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The paper focuses on mathematical modelling of a quadrotor and identification of parameters used in presented models. There are several models of the quadrotor that can be used to design a controller. The nonlinear model is presented with respect to the body-fixed frame and also to the inertial frame. The next model is defined in terms of quaternio...

## Contexts in source publication

**Context 1**

... direction of the rotation. Propellers with the angular speed ω1 and ω3 spin counter-clockwise and the other two spin clockwise. The alteration of the position and the orientation is reached by varying the thrust of a specific rotor. Angular velocities corresponding to the inertial frame EI ( ζ ) and the body- fixed frame EB (ƞ) are presented in Fig. ...

**Context 2**

... that the quadrotor is a rigid body, the dynamics of the quadrotor can be described using Newton-Euler equations. Several forms of mathematical models can be derived. In [1] a piecewise affine model was used to design a switching model predictive attitude controller. A linearized model of the quadrotor was used in [12] to design a linear quadratic (LQ) controller. This article focuses on the nonlinear model with respect to the inertial frame and also to the body-fixed frame, the model described by quaternions and the model of the quadrotor near the hover position. Each propeller rotates at the angular velocity ω i producing the corresponding force F i directed upwards and the counteracting torque directed opposite to the direction of the rotation. Propellers with the angular speed ω 1 and ω 3 spin counter-clockwise and the other two spin clockwise. The alteration of the position and the orientation is reached by varying the thrust of a specific rotor. Angular velocities corresponding to the inertial frame E I ( ζ ) and the body- fixed frame E B ( ƞ ) are presented in Fig. 1. The rotation matrix from E B to E I is an orthogonal matrix given by equation (1), where C angle and S angle designate cos(angle) and sin(angle) respectively [13, ...

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## Citations

... Furthermore, parameters such as the fiction of the actuators tend to change with respect to time; furthermore, not all variables can always be measured directly. This affects the performance of control algorithms based on the dynamic model [23]. ...

This paper shows a proposal for a control scheme for the trajectory tracking problem in a Two Degree of Freedom Helicopter (2DOFH). For this purpose, a control scheme based on a feedback linearization combined with a Generalized Proportional Integral (GPI) controller is used. In order to implement linearization by feedback, it is required to know and have access to all the physical 2DOFH parameters, however, angular velocity and viscous friction are often not available. Commonly, state observers are used to know the angular velocity, however, estimating friction results out to be more complex. Therefore, we propose the use of a Convolutional Neural Network (CNN) to estimate viscous friction and angular velocity. The variables estimated by the CNN are entered into both the GPI and feedforward controllers. Thus, the system is brought to a linear representation that directly relates the GPI control to the dynamics of perturbations and non-model parameters. Finally, results of numerical simulations are shown that validate the robustness of our scheme in the presence of disturbances in the tail rotor, as well as the advantages of using a feedforward control based on a CNN.

... where "^" is the hat operation to produce the skew matrix. We refer the readers to the literature 36,37 for the detail in modeling the coefficients of the thrust, drag moment, and inertia moment, I B . They are assumed to be constant in this research. ...

With the introduction of the laterally bounded forces, the tilt-rotor gains more flexibility in the controller design. Typical feedback linearization methods utilize all the inputs in controlling this vehicle; the magnitudes as well as the directions of the thrusts are maneuvered simultaneously based on a unified control rule. Although several promising results indicate that these controllers may track the desired complicated trajectories, the tilting angles are required to change relatively fast or in large scale during the flight, which turns to be a challenge in application. The recent gait plan for a tilt-rotor may solve this problem; the tilting angles are fixed or vary in a predetermined pattern without being maneuvered by the control algorithm. Carefully avoiding the singular decoupling matrix, several attitudes can be tracked without changing the tilting angles frequently. While the position was not directly regulated in that research, which left the position-tracking still an open question. In this research, we elucidate the coupling relationship between the position and the attitude. Based on this, we design the position-tracking controller, adopting feedback linearization. A cat-trot gait is further designed for a tilt-rotor to track the reference; three types of references are designed for our tracking experiments: set point, uniform rectilinear motion, and uniform circular motion. The significant improvement with less steady state error is witnessed after equipping with our modified attitude–position decoupler. It is also found that the frequency of the cat-trot gait highly influenced the steady state error.

... This process requires selecting a unique path to achieve the task and is directly related to the Non-Instrument Flight Rules (NIFR) during its execution. A complete symmetrical multirotor UAV structure understanding is required for the construction and modeling of the system; previous studies [8][9][10] confirm that the symmetrical multirotor UAV system is nonlinear. However, previously, researchers neglected many terms while converting the nonlinear into a linear system. ...

The design and implementation of a multi-stage PID (MS-PID) controller for non-inertial referenced UAVs are highly complex. Symmetrical multirotor UAVs are unstable systems, and it is thought that the kinematics of the symmetrical UAV rotor, such as the quadrotor and hexacopter resembles the kinematics of an inverted pendulum. Several researchers have investigated the structure and design of PID controllers for high-order systems during the last decade. The designs were always concerned with the enhanced response, robustness, model reduction and performance of PID controllers. An accurate tuning process of such a controller depends on the engineer's experience level. This is due to the number of variables and hyperparameters tuned during the process. An adaptive genetic algorithm (AGA) is utilized to optimize the MS-PID controllers for controlling the quadrotor in this study. The proposed method optimizes the offline-planned approach, providing several possibilities for adapting the controllers with various paths and or varying weather conditions. The MS-PID parameters are optimized in parallel, as every PID controller affects the other controller's behavior and performance. Furthermore, the proposed AGA generates new chromosomes for "new solutions" by randomly developing new solutions close to the previous best values, which will prevent any local minima solution. This study intends to investigate the design and development of a highly tuned robust multi-stage PID controller for a symmetrical multirotor UAV. The work presents a model for a non-referenced inertial frame multirotor UAV (quadcopter). Once the model is defined, a robust multi-stage PID controller for the non-inertial referenced frame symmetrical multirotor UAV is designed, tuned, and tested. A genetic algorithm (GA) will be used to tune the MS-PID controller. Finally, the performance comparison between the proposed and conventional methods is presented. The results show that the proposed method provides stability improvement , better transient response, and power consumption.

... This formulation does not contain any actions for heading changes. However, Equation (4) can be expanded if required by adding the rotation matrix in multi-copters [32]. An illustration of a problem formulation including the rotation matrix can be found in [6]. ...

Studies of past life forms on other planets are possible through the identification and localisation of desiccation cracks in ancient water bodies such as lakes, rivers and seas. Unmanned aerial vehicles (UAVs) are increasingly being used as a viable remote sensing solution for planetary exploration, as desiccation cracks are difficult to identify with the naked eye and are normally located in complex and unreachable environments. However, most UAVs have a strong reliance on human operators through their communication systems, as UAVs have limited onboard decision-making capabilities for autonomous navigation in such environments. UAV navigation in real-world scenarios is also challenging as data captured from their sensors is imperfect, and outputs from computer vision systems are, sometimes, inaccurate. These sensory and onboard vision limitations cause partial observability of the state of surveyed environments, inducing uncertainty in optimal path planning. This paper proposes a UAV system for autonomous onboard navigation, identification, and mapping of desiccation cracks for planetary exploration. The navigation problem is mathematically formulated as a partially observable Markov decision process (POMDP), where a motion strategy can be obtained by solving the POMDP in real time using the augmented belief tree (ABT) solver. The framework discussed in this work is validated with real flight tests using two desiccation crack patterns distributed across the surveyed area. Real-time segmentation from streamed camera frames of desiccation cracks is achieved through inference onboard the aircraft using a ResNet18 Con-volutional Neural Network (CNN) model, and an OpenCV AI Kit (OAK)-D camera. Results from real flight tests indicate that the system can reduce levels of object detection uncertainty to locate and map desiccation cracks in environments under partial observability. The system design allows further adaptation for similar time-critical applications requiring increased levels of UAV autonomy in unstructured environments under uncertainty and partial observability, such as humanitarian relief, wildlife monitoring, and surveillance.

... This formulation does not contain any actions for heading changes. However, Equation (4) can be expanded if required by adding the rotation matrix in multi-copters [32]. An illustration of a problem formulation including the rotation matrix can be found in [8]. ...

Recent advances in Unmanned Aerial Vehicles (UAVs) have resulted in their quick adoption for wide a range of civilian applications, including precision agriculture, biosecu-rity, disaster monitoring and surveillance. UAVs offer low-cost platforms with flexible hardware configurations, as well as an increasing number of autonomous capabilities, including takeoff , landing, object tracking and obstacle avoidance. However, little attention has been paid to how UAVs deal with object detection uncertainties caused by false readings from vision-based detectors, data noise, vibrations, and occlusion. In most situations, the relevance and understanding of these detections are delegated to human operators, as many UAVs have limited cognition power to interact autonomously with the environment. This paper presents a framework for autonomous navigation under uncertainty in outdoor scenarios for small UAVs using a probabilistic-based motion planner. The framework is evaluated with real flight tests using a sub 2 kg quadrotor UAV and illustrated in victim finding Search and Rescue (SAR) case study in a forest/bushland. The navigation problem is modelled using a Partially Observable Markov Decision Process (POMDP), and solved in real time onboard the small UAV using Augmented Belief Trees (ABT) and the TAPIR toolkit. Results from experiments using colour and thermal imagery show that the proposed motion planner provides accurate victim localisation coordinates, as the UAV has the flexibility to interact with the environment and obtain clearer visualisations of any potential victims compared to the baseline motion planner. Incorporating this system allows optimised UAV surveillance operations by diminishing false positive readings from vision-based object detectors.

... This neglects forces such as the ground effect (which occurs when the multirotor is less than one rotor length from the ground [41]) and rotor flapping (at high apparent wind speeds [42]) but are reasonably accurate otherwise. These coefficients may be identified either with a dedicated test bench, or with an identification based on in-flight data [13], [43], [44]. The control of the propeller speed is set by the electronic speed controllers and may be approximated as a first order system [45], although in practice for small UAVs this is often disregarded. ...

Bearing formation control allows groups of quadrotors to manoeuver in a desired geometry, using only visual measurements extractable from embedded monocular cameras. Prior works have treated quadrotors as single or double integrators, and as a result must operate slowly to compensate for unmodelled non-linearities. This thesis allows for faster bearing formations by developping higher-order controllers, considering the non-linear quadrotor and visual feature dynamics. A dynamic feedback controller based on second-order visual servoing and a model predictive controller are developped and tested in simulation and experiments, showing improved dynamic manoeuvering performance. The later is augmented with constraints such as field of view limitations and obstacle avoidance. All bearing formation algorithms depend on a sufficient degree of bearing rigidity to guarantee performance. This may be evaluated numerically, but as the rigidity is a function of the formation embedding, previous work could not guarantee rigidity in formations larger than a few robots. The second main contribution of this thesis is the evaluation of bearing rigidity singularities (i.e. embeddings where an otherwise rigid formation becomes flexible) by applying existing geometric analysis methods on an kinematic mechanism which is analoguous to the kinematic constraints imposed by the formation controller and robot models. This is extended to a novel classification system based on a contraction of constraint sets that can determine singular geometries for large formations, allowing for a formulation of a set of guaranteed rigid configurations without an ad-hoc kinematic analysis of individual formations.

... The identification of the SUAV mathematical model in the MatLab environment using the System Identification Toolbox package [11] The result of comparing the movement of "B-kopter" by height, obtained by simulating in Simulink MatLab identified model (7), with the experimental flight data is shown in Fig. 6, b (the transient in ms on the abscissa axis and the altitude in m on the ordinate axis are plotted). Here the line number 1 denote the experimental data, and the line number 2 indicates identification result. ...

The object of research in the article is various well-known approaches and methods of structural and parametric identification of dynamic controlled objects - unmanned aerial vehicles (UAVs). The subject of the research is the parameters of linear and nonlinear mathematical models of spatial and isolated movements, describing the dynamics and aerodynamic properties of the UAV and obtained both from the results of flight experiments and using computer object-oriented programs for 3-D UAV models. The goal is to obtain mathematical models of UAV flight dynamics in the form of differential equations or transfer functions, check them for reliability and the possibility of using them in problems of synthesis of algorithms for automatic control systems of UAVs. Tasks to be solved: evaluation of the analytical (parametric), direct (transient), as well as the identification method using the 3-D model of the control object. Methods used structural and parametric identification of dynamic objects; the determination of static and dynamic characteristics of mathematical models by the type of their transient process; the System Identification Toolbox package of the MatLab environment, the Flow Simulation subsystem of the SolidWorks software and the X-Plane software environment. The experimental parameters of UAV flights, as well as the results of modeling in three-dimensional environments, are the initial data for the identification of mathematical models. The following results were obtained: the possibility of analytical and computer identification of mathematical models by highly noisy parameters of the UAV flight was shown; the mathematical models of UAVs obtained after identification is reliable and adequately reproduce the dynamics of a real object. A comparative analysis of the considered UAV identification methods is conducted, their performance and efficiency are confirmed. Conclusions. The scientific novelty of the result obtained is as follows: good convergence, reliability and the possibility of using the considered identification methods for obtaining mathematical models of dynamic objects to synthesize algorithms for automatic control systems of UAVs is shown.

... The drone thrust force is usually computed using the pulse width modulation (PWM) signals, but such a computational formula is not usually provided by a drone manufacturer, and it is usually unique for each particular drone. Existing work for computing the motor thrust using a PWM signal has focused on identifying the coefficients of a highorder polynomial of the PWM signal using a load cell to measure the thrust force [31][32][33][34]. However, setting up such experiments by attaching load cells to the drone motors requires considerable efforts of disassembling drone components. ...

This paper considers the self-localization of a tethered drone without using a cable-tension force sensor in GPS-denied environments. The original problem is converted to a state-estimation problem, where the cable-tension force and the three-dimensional position of the drone with respect to a ground platform are estimated using an extended Kalman filter (EKF). The proposed approach uses the data reported by the onboard electric motors (i.e., the pulse width modulation (PWM) signals), accelerometers, gyroscopes, and altimeter, embedded in the commercial-of-the-shelf (COTS) inertial measurement units (IMU). A system-identification experiment was conducted to determine the model that computes the drone thrust force using the PWM signals. The proposed approach was compared with an existing work that assumes known cable-tension force. Simulation results show that the proposed approach produces estimates with less than 0.3-m errors when the actual cable-tension force is greater than 1 N.

... where p u (k) is the position of the UAV at time step k, and ∆p u (k) is the change in the UAV's position from time step k to time step k + 1. While this implementation does not incorporate the UAV Euler yaw angles in the action space, Equation (6) can be further expanded by including the multi-rotor rotation matrix [32] as presented in the problem formulation given in [26]. The procedure to model ∆p u (k) follows the system identification process presented in [27]. ...

Recent advances in autonomy of unmanned aerial vehicles (UAVs) have increased their use in remote sensing applications, such as precision agriculture, biosecurity, disaster monitoring, and surveillance. However, onboard UAV cognition capabilities for understanding and interacting in environments with imprecise or partial observations, for objects of interest within complex scenes, are limited, and have not yet been fully investigated. This limitation of onboard decision-making under uncertainty has delegated the motion planning strategy in complex environments to human pilots, which rely on communication subsystems and real-time telemetry from ground control stations. This paper presents a UAV-based autonomous motion planning and object finding system under uncertainty and partial observability in outdoor environments. The proposed system architecture follows a modular design, which allocates most of the computationally intensive tasks to a companion computer onboard the UAV to achieve high-fidelity results in simulated environments. We demonstrate the system with a search and rescue (SAR) case study, where a lost person (victim) in bushland needs to be found using a sub-2 kg quadrotor UAV. The navigation problem is mathematically formulated as a partially observable Markov decision process (POMDP). A motion strategy (or policy) is obtained once a POMDP is solved mid-flight and in real time using augmented belief trees (ABT) and the TAPIR toolkit. The system’s performance was assessed using three flight modes: (1) mission mode, which follows a survey plan and used here as the baseline motion planner; (2) offboard mode, which runs the POMDP-based planner across the flying area; and (3) hybrid mode, which combines mission and offboard modes for improved coverage in outdoor scenarios. Results suggest the increased cognitive power added by the proposed motion planner and flight modes allow UAVs to collect more accurate victim coordinates compared to the baseline planner. Adding the proposed system to UAVs results in improved robustness against potential false positive readings of detected objects caused by data noise, inaccurate detections, and elevated complexity to navigate in time-critical applications, such as SAR.

... The first kind uses a system model to capture the physical system's dynamics [13,24,46]. For example, a set of differential equations [14] or a machine learning model can be used to describe the motion of a quadcopter. The second kind of physical invariant refers to sensor correlation, where multiple (heterogeneous) sensors correlatively respond to the same physical aspect at the same time [2,28,54]. ...

... The load disturbance exists at each control step and is bounded by v max . If v max is larger, the reachable set overapproximation X k is bigger according to the Equation (14). Then, the reachable set is more likely to intersect with the unsafe state set, which leads to a shorter recovery length N . ...

The increasing autonomy and connectivity in cyber-physical systems (CPS) come with new security vulnerabilities that are easily exploitable by malicious attackers to spoof a system to perform dangerous actions. While the vast majority of existing works focus on attack prevention and detection, the key question is “what to do after detecting an attack?”. This problem attracts fairly rare attention though its significance is emphasized by the need to mitigate or even eliminate attack impacts on a system. In this article, we study this attack response problem and propose novel real-time recovery for securing CPS. First, this work’s core component is a recovery control calculator using a Linear-Quadratic Regulator (LQR) with timing and safety constraints. This component can smoothly steer back a physical system under control to a target state set before a safe deadline and maintain the system state in the set once it is driven to it. We further propose an Alternating Direction Method of Multipliers (ADMM) based algorithm that can fast solve the LQR-based recovery problem. Second, supporting components for the attack recovery computation include a checkpointer, a state reconstructor, and a deadline estimator. To realize these components respectively, we propose (i) a sliding-window-based checkpointing protocol that governs sufficient trustworthy data, (ii) a state reconstruction approach that uses the checkpointed data to estimate the current system state, and (iii) a reachability-based approach to conservatively estimate a safe deadline. Finally, we implement our approach and demonstrate its effectiveness in dealing with totally 15 experimental scenarios which are designed based on 5 CPS simulators and 3 types of sensor attacks.