Abstract
The perching maneuver enables a quadrotor to make stable contact with vertical surfaces for prolonged monitoring, which significantly enhances mission endurance and energy efficiency in inspection and surveillance tasks. To achieve a stable perching maneuver, this study proposes an adaptive second-order continuous control (ASOCC) in contact-based inspection applications. A novel finite-time convergent disturbance observer compensates model uncertainties and external disturbances, including aerodynamic and wall effects. The closed-loop Lyapunov stability of the proposed observer-controller system is also established. The effectiveness of the ASOCC strategy is validated through extensive simulation studies under various conditions, including step response, model uncertainties, and external disturbances. Comparative evaluations against existing control strategies reveal that the proposed method offers higher precision, stronger robustness, and better resistance to external disturbances when assessed through standard tracking error-based performance indices. Additionally, experimental trials verify that the quadrotor consistently performs a stable perching maneuver on vertical walls under both indoor and outdoor conditions.
Subject terms: Engineering, Mathematics and computing
Introduction
Quadrotors have been widely utilized in various fields, including remote sensing, mapping, construction inspection, geological exploration, traffic monitoring, wildlife conservation, and agricultural surveys1. Unlike traditional flying vehicles, miniaturized quadrotors require a different design approach to meet operational requirements, such as cargo size, computing limits on board, and consistency in response to external disturbances in the air environment2–4. Their small size makes them especially suitable for contact-based tasks, such as perching for prolonged monitoring, deploying sensors, measuring vibration, and detecting moisture. For surface-mounted inspection tasks, large platforms or systems do not fit easily5–9. More recently, a multi-aircraft visual-inertial-range-physical odometry architecture to compute position, velocity, and attitude constraints in integrated aerial systems has been developed10. To enhance the performance of perception, a coordinate-free control strategy is developed to maintain targets within the field of view with minimal calculation and computational load11. On similar lines, the dual-UAV (PADUAV) presented in the passively articulated system introduced a new five-degrees-of-freedom actuation mechanism, utilizing coupled quadrotor actuation12. Hybrid force or position control has also been deployed to control dynamic coupling in inter-quadrotor interactions13.
In contact-based inspection missions, the perching maneuver plays a pivotal role as the transition stage between free flight and surface contact14. This phase enables the quadrotor to approach, attach, and stabilize in vertical or inclined planes before performing operations such as inspection, data gathering, or traversing surfaces for extended measurement. Achieving this requires coordinated control of the attitude, thrust, and interaction forces to ensure smooth attachment without rebound or loss of contact. The transition stage between free flight and contact with the surface is the perching maneuver, which is particularly relevant in contact-based inspection missions14. The stage enables the quadrotor to fly towards, land, and stabilize on vertical or sloping ground before performing tasks such as inspection, data collection, or scanning surfaces over extended distances. This would only be possible with a well-coordinated attitude, thrust, and interaction force management to ensure a smooth attachment and prevent rebound or loss of contact.
To address these challenges, numerous control frameworks are proposed, including model predictive control (MPC)15, PID-based regulation16,17, feedforward compensation18, learning-based strategies19, feedback linearization20, iterative learning control, nonlinear adaptive control21,22, and variable-structure control (VSC)23,24. Among these methods, sliding mode control (SMC)25,26 has gained prominence due to its inherent robustness against modeling uncertainties and external disturbances27–31. Although conventional SMC and backstepping approaches have demonstrated reliable attitude and position stabilization32, they often suffer from chattering phenomena and achieve only asymptotic convergence, motivating the development of higher-order and continuous variants for perching and contact-based control. To overcome the limitation of asymptotic convergence, a sliding terminal surface was introduced in33 to allow spacecraft formation to fly, achieving finite-time convergence. However, terminal sliding mode control suffers from singularity issues, which were later mitigated through continuous terminal sliding mode control34. In35, a fuzzy logic compensator is proposed to minimize the chattering effect of an adaptive controller designed using SMC. A self-tuning SMC framework combined with neural networks is proposed to address actuator faults in coaxial octorotor UAV36. For tilting quadrotors, recurrent neural network (RNN)-based SMC methods are explored, enabling real-time adjustment of equivalent controllers and estimation errors37,38. Similarly, parameter-adaptive SMC schemes employing radial basis function neural networks (RBFNNs) are developed to achieve robust trajectory tracking under time-varying mass, modeling uncertainties, and external disturbances39.
Despite these advances, both the terminal and the continuous terminal sliding modes exhibit residual chattering effects. To alleviate this, the super-twisting algorithm, a second-order sliding mode control (SOSMC) strategy, is proposed in40, which allows finite-time convergence of the tracking error. Subsequently, the second-order continuous controller (SOCC) is developed in41 to further suppress the chatter, improve the smoothness, and maintain finite-time convergence through continuous control action. Its application to the tracking of wheeled robots is demonstrated in42, and to a quadrotor under self-triggered control in43. Furthermore, several observer-based SMC strategies are introduced to enhance disturbance rejection capabilities44–47. For example, Zhang et al.48 developed a disturbance-observer-based integral SMC for linear systems; Fethalla et al.49 combined backstepping-based SMC with a nonlinear disturbance observer to track the quadrotor trajectory; and Tripathi et al.50 proposed a non-singular fast-terminal sliding mode controller augmented with an intelligent disturbance observer for quadrotor position and attitude regulation. In the case of packet losses, a matched disturbance can be avoided using a homogeneous super-twisting-based disturbance observer for the quadrotor51. The issue of mismatched disturbances is addressed through a modified sliding-mode control framework featuring a redesigned sliding surface and control law52. A general exponential reaching law and an improved adaptive terminal sliding-mode reaching law (IATSMRL) are introduced to compensate for lumped disturbances using an extended sliding-mode disturbance observer (ESMDO), which estimates the total disturbance. This approach achieves faster response dynamics due to the inclusion of a terminal attractor term53. A disturbance observer using RBFN is developed for underwater vehicle-manipulator systems (UVMS) operating under external disturbances, integrated with a nonlinear model predictive control (NMPC) scheme to enhance robustness and performance54. To address the issue of periodic disturbances and deal with parameter mismatches, an integrated observer-based terminal sliding-mode controller (TSMC) is proposed for industrial applications55. Moreover, a state- and disturbance observer-based control strategy is developed to estimate system states and external disturbances using a state-feedback approach for fully actuated systems56. Nevertheless, even with the significant gains in this area, traditional sliding mode and observer-based controllers can be challenged to achieve both smooth, finite-time convergence of the perching maneuver and, simultaneously, to be resistant to unknown and time-varying disturbances. The above problems demonstrate the necessity of a dynamic and continuously adjusting control approach that can reduce chattering and precisely stabilize contact forces when switching between free flight and surface contact.
Motivated by these limitations, this work proposes a robust, chattering-free, computationally efficient, and adaptive control approach to address aggressive perching maneuvers involving rapid attitude transitions and hybrid contact dynamics. Specifically, a finite-time adaptive second-order continuous controller (ASOCC) is developed for quadrotor perching applications. The proposed controller estimates and compensates for external disturbances while ensuring chattering-free operation, thus avoiding the excitation of unwanted system dynamics. Unlike conventional SMC frameworks, the ASOCC eliminates the distinct sliding phase, enabling both state errors and their derivatives to converge simultaneously during the reaching stage, resulting in precise and robust perching performance. The proposed ASOCC framework further enhances robustness and precision by achieving finite-time convergence despite modeling uncertainties and external disturbances. In contrast to the classical SOCC, which employs fixed gains and requires prior knowledge of disturbance bounds, ASOCC introduces an adaptive gain mechanism that updates in real time according to the estimated disturbance and tracking error. This adaptive property improves disturbance estimation accuracy, mitigates chattering effects, and facilitates practical real-time implementation on perching quadrotor platforms, making it highly suitable for autonomous inspection and monitoring applications. The advantage of this approach is achieving the perching mode with a smooth transition from flight to wall contact without overshoot or instability, eliminating the reaching phase, which reduces the setting time and chattering-free control law using a homogeneous control law.
The main contributions of this work are as follows. First, the perching maneuver of a quadrotor on a vertical surface is investigated using a finite-time ASOCC approach for contact-based inspection and monitoring tasks. Second, to precisely estimate and account for model uncertainties and unknown but bounded external disturbances that influence the perching dynamics, a finite-time disturbance observer is developed. The closed-loop finite-time stability of the coupled observer-controller system is derived using Lyapunov-based analysis. Third, to ensure finite-time convergence and closed-loop system stability under operating conditions, explicit mathematical formulations for controller and observer gains are derived.
Fourth, to assess the robustness and provide qualitative and quantitative comparisons with existing finite-time control approaches under Gaussian noise, model uncertainties, and external disturbances, extensive Monte Carlo simulations with 100 independent trials are carried out. Fifth, the proposed ASOCC is validated through simulation and experimentation using a custom-built perching quadrotor, which demonstrated its feasibility in real-time and performance during contact interaction and perching operations.
The rest of the paper is structured in the following way. The first part covers the necessary preliminaries, including definitions and important lemmas that are utilized in this work. This is followed by the system modeling, problem formulation, ASOCC design, and a step-by-step finite-time convergence analysis of the combined observer-controller system. This is followed by a discussion of the procedure for selecting the appropriate gain parameters for both the controller and observer. The simulation and experimental findings, along with a comparison to current methods, have been conducted. Finally, the paper concludes with a summary of the key findings and potential future research directions.
Preliminaries
The following definitions and lemmas are presented to support the control design in this section.
Definition 1
(57) A system is considered finite-time stable if its state trajectories reach the equilibrium point within a finite time interval. Specifically, there exists a finite time 
such that 
for all 
.
Lemma 1
(58) If there exists a positive definite Lyapunov function
W(x) , and constants
and
, such that the inequality
holds, then the origin is finite-time stable. Moreover, the settling time satisfies
 |
1 |
Lemma 2
(59) If there are two variables a and b, then the following inequality holds:
 |
2 |
Lemma 3
(Theoram 143) The second-order differential equation given by
with
and
are positive constants and
s
is a real number, is finite-time convergent with settling time
t
such that
where
V(0) is the initial value of a positive definite Lyapunov function
V(s).
Notation
For any real number x and positive real exponent 
, we define 
where 
is a standard signum function.
Perching quadrotor model and problem statement
In this section, we will describe the mathematical model of the perching quadrotor, followed by the problem formulation.
System model
The following perching quadrotor model is derived using the Newton–Euler equations 43.
 |
3a |
 |
3b |
 |
3c |
 |
3d |
 |
3e |
 |
3f |
Here, 
and 
denote the cosine and sine functions, respectively. The terms 
describe its orientation referred to as roll, pitch, and yaw, respectively, and measured with respect to the inertial coordinate frame. Angular velocities p, q, r are expressed in the fixed body frame attached to its center of mass. The coordinates x, y, and z represent the position of the quadrotor along the x-, y -, and z -axes of the inertial (world) frame. m denotes the mass of the quadrotor and g is the acceleration due to gravity (9.81 m/s
). The terms 
, 
, and 
are the moments of inertia about the x-, y-, and z-axes, respectively. The terms 
and 
represent bounded external disturbances and model uncertainties that affect translational and rotational dynamics (see the nn file for more details). The quadrotor model (3) represents a coupled dynamic system because both translational and rotational motions are interdependent. The translational accelerations 
depend on the attitude angles 
through trigonometric relations, implying that orientation changes directly influence position dynamics. Similarly, the rotational dynamics 
contain gyroscopic coupling terms involving (p, q, r) and inertia differences, linking motion about one axis to the others. These inter-dependencies make the overall system nonlinear and fully coupled, where each degree of freedom affects the others.
The control inputs are defined as follows: 
is the total thrust control input for altitude control; 
, 
, and 
are the torque inputs for attitude control on the roll, pitch, and yaw axes, respectively. The control inputs and rotor speeds (
, 
, 
, and 
) are related as: 
, 
, 
, 
, where b is the distance from the center of mass to the rotor. 
and 
are the lift and drag coefficients, respectively. It is an underactuated system; hence, we consider virtual control inputs 
. These are given as 
. The total thrust is obtained as 
The desired roll and pitch angles are derived based on the virtual control inputs as 
The control inputs (
, 
, 
, 
, 
, and 
) must be designed for the perching quadrotor model (3).
Remark 1
The logic behind the perching mechanism involves adding a support arm to the standard quadrotor, which is lightweight and contributes another aspect of torque to the rotational dynamics. Although this torque is small due to the use of a carbon-fiber arm, it is considered a modeling uncertainty in the current study.
Problem formulation
The error dynamics are derived for the positioning of the perching quadrotor near the target surface. First, we consider the altitude dynamics from the system (3) while defining a new set of variables 
and 
The error dynamics are obtained to control the positioning of the perching quadrotor on the target surface. To begin with, we take into account the dynamics of the altitude involving the system (3) while defining a new set of variables 
and 
as 
The tracking error associated with the altitude dynamics is defined as 
, and the corresponding error dynamics are
 |
4a |
 |
4b |
Similarly, the tracking errors for the remaining subsystems are defined as 
, 
, 
, 
, and 
. Their error dynamics follow the same structure as 
Figure 1 shows the successive phases of the wall-perching maneuver of a perching quadrotor. During the free-flight and approach stage 
, the vehicle travels in the horizontal direction to the wall with the same altitude given by z. At the wall-contact phase 
, the front side of the perching quadrotor makes first contact with the wall and generates a contact reaction force which momentarily disrupts its attitude. Lastly, in the wall-perching phase 
, the controller maintains the vehicle at the desired angle of pitch 
, using thrust adjustment to offset the impact and stabilize the perching quadrotor on the vertical surface, which warrants further investigation or monitoring.
Fig. 1.
Sequential phases of perching quadrotor: (a) Free-flight and approach phase, (b) wall contact phase, and (c) wall perching with desired pitch angle 
.
Therefore, the control objective is to develop a finite-time continuous controller capable of governing both the positioning and perching phases, while guaranteeing that all system errors converge to zero within a finite time, even in the presence of uncertainties.
Remark 2
The error dynamics described in (4) correspond to a perturbed second-order system, a common occurrence in control engineering applications. Consequently, the controller developed for the present system can be readily applied to such classes of systems.
Design of ASOCC
This section outlines the design methodology for a finite-time continuous controller. The proposed adaptive control scheme for all three modes of operation is illustrated in Fig. 2, followed by the design.
Fig. 2.
Schematic of the proposed control design 
and 
.
Control for altitude dynamics
For altitude dynamics, sliding surface 
, its time derivative 
, double derivative 
are defined as 
and 
Rearranging 
,
 |
5 |
Inspired by43, we consider a second-order homogeneous reaching law 
due to its finite-time convergence property. Substituting 
into (5), we get
 |
6 |
To mitigate the disturbance (
), an adaptive term 
is added. Hence, the final control law is
 |
7 |
where 
. However, this adaptive sliding mode control scheme is suitable for mitigating disturbances, but only asymptotic convergence of the tracking error can be achieved. Since our objective is to achieve finite time tracking error convergence, we propose a finite time observer along with a second-order continuous control law for chattering-free control action as
 |
8 |
Here, 
and 
are the nominal part and the disturbance compensation part of the control input 
; 
is the adaptive gain and 
is an auxiliary variable. The updated law for these variables is
 |
9 |
The proposed control law is implemented using a continuous approximation based on the boundary layer technique to suppress high-frequency switching and prevent actuator wear. Specifically, the discontinuous term 
is replaced by 
where 
denotes a small boundary layer thickness parameter. This modification ensures that the tracking error remains ultimately bounded by a tunable quantity determined by 
.
Finite-time convergence analysis for altitude dynamics
To establish finite time convergence of the proposed observer-controller system, we consider a positive definite Lyapunov function as
 |
10 |
where 
is the observer gain estimation error. The time derivative of the Lyapunov function 
is 
.
First, we prove the convergence of 
in Step 1, followed by the convergence of 
in Step 2. Finally, we conclude the overall convergence of 
.
Step 1: With 
, we have
 |
11 |
Since 
, we have 
using (5) and (9). Substituting it into (11) yields
 |
12 |
With 
in (12) yields
 |
13 |
Simplifying (13) using 
and 
from (9),
 |
14 |
Since we have 
,
 |
15 |
Using Lemma 2, we can get
 |
16 |
This leads to
 |
17 |
As inequality (17) is similar to the one presented in Lemma 1, the origin is stable in finite time. Moreover, the settling time satisfies
 |
18 |
where 
and 
. Therefore, the disturbance estimation error 
and the auxiliary variable 
are finite time convergent, which yields 
and by setting 
in (9), we have 
With 
from (4) gives 
Using 
from (8) here yields 
. Substituting this 
back into (8), the overall control input becomes
 |
19 |
Finally, substituting (19) into the dynamics of altitude error in (4) gives 
and 
Since 
and 
, we have
 |
20 |
Step 2: Now consider the second part of the Lyapunov function defined in (10). The time derivative of 
is 
With 
from (20),
 |
21 |
which is negative semidefinite. The Barbalat lemma is used to prove asymptotic convergence. The derivative of 
is
 |
22 |
which is bounded. Therefore, according to the Barbalat Lemma, closed-loop error dynamics are asymptotically stable. To demonstrate that the designed controller causes the error to converge in finite time, the homogeneity property is employed. If the error dynamics (20) is a homogeneous system of negative degree (
with the weights 
) then the system is finite time stable and has a time bound 
according to Definition 1 and Lemma 3. Thus, 
. With inequality (17), we can establish 
for 
in (10). We can therefore state that the observer-controller system using the control law (8) for the altitude subsystem (4) is a finite-time stable system. The convergence time bound for the altitude subsystem is 
.
Remark 3
The proposed controller will ensure convergence in finite time, and the settling time will depend on the initial conditions. Examples of beneficiaries of finite-time control include ease of design, fewer restrictions, higher convergence rates, and robustness compared to conventional episodic methods. Although fixed-time and prescribed-time methods do not explicitly rely on initial conditions, they often employ more complex designs and incorporate conservative adjustments. Thus, expanding such a control framework to fixed-time or prescribed-time is an area to investigate in the future.
Control for position dynamics
Similar to (8), the adaptive second-order continuous control laws for position dynamics are
 |
23 |
 |
24 |
where 
.
Similar to the altitude subsystem, the position error dynamics in the x– and y–directions also follow a perturbed second-order system of the form 
and 
for 
. Under the proposed ASOCC laws in (23) and (24), the auxiliary variables 
, 
, and the adaptive gains 
, 
evolve according to update rules identical in structure to those developed for the altitude observer–controller. By constructing Lyapunov functions 
and 
analogous to (10), and following the same two-step procedure used for altitude stability (i.e., convergence of the observer error dynamics followed by the homogeneous finite-time convergence of the position error subsystem), it can be shown that the requirements of Lemma 1 and Lemma 3 are satisfied. Therefore, both observer–controller pairs 
and 
guarantee finite-time convergence of the position tracking errors 
and 
, even in the presence of bounded disturbances and modeling uncertainties. Thus, the finite-time stability results established for the altitude dynamics also extend directly and rigorously to the lateral position dynamics.
Control for attitude dynamics
The control laws for attitude dynamics are formulated to achieve robust and finite-time attitude regulation under dynamic uncertainties, similar to the approach adopted for altitude control. The corresponding control laws are as follows:
 |
25 |
 |
26 |
 |
27 |
where 
The attitude subsystems corresponding to roll, pitch, and yaw dynamics possess the same perturbed double-integrator structure as the altitude and position channels, with error dynamics given by 
, 
, and similarly for 
and 
. The proposed ASOCC laws in (25)-(27), together with the adaptive gain update rules 
, 
, and 
, ensure that each attitude channel incorporates an auxiliary observer variable and a disturbance compensation term identical in structure to those used for altitude control. By constructing Lyapunov functions 
, 
, and 
analogous to (10), and applying the same two-step analysis used for the altitude subsystem (observer error convergence followed by finite-time convergence of the homogeneous error system), it follows that the conditions of Lemma 1 and Lemma 3 are satisfied for all three rotational channels. Hence, the proposed observer–controller pairs 
, 
, and 
guarantee finite-time convergence of the attitude tracking errors 
, 
, and 
, respectively. This establishes that the same finite-time stability properties proven for the altitude and position dynamics extend rigorously to the attitude dynamics, even under bounded disturbances, gyroscopic coupling, and modeling uncertainties.
Remark 4
The proposed control structure adopts a higher-order finite-time formulation that inherently reduces chattering by smoothing the control effort near the equilibrium manifold. In contrast to conventional first-order sliding mode controllers, the higher-order design maintains continuity in the control input and its derivatives while preserving finite-time convergence. This results in smoother transient behavior, reduced oscillations, and enhanced actuator durability during prolonged operation.
Convergence analysis of the overall closed-loop system
The convergence properties of the overall perching quadrotor closed-loop system shall be presented in this subsection. The entire system comprises three subsystems: altitude, position, and attitude dynamics. Every subsystem has a perturbed double-integrator structure and a specific ASOCC controller with an augmentation of a finite-time disturbance observer. The coupled observer-controller dynamics are stabilized by integrating subsystem-level Lyapunov functions and leveraging the homogeneity of the error dynamics. Let the global Lyapunov function for the coupled closed-loop system be defined as
 |
where each term is positive definite and radially unbounded with respect to its corresponding subsystem error variables. Since each 
satisfies
 |
it follows that
 |
Because all subsystem error states are decoupled in the Lyapunov sense but evolve under a homogeneous control law of negative degree, standard results on the finite-time stability of interconnected homogeneous systems apply. In particular, based on finite-time Lyapunov theory, the origin of the combined state vector 
is finite-time stable, and the settling time satisfies 
where 
is the finite settling time associated with each subsystem (altitude, position, attitude). Therefore, the closed-loop quadrotor dynamics under the proposed ASOCC controller are globally finite-time stable, guaranteeing rapid and robust convergence during the perching maneuver.
Remark 5
The proposed ASOCC framework ensures several significant closed-loop stability characteristics. Firstly, estimated observer errors converge within a finite time in all the channels of translation and rotation. Second, the corresponding dynamics of the tracking errors of altitude, position, and attitude become a homogeneous negative-degree system, which converges to the origin in finite time. The second-order continuous nature of the controller also provides continuous, chattering-free control signals, which eliminate actuator wear and enhance the overall transient behavior. Lastly, the adaptive disturbance-compensation scheme gives excellent robustness to bounded disturbances, sensor noise, gyroscopic coupling phenomena, and modeling uncertainties. A combination of these properties demonstrates that the overall quadrotor closed-loop system is stable in finite time and can be relied upon to perform well in perching and contact-based tasks.
Selection of gain parameters for controller and observer systems
The requirement of selecting the parameters 
arises for finite-time stabilization of the double-integrator system, 
. The parameters 
and 
are chosen as 
where 
to satisfy the homogeneity condition ensuring finite-time convergence. This structure is derived from the homogeneous finite-time control design frameworks proposed in41,60,61. The gain parameters 
and 
are tuned to regulate convergence speed and damping characteristics. A proportional–derivative analogy is employed to initialize these gains, enabling intuitive tuning with guaranteed stability. Hence, the classical proportional-derivative (PD) tuning rules can be employed as 
where 
(rad/s) defines the desired convergence rate and 
determines damping characteristics. The nonlinear fractional powers (1/2, 2/3) in the actual control law guarantee finite-time convergence.
The observer contains a single tunable gain parameter 
from (16). Its selection is based on the Lyapunov inequality 
which yields
 |
28 |
The corresponding settling time bound is 
which matches the time bound derived in Lemma 1 as 
with 
and 
. To satisfy a desired time bound 
, the following condition ensures convergence within 
as 
. If only the upper bound 
is known, the inequality 
. It can be used for conservative tuning. In practice, excessively large 
may result in high control effort or chattering; therefore, a balanced value should be chosen to ensure both convergence and smooth operation.
Simulation and experimental results
This section presents the effectiveness of the proposed ASOCC approach through numerical simulations, real-time experiments, and comparative analysis. The results are analyzed using standard error indices and robustness against disturbances and model uncertainties. The proposed approach is also compared with recent state-of-the-art approaches to demonstrate its superiority.
Simulation setup
The simulations are conducted using the physical parameters of the quadrotor. The system parameters and control gains utilized in the simulation are as follows. The perching quadrotor has a total mass of 
and an arm length of 
. The principal moments of inertia in the roll, pitch, and yaw axes are 
, 
, and 
, respectively. These parameters correspond to a miniature platform designed for lightweight perching and surface interaction experiments. The control gains are set to 
and 
, while the position and attitude channels (x, y, roll, pitch, and yaw) used 
and 
. The adaptive disturbance observer parameters are chosen as 
and 
. These parameters collectively ensure finite-time convergence, smooth control action, and robustness against modeling uncertainties and external disturbances.
Simulation results
The dynamic model of the quadrotor is simulated in MATLAB/Simulink to run the simulations. Two simulations are taken into consideration; Case I analyzes the performance of the system in the presence of external disturbances when the perching angle is 
, and Case II analyzes the robustness of the controller in the presence of the external disturbances, and random Gaussian noise as well as model uncertainties, when the perching angle is 
. All these scenarios serve as indicators of the proposed control strategy’s strength and flexibility in various operating conditions.
Case I: set point tracking with external disturbances
The objective of Case I is to evaluate the robustness and stability of the proposed controller under bounded external disturbances. The quadrotor is initialized at a position of 
with Euler angles 
. The desired trajectory for the perching maneuver is defined as follows. During the initial phase (
s), the reference position and the yaw angle are set to 
, 
, 
, and 
. In the subsequent phase (
s), the reference position changes to 
, 
, 
, and 
, representing the hover stage before surface engagement. Additionally, the desired pitch angle is controlled as 
during 
s to achieve the nose-down orientation required for perching contact.
In this case, we apply a set of bounded external disturbances to the system dynamics to model real-world perturbations (e.g., wind gusts or modelling uncertainties). The disturbances are randomly selected to test the controller’s capability to track the trajectory and maintain attitude stability under external perturbations and system uncertainty. Specifically, translational disturbances along the x, y, and z axes are given by 
, 
, and 
, respectively. The corresponding rotational disturbances for roll, pitch, and yaw are defined as 
, 
, and 
, where 
and 
denote small bounded offsets that represent constant model uncertainties. These disturbance profiles are used to evaluate the robustness and finite-time convergence characteristics of the proposed controller.
The quadcopter performs a perching maneuver on a stationary wall located at 
. It performs the task in three modes: staging, approaching, and perching. The three modes and the variations in x, y, and 
are illustrated in Fig. 3a.
Fig. 3.
Simulation results for Case I: perching quadrotor modes and corresponding trajectory profiles.
In staging mode, it accelerates smoothly and reaches the desired altitude (
m) in 2.13 seconds, maintaining this altitude throughout the assigned task. In the approaching mode, it slowly moves along the x-axis to reach the desired separation distance from the stationary wall. The main objective during this mode is to move slowly toward the wall and remain close to it until the next phase begins.
Next, the perching mode begins, in which the quadcopter gradually changes its orientation with respect to the wall, attempting to align vertically. This is achieved by adjusting the pitch angle, which is set at a desired value of 
. During the staging mode, the desired pitch angle changes gradually, but in the approaching mode, it reaches zero and maintains that value until the perching mode begins. The profiles of the y-position, yaw and roll angles are shown in Fig. 3b. The position and angle responses are observed to converge to zero as the quadrotor approaches the wall. This behavior occurs due to the transition from normal flight mode to perching mode at 
, marking the onset of the perching phase.
Figure 4a(top and bottom) displays the profiles of the total thrust control input and the pitch control input. Initially, the thrust increases sharply to reduce the altitude error, then gradually decreases and settles to a steady value required for stable hovering during the staging and approach phases. During the perching phase, the quadrotor requires a high torque to achieve the desired pitch of 
, resulting in a noticeable increase in the control input.
Fig. 4.
Simulation results for Case I: control input profiles and adaptive gain tuning.
The control inputs for the roll and yaw are shown in Fig. 4b. These results demonstrate the controller’s ability to produce smooth and bounded control actions during the transition to the perching mode. These minor differences in the control input occur because the quadrotor is near the wall, as desired; hence, there are slight tracking errors that require minimal corrective measures.
Figure 4cdisplays the gain of the disturbance observer. The gain parameter is dynamic and flexible, and it changes depending on the required position. At the first stage, the value of both the values of 
and 
slowly declines and approaches zero. This is because the calculation of the tracking error leads to the calculation of the value of 
, which also approaches zero over time. The gain, denoted as 
, begins at zero and rises steadily, eventually reaching a constant value, as it is also contingent on the tracking error.
Case II: set point tracking with random Gaussian noise, external disturbances, and model uncertainties
The objective of Case II is to evaluate the robustness of the proposed controller under real-world situations where the quadrotor experiences sensor noise, environmental perturbations, and variations in its physical parameters. It is initialized at a position of 
with Euler angles 
. The desired trajectory is defined in three phases. During the initial phase (
s), the desired position and yaw angle are specified as 
, 
, 
, and 
. In the next phase (
s), the reference position changes to 
, 
, 
, and 
, representing the hover stage prior to surface engagement. Finally, during 
s, the desired pitch angle is 
to achieve the nose-down orientation necessary to initiate perching contact.
To simulate realistic conditions, a white Gaussian noise (0.01r), where 
is a random number, is added to the time-varying disturbances acting on the translational and rotational dynamics. Translational disturbances along the x, y, and z axes are expressed as 
, 
, and 
, respectively. Similarly, rotational disturbances for roll, pitch, and yaw are defined as 
, 
, and 
. Here, 
and 
represent bounded model uncertainties. In addition, model uncertainties are introduced in moments of inertia as 
. A Monte Carlo simulation with 100 iterations is performed to statistically evaluate the performance of the controller in terms of accuracy and stability of the trajectory tracking.
Figure 5aillustrates the vertical and forward motions (z, x, and 
), while Fig. 5bshows the lateral motion and attitude (y, 
, and 
). These results collectively represent the staging, approaching, and perching phases. The desired trajectories confirm that the quadrotor can perch at arbitrary pitch angles.
Fig. 5.
Simulation results for Case II: position and attitude profiles.
The simulation results demonstrate that the proposed controller maintains stability and achieves finite-time convergence despite the presence of noise, disturbances, and model uncertainties, confirming its robustness and practical feasibility.
Comparative analysis
This subsection presents a comparative analysis further to validate the effectiveness of the proposed control strategy. Performance is benchmarked using standard error indices. The initial parameters and constraints are kept identical to those used in Case II for consistency. Furthermore, comparisons with existing adaptive sliding mode and observer-based control approaches are included to highlight advantages.
The proposed ASOCC approach is compared against existing control strategies that include second-order continuous control (SOCC)43, super-twisting sliding mode control (STSMC)51, terminal sliding mode control (TSMC)55, and adaptive proportional-integral-derivative (APID)35 controllers. The comparison highlights precise trajectory tracking, improved robustness against external disturbances and uncertainties, and faster convergence during the perching maneuver. Quantitative indices, such as the integral of absolute error (IAE), the integral of squared error (ISE), and the integral of time-weighted absolute error (ITAE), are used to assess and validate the controller’s performance comprehensively. Figure 5aand 5billustrate the comparative time responses of the quadrotor under these control strategies for all three maneuvering phases-staging, approaching, and perching. The comparison is based on the average outcomes from Monte Carlo simulations (100 trials), ensuring statistical consistency. The corresponding results are summarized in Tables 1 and 2 for the position and attitude subsystems, respectively.
Table 1.
Position control performance using different performance indices.
| Method | Pos. | IAE | ISE | ITAE | MSE | RMSE |
|---|---|---|---|---|---|---|
| SOCC43 | x | 1.802 | 0.163 | 1.153 | 0.097 | 0.193 |
| y | 1.583 | 0.291 | 0.985 | 0.098 | 0.140 | |
| z | 2.316 | 1.928 | 0.891 | 0.019 | 0.297 | |
| STSMC51 | x | 0.209 | 0.120 | 1.711 | 0.011 | 0.093 |
| y | 0.516 | 0.512 | 0.672 | 0.013 | 0.130 | |
| z | 0.991 | 0.981 | 0.981 | 0.073 | 0.519 | |
| TSMC55 | x | 0.418 | 0.104 | 1.095 | 0.019 | 0.093 |
| y | 0.564 | 0.542 | 0.451 | 0.013 | 0.131 | |
| z | 1.714 | 1.136 | 1.052 | 0.193 | 0.528 | |
| APID35 | x | 0.901 | 0.084 | 2.112 | 0.018 | 0.123 |
| y | 0.886 | 0.712 | 0.985 | 0.098 | 0.140 | |
| z | 1.244 | 1.311 | 0.891 | 0.192 | 0.264 | |
|
ASOCC (Proposed) |
x | 0.209 | 0.037 | 0.979 | 0.011 | 0.043 |
| y | 0.497 | 0.177 | 0.296 | 0.009 | 0.094 | |
| z | 0.899 | 0.647 | 0.585 | 0.032 | 0.180 |
Table 2.
Attitude control performance using different performance indices.
| Method | Attitude | IAE | ISE | ITAE | MSE | RMSE |
|---|---|---|---|---|---|---|
| SOCC43 |  |
1.217 | 0.110 | 1.912 | 0.064 | 0.233 |
 |
1.365 | 0.201 | 1.927 | 0.092 | 0.317 | |
 |
1.113 | 0.304 | 1.925 | 0.091 | 0.443 | |
| STSMC51 |  |
0.291 | 0.090 | 1.761 | 0.011 | 0.104 |
 |
0.293 | 0.092 | 1.982 | 0.008 | 0.088 | |
 |
0.391 | 0.131 | 0.931 | 0.009 | 0.097 | |
| TSMC55 |  |
0.311 | 0.092 | 1.602 | 0.010 | 0.092 |
 |
0.493 | 0.091 | 1.495 | 0.008 | 0.085 | |
 |
0.695 | 0.109 | 0.927 | 0.009 | 0.082 | |
| APID35 |  |
1.061 | 0.081 | 1.392 | 0.010 | 0.023 |
 |
0.913 | 0.110 | 1.991 | 0.012 | 0.076 | |
 |
0.811 | 0.081 | 1.211 | 0.010 | 0.074 | |
|
ASOCC (Proposed) |
 |
0.232 | 0.010 | 0.991 | 0.001 | 0.023 |
 |
0.211 | 0.021 | 1.461 | 0.001 | 0.033 | |
 |
0.320 | 0.044 | 0.319 | 0.002 | 0.047 |
From the numerical results, it is evident that the proposed ASOCC consistently outperforms all other methods. For the position subsystem, ASOCC achieves approximately 35–
lower IAE and 40–
lower ISE compared to STSMC and TSMC, while maintaining smoother responses and reduced overshoot. Similarly, for the attitude subsystem, the ASOCC reduces ITAE by nearly 
and RMSE by 
compared to the best-performing benchmark (SOCC). A comparison between the proposed ASOCC and existing disturbance observer-based control methods is presented in Table 3 to highlight their key distinctions and advantages.
Table 3.
Comparison of proposed and existing observer-based control strategies.
| Methods | DO + intelligent control50 | Extended DO52 | IATSMRL + ESMDO53 | DO + NMPC54 | Observer-TSMC 55 | State and disturbance observer56 | Proposed approach |
|---|---|---|---|---|---|---|---|
| Disturbance observer (DO) | Matched | Mismatched | Lumped | Matched (varying) | Multi-source (periodic/ aperiodic) | Differentiable | Matched/ lumped |
| Robust to unknown bounded disturbance | Yes | No | No | Yes | No | Yes | Yes |
| Gain tuning mechanism | No | No | No | No | No | No | Yes |
| Smooth control action | Yes | Yes | No | No | No | No | Yes |
| Finite-time convergence | Yes | No | Yes | No | No | No | Yes |
| Experimental validation | Yes | No | Yes | Yes | Yes | No | Yes |
| Application domain | Quadrotor | General Systems | Permanent magnet motor | Underwater Vehicle | Permanent magnet motor | Fully actuated systems | Quadrotor Perching |
| Perching capability | No | No | No | No | No | No | Yes |
As per the Table 3, the proposed ASOCC controller demonstrates significant improvements in smooth control action, and robustness under noisy and uncertain conditions. The overall comparative analyses confirm its superior performance over state-of-the-art approaches, highlighting its suitability for real-world quadrotor perching and maneuvering applications.
Remark 6
In contrast to earlier SOCC-based method43, the proposed ASOCC is more robust and flexible in realistic scenarios with modeling uncertainties, Gaussian noise, and rapidly changing disturbances, without any predetermined control gains and disturbance limit knowledge. Moreover, the proposed Lyapunov-based design ensures global finite-time stability and sustained control effort, thereby eliminating the chattering effects commonly observed in traditional sliding mode methods.
Real-time experimental validation
Experimental setup
The perching quadrotor includes a custom-made quadrotor chassis with a support arm on the front side and Crazyflie (version 2.0) as a flight controller board. This chassis is 3D printed using PLA material. The flight controller board is equipped with an STM32F405 ARM Cortex-M4 processor operating at 168 MHz and a 10-DoF IMU comprising a gyroscope, accelerometer, magnetometer, and barometer. Communication between the perching quadrotor and the ground station is established using the Crazyradio PA link at 2.4 GHz. For communication purposes, a Crazyradio PA dongle is used. For distance measurement and altitude estimation, a range sensor deck (VL53L1x Time-of-Flight module) is mounted on the perching quadrotor chassis. The sampling is set at 100 Hz (i.e., every 0.01 s), while the average communication delay between the ground station and onboard controller was approximately 40-60 ms due to wireless latency.
Experimental results
The Crazyflie is programmed to execute an autonomous perching maneuver on a vertical surface inclined at 
, allowing the drone to stabilize and attach after wall contact. Figure 6 shows snapshots captured during the experimental validation, illustrating the performance of the proposed controller in real-time operation. Figure 7 shows the time-domain responses for altitude (z), horizontal position (x), and pitch angle (
) during the perching sequence. The first subplot shows the altitude trajectory, where the quadrotor ascends smoothly to 1 m and maintains stable hovering. The second subplot depicts the forward translation along the x-axis, followed by a controlled descent toward the wall. At the same time, the third one presents the attitude response, where the pitch angle approaches approximately 
to achieve the wall contact configuration. Discrete sensor noise, included to emulate real-time measurement effects, appears as small stair-like variations around the reference trajectories 
, 
, and 
.
Fig. 6.
Snapshots of perching quadrotor during the real-time experiment showing sequential stages of perching maneuver.
Fig. 7.
Experimental results: Perching response at a 
inclination showing altitude (z), horizontal position (x), and pitch angle (
) trajectories with smooth attitude transition and stable wall attachment.
The experimental results confirm that the proposed controller enabled the perching quadrotor to achieve a stable perching maneuver with accurate trajectory tracking, smooth attitude transition, and strong disturbance rejection.
Remark 7
In the real-time experiment, we have met various practical complications, i.e., sensor noise, actuator saturation, and low computation effectiveness. To address these issues, a complementary filter is applied to obtain a precise estimation of states, and a low-pass filter is used to eliminate sensor noise. A thrust saturation limit is applied to decrease the actuator saturation. The issue of computation is addressed by employing an external onboard computer to achieve stable and efficient operation. All these factors contribute to the increased stability of the system and demonstrate the practical feasibility of the control algorithm.
Remark 8
In the real-time experiment, we have met various practical complications, i.e., sensor noise, actuator saturation, and low computation effectiveness. To address these issues, a complementary filter is applied to obtain a precise estimation of states, and a low-pass filter is used to eliminate sensor noise. A thrust saturation limit is applied to decrease the actuator saturation. The issue of computation is addressed by employing an external onboard computer to achieve stable and efficient operation. All these factors contribute to the increased stability of the system and demonstrate the practical feasibility of the control algorithm.
Remark 9
The proposed ASOCC assumes bounded disturbances and smooth reference trajectories and is validated under moderate environmental conditions. Extreme impact dynamics, complex surface geometries, explicit force sensing, and fixed-time convergence guarantees are not considered in the current study and will be explored in future work.
Conclusion
This paper proposed an adaptive second-order continuous control approach for a perching quadrotor in the presence of disturbances and modeling uncertainty. The removal of the reaching phase in the sliding mode control framework and the addition of a finite-time disturbance observer enable the proposed coupled observer-controller system to achieve an improved control law with enhanced robustness during contact-based maneuvers. To tune the gain parameters, a selection criterion is also provided. The ASOCC performance is verified by Monte Carlo simulation in MATLAB software and real-time experiments using a custom-built perching quadrotor. The quantitative benchmarking indicated that tracking error indices reduced by 35-55
over existing control approaches. The experiment results demonstrated that the inclusion of an inclined-wall attachment, despite sensor noise, aerodynamic disturbances, and low-cost hardware requirements, did not result in rebound, overshoot, or attitude oscillation. These results confirm that perching on vertical surfaces, which are useful in long-term aerial monitoring, is safe and reliable with the help of ASOCC. This framework could be further extended to fixed-time convergence, prescribed-time convergence, combined-force or vision-based perching choice, and collaborative perching systems for distributed inspection. Accordingly, the proposed ASOCC controller provides a viable basis for next-generation energy-efficient UAV inspection platforms, which can both autonomously interact with the surface and perform extended monitoring.
Supplementary Information
Author contributions
Sandeep Gupta and Laxmidhar Behera conceived the experiment(s), Sandeep Gupta conducted the experiment(s), Anuj Nandanwar, Narendra Kumar Dhar and Suvendu Samanta analysed the results. All authors reviewed the manuscript.
Data availability
All data generated or analysed during this study are included in this published article.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
The online version contains supplementary material available at 10.1038/s41598-026-36857-9.
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Supplementary Materials
Data Availability Statement
All data generated or analysed during this study are included in this published article.