Coordinates and orientation

To describe the spatial configuration and dynamic behavior of the tiltable quadrotor system, three coordinate frames are defined. The inertial frame is fixed to the ground and used to present the global position and orientation of the UAV. The body frame is attached to the center of mass of the vehicle and moves with it. Each rotor is assigned a local rotor frame where X_R,i points outward along the arm and Z_R,i is aligns with the thrust direction of the rotor. The quadrotor arms are symmetrically distributed in a cross configuration, and the rotors are numbered counterclockwise starting from the front-left rotor R₁, as illustrated in Fig 1.

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Fig 1. Coordinate system for tiltable quadrotor.

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The orientation of the UAV with respect to the inertial frame is described using the ZYX Euler angle sequence, denoted as , where , and correspond to the roll, pitch and yaw angles, respectively. The rotation matrix from the body-fixed frame to the inertial frame is defined as:

(1)

The orientation of with respect to is denoted as:

(2)

where the rotation matrix about the z-axis is defined as:

(3)

The position vector at the origin of in the body frame is

(4)

where l is length of the quadrotor arm.

Rigid body model

To describe the motion characteristics of the tiltable quadrotor UAV, a six degree of freedom (6-DOF) rigid body dynamics model is established using the Newton-Euler formalism. It is assumed that the UAV performs unconstrained motion, with its position and orientation defined in the inertial frame as follows:

(5)

where represents the Euler angles; is the position vector; is the velocity vector in the inertial frame; denotes the angular velocity expressed in the body frame; The matrix satisfies:

(6)

The translation dynamics of the UAV in the inertial frame are described by:

(7)

where ; is the total thrust vector generated in the body frame; denotes the lumped disturbance force, which includes modeling errors and unknown external disturbances.

Then rotation dynamics described in are:

(8)

where is the inertia matrix of the quadrotor; is the control moment generated by the rotors; stands for external disturbances and unmodeled effects.

The above equations fully describe the translational and rotational dynamics of the tilt-rotor quadrotor UAV.

Aerodynamics

The aerodynamic model of the tiltable quadrotor establishes the mapping between each rotor’s variables and the corresponding aerodynamic forces and moments applied to the UAV body. The coordinate frames and reference axes involved in the modeling process are illustrated in Fig 2.

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Fig 2. Coordinate system for tiltable quadrotor.

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Let the angular velocity and tilting angle of the i-th rotor be denoted as n_i and , respectively. Assuming that the generated thrust and torque are proportional to the square of the rotor speed, the thrust vector and the counteracting torque in the local rotor frame are defined as:

(9)

where k_t is rotor thrust coefficient; c_d is the lift-to-drag ratio; is the active tilting joint angle of the i-th rotor about its arm axis x_R,i .

To obtain the resultant aerodynamic force and moment acting at the UAV’s center of mass in the body frame , all individual forces and moments are transformed from their respective rotor frames via the corresponding rotation matrices :

(10)

(11)

Combining the above expressions, the overall aerodynamic model can be compactly represented as a linear mapping between a generalized thrust vector and the combined wrench. Therefore, this model can be written as:

(12)

where is a static allocation matrix, as shown in (13), that only depends on the geometry of the quadrotor. is defined as the vector of all the components generated by the individual rotors, as shown in (14).

(13)

(14)

Problem formulation

To simplify subsequent controller design, the complete translational and rotational dynamics of the tiltable quadrotor UAV, as derived in Sections Rigid body model and Aerodynamics, are reformulated into a unified second-order nonlinear system:

(15)

with,

(16)

where is the state vector; is the input vector; represents the gravitational term; denotes the know nonlinear terms caused by gravity and Coriolis effects; is the control effectiveness matrix; denotes the lumped disturbance vector, and satisfies , where each d_i>0 is a known upper bound of the corresponding disturbance component D_i.

Remark 2. The proposed WRBF-FONTSMC strategy effectively relaxes the standard yet conservative assumption in conventional sliding mode control (SMC) that the upper bounds d_i of the lumped disturbance (where ) must be known a priori. This is achieved by employing a WRBF neural network to provide a real-time estimate of the total disturbance, thereby eliminating the controller’s explicit dependence on the precise knowledge of d_i.

By setting the desired state , the tracking error and its derivative are defined as:

(17)

The primary control objective of this work is to design a robust 6-DOF trajectory tracking controller to stabilize the vehicle subjected to the model uncertainties and external disturbances. The tracking errors given by Equation (17) are required to converge to zero in finite-time:

(18)

Introduction to fractional calculus

Fractional calculus generalizes integer-order calculus to cases where the order of differentiation or integration is a fraction. The commonly used basic operator in fractional calculus is denoted as , and its intuitive expression is:

(19)

In the above expression, represents the order; denotes its real part; t₀ and t are the lower and upper limits of the integral operator. In this article, is used instead of .

Currently, three definitions of fractional calculus are widely used in control theory: Grunwald-Letnikov(GL), Riemann-Liouville (RL), and Caputo [42]. This article adopts the Grunwald-Letnikov(GL) definition, which is defined as follows.

Definition 1. The Grunwald-Letnikov fractional calculus is defined as:

(20)

where [x] represents the integer part of x, h is the calculation step size, and is the binomial coefficient. The expression for the binomial coefficient is as follows:

(21)

Property 1 [53]. The fractional-order integration or differentiation calculus is a linear operation:

(22)

Property 2 [53]. For and , the following equality holds for the fractional derivative operator with

(23)

Remark 3. This work utilizes the Grünwald-Letnikov (GL) definition of the fractional derivative primarily for its computational advantages in digital control. The GL formulation is inherently discrete, expressed as a convolution that can be efficiently computed online using a recursive algorithm. This approach leverages a finite history of past values, significantly reducing the computational burden at each time step. In contrast, the Caputo definition often requires the more intensive numerical evaluation of an integral. For the real-time control of our tilt-rotor UAV, where the fractional-order derivative in the sliding surface (Eq 24) must be updated rapidly, the GL definition provides a more feasible and efficient implementation without sacrificing the performance benefits of fractional-order control.

FONTSM controller design

To ensure accurate and fast tracking under the presence of external disturbances and model uncertainties, a FONTSMC scheme is developed for all six degrees of freedom of the tiltable quadrotor system. According to [54], each channel , the sliding surface is defined as:

(24)

where is the fractional order, is a designed constant, and with p, q being odd positive integers, .

By differentiating Equation (24) with respect to time and utilizing Equations (7) and (17), the following expression is obtained:

(25)

The exponential reaching law is introduced and designed as follows to prevent singularity issue:

(26)

where are design parameters that control the convergence speed and damping of the system.

Remark 4. The exponential reaching law is selected considering the integrated control architecture. The primary role of disturbance rejection is delegated to the WRBF neural network, which accurately estimates and compensates for lumped uncertainties in real-time. Consequently, the reaching law is primarily responsible for governing the convergence dynamics, a task for which the exponential law is well-suited due to its simplicity and effectiveness. This approach, combined with the continuous sigmoid function and the chattering-suppression properties of fractional-order calculus, forms a comprehensive strategy that ensures smooth and robust performance without the added complexity of higher-order algorithms like super-twisting.

By combining Equation (25) with the fractional-order exponential reaching law Equation (26), the control law is derived as follows:

(27)

where is the estimated disturbance provided by the WRBF neural network (to be detailed in Section Wavelet RBF Neural Network.

Finally, the full control input vector is composed as:

(28)

Remark 2. The discontinuity of the function often causes significant chattering issues [55]. To resolve this, we propose substituting the function with a continuous function. This substitution effectively reduces chattering [56]. The definition of the function is provided below:

(29)

Here, b is an adjustable parameter that controls the steepness of the sigmoid function, allowing for fine-tuning of the control input’s smoothness.

Wavelet RBF neural network

To estimate the unknown lumped disturbance in real time, an adaptive Wavelet Radial Basis Function Neural Network (WRBFNN) is introduced as the core disturbance observer. The network output serves as an estimate of , which is a directly injected into the control law for active cancellation

The WRBFNN is a three-layer feedforward network composed of an input layer, hidden layer, and output layer, as shown in Fig 4.

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Fig 4. Structure of WRBF neural network.

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Input Layer: The input vector is , representing the the position and attitude tracking errors and their derivatives. This choice provides the network with direct access to the system’s dynamic performance deviation, enabling it to generate a state-dependent disturbance estimate.

Hidden Layer: Each neuron utilizes a Mexican Hat wavelet function as its activation:

(30)

where is the center and b_i is the scale parameter for the i-th neuron. Compared to the conventional Gaussian function, the Mexican Hat wavelet offers superior local feature extraction capabilities and faster convergence, leading to enhanced approximation accuracy and noise robustness. [57]

Output Layer: The output of the network is the estimated lumped disturbance vector, calculated as a linear combination of the activated wavelet functions:

(31)

This WRBFNN structure leverages the Mexican Hat wavelets as its activation function, building upon the framework established in [57]. In our work, this network is specifically tailored and integrated as an adaptive disturbance observer within the control loop. Here, is the estimated weight matrix; is the vector of Mexican Hat basis functions; and represent the estimated force and moment components, respectively.

By integrating the estimated disturbance from Equation (31) into the FONTSMC control law, we derive the complete control input as:

(32)

This six-dimensional control input is composed of desired force vector and moment . They are respectively expressed as:

(33)

(34)

where , are estimated force and moments vector; are the desired accelerations for position and attitude.

Stability analysis

Lemma 1. [58] Suppose there exists a positive definite Lyapunov function V(x) that satisfies the inequality where and 0<k₃<1. Let denote the initial value of V(x). Then the system state converges to zero in finite time, and the upper bound of the convergence time T is given by:

(35)

Theorem 1. For the nonlinear system described by Equation (7), under the proposed control law (Equation (32)) which integrates the FONTSMC with WRBFNN using the Mexican Hat basis function, all system states converge to zero in finite time.

Proof of Theorem 1. Define the Lyapunov function for each channel as:

(36)

where s_i is the sliding surface, is the weight estimation error, and is a positive constant.

Taking its derivative yields:

(37)

Substituting the control law (Equation 32) into the dynamic expression for , we get:

(38)

where denotes the approximation error of the WRBFNN, and is bounded.

Substituting (38) into (37) gives:

(39)

Choose the adaptive update law as:

(40)

Then, the Lyapunov derivative becomes:

(41)

Assuming , , and bounded, it follows that , thus satisfying the Lyapunov stability condition. Therefore, the closed-loop system is globally asymptotically stable.

To establish finite-time convergence, consider the dynamics of the sliding variable s_i. From (38) and the boundedness of , there exists a positive constant such that . Since is a smooth approximation of satisfying (where is small), and noting that , the reaching dynamics can be upper bounded by:

(42)

Defining the positive constants: , we obtain the differential inequality:

(43)

Equation (43) corresponds to a standard finite-time stable system [58]. By integrating this inequality, the finite-time settling bound of s_i can be derived as:

(44)

Hence, the sliding variable s_i converges to zero within the finite time T_s. Once the system reaches the manifold s_i = 0, the sliding surface dynamics reduce to:

(45)

which satisfies the finite-time convergence condition according to Lemma 1. Therefore, both e_i and converge to zero in finite time.

Remark 5. The above proof demonstrates that the proposed WRBF-FONTSMC scheme guarantees finite-time stability. The upper bound of the settling time T_s for s_i to reach zero from its initial value s_i(0) is explicitly given by:

(46)

where denotes the upper bound of the neural network approximation error . This expression provides a conservative estimate of the convergence speed, which can be tuned through the controller parameters k_i, , and . During the reaching phase, is slowly varying and can be approximated as constant, ensuring the analytical validity of (46).

Control allocation

In the context of control allocation, the six-degree-of-freedom (6-DOF) control wrench is mapped to eight actuator commands. This problem is typically challenging due to the non-uniqueness of solutions arising from the system’s over-actuation. Furthermore, without proper selection, the allocation matrix can nonlinearly depend on the controllable inputs, necessitating a computationally expensive optimization process. To address these challenges, we propose a linear solution that first transforms the forces and moments generated by each rotor into vertical and lateral components, as shown in (12). Since the allocation matrix is static, the vertical and lateral components can be efficiently computed by inverting (12) as follows:

(47)

where is the Moore–Penrose pseudo-inverse matrix of .

(48)

And then the tilting angles and motor speeds can be computed directly.

(49)

This approach allows for the utilization of a static allocation matrix that remains independent of the controllable inputs, thereby enabling a computationally efficient solution. By eliminating the need for nonlinear optimization, the method significantly reduces the computational complexity typically associated with control allocation problems.

To ensure that the control allocation results are physically realizable, actuator constraints were explicitly modeled in the simulation. The tilting servo angles of each rotor were restricted to , consistent with the mechanical range of the tilt-rotor joints. The generated thrust commands were bounded as for vertical lift, and the lateral thrust components were saturated within to reflect aerodynamic limitations. Similarly, the control moments were constrained by to guarantee feasible torque generation.

These limits were implemented via nonlinear saturation functions applied after the Moore–Penrose pseudo-inverse allocation. Whenever the commanded values exceeded their physical boundaries, they were projected back into the admissible range.

Parameter tuning

Given that the control performance in practical applications is susceptible to factors like model uncertainties and actuator limitations, the parameter set must be carefully engineered to ensure reliable operation. In this section, we elucidate the function of each parameter and offer a set of well-tuned, suggested values that have been verified to yield robust performance across various flight scenarios.

1) Selections of p, q, : These parameters define the fractional-order non-singular terminal sliding surface in Eq (24). The ratio ensures finite-time convergence without singularity, with larger values accelerating convergence but potentially increasing control effort. Parameter weights the terminal attractor term, where increasing enhances dynamic response but may amplify noise. Recommended values are p = 9, q = 7, and per channel requirements.

2) Selections of k, : These parameters govern convergence and disturbance rejection. Gain k controls sliding variable convergence, where larger values accelerate response but may cause oscillations. Switching gain provides robustness but causes chattering if excessive. The sign function is replaced by a continuous sigmoid function Eq (29) to mitigate this effect. Recommended ranges are k = 10 and .

3) Selections of : The fractional order in Eq (24) introduces memory effects that smooth control signals and reduce chattering. Higher values improve smoothness but may slow initial response. Empirical results suggest provides an effective balance between response speed and signal smoothness.

4) Selections of c_ij, b_j, : The WRBF neural network employs Mexican Hat wavelets Eq (30) for disturbance estimation in the control law Eq (27). The centers c_ij are uniformly distributed over [–1, 1] to cover the normalized error range. Widths b_j = 3 balance localization and generalization, while the learning rate ensures stable weight adaptation in Eq (39) across all channels.

Theoretical comparison and analysis

Having established the finite-time stability and boundedness of the proposed WRBF-FONTSMC controller, this section further compares its theoretical characteristics with other existing control schemes. A detailed comparison of key control attributes is summarized in Table 1.

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Table 1. Theoretical comparison among different control schemes.

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Table 1 provides a theoretical comparison of five controllers, including NN-PID, NTSMC, FONTSMC, RBF-FONTSMC, and the proposed WRBF-FONTSMC. The NN-PID offers simple implementation but limited robustness, while NTSMC achieves finite-time convergence at the cost of chattering due to its discontinuous control law. The FONTSMC introduces fractional-order dynamics to improve convergence speed and memory characteristics. The RBF-FONTSMC enhances disturbance compensation by using a Gaussian RBF neural estimator, though its global approximation can lead to slower adaptation in nonstationary conditions. In contrast, the proposed WRBF-FONTSMC integrates a wavelet-based neural estimator with a fractional-order nonsingular terminal sliding surface and a smooth sigmoid reaching law. This synergistic design achieves fast finite-time convergence, improved robustness, and chattering-free performance, making it more suitable for six-degree-of-freedom UAV control under complex disturbances.

Fixed-attitude trajectory tracking

As a fundamental test of tracking performance, this experiment evaluates the controller’s ability to follow a 3D trajectory while maintaining a fixed attitude. The desired trajectory is:

The proposed WRBF-FONTSMC controller demonstrates exceptional tracking performance in this basic test. As shown in Figs 6 and 7a, WRBF-FONTSMC accurately tracks the desired trajectory. The position error in Fig 7b converge rapidly within 2 seconds. Fig 7c shows perfect attitude maintenance at , with attitude errors remaining near zero in Fig 7d. The control forces and moments, presented in Fig 7e and 7f, display smooth profiles with no noticeable high-frequency chattering, highlighting the combined advantages of fractional-order operators and neural network compensation. Accordingly, the sliding surfaces in Figs 7g and 7h converge to zero within about 2 seconds and remain stable afterwards. These results collectively confirm that the WRBF-FONTSMC controller achieves fast convergence, minimal chattering, and strong robustness in both position and attitude control.

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Fig 6. 3D trajectory tracking.

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Fig 7. Trajectory tracking with fixed attitude.

a: Position tracking. b: Position tracking error. c: Attitude tracking. d: Angular tracking errors. e: Control force. f: Control moment. g: Position sliding surface. h: Attitude sliding surface.

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Based on the quantitative analysis in Table 4, the WRBF-FONTSMC controller demonstrates outstanding performance. In terms of convergence speed, all channels achieve convergence within 1.8 seconds, with the Z-axis responding particularly rapidly (0.0003 s). Regarding control quality, the overshoot in all channels remains below 1%, indicating a smooth and oscillation-free system response. Furthermore, the controller achieves remarkably high steady-state accuracy: the steady-state position error reaches as low as the order of 10⁻⁴ m, while the steady-state attitude error remains stable at the order of 10⁻³ rad. This series of data fully confirms that the control scheme ensures both rapid dynamic response and high control precision and stability.

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Table 4. Key performance indexes of fix-attitude trajectory tracking.

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Fixed point rotation

The tiltable quadcopter is capable of omnidirectional flight. To highlight its advantages over the traditional quadcopter design, a fixed-point rotation experiment is conducted. This experiment primarily examines the attitude tracking ability of the quadcopter during hovering. The simulation results are shown in Fig 8. The reference trajectory is:

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Fig 8. Fixed-point rotation performance.

a: Position tracking. b: Position tracking error. c: Attitude tracking. d: Angular tracking errors. e: Control force. f: Control moment.

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The proposed WRBF-FONTSMC controller demonstrates exceptional decoupling control capability in this challenging scenario. As shown in Fig 8a, WRBF-FONTSMC accurately maintains the designated position at [0,0,2] meters while performing simultaneous roll and pitch rotations following radians. The position tracking errors in Fig 8b remain within strict bounds of 0.02 m throughout the maneuver. Fig 8c shows precise attitude tracking of the reference commands, with attitude errors maintained within radians in Fig 8d. The control forces and moments in Fig 8e, 8f exhibit well-regulated characteristics without excessive actuation, demonstrating effective management of coupled dynamics through the integrated control framework.

To quantitatively assess the controller’s decoupling performance, key metrics summarized in Table 5. The controller exhibits notably fast convergence across all degrees of freedom, with convergence times not exceeding 1.5 seconds. In particular, the roll, the pitch, and yaw channels each stabilize within 0.3 seconds, indicating excellent dynamic response in attitude control. All overshoot values remain below 1.6%, reflecting well-damped and smooth transients. In terms of steady-state accuracy, the position errors in the x, y, and z axes are consistently on the order of 10⁻⁴ m, demonstrating high-precision tracking performance. Meanwhile, the steady-state attitude error remains stable on the order of 10⁻¹ rad. In robotic dynamic control, this level of accuracy, combined with rapid convergence characteristics, is fully adequate to meet the core requirements for attitude stabilization in various complex tasks.

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Table 5. Key performance indexes of fix point rotation.

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Subsequent simulation experiments further investigate the disturbance rejection and robustness of the WRBF-FONTSM controller.

Anti-interference test

This study establishes a composite wind field environment based on the Dryden turbulence model, generating disturbance signals that conform to atmospheric turbulence characteristics through shaped filtering of Gaussian white noise. It systematically evaluates the trajectory tracking accuracy, disturbance rejection capability, and dynamic response characteristics of five controllers under wind disturbance conditions. The experimental setup adopts a composite working condition combining a steady wind field of with random turbulence. Key turbulence parameters (including turbulence intensities L_u, , L_w and turbulence scales , , ) are configured strictly according to the transfer function model in reference [59], with specific parameters provided in Table 6. The time-domain characteristics of the generated turbulent wind field are shown in Fig 9, providing a reliable disturbance environment benchmark for controller performance evaluation.

(50)

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Fig 9. Turbulent wind field diagram.

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Table 6. Turbulent wind field parameters. Key parameter settings used for modeling the turbulent wind field.

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The reference trajectory is:

This simulation conducted a systematic performance evaluation of five typical controllers (NTSMC, FONTSMC, RBF-FONTSMC, WRBF-FONTSMC, and NN-PID) under combined steady-state wind field and turbulent conditions. The trajectory tracking results shown in Figs 10 and 11a demonstrate that: NN-PID exhibits significant steady-state error with incomplete convergence; NTSMC demonstrates initial deviation and slow convergence; both FONTSMC and RBF-FONTSMC achieve effective tracking but with overshoot, though RBF neural network integration significantly improves FONTSMC’s setting time. The proposed WRBF-FONTSMC achieves optimal comprehensive performance with the fastest settling time, minimal overshoot, and highest tracking accuracy.

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Fig 10. 3D trajectory tracking.

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Fig 11. Trajectory tracking under the action of wind field.

a: Position tracking. b: Position tracking error. c: Attitude tracking. d: Angular tracking errors. e: Control force. f: Control moment.

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Further analysis of position tracking errors in Fig 11b confirms these findings. WRBF-FONTSMC maintains position errors within meters with minimal oscillation, reducing error peaks by 50% compared to RBF-FONTSMC. Attitude control performance (Fig 11c) shows NN-PID produces smooth responses but with steady-state error; NTSMC confines errors within but with significant chattering; FONTSMC exhibits oscillations during gusts, which RBF compensation alleviates but doesn’t eliminate; while WRBF-FONTSMC achieves optimal tracking accuracy with smooth responses. The attitude error statistics in Fig 11d demonstrate WRBF-FONTSMC’s superior performance, strictly limiting errors to .

The control inputs in Figs 11e and 11f reveal distinct performance characteristics. In terms of smoothness, both the NN-PID and the proposed WRBF-FONTSMC controllers generate signals free of chattering, while the NTSMC and other benchmark controllers exhibit significant or minor chattering, respectively. Regarding performance, the NN-PID’s capabilities are limited by its basic structure, making it far less effective than the WRBF-FONTSMC. Furthermore, the compensatory effect of the RBF and WRBF networks on the FONTSMC controller is clearly visible in Fig 11f around 5s and 8s. These observations collectively validate the superior control quality of the WRBF-FONTSMC approach.

Quantitative performance (Table 7) provide additional validation. WRBF-FONTSMC achieves the optimal RMSE of 0.2923, showing improvements of 91.2%, 70.0%, 32.3%, and 9.4% over NN-PID, NTSM, FONTSM, and RBF-FONTSM respectively. Its ITAE value of 3.7348 represents reductions of 97.4%, 78.2%, 51.7%, and 39.1% compared to the other controllers. While WRBF-FONTSMC’s ISU index 704 is slightly higher than FONTSM’s optimal value, it demonstrates reductions of 1.7% and 18.5% compared to NN-PID and NTSM respectively. Most notably, WRBF-FONTSMC maintains similar control energy consumption to RBF-FONTSM while achieving significantly improved control accuracy, highlighting the unique advantages of wavelet radial basis functions in control performance optimization.

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Table 7. Comparative analysis of control performance metrics.

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To evaluate the performance advantages of the Wavelet Radial Basis Function (WRBF) over conventional RBF networks, we analyzed output responses and weight evolution in position and attitude channels (Figs 12 and 13), using 10 basis nodes for both networks.

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Fig 12. WRBF output and weight evolution analysis.

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Fig 13. RBF output and weight evolution analysis.

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In the position channel, both networks achieved comparable tracking accuracy within [–10,0], though RBF exhibited chattering. WRBF demonstrated superior weight efficiency: its weights spanned 6V (–4V to +2V) versus RBF’s 8V (–6V to +2V), representing a 25% reduction in variation range, indicating more efficient parameter utilization.

The performance gap widened significantly in the attitude channel. RBF output oscillated violently within [–8, 2] with aggressive weight variations in [–2, 1]. In contrast, WRBF suppressed oscillations to [–0.5, 0.5] (90% reduction) and maintained weights in a narrower [–1, 1] range with faster convergence and smoother evolution. These results validate WRBF’s superiority in adaptive control of complex nonlinear systems.

Computation efficiency analysis

The computational load of the proposed WRBF-FONTSM controller primarily comes from three components: the fractional-order sliding mode control law with O(n) complexity, the WRBF neural network requiring O(Mn) operations, and the adaptive update law with O(Mn) complexity. With parameter configuration (n=6, M=10), the average execution time per control step measured in MATLAB/Simulink (Intel i7) is 0.24 ms, which is well below the typical 5 ms sampling period for UAV systems. This demonstrates sufficient computational margin for real-time implementation, while the linear complexity of all components ensures good scalability for embedded platforms.