Simulation and control design for uav Simulation and control design for uav Simulation and control design for uav
INTRODUCTION
Overview
1.1.1 Reason for choosing the topic
Unmanned aerial vehicles (UAVs), or drones, have become essential across various industries due to their ability to operate autonomously or be remotely controlled Their versatility allows for applications in agriculture, logistics, surveillance, infrastructure inspection, and emergency response UAVs excel in accessing challenging or hazardous areas, making them invaluable for tasks like crop monitoring and precision spraying in agriculture, which optimize yields and conserve resources In the energy sector, drones enhance safety and efficiency by inspecting power lines, wind turbines, and pipelines, reducing the reliance on manual inspections in dangerous environments.
UAVs are revolutionizing traditional workflows and fostering innovation across various sectors Their significant impact highlights the technology's ability to address challenges and generate new opportunities, establishing them as essential elements in contemporary industries and beyond.
Figure 1.1 Image of the drone in working mode
Quadrotors have gained significant popularity among Unmanned Aerial Vehicles (UAVs) due to their unique advantages over other configurations like hexacopters and fixed-wing drones Their compact size and lightweight design make them ideal for operations in confined spaces, such as indoor inspections and urban surveillance Additionally, quadrotors are more affordable to manufacture and maintain, making them accessible to a diverse range of users, including enthusiasts and commercial enterprises Known for their durability and reliability, quadrotors offer a balance of stability and maneuverability, suitable for various missions like aerial photography and cargo delivery Their simple mechanical structure reduces the risk of failure, enhancing their reliability in critical operations Furthermore, the straightforward design facilitates analysis and testing, allowing engineers to easily modify configurations for research and development Overall, quadrotors are versatile tools that are central to innovation in UAV technology, proving their value across academic, industrial, and recreational applications.
Research methods
From a practical perspective, quadrotors are particularly well suited for operations in confined environments, such as greenhouses, warehouses or urban environments,
Quadrotors are highly effective in tight spaces due to their compact design and precise maneuverability, making them ideal for tasks that require high accuracy, such as inspecting delicate structures and conducting targeted agricultural treatments However, their deployment faces challenges, particularly from unpredictable operating conditions like wind gusts and turbulence, which can affect stability and flight performance To overcome these obstacles, the development of effective path tracking controllers is essential Path tracking is crucial for autonomous flight, ensuring that the quadrotor adheres to a predetermined trajectory while maintaining stability This involves two key control elements: position tracking control, which keeps the quadrotor on its designated path in three-dimensional space, and attitude tracking control, which adjusts the quadrotor's orientation and angular motion to align with the required trajectory.
Various control strategies have been introduced to tackle challenges and achieve precise tracking performance Among these, PID control has been extensively analyzed
The evaluation of the closed-loop trajectory under PID control highlights the importance of tuning control gains for optimal performance Real-time experiments demonstrate the effectiveness of PID-based control, sliding mode controllers, and an innovative approach to enhance system response.
To ensure consistent power usage in experiments, control gains for all controllers are systematically adjusted Robustness testing is conducted by introducing disturbances, like reduced actuator force, to evaluate performance Among the various methods tested, a nonlinear PID-based algorithm stands out for its superior tracking accuracy.
Robust control techniques, as detailed by Aykut C Satici et al (2016), aim to design an L1-optimal controller for quadcopter UAVs, enabling precise tracking of position and orientation through propeller-generated forces In ideal conditions with accurate parameters and no noise, the tracking error diminishes exponentially to zero Even in the presence of parameter uncertainties and noise, the controller guarantees exponential error reduction while adhering to L1-optimal criteria Additionally, it minimizes the L∞ gain response to disturbances, with experimental results supporting its effectiveness.
4 its disturbance rejection capabilities When compared to a prior robust nonlinear controller, the L1-optimal controller achieves a reduced average error margin
An adaptive terminal integral sliding mode (AITSM) control scheme with an uncertainty observer (UO) has been developed for automotive electronic throttle (AET) systems, enhancing error convergence and robustness against uncertainties such as friction and gear play This innovative approach utilizes a terminal integral sliding surface, ensuring the closed-loop system initiates on this surface, while dynamically tuning sliding parameters through adaptive laws based on Lyapunov theory for stable performance across various commands To reduce control oscillations from high switching gain, the SM-based UO provides real-time uncertainty estimates and incorporates pre-compensation Additionally, model-free robust discriminators (MREDs) effectively calculate angular velocity and acceleration despite measurement noise Experimental results on a DSP-based AET plant confirm the controller's effectiveness, showcasing significant reductions in RMS and MAX errors compared to traditional methods, especially during disturbance rejection with trapezoidal and sinusoidal commands.
A recent study investigates a robust end-to-end integral sliding mode (AITSM) control scheme for AET systems that integrates uncertainty observation (UO) This innovative design enhances tracking accuracy and system resilience against nonlinear challenges such as spring-back and gear play By dynamically adjusting sliding parameters through adaptive laws, consistent performance is maintained The sliding mode-based UO effectively mitigates jitter caused by high switching gains by providing real-time uncertainty estimation and compensation Additionally, MREDs significantly improve measurement accuracy in noisy environments When validated on a DSP-based AET system, the proposed control scheme demonstrated lower RMS and MAX tracking errors compared to alternative controllers, underscoring its superior performance.
A robust discrete terminal sliding mode recurrent controller has been developed for nonlinear positioning systems, significantly improving transient response and resilience to parameter variations and nonlinear disturbances This controller employs a recurrent control mechanism to effectively manage periodic uncertainties and incorporates a phase compensator for enhanced high-frequency tracking Rigorous analysis confirms the controller's stability and finite-time convergence, while experimental results on nonlinear systems showcase its exceptional disturbance rejection and tracking capabilities.
A modified repetitive sliding mode controller has been developed for uncertain linear systems, integrating a discrete-time sliding mode controller with repetitive control to enhance response speed and robustness Stability analysis supports the effectiveness of this approach.
5 bounded control and error convergence, while simulation results show effective reference tracking and disturbance rejection under parameter variations
Fixed-time control strategies enhance robotic collective behaviors by introducing innovative interaction rules based on acute-angle tests (AATs) This AAT-based method allows for interactions with specific distant neighbors while disregarding unnecessary close ones, significantly improving scalability and robustness compared to traditional neighbor-based rules.
A spring-like controller guarantees collision-free scattering, and a fixed-time energy consensus function enhances system stability Additionally, a co-optimization problem refines communication and collective behaviors Simulations confirm the effectiveness of this approach in improving coordination and stability.
Adaptive control utilizes fuzzy logic systems to approximate nonlinear functions in switched uncertain systems, employing adaptive backstepping and Lyapunov-based switching laws for smooth transitions and bounded error convergence Modified switching laws are implemented to avoid rapid switching or Zeno behavior, and validation through various examples showcases the effectiveness of this approach.
Neural network-based control employs a leader-following consensus approach for nonlinear multi-agent systems, effectively addressing actuator errors By utilizing adaptive neural networks, this method dynamically approximates unknown nonlinear functions and uncertainty bounds, ensuring robust fault-tolerant control The Lyapunov stability theorem supports the boundedness of signals and convergence of errors Simulations involving forced pendulum systems demonstrate the controller's exceptional tracking and disturbance rejection abilities, showcasing the rapid response and zero-error convergence of sliding mode control in nonlinear applications.
Numerous studies highlight the effectiveness of sliding mode control (SMC) in managing quadrotor systems, particularly through the implementation of second-order sliding mode control (2-SMC) for small UAVs The design of the switching slip distributor, which is complex due to its nonlinear characteristics, requires careful selection of specific coefficients The quadrotor's dynamic model is divided into a fully actuated subsystem and an under-actuated subsystem In the fully actuated subsystem, the slip distributor integrates position and velocity tracking errors from a single state variable, resulting in two coefficients Conversely, the under-actuated subsystem employs a linear combination of position and velocity tracking errors from two state variables, yielding four coefficients Nonlinear coefficients are established through Hurwitz stability analysis, while Lyapunov theory is utilized to maintain system stability by ensuring the trajectory remains within the sliding mode.
6 surfaces Simulations validate the effectiveness of this method in achieving constrained tracking and positioning
A sliding mode disturbance observer (SMDO) has been developed for vertical takeoff and landing (VTOL) aircraft, particularly enhancing the robust control of small quadrotors This innovative control technique effectively addresses external disturbances and model uncertainties without the need for high computational resources or control gains, thus minimizing extensive pre-flight analysis and reducing design costs The multi-loop, multi-timescale SMDO controller achieves robust position and attitude control by utilizing disturbance bounds Simulations using a six-degree-of-freedom model demonstrate the controller's capability to manage various disturbances, including wind, collisions, actuator failures, and model uncertainties, ensuring the quadrotor's overall robustness.
Integral sliding mode control (ISMC) is a robust method introduced to enhance quadrotor tracking by eliminating the reaching phase and effectively addressing model uncertainty and external disturbances This approach utilizes an inner-outer loop structure, where reference signals for pitch and yaw angles are generated in the outer loop, while the inner loop employs ISMC for precise tracking of positions and angles Lyapunov stability analysis demonstrates the controller's ability to mitigate disturbances and uncertainties Additionally, ISMC is applied to a heterogeneous multi-agent system (MAS) comprising a quadrotor and a two-wheeled mobile robot (2WMR), successfully solving a consensus problem while maintaining a spanning tree Experimental results further validate the controller's performance.
Layout of the thesis
This project enhances existing studies by developing a sliding mode control (SMC) scheme integrated with a Radial Basis Function Neural Network (RBFNN) The innovative control method features adaptive compensation to effectively manage externally induced disturbances, ensuring robust system performance The quadrotor, chosen for its dynamic and nonlinear characteristics, serves as an ideal platform for exploring advanced control techniques Its high maneuverability, versatility, and growing importance across various industries further validate its selection as the primary experimental platform for this research Key contributions of this study are summarized as follows.
Chapter 2 will delve into the basic concepts needed to understand and analyze quadrotor systems This chapter introduces the basic principles, considers the relevant coordinate systems, and builds a mathematical model of a quadrotor
Chapter 3 presents some typical related problems of quadrotor, introduces radial symmetric network, thereby designing position controller and state controller, and calculating stability by considering Lyapunov candidate function
Chapter 4 presents some simulation scenarios to verify and compare the results of the above-mentioned controller The parameters of the quadrotor model used are based on the Quanser aircraft model.
Conclusion of chapter 1
Chapter 1 provides an overview of the practical applications of unmanned aerial vehicles (UAVs), highlighting the different types of UAVs and their roles in various domains This chapter introduces the UAV and explores its relevance in modern
Chapter 1 provides a comprehensive overview of the study's scope and objectives, establishing essential insights that inform the algorithmic model in Chapter 2 and the controller design in Chapter 3 By outlining the context and goals, this chapter sets a structured framework for tackling the challenges of UAV control and operation, creating a clear roadmap for both research and practical applications.
MATHEMATICAL MODEL
Principal of operation
A quadrotor, commonly known as a quadcopter, is a helicopter variant featuring four engines and propellers Its structure includes a body frame with four rotors arranged symmetrically around the center, generating angular speed 𝜔 𝑖 and force 𝐹 𝑖 from each motor (i = 1, 2, 3, 4) The quadrotor's mass is denoted as m, while g represents the acceleration due to gravity The roll, pitch, and yaw angles describe the quadrotor's rotational movements, as illustrated in Figure 2.1.
The quadrotor operates by varying the speeds of four brushless motors, enabling control over its roll, pitch, and yaw angles To adjust the roll angle, the speeds of rotors 2 and 4 are symmetrically modified, while rotors 1 and 3 are adjusted for pitch control Yaw angle control is achieved through the counter-rotation of rotor pairs 1-3 and 2-4, which creates opposing torque and stabilizes the quadrotor's orientation This steering mechanism is visually represented in the accompanying diagram (Figure 2.2).
The quadrotor steering principle is illustrated through the figure below, the level of darkness on the figure represents the rotational speed of the large or small quadrotor motor respectively:
Standard notations
Table 2.1 Table of symbols used in the thesis
Symbols Description m(kg) Total mass g(m/𝑆 2 ) Gravity Force
𝑘 𝑥 𝑘 𝑦 𝑘 𝑧 (N s/m), 𝑘 𝜙 𝑘 𝜃 𝑘 Ψ (N s/rad) Drag coefficients l (m) Distance Between The Rotor And the body coordinate origin
𝜏 𝜙 , 𝜏 𝜃 , 𝜏 𝛹 Torque of pitch axis, roll axis and yaw axis
𝑑 𝑥 (t), 𝑑 𝑦 (t), 𝑑 𝑧 (t), 𝑑 𝜙 (t), 𝑑 𝜃 (t), 𝑑 𝛹 (t) Represent the time-dependent external noise, environmental factors
𝜔 𝑘 The angular velocity of each motor
𝑒⃗ 𝑥 , 𝑒⃗ 𝑦 , 𝑒⃗ 𝑧 Unit vector in a coordinate system
K Proportional factor (or amplification factor) s Laplace transform
𝑣 𝑘 Fan speed or control signal for motor 𝑘
D, 𝑑 𝑧 External interference d Unknown positive constants
∆(x) Continuous and differentiable intercept function
V Quantity representing the Lyapunov function
Mathematical expression
To analyze the quadrotor's movement in space, we utilize two coordinate systems: the inertial coordinate system (fixed to the Earth) denoted as 𝑂 𝑥𝑦𝑧, and the non-inertial coordinate system (which is moving).
To establish the relationship between the coordinates in the two referenced coordinate systems at point G, which is located at the center of gravity of the quadrotor, we execute a series of sequential rotations.
Consider the rotation from the 𝑂 𝑥𝑦𝑧 axis system to the 𝑂 𝑥 1 𝑦 1 𝑧 1 axis system with the rotation angle ψ around the z axis as illustrated in (Figure 2.3.): y
(2.1) also can be rewritten as follows
Therefore, the cosine orientation matrix represents the relationship between the unit vectors between the system 𝑂 𝑥𝑦𝑧 axis with 𝑂 𝑥1𝑦1𝑧1 axis system when rotating around z axis with angle 𝛹 as follows:
Consider the rotation from the 𝑂 𝑥𝑦𝑧 axis system to the 𝑂 𝑥 2 𝑦 2 𝑧 2 axis system with the rotation angle 𝜙 around the z axis as illustrated in (Figure 2.4.):
Therefore, the cosine orientation matrix represents the relationship between the unit vectors between the system 𝑂 𝑥𝑦𝑧 axis with 𝑂 𝑥 2 𝑦 2 𝑧 2 axis system when rotating around z axis with angle 𝜙 as follows:
Consider the rotation from the 𝑂 𝑥𝑦𝑧 axis system to the 𝑂 𝑥 3 𝑦 3 𝑧 3 axis system with the rotation angle 𝜃 around the z axis as illustrated in (Figure 2.5.):
Therefore, the cosine orientation matrix represents the relationship between the unit vectors between the system 𝑂 𝑥𝑦𝑧 axis with 𝑂 𝑥 3 𝑦 3 𝑧 3 axis system when rotating around z axis with angle 𝜃 as follows:
After executing the three sequential rotations, the resulting coordinate system (O x₃ y₃ z₃) aligns its unit vectors with those of the coordinate system (O B x_B y_B z_B) Consequently, the matrix R serves as the coordinate transformation matrix that encapsulates the rotational relationship between these two coordinate systems.
The non-inertia associated with the quadrotor to the inertial coordinate system (also known as the rotation matrix) is determined by:
H 2 = sin(𝜙) sin(𝜃) cos(𝛹) − cos(𝜙) sin(𝛹)
H 3 = cos(𝜙) sin(𝜃) cos(𝛹) + sin(𝜙) sin(𝛹)
K 1 = cos(𝜃) sin(𝛹) sin(𝜙) sin(𝜃) sin(𝛹)
K 2 = sin(𝜙) sin(𝜃) sin(𝛹) + cos(𝜙) cos(𝛹)
K 3 = cos(𝜙) sin(𝜃) sin(𝛹) − sin(𝜙) cos(𝛹)
Then the rotation R is an orthogonal matrix, so 𝑹 −1 = 𝑹 𝑇
The absolute position of the quadrotor in the 𝑂 𝑥𝑦𝑧 coordinate system is determined by P and position angular position according to the 3 Euler angles determined by A:
Dynamic model
The Quanser Innovate Educate research platform features a quadrotor UAV equipped with four symmetrically mounted brushless motors In this configuration, motors 2 and 4 rotate counterclockwise, while motors 1 and 3 rotate clockwise The quadrotor's motion control is effectively managed by varying the rotational speeds of these four motors.
Assumption 1: The structure of the quadrotor is a solid symmetrical structure
Consider the Quadrotor according to the coordinate axis of the inertial reference system 𝐹 𝑖
(𝑂 𝑖 𝑋 𝑖 𝑌 𝑖 𝑍 𝑖 ) and the coordinate axis of the reference system attached to the frame of the object
𝐹 𝑏 with the initial coordinate origin 𝑂 𝑏 is the center of gravity of the UAV quadrotor mass
The thrust formula of each engine is in the form:
𝑠+ 𝐵 𝑤 𝑣 𝑘 (2.8) where k = 1, 2, 3, 4; K is the positive gain, 𝐵 𝑤 is the bandwidth of each brushless motor, the input signal is a PWM pulse represented by the 𝑣 𝑘 value
Therefore, the resultant force and torque of the x-axis, y-axis and z-axis is written as:
The total force (F) in a system is influenced by the torques acting on the pitch axis (𝜏 𝜙), roll axis (𝜏 𝜃), and yaw axis (𝜏 𝛹) The distance (l) from the center of point O to each motor plays a crucial role in this dynamic Additionally, the torque produced by each motor (𝜏 𝑘) is directly related to the thrust force (𝐹 𝑘) through the equation 𝜏 𝑘 = h.𝐹 𝑘, where h is a constant.
Furthermore, according to equation (2.8), it can be assumed that 𝐹 𝑘 ≈ K.𝑣 𝑘 , so (2.9) can be rewritten as:
Furthermore, by applying the small angle approximation, the height dynamics and position dynamics are placed on the quadrotor frame and the inertial frame
𝑚𝑥̈ = 𝐹(cos 𝜙 sin 𝜃 cos 𝛹 + sin 𝜃 sin 𝛹) − 𝑘 𝑥 𝑥̇ + 𝑑 𝑥 (𝑡) 𝑚𝑦̈ = 𝐹(cos 𝜙 sin 𝜃 sin 𝛹 − sin 𝜃 cos 𝛹) − 𝑘 𝑦 𝑦̇ + 𝑑 𝑦 (𝑡)
𝑢 𝑥 = 𝐹(cos 𝜙 sin 𝜃 cos 𝛹 + sin 𝜃 sin 𝛹)
𝑢 𝑦 = 𝐹(cos 𝜙 sin 𝜃 sin 𝛹 − sin 𝜃 cos 𝛹)
The equation \( G_3 = (J_x - J_y) \dot{\theta} \dot{\phi} \) describes the dynamics of a UAV quadrotor, where \( \phi, \theta, \) and \( \psi \) denote the roll, pitch, and yaw angles, respectively The coordinates \( x, y, \) and \( z \) originate from the quadrotor's center in the inertial reference frame, while \( J_x, J_y, \) and \( J_z \) represent the moments of inertia along these axes, with \( J_T \) indicating the moment of inertia for each motor The state vector of the quadrotor is given by \( [\phi, \dot{\phi}, \theta, \dot{\theta}, \psi, \dot{\psi}]^T \), and \( g \) signifies gravitational acceleration The angular velocity of each motor is expressed as \( \omega = -\omega_1 - \omega_2 + \omega_3 + \omega_4 \) Additionally, \( d_x(t), d_y(t), d_z(t), d_{\phi}(t), d_{\theta}(t), \) and \( d_{\psi}(t) \) account for time-dependent external noise and environmental factors affecting the quadrotor's performance.
Based on equation (2.11), the kinematic equation then has the following form:
In addition, the quadrotor dynamic equation (2.12) is also represented in the form:
The reference signal with the desired angular state 𝐀 𝑑 =[
𝛹 𝑑 ] and the reference signal with the desired position 𝐏 𝑑 =[
𝑧 𝑑 ] The control goal is to design a control method so that the altitude and position of the quadrotor can track the reference signal with very small errors
To facilitate the construction of a quadrotor dynamics model for algorithmic control, the following assumptions are made
Assumption 2: The external disturbances affecting 𝑫 𝑝 and 𝑫 𝑎 are bounded by ∥𝑫 𝑝 ∥ ≤ 𝑑 𝑝 and ∥𝑫 𝑎 ∥ ≤ 𝑑𝑎 where 𝑑 𝑝 , 𝑑 𝑎 are unknown positive constants
Lemma 1 : Define a continuous and differentiable bounding function ∆(x), ∀(x) ∈ [𝑡 0 ,𝑡 1 ]
If ∆(x) satisfies |∆(x)| ≤ π, where π is a positive constant, ∆(x) is bounded
Lemma 2: Given a continuous and positively defined Lyapunov function V (x) and a bounded initial condition satisfying 𝜋 1 (∥x∥) ≤ V (x) ≤ 𝜋2 (∥x∥) such that 𝑉̇(𝑥)≤ −𝑐1 V (x) +
𝑐 2 where 𝜋 1 , 𝜋 2 are rank functions, 𝑐 1 , 𝑐 2 are positive constants.
Discussion
Chapter 2 of this project presents the coordinate reference systems required to describe the quadrotor dynamics This chapter also presents the process of building a mathematical model of the quadrotor, combining translational and rotational dynamics derived from fundamental physical principles The established mathematical framework serves as a foundation for the development of advanced control strategies In the upcoming Chapter 3, the focus will shift to the design of controllers for both position and attitude control These controllers will be developed using modern control techniques to ensure accurate and stable navigation of the quadrotor under a wide range of operating conditions
CONTROLLER DESIGN
Some control problems of quadrotor object
Flight stability control is essential for maintaining a quadcopter's stability by preventing unwanted pitch or rotation The primary objective is to keep roll, pitch, and yaw angles at zero or set values, ensuring the aircraft remains level and oriented correctly This control is crucial for stabilizing the quadcopter during flight and lays the groundwork for more advanced control tasks It is especially important during manual operation, where the pilot directs the vehicle while ensuring stability For instance, when hovering, flight stability control prevents drifting or pitching caused by external factors like wind, and during any directional movement, it maintains alignment and stability for accurate trajectory tracking.
Position control is a critical challenge that aims to direct a quadrotor to a designated location in three-dimensional space Unlike other control tasks that emphasize speed or trajectory optimization, the primary goal of position control is to ensure the quadrotor accurately reaches its specified target position.
Position control prioritizes accuracy in final positioning over movement efficiency, making it ideal for tasks like cargo transport in open environments In flight missions, quadrotors continuously monitor their current position against the desired target, adjusting motor speed and direction to navigate accurately This control strategy also addresses external disturbances, such as wind and load variations, to ensure stability and precision during flight.
Position control serves as a foundational capability for many advanced applications, enabling quadrotors to perform autonomous tasks with high precision and reliability in a wide range of operating scenarios.
Figure 3.1 Illustration of position control problem
The path-following problem is a specialized control challenge aimed at guiding a quadcopter along a specific trajectory in three-dimensional space Unlike position control, which targets a precise location, path-following control ensures the quadcopter adheres to a predetermined path defined by mathematical equations or geometric descriptions throughout its flight This method is crucial for maintaining the quadcopter's trajectory, regardless of the duration of the journey, and is particularly beneficial in scenarios requiring accurate navigation through complex or obstacle-laden environments, such as urban cargo delivery missions.
In environments with narrow corridors formed by buildings and structures, quadcopters can be programmed to navigate predetermined safe paths, emphasizing the significance of precision, adaptability, and safety in their navigation This reliable path-following capability enables diverse applications, including automated delivery in crowded areas and systematic exploration of unfamiliar terrain Consequently, path following serves as the cornerstone of autonomous drone operations, enhancing the efficiency and success of missions across various environments.
Figure 3.2 Illustration of the path control problem
The trajectory tracking control problem is a complex challenge that ensures a quadrotor follows a predetermined trajectory in both space and time Unlike traditional trajectory tracking, this approach requires the quadrotor to synchronize its motion with a moving reference point, converging its coordinates to the desired reference point q(t) → 𝑞 𝑟𝑒∫ (𝑡) as closely as possible This control problem is crucial in dynamic applications, such as target surveillance, where the quadrotor must track moving objects while maintaining specific distances or angles The control algorithm continuously compares the quadrotor's current state with the desired trajectory, adjusting motor speed and thrust to minimize errors while accounting for external disturbances.
To achieve precise tracking performance despite challenges like wind and varying payload dynamics, advanced control techniques such as proportional–integral controllers (PIDs), model predictive control (MPCs), and neural network-based adaptive controllers are employed These methods enable quadrotors to accurately follow designated spatial paths while ensuring timely execution, facilitating highly accurate and dynamic missions in diverse real-world applications.
Figure 3.3 Illustration of trajectory tracking control problem
Backstepping Slide Controller Design
3.2.1 Backstepping slider controller for quadrotor position control
According to the dynamic equation 2.11, we consider the height subsystem of the quadrotor:
The first derivative with respect to time of equation (3.4)
Then we select the virtual controller:
𝑧̇ = 𝑎 𝑧 = 𝑆 𝑧 − 𝑐 0𝑧 𝑒 𝑧 + 𝑧̇ 𝑑 (3.7) Substitute into equation (3.5) we get:
𝑉̇ 1𝑧 = −𝑐 0𝑧 𝑐 𝑧 2 + 𝑆 𝑧 𝑒 𝑧 (3.8) if 𝑆 𝑧 tends to zero then 𝑉̇ 1𝑧 = −𝑐 0𝑧 𝑐 𝑧 2 ≤ 0
From equation (3.6), we have the first derivative with respect to time of the sliding surface:
First derivative with respect to time of 𝑉 2𝑧 :
When the sliding surface 𝑆 𝑧 = 𝑒̇ 𝑧 + 𝑐 0𝑧 𝑒 𝑧 = 0, the error 𝑒 𝑧 will approach 0 along the sliding surface Then the control signal required to maintain the system on the sliding surface is:
𝑚𝑧̇ + 𝑧̈ 𝑑 (3.12) Control signal to pull the system to the sliding surface:
Choose 𝑐 1𝑧 = 𝐷 𝑧 + 𝜆 𝑧 , where 𝐷 𝑧 is the upper limit of 𝑑 𝑧 , 𝜆 𝑧 is an arbitrary positive constant Then 𝑉̇ 2𝑧 is negative and the system is asymptotically stable according to Lyapunov
Similar to the controller for the height subsystem, we have the following control laws for the x and y coordinate subsystems:
𝑚𝑦̇ + 𝑦̈ 𝑑 − 𝑐 1𝑦 𝑠𝑖𝑔𝑛(𝑆 𝑦 ) (3.16) in there, 𝑐 1𝑥 = 𝐷 𝑥 + 𝜆 𝑥 , 𝑐 1𝑦 = 𝐷 𝑦 + 𝜆 𝑦 (with 𝐷 𝑥 , 𝐷 𝑦 is the upper limit of external noise
3.2.2 Backstepping sliding controller for Euler angle system
From the control signals𝑣 𝑥 , 𝑣 𝑦 and 𝑣 𝑧 and the angle 𝛹 𝑑 we can determine the angles
Consider the subsystem of Euler angles according to the dynamic equation 2.11:
Consider the tracking error of the subsystem:
Proving similarly to the above section on the controller for the elevation subsystem, we obtain the backstepping sliding control laws for the Euler angle subsystem:
𝐽 𝑧 ) − 𝐶 1𝛹 𝑠𝑖𝑔𝑛(𝑆 𝛹 )] where, 𝑐 1𝜙 = 𝐷 𝜙 + 𝜆 𝜙 , 𝑐 1𝜃 = 𝐷 𝜃 + 𝜆 𝜃 and 𝑐 1𝛹 = 𝐷 𝛹 + 𝜆 𝛹 (where 𝐷 𝜙 , 𝐷 𝜃 and 𝐷 𝛹 are the upper bounds of the external noise 𝑑 𝜙 , 𝑑 𝜃 and 𝑑 𝛹 respectively) and 𝑐 0𝜙 , 𝑐 0𝜃 , 𝑐 0𝛹 , 𝜆 𝜙 ,
Remark 1 With the controllers proposed in equations (3.14), (3.16) and (3.20) the system will be globally asymptotically stable However, from the formulas for determining the control signals, we can see that, in order to produce accurate control signals, we need to know all the model parameters of the system, but this is practically impossible, especially the parameters of the moment of inertia of the system, or the coefficient of air resistance
The upper limits of external disturbances affecting the system, represented as Di (where i = x, y, z, 𝜙, 𝜃, 𝛹), are unknown When disturbances push the system away from its equilibrium point, the ability to return to equilibrium heavily relies on the system's control parameters In scenarios with significant disturbance intensity, the controller may struggle to restore equilibrium promptly after a set value change To address these challenges, the next section will explore an adaptive sliding control algorithm integrated with a radial neural network, along with a comparative simulation to assess its effectiveness against the designed backstepping sliding controller.
Introduction to Radial Neural Networks
Robust radial basis function neural networks (RBFNN) are essential tools in approximation theory, effectively approximating unknown functions from input-output pairs These systems can be represented as perceptron networks in model recognition, making RBFNN particularly adept at approximating any nonlinear system Essentially, RBFNN operates as a three-layer neural network with one hidden layer, where the output consistently varies linearly in relation to the link weights.
Figure 3.4 Radial neural network structure diagram
The multi-input, single-output Radial Basis Function Neural Network (RBFNN) depicted in Figure 3.4 features an input layer that transmits each element of the input vector to all hidden nodes Each hidden node corresponds to a specific center of the RBFNN and utilizes basic functions Ψ to calculate the Euclidean distance between the input vector and its designated center Consequently, each hidden node produces a scalar output based on its associated center.
3.3.2 Some basic functions in RBFNN
• Radial function: The radius function depends only on the distance from argument x to a given point 𝜇 (called the center)
𝛹(𝑥) = 𝛹(‖𝑥 − 𝜇‖) = 𝛹(𝑟) with r = ‖𝑥 − 𝜇‖, 𝜇 - Vector containing the RBFNN centers, where: 𝛹 - Basis function or activation function of the network
Training the RBFNN network depends on how the centers are chosen There are two techniques for training the RBFNN network:
• Choose fixed center values Then use adaptive techniques to train the network to find the optimal bi weights
During the training process, center values are dynamically selected rather than predetermined Specifically, both 𝑎 𝑖, which represents the center vector of the i-th neuron in the hidden layer, and 𝑏 𝑖, which indicates the width of the i-th neuron, are determined using gradient descent equations.
In this paper, we assume that f(.) represents the unknown continuous function of a nonlinear system, which is constructed by a linear combination of basis functions
W is the weight of the ideal matrix, 𝜀 is considered as the error of approximation, 𝛹(𝜒) is the vector of the Gaussian function
The input vector is represented by 𝜒 = (𝑥 1 𝑥 2 … 𝑥 𝑛 ) The closer the input is to the center, the larger the value
Assumption 4 The approximation error 𝜀 is bounded by ∥ 𝜀 ∥ ≤ 𝜀 𝑁 , where 𝜀 𝑁 is an undefined positive constant.
Design of sliding mode controller based on radial neural network
This article proposes a method for controlling a quadrotor's dynamic model and tracking objects using Radial Basis Function Neural Networks (RBFNN), ensuring precise control of the six-degree-of-freedom system The system's block diagram is illustrated in Figure 3.5.
Figure 3.5 Block diagram of the system
Consider the definition of position and state deviation:
With 𝑒 𝑝 = [𝑒 𝑥 𝑒 𝑦 𝑒 𝑧 ] 𝑇 , 𝑒 𝑎 = [𝑒 𝜙 𝑒 𝜃 𝑒 𝛹 ] 𝑇 The first and second derivatives of
𝑒 𝑝 and 𝑒 𝑎 are considered according to the differential function of equation (3.23), substituted into equation (2.12):
The control goal can be achieved when the defined errors converge to 0, lim𝑡→∞𝑒 𝑙 = 0, 𝑙 ∈ (𝑥, 𝑦, 𝑧, 𝜙, 𝜃, 𝛹 )
In this section, the position tracking controller for quadrotor is designed in the case of variable quadrotor mass and uncertain external disturbance
Based on the elements 𝑒 𝑝 and 𝑒̇ 𝑝 , the sliding function is designed:
Here 𝑆 𝑝 = [𝑆 𝑝1 𝑆 𝑝2 𝑆 𝑝3 ] 𝑇 and 𝐶 𝑝 =diag[𝑐 𝑝1 𝑐 𝑝2 𝑐 𝑝3 ] positive constant matrix Take derivative 3.26 multiplied by m, combine formula (3.25), dynamic equation of the slip surface:
𝑚𝑆̇ 𝑝 = 𝑈 𝑝 + 𝑓(𝜒 𝑝 ) + 𝐷 𝑝 − 𝑚𝑃̈ 𝑑 + 𝑚𝐶 𝑝 𝑒̇ 𝑝 (3.27) where 𝑓(𝜒 𝑝 ) = −𝐾 𝑝 𝑃̇ and 𝜒 𝑝 = [𝑒 𝑝 𝑇 𝑒̇ 𝑝 𝑇 𝑃 𝑇 𝑃̇ 𝑇 ] 𝑇 ∈ 𝑅 12 Specifically, the function 𝑓(𝜒 𝑝 ) is an uncertain parameter, according to the RBFNN mentioned above, there exists an ideal weight 𝑊 𝑝 that makes the network approximate 𝑓(𝜒 𝑝 ), we have:
𝑓(𝜒 𝑝 ) = 𝑊 𝑝 𝑇 𝛹(𝜒 𝑝 ) + 𝜀 𝑝 (3.28) Here 𝑊 𝑝 is an unknown constant matrix, the actual output of the network is denoted:
𝑤ℎ𝑒𝑟𝑒𝑓̂(𝜒 𝑝 ) 𝑖𝑠 𝑎𝑛 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓(𝜒 𝑝 ) 𝑎𝑛𝑑 𝑊̂ 𝑝 [𝑊̂ 𝑝1 𝑇 𝑊̂ 𝑝2 𝑇 𝑊̂ 𝑝3 𝑇 ] 𝑇 𝑖𝑠 considered as the estimated weight of the matrix The error of the matrix weight is expressed as:
34 where 𝛿 𝑝 = 𝜀 𝑝 + 𝐷 𝑝 is an unknown vector bounded by an unknown constant‖𝛿 𝑝 ‖ ≤ 𝛿 𝑁𝑝
𝑈 𝑝 = −𝑓̂ 𝑝 (𝜒 𝑝 ) + 𝑚̂𝐻 − 𝑘 𝑝 𝑠𝑖𝑔𝑛(𝑆 𝑝 ) − 𝜂 𝑝 𝑆 𝑝 − 𝛿̂ 𝑝 (3.32) to ensure asymptotic convergence of the slip surface 𝑆 𝑝 , where𝐻 = 𝑃̈ 𝑑 − 𝐶 𝑝 𝑒̇ 𝑝 ; 𝑘 𝑝 >0; 𝜂 𝑝
>0, the estimate of 𝛿 𝑝 is expressed as 𝛿̂ 𝑝 Let the error 𝛿̃ 𝑝 = 𝛿 𝑝 − 𝛿̂ 𝑝
The adaptive law 𝑊̂ 𝑝𝑖 is given as follows:
The Adaptation Law 𝛿̂ 𝑝 and 𝑚̂ is defined:
Theorem 1 Consider the position subsystem of the quadrotor constructed in (3.24) which satisfies assumptions 2 and 4 Using the controller (3.32) with the control laws (3.33) and (3.34), it can be concluded that all the signals of the quadrotor system are bounded, and the sliding function equations 𝑆 𝑝 and 𝑒 𝑝 can converge to 0 for any initial values 𝑥(𝑡 0 ), 𝑦(𝑡 0 ), 𝑧(𝑡 0 )
Proof 1 According to Lyapunov stability theory, choose the candidate function:
Derivative 𝑉 1 with respect to time:
According to assumption 3 we have 𝑚̇< 0:
𝛾 3𝑚̃(𝑚̇ − 𝑚̂̇) (3.39) Applying the control design equation (3.32) to (3.39):
𝛾 3 𝑚̃(𝑚̇ − 𝑚̂̇) (3.42) Replace the adaptive laws (3.33) and (3.34):
Then 𝑉̇ 1 < 0 the 𝑆 𝑝 slip surface converges to 0 and is asymptotically stable
Remark 2 When considering some special cases where the quadrotor load increases or decreases suddenly, then considering that the change is finite, let 𝑚̇ = 0 to simplify In this
36 case, consider Corollary 1 below to improve the result of Assumption 4 without knowing the information of 𝑚̇
Corollary 1 Consider the quadrotor position system constructed according to (3.24) satisfying the conditions: ‖𝐷 𝑝 ‖ ≤ 𝑑 𝑝 and ‖𝐷 𝑎 ‖ ≤ 𝑑 𝑎 in assumption 2 By using the controller (3.32) with the adaptive laws (3.33) and (3.34) and adjusting the adaptive law𝑚̂̇ = −𝛾 3 𝑆 𝑝 𝑇 𝐻, it can be concluded that all quadrotor signals are bounded, the sliding surface function 𝑆 𝑝 and the error 𝑒 𝑝 converge to zero for any initial values 𝑥(𝑡 0 ), 𝑦(𝑡 0 )and 𝑧(𝑡 0 )
The pitch and roll angles need to be calculated with the virtual position control input
𝑈 𝑝 = [𝑢 𝑥 𝑢 𝑦 𝑢 𝑧 ]obtained from (3.32) the desired state trajectory represents:
Based on the elements 𝑒 𝑎 and 𝑒 𝑎 determined by equations (3.23) and (3.25), the slip function is designed:
Here 𝑆 𝑎 = [𝑆 𝑎1 𝑆 𝑎2 𝑆 𝑎3 ] 𝑇 and 𝐶 𝑝 =diag[𝑐 𝑎1 𝑐 𝑎2 𝑐 𝑎3 ] positive constant matrix Take derivative (3.46) multiplied by J, combine formula (3.25), dynamic equations of the sliding surface:
𝐽𝑆̇ 𝑎 = 𝑈 𝑎 + 𝐺 + 𝑓(𝜒 𝑎 ) + 𝛿 𝑎 − 𝐽𝐴̈ 𝑑 + 𝐽𝐶 𝑎 𝑒̇ 𝑎 (3.47) where 𝑓̂(𝜒 𝑎 ) = −𝐾 𝑎 𝐴̇ and 𝜒 𝑎= [𝑒 𝑎 𝑇 𝑒̇ 𝑎 𝑇 𝐴 𝑇 𝐴̇ 𝑇 ] 𝑇 ∈ 𝑅 12 Specifically, the function 𝑓(𝜒 𝑎 ) is an uncertain parameter, according to the radial symmetry function mentioned above, there exists an ideal weight 𝑊 𝑎 that approximates the network 𝑓(𝜒 𝑎 ), we have:
Considering 𝑊 𝑎 to be an unknown constant matrix, the actual output of the network is expressed as:
37 where 𝑓̂(𝜒 𝑎 )is an approximation function of 𝑓(𝜒 𝑎 ) and 𝑊̂ 𝑎 = [𝑊̂ 𝑎1 𝑇 𝑊̂ 𝑎2 𝑇 𝑊̂ 𝑎3 𝑇 ] 𝑇 is considered the estimated weight of the matrix The error of the matrix weight is expressed as:
𝐽𝑆̇ 𝑎 = 𝑈 𝑎 + 𝐺 + 𝑊̂ 𝑎 𝑇 𝛹(𝜒 𝑎 ) + 𝛿 𝑎 − 𝐽𝐴̈ 𝑑 + 𝐽𝐶 𝑎 𝑒̇ 𝑎 (3.51) where 𝛿 𝑎 = 𝜀 𝑎 + 𝐷 𝑎 is an unknown vector bounded by an unknown constant‖𝛿 𝑎 ‖ ≤ 𝛿 𝑁𝑎
State trajectory tracking control design:
To ensure asymptotic convergence of the slip surface 𝑆 𝑎 , 𝑘 𝑎 > 0; 𝜂 𝑎 > 0 >, the estimate of
𝛿 𝑎 is expressed as 𝛿̂ 𝑎 The error 𝛿̃ 𝑎 = 𝛿 𝑎 − 𝛿̂ 𝑎
Theorem 2 According to the state model in equation (3.25), assumptions 2 and 4 are satisfied By using the controller (3.49) combined with the adaptive rules (3.50) and (3.51), it means that all signals in the state model are bounded, the sliding surface value 𝑆 𝑎 and the error 𝑒 𝑎 converge to 0 for any initial values 𝜙(𝑡 0 ), 𝜃(𝑡 0 ) and 𝛹(𝑡 0 )
Proof 2 According to Lyapunov stability theory, choose the candidate function:
2𝜁 2 𝛿̃ 𝑎 𝑇 𝛿̃̇ 𝑎 (3.55) Derivative of V2 with respect to time:
𝜁 2𝛿̃ 𝑎 𝑇 𝛿̂̇ 𝑎 (3.57) Substituting equation (3.51) into (3.57) we get:
𝜁 2𝛿̃ 𝑎 𝑇 𝛿̂̇ 𝑎 (3.58) Substitute 𝑈 𝑎 from the control formula (3.52) into (3.58):
Replace the adaptive laws (3.53) and (3.54):
= −𝑘 𝑎 ‖𝑆 𝑎 ‖ − 𝜂 𝑎 𝑆 𝑎 𝑇 𝑆 𝑎 where −𝑘 𝑎 ‖𝑆 𝑎 ‖ − 𝜂 𝑎 𝑆 𝑎 𝑇 𝑆 𝑎 < 0 when 𝑘 𝑎 > 0 and 𝜂 𝑎 > 0 Then 𝑉̇ 2 < 0 the 𝑆 𝑝 slip surface converges to 0 and is asymptotically stable
Remark 3 In order to reduce the chattering phenomenon caused by the sliding controller, in this paper we consider that the saturation function sat(S) can be used to replace the sign(S) function The saturation function is expressed as:
Conclusion of chapter 3
Chapter 3 has presented the basic concept of radial symmetric networks This chapter also delves into their application in the construction and design of both positional and rotational controllers These controllers serve as essential components to ensure the accurate and stable operation of systems based on rotational and positional adjustments
This chapter showcases various designs while validating their stability through Lyapunov theory principles This theoretical framework not only establishes the stability of the controllers but also guarantees their robustness in dynamic and uncertain environments.
Chapter 4 transitions from theoretical concepts to practical verification of the proposed controllers, utilizing MATLAB simulations to confirm the design and stability criteria established in Chapter 3.
CHAPER4 RESULTS AND SIMULATION FOR UAV CONTROLLER
This chapter presents the effectiveness of the UAV control system through simulations that demonstrate minimal trajectory errors in optimal scenarios It compares the performance of the Backstepping Sliding Mode Controller (BSMC) and the Adaptive Neural Network Sliding Mode Controller (ANNSMC), providing valuable insights into their reliability and robustness under various conditions The simulations assess each controller's ability to manage disturbances, minimize trajectory deviations, and maintain stable flight across four separate scenarios This comparative analysis highlights their strengths and limitations in tracking accuracy, disturbance robustness, and adaptability to dynamic environments The results are crucial for validating the proposed control strategies, ensuring the UAV system meets the demands of real-world applications.
Tool
To simulate the flight motion of the UAV controller in MATLAB, several tools and methods are used to ensure accurate modeling and analysis of the system:
The C programming language is essential for custom simulation as it efficiently implements core control algorithms, enabling swift processing speeds This efficiency allows for precise execution of control loops and real-time adjustments in flight motion.
MATLAB enhances the C programming language by offering a robust visual environment for simulating and analyzing UAV dynamics, facilitating effective model verification This integration creates a seamless workflow that streamlines the development and refinement of UAV flight controllers.
The ode15s solver is essential for addressing systems of differential equations in UAV dynamics and control, effectively managing both rigid and non-rigid problems Its capability to utilize variable-order implicit methods enhances its suitability for these complex applications.
Simulink, a graphical extension of MATLAB, enables system-level simulation of UAVs by allowing users to model and connect various components like sensors, actuators, and controllers in a block diagram format This capability supports thorough testing of UAV behavior across diverse conditions.
• Aerospace Toolbox and Blocks: MATLAB's Aerospace Toolbox and Blocks provide pre-built models and functions for simulation and analysis of flight dynamics These
41 tools include features for modeling aerodynamics, control surfaces, and environmental conditions, enhancing the realism of simulations
The Control Systems Toolbox is essential for designing, analyzing, and simulating control systems, particularly in UAV operations It allows for the creation of PID controllers, state-space models, and advanced control strategies By providing efficient and accurate simulation capabilities, this toolbox lays a strong foundation for developing robust UAV controllers, enabling iterative improvements and thorough performance testing.
Simulation scenarios
This chapter presents simulation scenarios to rigorously evaluate and compare the performance of Backstepping Sliding Mode Control (BSMC) and Adaptive Neural Network Sliding Mode Control (ANNSMC) under various flight conditions and disturbances The simulations aim to assess each controller's effectiveness in tracking desired flight trajectories and adapting to dynamic changes during flight To ensure realism, the quadrotor model parameters are derived from the well-established Quanser aircraft model, recognized for its high fidelity and relevance to practical applications.
In this case, consider the quadrotor position control problem of 2 points and no disturbance
The set value of the scenario is shown in the following table:
An analysis of the flight trajectory in Figure 4.1 reveals that both controllers, despite their differing designs, effectively guide the UAV along the designated path between points A and B This trajectory illustrates the controllers' ability to adhere to the specified route successfully.
Figure 4.1 Quadrotor flight trajectory according to scenario 1
Backstepping Sliding Mode Control (BSMC) and Adaptive Neural Network Sliding Mode Control (ANNSMC) effectively guide UAVs from their starting positions to target points with minimal deviation from the reference path Both control strategies successfully navigate through critical waypoints, demonstrating their capability in trajectory tracking and ensuring high positioning accuracy under normal conditions.
Figure 4.2 Output Quadrotor position according to scenario 1 on x-axis
Figure 4.3 Output Quadrotor position according to scenario 1 on y-axis
Figure 4.4 Output Quadrotor position according to scenario 1 on z-axis
Figure 4.5 Output of Euler angle in scenario 1 in roll angle
Figure 4.6 Output of Euler angle in scenario 1 in pitch angle
Figure 4.7 Output of Euler angle in scenario 1 in yaw angle
The visual outputs of the Quadrotor's position and Euler angles from both controllers illustrate their effective flight performance, demonstrating stability and responsiveness to trajectory parameters While minor variations may arise, they do not substantially impact trajectory tracking accuracy, indicating that both control systems function within acceptable limits This consistency in trajectory tracking underscores the reliability and precision of both controllers in navigating complex paths.
This article addresses the challenge of managing the quadrotor's position at four designated points within a square, focusing on the influence of external disturbances over a specific time frame.
The scenario settings are shown in the following table:
External noise is represented as:
Figure 4.8 Quadrotor trajectory according to scenario 2
When subjected to minor disturbances, both controllers exhibit strong performance, with the input signal experiencing slight fluctuations while still accurately following the desired set value These minor variations are anticipated and showcase the system's resilience to low-level disturbances, maintaining overall stability The controllers effectively reject these small disturbances, ensuring that the UAV consistently tracks the reference trajectory with minimal deviation.
Figure 4.9 External disturbances affecting the system according to scenario 2
Figure 4.10 Quadrotor position outputs according to scenario 2
Figure 4.11 Deviation of positions according to scenario 2
Figure 4.12 Outputs of Euler angles according to scenario
Under severe disturbance conditions, the Backstep Sliding Mode Control (BSMC) controller reveals its limitations, particularly between 10 and 30 seconds when a substantial disturbance along the x-axis is introduced This disturbance significantly affects the system's performance, and despite initial attempts to maintain trajectory control, the BSMC controller fails to ensure that the output signal returns to the desired set value after the disturbance cycle As the disturbance persists, the controller experiences considerable deviations from the reference trajectory.
The performance limitations of the backstepping controller (BSMC) become evident when faced with high-intensity disturbances, as it is primarily effective under ideal conditions and small disturbances In scenarios involving significant disturbances or uncertainties, the BSMC struggles, resulting in performance degradation and a failure to maintain trajectory adherence This inability to adequately recover from intense disturbances highlights a critical weakness in its robustness, particularly in real-world applications where disturbances can be both unpredictable and severe.
Adaptive controllers, like the Adaptive Neural Network Sliding Mode Control (ANNSMC), excel at managing disturbances by utilizing real-time adjustments This capability allows them to maintain accurate trajectory tracking, even when faced with larger and more complex disturbances.
In this scenario, consider the spring-shaped trajectory tracking problem under the influence of random noise The coordinate set trajectory in this scenario shows:
Figure 4.13 Quadrotor trajectory according to scenario 3
The adaptive sliding neural network controller (ANNSMC) ensures impressive stability and trajectory tracking for UAVs, even under the influence of random noise The UAV maintains a flight path closely aligned with the desired trajectory, keeping tracking errors within acceptable limits despite minor fluctuations While larger noise may introduce slight deviations, overall control performance remains unaffected, allowing the UAV to navigate its predetermined path with high accuracy ANNSMC's adaptability to noise guarantees effective operation in real-world environments, where noise and uncertainty are common challenges.
Figure 4.14 External disturbances affecting the system according to scenario 3
Figure 4.15 Quadrotor position output according to scenario 3 on x-axis
Figure 4.16 Quadrotor position output according to scenario 3 on y-axis
Figure 4.17 Quadrotor position output according to scenario 3 on z-axis
Figure 4.18 Deviation of positions according to scenario 2
Figure 4.19 Output of Euler angle in scenario 3 according to roll angle
Figure 4.20 Output of Euler angle in scenario 3 according to pitch angle
Figure 4.21 Output of Euler angle in scenario 3 according to yaw angle
Both Backstepping Sliding Mode Control (BSMC) and Adaptive Neural Network Sliding Mode Control (ANNSMC) demonstrate strong performance under random noise, particularly with small disturbances, maintaining stable trajectory tracking with minimal deviations To combat chattering, a common issue in sliding mode control, both controllers replace the traditional signum function with a saturation function (sat(s)), which reduces high-frequency oscillations and enhances control smoothness Although this modification significantly lessens chattering, it cannot be completely eliminated, especially under varying noise conditions; however, the residual chattering remains manageable, particularly during larger or sudden disturbances.
The ANNSMC controller demonstrates commendable performance in UAV trajectory tracking, despite a slight discrepancy in yaw angle adherence to the desired reference This minimal deviation does not significantly affect the UAV's overall path accuracy Notably, the ANNSMC controller outperforms the BSMC controller in trajectory adherence, particularly in the presence of random noise and minor disturbances Its ability to maintain control stability while adapting to external uncertainties underscores its suitability for real-world applications.
This section evaluates the control quality of controllers by examining four-leaf clover trajectory tracking under noise effects The noise is synthesized from two components: random noise in scenario 2 and continuous noise in scenario 3 The trajectory of the four-leaf clover is defined as follows:
Figure 4.22 Quadrotor trajectory according to scenario 4
Simulation results indicate that while both controllers demonstrate effective performance under nominal conditions, the Backstepping Sliding Mode Control (BSMC) controller reveals its limitations when faced with long-term and dynamic disturbances This is evident in the 3D trajectory tracking graph, which shows substantial deviations from the desired path, reflecting a decline in controller stability and tracking accuracy The observed oscillations and drifts during these challenging conditions underscore the BSMC controller's limited robustness.
Figure 4.23 External disturbances affecting the system according to scenario 4
Figure 4.24 Quadrotor position output according to scenario 4 on x-axis
Figure 4.25 Quadrotor position output according to scenario 4 on y-axis
Figure 4.26 Quadrotor position output according to scenario 4 on z-axis
Figure 4.27 Deviation of positions according to scenario 4
Figure 4.28 Output of Euler angle in scenario 4 according to roll angle
Figure 4.29 Output of Euler angle in scenario 4 according to pitch angle
Figure 4.30 Output of Euler angle in scenario 4 according to yaw angle
The Backstepping Sliding Mode Control (BSMC) controller demonstrates commendable performance; however, it struggles with high-intensity disturbances, particularly evident from 10s to 30s in the simulation During this interval, a strong disturbance along the x-axis hampers the BSMC controller's ability to maintain precise trajectory tracking, leading to significant deviations from the intended path This loss of control stability and accuracy is clearly illustrated in the 3D trajectory tracking graph, where the UAV's trajectory diverges from the reference trajectory under challenging conditions.
Conclusion of Chapter 4
Chapter 4 of the project introduces a total of four distinct simulation scenarios that have been carefully designed to compare and assess the performance of the controllers discussed in Chapter 3 Each scenario was selected to challenge the controllers under various conditions, including changes in key parameters and the presence of disturbances The aim was to thoroughly evaluate the robustness and reliability of the proposed controllers, particularly in situations where exact parameter values may not be available Among the different controllers tested, the Adaptive Neural Network Sliding Mode Control (ANNSMC) controller consistently demonstrated strong performance across all scenarios Even when the precise values of critical parameters, such as the drag coefficient and the air resistance moment, were not known or were subject to variations, the ANNSMC controller still managed to maintain an acceptable level of control Moreover, its effectiveness remained intact despite external disturbances acting on the object, as well as internal disturbances within the system These findings suggest that the ANNSMC controller is not only capable of adapting to parameter uncertainties but also robust enough to handle real- world complexities, making it a reliable choice for applications where precision is difficult to achieve or maintain
Conclusion
The project successfully achieved its overall goal by developing a non-linear control algorithm that ensures system stability and accurately tracks the desired trajectory.
The simulation results clearly demonstrate the algorithm's effectiveness, showcasing its highly accurate tracking performance even in challenging conditions such as environmental noise disturbances, uncertainties in drag coefficient parameters, and variations in aerodynamic drag moments This system's ability to maintain stability and adapt to these uncertainties underscores its robustness and reliability.
This achievement marks a crucial advancement in UAV control systems, demonstrating their practical applicability in real-world scenarios The findings validate that the developed algorithm is both theoretically robust and capable of effectively managing dynamic and uncertain environments, positioning it as a promising solution for future UAV applications.
Future development direction
Based on the results obtained, several opportunities for future project improvement and development have been identified:
The nonlinear control algorithms developed in Chapter 3 are implemented on hardware systems to experimentally validate the proposed framework and demonstrate its practical applicability.
• Extension to multi-UAV systems: Research and development of advanced control algorithms for coordinating multi-UAV systems, enabling efficient and synchronized operation in multiple domains
These directions aim to improve the quality of current work while exploring new possibilities for broader and more impactful applications of UAV control systems
UAV specifications in the thesis [25]:
Mass of UAV(m) 1.66 Kg Rotor diameter 0.228 m
Mass of the battery 0.5 Kg Power of consumption 250 W
Number of rotors 4 Endurance 50 sec
APPENDIX Attitude controller code for ANNbSMC
Sa = Ea_dot + Ca*Ea;
Xa = [Ea;Ea_dot;q(4:6); q_dot(4:6) ]; f_hat_a = Wa_hat '*gauss (Xa);
Ua = - f_hat_a + J*ddqar (:,i) - J*Ca*Ea_dot - ka*tanh(Sa)
- eta_a*Sa - dentaNa_hat; taul = Ua(1) + du(2,1) ;tau2 = Ua(2) + du(3,i); tau3 = Ua(3) + du(4,i);
Wal_hat_dot = zeta_1*Sa (1) *gauss (Xa);
Wa2_hat_dot = zeta_1*Sa (2) *gauss (Xa);
Wa3_hat_dot = zeta_1*Sp (3) *gauss (Xa); dentaNa hat dot = zeta 2*5a;
Position controller code for ANNbSMC
Sp = Ep_dot + Cp*Ep;
Xp = [Ep; Ep_dot; q(1:3); q_dot (1:3)]; f_hat_p = Wp_hat**gauss (Xp) ;
Up = m*H - f_hat_p - kp*tanh(Sp) - eta_p*Sp - dentaNp_hat;
F = sqrt(Up(1)^2 + Up(2)^2 + (Up(3)+m*g)^2) + du(1,i);
Wp1_hat_dot = gamma_1*Sp(1) *gauss (Xp);
Wp2_hat_dot = gamma_1*Sp(2) *gauss (Xp);
Wp3 _hat_dot = gamma_1*Sp(3) *gauss (Xp); dentaNp_ hat_dot = gamma_ 2*5p;
Outer controller code for BSMC
% parameter cox = 5; cOy = 5; cOz = 5; clx = 2; cly = 2; clz = 2;
Sz = cOz*e (3) + de (3); v1 = - e (1) - cOx*de(1) + Kfx/m*dq(1) + ddqr (1,i) - c1x*(Sx); v2 = - e (2) - cOy*de(2) + Kfy/m*dq(2) + ddqr (2,i) - cly*(Sy); v3 = - e (3) - cOz*de(3) + Kfz/m*da(3) + ddar(3,i) - c1z*(Sz);
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SOCIALIST REPUBLIC OF VIETNAM Independence – Freedom - Happiness
EXPLANATORY REPORT ON CHANGES/ADDITIONS BASED ON THE DECISION OF GRADUATION THESIS COMMITTEE
FOR UNDERGRADUATE PROGRAMS WITH DEGREE AWARDED BY
Student’s full name: Le Quang Huy
Graduation thesis topic: SIMULATION AND CONTROL DESIGN FOR UAV
Major: INFORMATICS AND COMPUTER ENGINEERING
According to VNU-IS's decision no …… QĐ/TQT, dated … / … / ……., a Graduation Thesis Committee has been established for Bachelor programs at Vietnam National University, Hanoi, overseeing the defense and subsequent modifications of the thesis in the specified sections.
No Change/Addition Suggestions by the Committee Detailed Changes/ Additions Page
1 Correct the controller diagram - Controller diagram has been modified as requested 32
2 Add Matlab code in the appendix
- In the appendix, the code for the two controllers ANNbSMC and BSMC has been added
Student needs to check carefully the document’ format and organization, should not capture the equations, number if figures…
- The format and arrangement of the document are satisfactory
- Equations and figures are all edited directly on Word