cyclic-to-torque gains and $k_{\Delta }$ (N · m/${\mathrm{rpm}}^2$) is the yaw torque gain, all of which shall be determined by experiments.

3.3. High-Fidelity Dynamic Model

To achieve high-fidelity simulation of a coaxial UAV, we model the propulsion subsystem by explicitly coupling the motor dynamics, the rotor aerodynamics computed from blade element momentum theory (BEMT), and the cyclic pitch control mechanism, which governs the mean thrust and torque as well as the azimuth-dependent aerodynamic variations induced by cyclic pitch actuation.

The brushless direct current (BLDC) motor applied in our design is driven by a closed-loop field-oriented control (FOC)-based electronic speed controller (ESC), and can therefore be reasonably approximated as a first-order system. The rotational dynamics of the rotor assembly can be described by

$$\dot{\omega }={\displaystyle \frac{1}{{\tau }_m}}\left({\omega }_{\mathrm{cmd}}-\omega \right),$$

(5)

where $\omega$ is the actual rotor angular speed and ${\tau }_m>0$ is the time constant. The target angular velocity of the motor ${\omega }_{cmd}$ can be mapped from the normalized input thrust.

Consider a rotor of radius R and disk area $A=\pi R^2$ with B blades. The blade is discretized along the radius $r\in [r_0,R]$, where $r_0$ is the hub cutout, and aerodynamic loads are computed element-wise. At each blade element $(r,\psi )$, where r denotes the radial position and $\psi$ denotes the azimuth angle, the relative velocity magnitude is

$$W(r,\psi )=\sqrt{U_T^2(r,\psi )+U_A^2(r,\psi )},$$

(6)

where $U_T$ is the tangential component and $U_A$ is the axial component. A common kinematic decomposition is

$$U_T(r,\psi )=\omega r+u_{\theta }(r,\psi ),U_A(r,\psi )=V_z+v_i(r,\psi ),$$

(7)

with body velocity $V_z$, induced velocity $v_i$, and tangential induced component $u_{\theta }$. The inflow angle $\varphi$ is

$$\varphi (r,\psi )=arctan\left({\displaystyle \frac{U_A(r,\psi )}{U_T(r,\psi )}}\right).$$

(8)

Let $c\left(r\right)$ denote the local chord length. The sectional lift and drag magnitudes for each blade element are

$$dL={\displaystyle \frac{1}{2}}\rho W^{2}c\left(r\right)C_L\left(\alpha \right)dr,dD={\displaystyle \frac{1}{2}}\rho W^{2}c\left(r\right)C_D\left(\alpha \right)dr,$$

(9)

where $\rho$ is air density and $C_L\left(\alpha \right)$, $C_D\left(\alpha \right)$ are airfoil coefficients with respect to angle of attack $\alpha$. $dL$ and $dD$ are then projected to the axis and rotor-disk plane, which account for the elemental thrust $dT$ and elemental torque $dQ$:

$$\begin{array}{cc}\hfill dT(r,\psi )& =B\left(dLcos\varphi -dDsin\varphi \right),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill dQ(r,\psi )& =Br\left(dLsin\varphi +dDcos\varphi \right).\hfill \end{array}$$

(11)

The total thrust and aerodynamic torque are obtained by integration over radius and azimuth:

$$T={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{\int }_{r_0}^{R}dT(r,\psi )drd\psi ,Q={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }{\int }_{r_0}^{R}dQ(r,\psi )drd\psi .$$

(12)

The reaction torque exerted on the airframe is opposite to the rotor spin direction and has a magnitude equal to $\left|Q\right|$.

Consider the cyclic pitch control of the blades. Let $\theta (r,\psi )$ denote the geometric pitch for the blade element at $(r,\psi )$, the local angle of attack is

$$\alpha (r,\psi )=\theta (r,\psi )-\varphi (r,\psi ),$$

(13)

where $\varphi (r,\psi )$ is given by (8). For a variable-pitch rotor, by means of substituting (13) into (9), $\theta (r,\psi )$ is decomposed as

$$\theta (r,\psi )={\theta }_0\left(r\right)+{\theta }_{1c}\left(r\right)cos(\psi +\phi )+{\theta }_{1s}\left(r\right)sin(\psi +\phi ),$$

(14)

where ${\theta }_0$ is the collective pitch distribution, ${\theta }_{1c}$ and ${\theta }_{1s}$ represent the longitudinal and lateral cyclic components, respectively, and $\phi$ is an effective phase offset that accounts for rotor phase lag, which is introduced by flapping hinge and aeroelastic effects of the blades.

In simulation, ${\theta }_{1c}$ and ${\theta }_{1s}$ are computed from the swashplate tilt commands, as

$${\theta }_{1c}=k_{\theta }{\delta }_{\mathrm{long}},{\theta }_{1s}=k_{\theta }{\delta }_{\mathrm{lat}},$$

(15)

where $k_{\theta }$ is a gain factor relating swashplate tilt and blade pitch. In our configuration, $k_{\theta }\approx 1$. By substituting the variations in $\alpha (r,\psi )$ into Equations (9)–(12), the control torque $\mathit{\tau }$ can be computed based on the output of the actuators, and this mapping relationship is significantly more accurate and realistic than the simplified model given in Equation (4).

4. Cascaded ADRC Strategy for Robust Flight

4.1. Overall Framework

Conventional coaxial UAVs typically employ cascaded PID control, following the same hierarchical architecture widely used in quadrotor flight controllers [36]. Although this approach is simple and effective under nominal conditions, it becomes inadequate when confronted with the aerodynamic coupling, actuator delay, and disturbance sensitivity inherent to coaxial platforms. These effects often lead to sluggish transient responses, amplified oscillations, and unstable flight performance in practical or even unfavorable flight scenarios.

To address these challenges, ADRC is adopted as an effective solution. However, unlike conventional ADRC schemes that estimate the total disturbance as a single lumped term, the proposed approach retains the cascaded control structure and separates external disturbances into force and torque components, which are handled separately as shown in Figure 4. This distinction aligns with the inherent characteristics of the coaxial UAV dynamics, where the outer loop exhibits relatively slow translational responses, while the inner loop deals with fast and noise-sensitive rotational dynamics.

Compared with cascaded PID control and standard ADRC, the proposed design offers several advantages. First, it provides faster recovery from unexpected perturbations, as the inner loop reacts instantly to attitude errors while the outer loop gradually compensates for position deviations. Second, by decoupling disturbance handling tasks, it prevents the accumulation of tracking errors across loops, which is a common issue in coaxial systems. Third, the two-level observer structure reduces the complexity of parameter tuning and facilitates agile deployment of the control system. These improvements enhance both control consistency and engineering practicality, making the strategy particularly suitable for compact coaxial UAVs.

4.2. Principles for Outer Loop Control

In coaxial UAVs, the reference commands (e.g., thrust or attitude angles) often contain high-frequency components due to sensor quantization or operator inputs. A tracking differentiator (TD) is adopted to generate a smooth trajectory and its derivative, thereby attenuating high-frequency disturbances while preserving fast transient response. The TD dynamics are given by

$$\dot{\tilde{\mathit{p}}}=\tilde{\mathit{v}},\dot{\tilde{\mathit{v}}}=\mathrm{\Phi }(\tilde{\mathit{p}}-{\mathit{p}}_{ref},\tilde{\mathit{v}}),$$

where ${\mathit{p}}_{ref}$ is the reference input, $\tilde{\mathit{p}},\tilde{\mathit{v}}\in {\mathbb{R}}^3$ are the TD states, and $\mathrm{\Phi }(·)$ is a nonlinear function that switches between a linear tracking mode and a saturated mode. In this work, $\mathrm{\Phi }(·)$ is defined as

$$\mathrm{\Phi }(x_1,x_2)=\left\{\begin{array}{cc}-{\displaystyle \frac{r_0}{d}}a,\hfill & \left|a\right|\le d,\hfill \\ -r_0\mathrm{sign}\left(a\right),\hfill & \left|a\right|>d,\hfill \end{array}\right.a=x_1+h_0{x}_2,d=r_0{h}_0^2,$$

(16)

and $r_0,h_0$ are design parameters controlling the trade-off between response speed and filtering. Through this mechanism, $\tilde{\mathit{p}}$ provides a noise-attenuated version of the reference position, while $\tilde{\mathit{v}}$ yields a stable derivative estimate for use in subsequent control loops.

Coaxial UAVs suffer from complex aerodynamic disturbances, including rotor–wake interference, torque asymmetry, and unmodeled ground effects. The ESO is employed to estimate both the system states and these lumped disturbances in real time. Let ${\mathit{p}}_m$ be the measured positional output, and $\widehat{\mathit{p}},\widehat{\mathit{v}},\widehat{\mathit{a}}\in {\mathbb{R}}^3$ are the ESO states. The ESO dynamics are given by

$$\begin{array}{cc}\hfill \dot{\widehat{\mathit{p}}}& =\widehat{\mathit{v}}-{\mathit{\beta }}_1{\mathit{e}}_1,\hfill \\ \hfill \dot{\widehat{\mathit{v}}}& =\widehat{\mathit{a}}+{\displaystyle \frac{1}{m}}\mathit{u}-{\mathit{\beta }}_2\mathrm{fal}({\mathit{e}}_1,{\mathit{\sigma }}_1),\hfill \\ \hfill \dot{\widehat{\mathit{a}}}& =-{\mathit{\beta }}_3\mathrm{fal}({\mathit{e}}_1,{\mathit{\sigma }}_2).\hfill \end{array}$$

(17)

where ${\mathit{e}}_1=\widehat{\mathit{p}}-{\mathit{p}}_m$ is the estimation error, m is the mass of the whole system, and $\mathit{u}$ is the control input. ${\mathit{\sigma }}_{\mathbf{1}}$ and ${\mathit{\sigma }}_{\mathbf{2}}$ represent observer gain factors in the nonlinear function $\mathrm{fal}(·)$, which is defined as

$$\begin{array}{c}\hfill \mathrm{fal}(e,\alpha )=\left\{\begin{array}{cc}{\left|e\right|}^{\alpha }\mathrm{sign}\left(e\right),\hfill & \left|e\right|>\delta ,\hfill \\ {\displaystyle \frac{e}{{\delta }^{1-\alpha }}},\hfill & \left|e\right|\le \delta ,\hfill \end{array}\right.\alpha \in (0,1].\end{array}$$

(18)

$\mathrm{fal}(·)$ implements a piecewise nonlinear error feedback, which adapts the observer gain depending on the error magnitude, improving robustness to both small-signal noise and large transient deviations. The parameter $\delta$ is a common threshold that determines the width of the linear region of $\mathrm{fal}(·)$. With proper tuning of the bandwidth $w_0$, the ESO provides accurate disturbance estimation for coaxial UAVs even under strong aerodynamic coupling.

Based on the TD reference $\tilde{\mathit{p}},\tilde{\mathit{v}}$ and ESO estimates $\widehat{\mathit{p}},\widehat{\mathit{v}},\widehat{\mathit{a}}$, the NLSEF constructs the control law as

$$\begin{array}{cc}\hfill \dot{{\mathit{u}}_0}& ={\mathit{k}}_1\mathrm{fal}(\tilde{\mathit{p}}-\widehat{\mathit{p}},{\mathit{d}}_3)+{\mathit{k}}_2\mathrm{fal}(\tilde{\mathit{v}}-\widehat{\mathit{v}},{\mathit{d}}_4),\hfill \\ \hfill \dot{\mathit{u}}& =m\left({\mathit{u}}_0-\widehat{\mathit{a}}\right).\hfill \end{array}$$

(19)

Here ${\mathit{k}}_1,{\mathit{k}}_2\in {\mathbb{R}}^3$ are feedback gains, ${\mathit{d}}_3,{\mathit{d}}_4\in {\mathbb{R}}^3$ are nonlinear exponents in $\mathrm{fal}(·)$, and $\widehat{\mathit{a}}$ serves as the online disturbance compensation. For the coaxial UAV, this design allows fast convergence of attitude dynamics while actively rejecting unmodeled rotor interactions and external disturbances, thereby enhancing stability and robustness in aggressive flight conditions.

4.3. Principles for Inner Loop Control

Although the inner-loop dynamics have been simplified for controller design, the system still experiences complex and unmodeled disturbances arising from aerodynamic coupling, structural vibration, and actuator imperfections. To focus on the rotational control, only disturbances as torques are explicitly considered in this layer. Accordingly, an ESO formulated on $SO\left(3\right)$ is developed to estimate and compensate for these disturbances in real time.

Let $\widehat{R}$ denote the estimated attitude, $\widehat{\mathrm{\Omega }}$ denote the estimated angular velocity, and ${\widehat{\mathit{\tau }}}_d$ denote the estimated disturbance torque obtained from the ESO. The attitude tracking error is defined using the right-invariant form on $SO\left(3\right)$ as

$$\begin{array}{c}\hfill {\mathit{e}}_R=\widehat{R}{R}^T\end{array}$$

(20)

or equivalently

$$\begin{array}{c}\hfill {\mathit{e}}_R^{\vee }={\displaystyle \frac{1}{2}}{({\mathit{e}}_R-{\mathit{e}}_R^T)}^{\vee }={\displaystyle \frac{1}{2}}{(\widehat{R}{R}^T-R{\widehat{R}}^T)}^{\vee }\end{array}$$

(21)

where ${\mathit{e}}_R\in SO\left(3\right)$ represents the right-invariant error between the estimated and actual attitude, while ${\mathit{e}}_R^{\vee }\in {\mathbb{R}}^3$ is its vector form. This right-invariant definition ensures that the error dynamics are independent of the inertial frame, providing a geometrically consistent formulation for observer and controller design.

Correspondingly, the estimation error of angular velocity is defined as follows under the right-invariant definition:

$$\begin{array}{c}\hfill {\mathit{e}}_{\mathbf{\Omega }}=\mathbf{\Omega }-\widehat{\mathbf{\Omega }},\end{array}$$

(22)

and error of estimated disturbance is

$$\begin{array}{c}\hfill {\mathit{e}}_{{\mathit{\tau }}_d}={\widehat{\mathit{\tau }}}_d-{\mathit{\tau }}_d.\end{array}$$

(23)

Note that since ${\mathit{\tau }}_d$ is introduced as an extended state, the corresponding estimation error in the ESO generally follows an opposite sign convention.

Considering the practical operating environment of the observer, the gyroscope is subject to strong interference caused by the cyclic pitch modulation and structural vibrations. To handle gyroscope noise while maintaining robust disturbance estimation, a weighted geometric ESO is designed as follows:

$$\begin{array}{cc}\hfill \dot{\widehat{R}}& =\widehat{R}\left({\left[\widehat{\mathbf{\Omega }}\right]}_{\times }-K_R{\left[{\mathit{e}}_R^{\vee }\right]}_{\times }\right),\hfill \\ \hfill \dot{\widehat{\mathbf{\Omega }}}& ={\mathit{J}}^{-1}\left({\mathit{\tau }}_c+{\widehat{\mathit{\tau }}}_d-\widehat{\mathbf{\Omega }}\times \left(\mathit{J}\widehat{\mathbf{\Omega }}\right)\right)-K_R{\mathit{e}}_R^{\vee }+\beta K_{\omega }({\mathbf{\Omega }}_m-\widehat{\mathbf{\Omega }}),\hfill \\ \hfill {\dot{\widehat{\mathit{\tau }}}}_d& =K_{\tau }\left(\beta ({\mathbf{\Omega }}_m-\widehat{\mathbf{\Omega }})+(1-\beta ){\mathrm{\Lambda }}_R{\mathit{e}}_R^{\vee }\right)-{\alpha }_{\tau }{\widehat{\mathit{\tau }}}_d.\hfill \end{array}$$

(24)

where ${\mathrm{\Omega }}_m$ denotes the measured angular velocity in $\left\{\mathcal{B}\right\}$, ${\alpha }_{\tau }$ denotes the parameter for leakage term, and $K_R,K_{\omega },K_{\tau }$ are the attitude-error injection gain, angular-velocity correction gain, and extended-state gain, respectively.

The observer incorporates both attitude and angular velocity information, and the strength of the gyroscope feedback is scaled by a tunable weighting factor $\beta \in [0,1]$ which allows continuous tuning between purely attitude–based correction ($\beta =0$) and full gyroscope–aided feedback ($\beta =1$).

4.4. Proof of Stability

Regarding the stability of the proposed method, since the ADRC framework in the outer loop is classical, the discussion in this section is focused on the inner loop, which employs a geometric ESO defined on $SO\left(3\right)$. Prior to the proof of stability, the following assumptions should be justified:

• Assumption 1: Bounded and slowly varying disturbance. The lumped disturbance torque ${\mathit{\tau }}_d$ and its time derivative are both supposed to be bounded, ${\mathit{\tau }}_d\in {\mathcal{L}}_{\infty }$ and ${\dot{\mathit{\tau }}}_d\in {\mathcal{L}}_{\infty }$. This assumption implies that the disturbance varies sufficiently slowly with respect to the observer bandwidth and can therefore be regarded as an extended state to be estimated by the ESO. Such a condition is commonly satisfied in practical coaxial UAVs, where aerodynamic coupling, rotor interference, and structural effects evolve on a slower time scale than the inner-loop attitude dynamics. Note that if part of the disturbance does not satisfy this assumption, it implies that it also lies far beyond the bandwidth of the actuator. Consequently, it cannot be compensated for from the perspective of control and should therefore be treated as high-frequency vibrations and be properly filtered out.
• Assumption 2: Avoidance of topological singularity. The initial attitude estimation error is supposed to satisfy ${\mathit{e}}_{\theta }\left(0\right)\in [0,\pi )$, where ${\mathit{e}}_{\theta }$ denotes the rotation angles corresponding to the right-invariant attitude error ${\mathit{e}}_R\in SO\left(3\right)$. This assumption excludes the singular case of a ${180}^{\circ }$ rotation. In practice, it is naturally satisfied since the initial attitude is provided by inertial sensors and coarse alignment algorithms.
• Assumption 3: Bounded angular velocity measurement. The measured angular velocity ${\mathbf{\Omega }}_m$ is supposed to be bounded, and the measurement noise is bounded or square-integrable. This assumption ensures that the feedback injection from gyroscope measurements does not destabilize the observer dynamics and is consistent with the physical limitations of inertial sensors.

Under the above assumptions, consider the weighted geometric ESO given in (24). With the right-invariant error definition given in (20), the kinematic error satisfies

$${\dot{\mathit{e}}}_R={\mathit{e}}_R{\left[\mathrm{\Omega }\right]}_{\times }-{\left[\widehat{\mathrm{\Omega }}\right]}_{\times }{\mathit{e}}_R+{\mathit{e}}_R{K}_R{\left[{\mathit{e}}_R^{\vee }\right]}_{\times },$$

(25)

where the additional term ${\mathit{e}}_R{K}_R{\left[{\mathit{e}}_R^{\vee }\right]}_{\times }$ is introduced by the attitude injection.

Subtracting the rotational dynamics from the observer dynamics yields

$${\dot{\mathit{e}}}_{\mathrm{\Omega }}=-{\mathit{J}}^{-1}{\mathit{e}}_{{\mathit{\tau }}_d}-{\mathit{J}}^{-1}{\Delta }_{\times }+K_R{\mathit{e}}_R^{\vee }-\beta K_{\omega }({\mathit{e}}_{\mathrm{\Omega }}+{\mathit{n}}_{\mathrm{\Omega }}),$$

(26)

where ${\Delta }_{\times }=\mathrm{\Omega }\times \left(\mathit{J}\mathrm{\Omega }\right)-\widehat{\mathrm{\Omega }}\times \left(\mathit{J}\widehat{\mathrm{\Omega }}\right)$ and ${\mathrm{\Omega }}_m=\mathrm{\Omega }+{\mathit{n}}_{\mathrm{\Omega }}$, with ${\mathit{n}}_{\mathrm{\Omega }}$ denoting the measurement noise. Thereby, let ${\mathrm{\Lambda }}_R={\displaystyle \frac{1}{2}}(tr\left({\mathit{e}}_R\right)\mathit{I}-{\mathit{e}}_R),{\mathrm{\Lambda }}_R\in {\mathbb{R}}^{3\times 3}$, the dynamics of disturbance error observation follows ${\dot{\mathit{e}}}_{{\mathit{\tau }}_d}={\dot{\widehat{\mathit{\tau }}}}_d-{\dot{\mathit{\tau }}}_d$, which is

$${\dot{\mathit{e}}}_{{\tau }_d}=\beta K_{\tau }{\mathit{e}}_{\mathrm{\Omega }}+\beta K_{\tau }{\mathit{n}}_{\mathrm{\Omega }}+(1-\beta )K_{\tau }{\mathrm{\Lambda }}_R{\mathit{e}}_R^{\vee }-{\alpha }_{\tau }{\mathit{e}}_{{\tau }_d}-({\dot{\mathit{\tau }}}_d+{\alpha }_{\tau }{\mathit{\tau }}_d).$$

(27)

In order to prove the convergence of ${\mathit{e}}_{{\mathit{\tau }}_d}$, a Lyapunov function can be constructed as follows:

$$V={\displaystyle \frac{1}{2}}{\parallel {\mathit{e}}_R^{\vee }\parallel }^2+{\displaystyle \frac{1}{2}}{\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{\mathit{e}}_{\mathrm{\Omega }}+{\displaystyle \frac{1}{2}}{\mathit{e}}_{{\tau }_d}^T{K}^{-1}{\mathit{e}}_{{\tau }_d}.$$

(28)

The derivative of the attitude-related terms can be expressed as

$${\displaystyle \frac{d}{dt}}({\displaystyle \frac{1}{2}}\parallel {\mathit{e}}_R^{\vee }{\parallel }^2)={\left({\mathit{e}}_R^{\vee }\right)}^T{\dot{\mathit{e}}}_R^{\vee }={\left({\mathit{e}}_R^{\vee }\right)}^T\mathcal{E}\left({\mathit{e}}_R\right){\mathit{e}}_{\mathrm{\Omega }}-{\left({\mathit{e}}_R^{\vee }\right)}^T{K}_R{\mathit{e}}_R^{\vee }.$$

(29)

where $\mathcal{E}\left({\mathit{e}}_R\right)\in {\mathbb{R}}^{3x3}$ denotes the Jacobian matrix of the logarithmic map from ${\mathit{e}}_R$ to ${\mathit{e}}_R^{\vee }$ on $SO\left(3\right)$. Since $\mathcal{E}\left({\mathit{e}}_R\right)$ is bounded and satisfies $\mathcal{E}\left({\mathit{e}}_R\right)\to \mathit{I}$ as ${\mathit{e}}_R\to \mathit{I}$, it can be absorbed into the gain design without affecting the results of the stability analysis, which indicates

$${\displaystyle \frac{d}{dt}}({\displaystyle \frac{1}{2}}\parallel {\mathit{e}}_R^{\vee }{\parallel }^2)\approx {\left({\mathit{e}}_R^{\vee }\right)}^T{\mathit{e}}_{\mathrm{\Omega }}-K_R{\parallel {\mathit{e}}_R^{\vee }\parallel }^2.$$

(30)

The derivative of the angular velocity-related terms can be written as

$${\displaystyle \frac{d}{dt}}({\displaystyle \frac{1}{2}}{\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{\mathit{e}}_{\mathrm{\Omega }})={\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{\dot{\mathit{e}}}_{\mathrm{\Omega }}=-{\mathit{e}}_{\mathrm{\Omega }}^T{\mathit{e}}_{{\tau }_d}-{\mathit{e}}_{\mathrm{\Omega }}^T{\Delta }_{\times }+\beta {\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{K}_R{\mathit{e}}_{\mathrm{\Omega }}-\beta {\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{K}_{\omega }{\mathit{n}}_{\mathrm{\Omega }}.$$

(31)

The derivative of the disturbance-related terms can be written as

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}({\displaystyle \frac{1}{2}}{\mathit{e}}_{{\tau }_d}^T{K}_{\tau }^{-1}{e}_{{\tau }_d})& ={\mathit{e}}_{{\tau }_d}^T{K}_{\tau }^{-1}{\dot{\mathit{e}}}_{{\tau }_d}\hfill \\ \hfill & ={\mathit{e}}_{{\tau }_d}^T\left(\beta K_{\tau }{\mathit{e}}_{\mathrm{\Omega }}+\beta K_{\tau }{\mathit{n}}_{\mathrm{\Omega }}+(1-\beta )K_{\tau }{\mathrm{\Lambda }}_R{\mathit{e}}_R^{\vee }-{\alpha }_{\tau }{\mathit{e}}_{\tau }-{\dot{\tau }}_d-{\omega }_{\tau }\right)\hfill \end{array}$$

(32)

where ${\omega }_{\tau }=\dot{\mathit{\tau }}+{\alpha }_{\tau }{\mathit{\tau }}_d$, which represents the effect of time-varying disturbances and the leakage term.

By means of combining the derivatives of these components, we obtain

$$\begin{array}{cc}\hfill \dot{V}=& -K_R\parallel {\mathit{e}}_R^{\vee }{\parallel }^2-\beta \mathit{J}{K}_{\omega }\parallel {\mathit{e}}_{\mathrm{\Omega }}{\parallel }^2-{\alpha }_{\tau }{K}_{\tau }^{-1}{\parallel e_{{\tau }_d}\parallel }^2\underset{\left(a\right)}{\underbrace{-{\mathit{e}}_{\mathrm{\Omega }}^T{\Delta }_{\times }}}\hfill \\ \hfill & +\underset{\left(b\right)}{\underbrace{{\left({\mathit{e}}_R^{\vee }\right)}^T{\mathit{e}}_{\mathrm{\Omega }}}}+\underset{\left(c\right)}{\underbrace{K_R{\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{\mathit{e}}_R^{\vee }}}\underset{\left(d\right)}{\underbrace{-(1-\beta ){\mathit{e}}_{\mathrm{\Omega }}^T{\mathit{e}}_{{\tau }_d}}}+\underset{\left(e\right)}{\underbrace{(1-\beta ){\mathit{e}}_{{\tau }_d}^T{\mathrm{\Lambda }}_R{\mathit{e}}_R^{\vee }}}\hfill \\ \hfill & +\underset{\left(f\right)}{\underbrace{\beta {\mathit{e}}_{{\tau }_d}^T{\mathit{n}}_{\mathrm{\Omega }}}}\underset{\left(g\right)}{\underbrace{-\beta {\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{K}_{\omega }{\mathit{n}}_{\mathrm{\Omega }}}}\underset{\left(h\right)}{\underbrace{-{\mathit{e}}_{{\tau }_d}^T{K}_{\tau }^{-1}{\omega }_{\tau }}}.\hfill \end{array}$$

(33)

As for term (a), which arises from the mismatch between the rigid-body gyroscopic coupling in the system and that in the observer, by substituting $\mathrm{\Omega }=\widehat{\mathrm{\Omega }}+e_{\mathrm{\Omega }}$ and expanding, one obtains

$${\Delta }_{\times }=\widehat{\mathrm{\Omega }}\times \left(\mathit{J}{e}_{\mathrm{\Omega }}\right)+e_{\mathrm{\Omega }}\times \left(\mathit{J}\widehat{\mathrm{\Omega }}\right)+e_{\mathrm{\Omega }}\times \left(\mathit{J}{e}_{\mathrm{\Omega }}\right).$$

(34)

For bounded $\widehat{\mathrm{\Omega }}$ and sufficiently small $e_{\mathrm{\Omega }}$, the nonlinear term satisfies a local Lipschitz condition

$$\parallel {\Delta }_{\times }\parallel \le L_{\mathrm{\Omega }}\parallel e_{\mathrm{\Omega }}\parallel ,$$

(35)

with some constant $L_{\mathrm{\Omega }}>0$. Consequently, the contribution of this term to the Lyapunov derivative can be upper bounded as

$$-{\mathit{e}}_{\mathrm{\Omega }}^T{\Delta }_{\times }\le L_{\mathrm{\Omega }}{\parallel {\mathit{e}}_{\mathrm{\Omega }}\parallel }^2,$$

(36)

which can be dominated by the negative definite angular-velocity error term through a proper choice of the observer gain.

There are several cross terms remained in $\dot{V}$, which can be handled using Young’s inequality as follows:

$$a^{T}b\le {\displaystyle \frac{\epsilon }{2}}{\parallel a\parallel }^2+{\displaystyle \frac{1}{2\epsilon }}{\parallel b\parallel }^2,\forall \epsilon >0.$$

(37)

According to Young’s inequality, term (b) satisfies

$${\left({\mathit{e}}_R^{\vee }\right)}^T{\mathit{e}}_{\mathrm{\Omega }}\le {\displaystyle \frac{{\epsilon }_b}{2}}\parallel {\mathit{e}}_R^{\vee }{\parallel }^2+{\displaystyle \frac{1}{2{\epsilon }_b}}{\parallel {\mathit{e}}_{\mathrm{\Omega }}\parallel }^2.$$

(38)

And as for term (c),

$$K_R{\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{\mathit{e}}_R^{\vee }\le {\displaystyle \frac{{\epsilon }_c}{2}}\parallel {\mathit{e}}_{\mathrm{\Omega }}{\parallel }^2+{\displaystyle \frac{K_R^2{\parallel \mathit{J}\parallel }^2}{2{\epsilon }_c}}{\parallel {\mathit{e}}_R^{\vee }\parallel }^2.$$

(39)

Given $\beta \in [0,1]$, term (d) satisfies

$$(\beta -1){\mathit{e}}_{\mathrm{\Omega }}^T{\mathit{e}}_{{\tau }_d}\le (1-\beta )({\displaystyle \frac{{\epsilon }_d}{2}}\parallel {\mathit{e}}_{\mathrm{\Omega }}{\parallel }^2+{\displaystyle \frac{1}{2{\epsilon }_d}}\parallel {\mathit{e}}_{{\tau }_d}{\parallel }^2).$$

(40)

And term (e) satisfies

$$(1-\beta ){\mathit{e}}_{{\tau }_d}^T{\mathrm{\Lambda }}_R{\mathit{e}}_R^{\vee }\le (1-\beta )({\displaystyle \frac{{\epsilon }_e}{2}}\parallel {\mathit{e}}_{{\tau }_d}{\parallel }^2+{\displaystyle \frac{\parallel {\mathrm{\Lambda }}_R{\parallel }^2}{2{\epsilon }_e}}\parallel {\mathit{e}}_R^{\vee }{\parallel }^2).$$

(41)

Term (f) satisfies

$$\beta {\mathit{e}}_{{\tau }_d}^T{\mathit{n}}_{\mathrm{\Omega }}\le \beta {\displaystyle \frac{{\epsilon }_f}{2}}\parallel {\mathit{e}}_{\mathrm{\Omega }}{\parallel }^2+{\displaystyle \frac{\beta }{2{\epsilon }_f}}{\parallel {\mathit{n}}_{\mathrm{\Omega }}\parallel }^2.$$

(42)

Term (g) satisfies

$$-\beta {\mathit{e}}_{\mathrm{\Omega }}^T\mathit{J}{K}_{\omega }{\mathit{n}}_{\mathrm{\Omega }}\le {\displaystyle \frac{{\epsilon }_g}{2}}\parallel {\mathit{e}}_{\mathrm{\Omega }}{\parallel }^2+{\displaystyle \frac{{\beta }^2{K}_{\omega }^2{\parallel J\parallel }^2}{2{\epsilon }_g}}{\parallel {\mathit{n}}_{\mathrm{\Omega }}\parallel }^2.$$

(43)

And finally, term (h) satisfies

$$-{\mathit{e}}_{{\tau }_d}^T{K}_{\tau }^{-1}{\mathit{\omega }}_{\tau }\le {\displaystyle \frac{{\epsilon }_h}{2}}\parallel {\mathit{e}}_{{\tau }_d}{\parallel }^2+{\displaystyle \frac{1}{2{\epsilon }_h{\parallel K_{\tau }\parallel }^2}}{\parallel {\mathit{\omega }}_{\tau }\parallel }^2.$$

(44)

By applying (36)–(44), the upper bound of $\dot{V}$ can be rewritten in the following form:

$$\dot{V}\le -{\kappa }_1\parallel e_R^{\vee }{\parallel }^2-{\kappa }_2\parallel e_{\mathrm{\Omega }}{\parallel }^2-{\kappa }_3\parallel {\mathit{e}}_{{\tau }_d}{\parallel }^2+{\kappa }_4\parallel {\mathit{n}}_{\mathrm{\Omega }}{\parallel }^2+{\kappa }_4{\parallel {\mathit{\omega }}_{\tau }\parallel }^2$$

(45)

As a result, the Lyapunov derivative can be rendered negative definite up to bounded input terms by appropriately selecting the observer gains, leading to an input-to-state stable (ISS) or uniformly ultimately bounded (UUB) estimation error dynamics under general operating conditions.

4.5. Control Allocation

The control allocation module maps the virtual inputs generated by the cascaded ADRC (total thrust and body torques) to the physical actuators of the coaxial UAV. Unlike quadrotors, the coaxial configuration introduces stronger rotor–wake coupling and uncertain phase lag in cyclic pitch control, making explicit allocation essential to ensure realizable commands within actuator limits. In practice, this mapping can be decomposed into three stages.

First, the upper and lower rotor speeds are obtained by solving a linear system of two equations that relates total thrust and commanded yaw torque:

$$\left[\begin{array}{cc}{\kappa }_{Tu}& {\kappa }_{Tl}\\ k_{\Delta }& -k_{\Delta }\end{array}\right]\left[\begin{array}{c}{n}_u^2\\ n_l^2\end{array}\right]=\left[\begin{array}{c}{T}_{\mathrm{cmd}}\\ {\tau }_{z,\mathrm{cmd}}\end{array}\right].$$

If the system becomes ill-conditioned, thrust is evenly distributed between the two rotors and yaw torque is relaxed. The resulting solutions are then saturated within the admissible rpm range $[n_{min},n_{max}]$.

Second, the roll and pitch torques $({\tau }_x,{\tau }_y)$ are realized by cyclic inputs applied exclusively to the lower rotor. The achievable cyclic torques are bounded by

$${\tau }_{x,max}=k_x{n}_l^2{\delta }_{max},{\tau }_{y,max}=k_y{n}_l^2{\delta }_{max},$$

where ${\delta }_{max}$ is the maximum cyclic deflection.

To avoid axis-wise saturation distortions, an elliptical limiting strategy is employed. The commanded torques are first normalized with respect to their bounds:

$${\widehat{\tau }}_x={\displaystyle \frac{{\tau }_x^{\mathrm{cmd}}}{{\tau }_{x,max}}},{\widehat{\tau }}_y={\displaystyle \frac{{\tau }_y^{\mathrm{cmd}}}{{\tau }_{y,max}}}.$$

If the vector $({\widehat{\tau }}_x,{\widehat{\tau }}_y)$ lies outside the unit circle, i.e.,

$${\widehat{\tau }}_x^2+{\widehat{\tau }}_y^2>1,$$

it is radially scaled back onto the circle:

$$({\widehat{\tau }}_x^{\prime },{\widehat{\tau }}_y^{\prime })={\displaystyle \frac{1}{\sqrt{{\widehat{\tau }}_x^2+{\widehat{\tau }}_y^2}}}({\widehat{\tau }}_x,{\widehat{\tau }}_y).$$

The physically feasible torques are then reconstructed as

$${\tau }_x={\widehat{\tau }}_x^{\prime }{\tau }_{x,max},{\tau }_y={\widehat{\tau }}_y^{\prime }{\tau }_{y,max}.$$

Finally, the corresponding cyclic deflections are obtained as

$${\delta }_{long}={\displaystyle \frac{{\tau }_x}{k_x{n}_l^2}},{\delta }_{lat}={\displaystyle \frac{{\tau }_y}{k_y{n}_l^2}},$$

with an additional saturation applied in

$${\delta }_{long},{\delta }_{lat}\in [-{\delta }_{max},{\delta }_{max}].$$

5. Simulation and Validation

To demonstrate the effectiveness of the proposed cascaded ADRC framework, numerical and hardware-in-the-loop (HIL) simulations are carried out. The simulations provide a controlled environment to compare different control strategies under identical conditions, which confirm the capability of the proposed method to achieve accurate trajectory tracking and robust disturbance rejection.

5.1. Simulation Results for Different Control Strategies

The effectiveness of the cascaded ADRC framework is first examined through numerical simulations with MATLAB/Simulink R2023b. Several control strategies are implemented and compared under identical conditions to assess their tracking performance and robustness. Two types of scenarios are considered: (i) a baseline case without external disturbances to evaluate nominal tracking performance, and (ii) a case with model uncertainties and external disturbances to assess robustness.

The physical and aerodynamic parameters used in the simulations are summarized in

Table 2. Simulation Parameters.

Parameter	Value
m	$9\mathrm{kg}$
g	$9.81{\mathrm{m}·\mathrm{s}}^{-2}$
$J=\mathrm{diag}(J_x,J_y,J_z)$	$0.81,0.81,0.405{\mathrm{kg}·\mathrm{m}}^2$
${\kappa }_{Tu}$	$4.4\times {10}^{-5}$
${\kappa }_{Tl}$	$4.18\times {10}^{-5}$
$k_{\Delta }$	$1.92\times {10}^{-6}$
$k_x,k_y$	$2.86\times {10}^{-3},2.86\times {10}^{-3}$
$C_v=\mathrm{diag}\left(C_{dv}\right)$	$1.0,1.0,1.5$
$C_{\omega }=\mathrm{diag}\left(C_{dw}\right)$	$0.27,0.27,0.101$

. These values represent the nominal model of the coaxial UAV, including mass, inertia, propeller thrust/torque coefficients, periodic torque coefficients, and translational and rotational damping terms.

In both scenarios, all controllers are required to follow a predefined three-dimensional reference trajectory in order to excite both translational and rotational dynamics of the UAV. The reference trajectory is defined as a three-dimensional lemniscate with a gentle altitude variation:

$$\begin{array}{cc}\hfill x_d\left(t\right)& =Asin\left(\omega t\right),\hfill \\ \hfill y_d\left(t\right)& =Bsin(2\omega t+\varphi ),\hfill \\ \hfill z_d\left(t\right)& =z_0+A_{z}sin\left({\omega }_{z}t\right),\hfill \end{array}$$

(46)

where $x_d\left(t\right)$, $y_d\left(t\right)$, and $z_d\left(t\right)$ denote the desired positions along the three axes, with $A=10$ m and $B=6$ m representing the horizontal and vertical scales of the trajectory, $\omega =0.15$ rad/s the angular frequency (corresponding to a period of $2\pi /\omega$), $\varphi =0$ the phase shift, and $z_0=4$ m with $A_z=2$ m the altitude offset and oscillation amplitude. The UAV is initialized at the origin with zero velocity and attitude, and the control objective is to accurately follow this reference trajectory throughout the simulation.

5.1.1. Baseline Tracking Without Disturbances

To demonstrate the advantages of the proposed cascaded ADRC framework, this case conducts a comparative study under nominal conditions. The evaluation involves four controllers: the proposed cascaded ADRC (ADRC-C), ADRC applied only in the outer loop (ADRC-O), ADRC applied only in the inner loop (ADRC-I), and a conventional cascaded PID controller. Figure 5 presents the three-dimensional reference trajectory together with the tracking responses of the controllers, showing their overall ability to follow the desired path.

Since the overall trajectories show only minor visible differences, the analysis is extended to the position tracking errors (Figure 6), which provide clearer insight into the performance differences among the controllers.

As shown in Figure 6, the position trajectories on the left indicate that all controllers are capable of following the reference path. However, the proposed cascaded ADRC (ADRC-C) exhibits trajectories that are more closely aligned with the reference, reflecting its superior transient response. This observation is further supported by the tracking error plots on the right, where ADRC-C consistently achieves smaller deviations compared with the other controllers. It should be noted that although the trajectories in the left column appear nearly overlapped, the error plots in the right column reveal larger discrepancies. This is mainly due to time delays in the responses, which are not visually apparent in the trajectory plots but are captured in the error curves. In particular, the PID controller exhibits significant delay effects, which stem from its intrinsic “after-the-fact correction” mechanism based purely on error feedback.

Since the cascaded ADRC (ADRC-C) demonstrated the best overall performance and the main differences among controllers are already reflected in the position results, only its attitude tracking behavior is further analyzed here. Figure 7 shows the roll, pitch, and yaw tracking performance, including both angle responses and the corresponding errors, confirming the stability and convergence of the proposed controller. ADRC-C achieves high-accuracy tracking of the reference angles, with only small initial deviations caused by the settling of the ESO, which requires a short time to accurately estimate the lumped disturbances. As illustrated in Figure 8, this settling process produces small oscillations in the disturbance estimates, which in turn induce oscillatory behavior in the control response before the observer stabilizes. It should also be noted that the yaw angle remains at zero throughout the simulation. This is because, in the cascaded control framework shown in Figure 4, the yaw dynamics are decoupled and no specific yaw command was imposed in this test scenario.

5.1.2. Tracking Under Uncertainties and Disturbances

To further evaluate the robustness of the proposed cascaded ADRC framework, a second scenario is investigated in which the UAV is subject to both model uncertainties and external disturbances. The model uncertainties are introduced by increasing the nominal mass and moments of inertia by 11%, while the external disturbances consist of additional force and torque inputs applied to the translational and rotational dynamics. For simulation purposes, the external disturbances are modeled with sinusoidal, constant, and step-like components: ${\mathit{\tau }}_d\left(t\right)={\left[\begin{array}{c}2sin\left(t\right),D_2\left(t\right),0.1\end{array}\right]}^T$, ${\mathit{f}}_d\left(t\right)={\left[\begin{array}{c}{D}_4\left(t\right),0.5sin\left(0.5t\right),0.2sin\left(0.5t\right)\end{array}\right]}^T.$ The step-like components specified as

$$\begin{array}{cc}\hfill D_2\left(t\right)& =\left\{\begin{array}{cc}0,\hfill & t<10,\hfill \\ 2,\hfill & t\ge 10,\hfill \end{array}\right.\hfill \\ \hfill D_4\left(t\right)& =\left\{\begin{array}{cc}0,\hfill & t<10,\hfill \\ -0.5,\hfill & t\ge 10.\hfill \end{array}\right.\hfill \end{array}$$

(47)

The tracking performance under uncertain and disturbed conditions is illustrated in Figure 9 and Figure 10. Figure 9 presents the position responses and errors along the x, y, and z axes. All controllers exhibit oscillatory errors due to the sinusoidal disturbances, yet the cascaded ADRC (ADRC-C) consistently achieves the smallest deviations. It is worth noting that in the nominal case without disturbances (Figure 6), ADRC-O and ADRC-I perform similarly and both outperform the conventional PID controller. When disturbances are introduced, however, ADRC-O degrades considerably, with error levels approaching those of PID, while ADRC-I maintains intermediate performance between ADRC-C and PID.

This behavior can be explained by the way each control structure addresses disturbances and model uncertainties. In ADRC-O, the outer-loop ESO estimates low-frequency translational disturbances and mass mismatch, but the inner PID loop introduces phase lag and lacks disturbance rejection, allowing torque-induced oscillations and inertia mismatches to propagate into position errors. In ADRC-I, the inner-loop ESO compensates for torque disturbances and inertia mismatches, leading to well-damped attitude tracking; however, translational disturbances and mass variations remain unaddressed in the outer PID, leaving residual errors. By contrast, the cascaded ADRC combines the advantages of both configurations: the inner loop suppresses torque-driven oscillations, while the outer loop estimates and compensates translational disturbances and mass uncertainties. Furthermore, the fast inner loop effectively linearizes the plant dynamics, improving the accuracy of the outer-loop disturbance estimation. As a result, ADRC-C achieves the smallest error amplitude at the disturbance frequency and delivers the most stable overall tracking performance.

Figure 10 illustrates the attitude tracking performance of the cascaded ADRC under combined model uncertainties and external disturbances. The roll response follows the reference almost perfectly, with tracking errors close to zero except for a sharp transient fluctuation at $t=10$ s, caused by the activation of the step-like disturbances $D_2\left(t\right)$ and $D_4\left(t\right)$. This behavior is consistent with the ESO disturbance estimates shown in Figure 11, where the pitch channel remains nearly unaffected, while the roll channel exhibits a sudden increase at $t=10$ s. This confirms that the fluctuation in roll originates from the activation of $D_2\left(t\right)$, and the cascaded ADRC rapidly compensates for it once the ESO converges.

In contrast, the pitch and yaw responses exhibit small oscillations throughout the simulation due to the sinusoidal torque disturbances. For the yaw channel in particular, the response converges to zero before $t=10$ s, but after the step disturbances are introduced, oscillations of about $3^{\circ }$ appear. These deviations, however, remain bounded and quickly attenuate, demonstrating that the cascaded ADRC can effectively suppress disturbance-induced errors across all attitude channels. Overall, the results confirm that ADRC-C achieves stable and accurate attitude regulation even under combined parametric uncertainties and external disturbances.

After analyzing the attitude responses and the ESO disturbance estimates, the actuator behaviors are compared in Figure 12. In the nominal case, both rotors operate within 900–1000 rpm, while under disturbances they exceed 1100 r/min, with a sharp fluctuation at $t=10$ s caused by the activation of $D_2\left(t\right)$. On the servo side, the outputs remain within $\pm {15}^{°}$ in both scenarios, verifying effective constraint enforcement. Notably, ${\theta }_c$ varies periodically in the nominal case but saturates at ${15}^{°}$ under disturbances to counteract roll perturbations, whereas ${\theta }_s$ maintains periodic oscillations. These results show that actuator saturation may occur in the roll channel under disturbances, but overall stability is preserved.

5.2. High-Fidelity HIL Experiment

The previous section evaluated the proposed ADRC-C algorithm through numerical simulations conducted in MATLAB/Simulink. While such simulations provide an efficient environment for performance analysis, they do not fully reflect embedded execution characteristics of practical flight-control systems. However, since our coaxial UAV platform features a relatively large overall weight and dimensions, its physical experiments, particularly those involving trajectory tracking, are highly demanding in terms of site requirements and safety assurance. Therefore, a high-fidelity HIL simulation is adopted as a necessary intermediate validation step between numerical simulation and real flight testing. In the HIL framework, vehicle dynamics are simulated in real time using FastSim, which is a high-fidelity simulator based on Unity [37]. As shown in Figure 13, both state estimation and control algorithms are executed on a CUAV X7+ flight controller, which is the same onboard control device of the coaxial UAV platform shown in Figure 1. The CUAV X7+ runs a customized firmware modified from PX4 v1.14 with the same parameters as those employed in real flight tasks. The simulation software runs on a laptop with an Ryzen 5900X processor, 32 GB memory and an RTX 3070Ti graphics processing unit (GPU).

FastSim can substitute for the real actuators by receiving command signals directly from the flight controller and reproducing the corresponding actuator effects based on the high-fidelity modeling approach described in Section 3.3. Furthermore, arbitrary disturbances can be injected into the vehicle dynamics, enabling systematic evaluation of the control performance under controlled and repeatable disturbance conditions.

5.2.1. Disturbance Modeling

To benchmark the disturbance-rejection capability of the proposed controller in Unity, bounded and band-limited external disturbances as a pair of force and torque are applied at the vehicle center of mass, which are

$${\mathit{f}}_d\left(t\right)={\mathit{f}}_{bias}+{\mathit{f}}_{per}\left(t\right)+{\mathit{f}}_{col}\left(t\right),{\mathit{\tau }}_d\left(t\right)={\mathit{\tau }}_{bias}+{\mathit{\tau }}_{per}\left(t\right)+{\mathit{\tau }}_{col}\left(t\right),$$

(48)

where, ${\mathit{f}}_{bias}$ and ${\mathit{\tau }}_{bias}$ represent constant offsets, ${\mathit{f}}_{per}$ and ${\mathit{\tau }}_{per}$ provide repeatable gust excitation, while ${\mathit{f}}_{col}$ and ${\mathit{\tau }}_{col}$ introduce stochastic yet smooth perturbations.

Finite sinusoidal series are used to generate reproducible disturbances which are

$${\mathit{f}}_{per}\left(t\right)=\sum _{k=1}^{N_f}{\mathit{a}}_k\odot sin(2\pi f_{k}t+{\mathbf{\Phi }}_k),{\mathit{\tau }}_{per}\left(t\right)=\sum _{k=1}^{N_{\mathit{\tau }}}{\mathit{b}}_k\odot sin(2\pi {\nu }_{k}t+{\mathbf{\Psi }}_k),$$

(49)

where ⊙ denotes element-wise multiplication, and $\{{\mathit{a}}_k,{\mathit{b}}_k,f_k,{\nu }_k,{\mathbf{\Phi }}_k,{\mathbf{\Psi }}_k\}$ determine axis-wise amplitudes, frequencies, and phases, respectively, and $N_f,N_{\mathit{\tau }}\in {\mathbb{N}}^{+}$. The terms ${\mathit{f}}_{col}\left(t\right)$ and ${\mathit{\tau }}_{col}\left(t\right)$ denote colored stochastic disturbances obtained by low-pass shaping of zero-mean white-noise samples, yielding perturbations that avoid unrealistically discontinuous inputs. In practice, ${\mathit{f}}_{col}\left(t\right)$ and ${\mathit{\tau }}_{col}\left(t\right)$ are generated as smooth random processes with tunable correlation time and bounded magnitude. Furthermore, in addition to increasing the complexity of the disturbances, their magnitude was also substantially amplified as shown in Figure 14 and Figure 15 in order to push the controller to its operational limits.

5.2.2. Trajectory Tracking Performance

In the HIL simulation, we investigated the performance of the ADRC-C scheme in trajectory tracking tasks.

Figure 16 presents the HIL simulation results of a coaxial UAV following a relatively complex reference trajectory. The trajectory is parameterized by polynomial curves, and the command velocity along the path is visualized using a color gradient. The maximum speed is approximately $3.8\mathrm{m}·{\mathrm{s}}^{-1}$ and the peak acceleration reaches $5.6\mathrm{m}·{\mathrm{s}}^{-2}$, which can be considered a challenging maneuver for a coaxial UAV, especially under strong disturbances. During this tracking process, the commanded yaw angle is continuously aligned with the instantaneous velocity direction of the reference trajectory. Since the tracking controller adopts a cascaded PID structure and the achievable control gains in the simulation are actually higher than those in the real-world experiments, the UAV is able to accomplish the tracking task, even though with noticeable tracking errors remaining.

Figure 17 shows the tracking trajectory of the coaxial UAV with the inner-loop disturbance compensation enabled. As can be observed, although a certain amount of tracking error persists, the overall flight path becomes substantially less affected by external disturbances. This improvement is because the inner loop primarily compensates for disturbance torques, thereby enhancing attitude control accuracy. However, inner-loop disturbance suppression cannot proactively counteract external forces. Consequently, during the tracking task in Figure 17, the position error is corrected mainly through PID feedback, which is not sufficiently sensitive to time-varying external forces.

Figure 18 illustrates the catastrophic outcome when only the outer-loop ADRC is enabled. Although the outer-loop disturbance rejection can effectively compensate for external forces, in this particular experiment it caused the UAV to completely lose flight stability. Such instability does not necessarily occur in every test. Nevertheless, the result clearly indicates the potential stability risk of relying solely on outer-loop disturbance suppression. Note that the disturbances injected in the HIL simulation are relatively aggressive, and real flight conditions typically do not involve torque disturbances of comparable magnitude. Even so, this experiment demonstrates that suppressing fast disturbance torques in the inner loop is crucial for maintaining stable flight.

Figure 19 presents the results obtained with the complete ADRC-C scheme. The outer-loop parameters remain unchanged compared with Figure 18, yet the overall trajectory-tracking performance is the best among all cases. In fact, the average tracking error is, to some extent, even smaller than that in the disturbance-free case. This is because the position tracking error is also interpreted as an external disturbance and is incorporated into the outer-loop observer for online compensation. Although small oscillations in position can be observed, which is indicated in Figure 20, they do not degrade the overall tracking performance. Furthermore, while such rapid yaw maneuvers are uncommon in most industrial applications, the simulation results indicate that ADRC-C provides effective suppression of disturbances along the yaw channel as well.

6. Conclusions

This paper presented a cascaded ADRC-based control framework for a newly designed coaxial UAV platform. The platform emphasizes structural compactness and feasibility for long-duration missions, while the proposed control framework addresses the inherent challenges of coaxial configurations, including rotor coupling, phase delay, and external disturbances. By replacing conventional PID controllers with ADRC in both inner and outer loops, the framework enables disturbance estimation and compensation at multiple time scales, thereby improving robustness and tracking performance.

The effectiveness of the proposed approach was validated through numerical and high-fidelity HIL simulations. The results demonstrate that the cascaded ADRC framework provides faster recovery from disturbances, reduced tracking errors, and more consistent control performance compared with conventional PID control. These findings confirm the practical applicability of ADRC in coaxial UAVs, where aerodynamic complexity often limits traditional methods.

Future work will focus on extending the framework to more complex scenarios, such as outdoor environments with stronger wind fields, formation flight of multiple coaxial UAVs, and integration with high-level planning algorithms. In addition, further optimization of the allocation strategy and observer design will be explored to improve efficiency and scalability.