Methods of a Priori Statistical Analysis of Disturbed Motion of Aircraft in Turbulent Environments

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Introduction Disturbed motion of aircraft in turbulent environments is one of the key problems of aero- dynamics, flight dynamics and control. Turbulent flows arising in the atmosphere significantly complicate the prediction of the aircraft trajectory, causing random disturbances that can lead to a change in its motion and deterioration in controllability. The problem lies not only in modeling such motion, but also in developing analysis methods that allow one to estimate potential deviations of motion from a given trajectory and predict them with a high accuracy. Understanding turbulent environments is im- portant in the design and operation of aircraft, as it can have a significant impact on flight safety and the efficiency of control systems. To solve this problem, it is important to use methods of a priori statistical analysis, which allows one to make probabilistic forecasts about the behavior of the system before receiving the observed data. The aim of this work is to study the methods of a priori statistical analysis of the disturbed motion of aircraft in turbulent environments and to assess their applicability for modeling and pre- dicting the behavior of aircraft under uncertainty. 1. Mathematical models of disturbed motion Technical devices designed to move in the atmosphere or outer space by creating lift or jet propulsion are called aircrafts. They include both manned and unmanned vehicles, such as airplanes, helicopters, airships, rockets, spacecraft, and drones. Depending on the flight environment and operating principle, aircraft can use aerodynamic forces (e.g. airplanes and helicopters) or jet propulsion (e.g. rockets) to maintain flight and maneuver. Turbulence is a chaotic and unpredictable movement of air flows that occurs in the atmo- sphere, which has a significant impact on flight dynamics [1]. Turbulent flows are characterized by rapid changes in wind speed and direction at different points in space, which leads to distur- bances in the trajectory, stability, and control- ability of the aircraft (Figure 1). Figure 1. Schematic of dynamic turbulence S o u r c e: made by A.S. Ermilov, O.A. Saltykova Turbulence near the ground, especially in urban environments and rough terrain, occurs due to the interaction of air flows with various obstacles, such as buildings, trees, and other artificial or natural structures. These objects create zones of disturbed flow, where the air movement becomes chaotic, therefore vortices and sharp changes in wind speed and direction are formed (Figure 2). Figure 2. Formation of perturbed air flows near the ground S o u r c e: made by A.S. Ermilov, O.A. Saltykova Under conditions where an aircraft is subject to random disturbances, its trajectory deviates from the calculated one, which requires the develop-ment of special methods for describing and pre- dicting such deviations. The dynamics of an aircraft under disturbed conditions is characterized by com- plex nonlinear processes that require taking into account not only traditional aerodynamic forces and moments, but also random changes in these para- meters under the influence of the environment [2]. To describe the motion of an aircraft under turbulent flow conditions, mathematical models are used that include both deterministic and sto- chastic components. The main parameters that determine the motion are linear and angular velo- cities, the position of the center of mass, orientation angles (roll, pitch, yaw) and the forces acting on the apparatus. The classical model of aircraft flight dynamics includes two types of equations: equations des- cribing translational motion (based on Newton’s law) and equations describing rotational motion (based on Euler’s equations). Translational motion is described by Newton’s second law in vector form: (1) where m - mass of an aircraft; - acceleration vector representing the derivative of the velocity vector with respect to time; - the resulting force acting on the aircraft (including aerodynamic force, gravity and thrust). The rotational motion of the aircraft is described by Euler’s equations, which relate the moment of force to angular accelerations: (2) where - angular momentum of the aircraft relative to the center of mass; - the derivative of angular momentum with respect to time, des- cribing the angular acceleration; - resultant moment of forces acting on the aircraft. Together, these equations define a complete dynamic model of aircraft motion that takes into account both its translational and rotational motion. However, in a turbulent environment, these algo-rithms must be modified by adding stochastic perturbations to parameters such as drag force, lift force, and moments of inertia [3]. Models of motion in a turbulent environment can be divided into two types: deterministic and stochastic. Deterministic models describe motion based on known initial conditions and environ-mental parameters [4]. It is assumed that all external influences on the aircraft, including turbulent flows, are known and can be accurately described, which is unlikely in real conditions. Stochastic models, such as random process or Gaussian disturbance models, allow the uncertainty associated with the effects of turbulent flows to be taken into account. For example, the von Kármán wind turbulence model and the Iver model are widely used to describe the structure of turbulent flows in the atmosphere [5]. They allow the statistical characteristics of turbulence, such as the intensity and spectrum of disturbances, to be calculated, which is the basis for predicting the effects of turbulence on aircraft motion. When modeling the disturbed motion of an aircraft in turbulent environments, it is important to take into account many parameters that can affect the trajectory and dynamics of the flight. They characterize both external factors, such as atmo-spheric turbulence and meteorological conditions, and the internal properties of the aircraft itself, including its aerodynamic characteristics and mass. Correctly taking these parameters into account allows us to create more accurate mathematical models that predict the behavior of the apparatus in complex conditions (Table). Parameters determining the dynamics of perturbed motion of an aircraft in a turbulent environment Parameter Description Impact on Flight Dynamics Linear velocities Velocities along the X, Y, Z axes, affected by external forces. Determine the flight trajectory and rate of position change. Angular velocities Rotational speeds around the X, Y, Z axes. Influence the orientation and stability of the aircraft. Turbulence intensity Amplitude and frequency of air mass disturbances, determining the force acting on the aircraft. Lead to deviations in trajectory and orientation. Aerodynamic forces Lift and drag, dependent on the angle of attack. Affect lift and drag, influencing altitude and flight speed. Moments of inertia Resistance to changes in angular velocities. Influence rotational stability. Mass of the aircraft Affects inertia and susceptibility to external disturbances. Greater mass reduces trajectory deviation but increases inertia. Thrust forces Engine forces, varying under external influences. Affect speed and trajectory. Meteorological conditions Pressure, temperature and air density. Affect aerodynamics and controllability. S o u r c e: made by A.S. Ermilov, O.A. Saltykova, data from [6; 7]. Mathematical models of disturbed aircraft motion are a combination of deterministic equations of motion and stochastic models of turbulent effects. This allows one to describe both short-term changes in the aircraft trajectory under the influence of random disturbances and long-term changes in the stability and controllability of the device. To improve the accuracy of mathematical models, it is necessary to use a priori statistical analysis. 2. Basic approaches to a priori analysis A priori analysis is a statistical method that is based on the use of previously known data or assumptions to construct mathematical models and forecasts. In the context of aircraft dynamics in a turbulent environment, a priori analysis allows for uncertainties in motion parameters and external influences, such as turbulent flows, to be taken into account long before direct measurements or experiments are carried out. This approach makes it possible to predict the behavior of a system in conditions where precise data have not yet been obtained, but there is enough information to form reasonable hypotheses. There are several main approaches to a priori analysis that are used to estimate the disturbed motion of an aircraft. One of the most well-known methods is the Bayesian approach (application of Bayes’ theorem) to update a priori assumptions based on the data obtained: (3) where - posterior probability of a para- meter after receiving the data ; - likelihood function describing the probability of observing data D at a given parameter value ; - prior probability of the parameter before receiving data; - normalizing constant called the total probability of the data. In the context of turbulence and disturbance modeling, may represent parameters describing turbulent flows, such as intensity, frequency of disturbances, or other physical characteristics of the medium [8]. Prior distribution can be given on the basis of previous experiments, numerical simulations or theoretical estimates. The likeli- hood function reflects the probability of observing real data D (e.g. wind speed measure- ments or aircraft trajectories) given known values of the turbulence parameters. After updating the prior distribution with data, a posterior distribution is obtained, which provides a more accurate estimate of the turbulence and disturbance para- meters. This process can be repeated as more data becomes available, gradually refining the model and making the forecasts more accurate. The Bayesian approach allows one to take into account the initial uncertainty regarding the para- meters of the disturbed motion and to correct them based on incoming information, which is especially important in conditions of complex and uncertain external influences, such as turbulent flows. Maximum Likelihood Estimation (MLE) is used to estimate model parameters based on known data and prior assumptions [9]. In this method, the task is to find parameters θ that maximize the likelihood of the data D, that is, maximize the probability that the observation data could have been obtained with the given parameter values. This is expressed through the likelihood function: . (4) In turbulent conditions, the maximum likelihood method is used to estimate the parameters of the disturbance model, such as the intensity and frequency of turbulent flows. For example, if the observational data D describe the deviations of the aircraft trajectory in turbulent conditions, then θ can represent the parameters of the turbulent effects, such as the mean velocity and variance of the disturbances. Another effective tool for a priori analysis is the Monte Carlo method, especially in the case of complex stochastic systems [10]. In modeling the disturbed motion of an aircraft, it allows for statistical analysis of various flight trajectories in a turbulent environment, assessing the probability of various deviations from the calculated trajectories (Figure 3). Figure 3. Scheme of the Monte Carlo method S o u r c e: made by A.S. Ermilov, O.A. Saltykova The factors x, y and z represent various external parameters such as wind speed, direction of turbulent flows and other disturbing forces. Probability distributions are modeled for each of these factors, which are then used to estimate the deviation of the aircraft from the calculated trajectory. The points on the graphs represent a set of possible outcomes obtained using the Monte Carlo method, which allows us to estimate the influence of each factor on the resulting motion of the apparatus. Regularization is used to solve ill-conditioned problems when there is an excessive amount of a priori data or there is high uncertainty in the initial parameters (Figure 4). Figure 4. Regularization scheme S o u r c e: made by A.S. Ermilov, O.A. Saltykova In aircraft control, regularization methods play an important role when working with data obtained in real time from sensors, for example, about the position and speed of the aircraft. In conditions of turbulence or other external in- fluences, the readings may contain significant noise, which complicates the calculation of correct control actions [11]. Regularization helps to smooth out such fluctuations, making the data more reli- able for decision-making [12]. In some cases, a priori analysis can be based on deterministic approaches, when the a priori values of the parameters are assumed to be known and unchanged [13]. These methods make it possible to significantly simplify calculations, but their use is justified only under conditions of a low degree of uncertainty. For example, deterministic a priori methods can be useful for analyzing aircraft motion in weak turbulence or in conditions when the nature of the disturbances is well understood [14]. 3. Application of a priori models and approaches to the analysis of aircraft dynamics One example of the application of a priori models is the analysis of the motion of unmanned aerial vehicles (UAVs) in turbulent flows at low altitudes [15]. Turbulence near the earth’s surface can be intense and unpredictable, which complicates control and trajectory prediction. In such situations, a priori probability models are used to describe the parameters of turbulent flows (for example, the average value of wind speed and its dispersion). These models allow calculating deviations from the calculated trajectory and assessing the probability of significant disturbances. The main equations describing the dynamics of aircraft motion include Newton’s equations for translational motion: , (5) where - mass of the device; - velocity vector; - aerodynamic forces; - disturbances caused by turbulent flows; - gravity. After applying Newton’s equations to analyze the aircraft dynamics in turbulent conditions, the influence of turbulent disturbances on the trajectory of motion can be assessed. Value , which is a random force caused by turbulence, can fluctuate depending on the characteristics of the atmosphere and the flight altitude. This leads to the aircraft trajectory deviating from the calculated ones, causing additional maneuvers to stabilize the flight. Using a priori models, it is possible to predict the most probable deviations and adjust the control systems in advance to minimize the effects of turbulence. This approach improves the stability of the aircraft and prevents abrupt changes in trajectory, which is especially important for unmanned systems or when flying at low altitudes, where turbulence is more pronounced. One of the effective methods of a priori analysis in aircraft control is the Bayesian approach, which allows dynamically updating turbulence forecasts as new data arrives. This method is actively used in high-altitude flight conditions, where turbulence can suddenly occur and have a significant impact on the trajectory. At cruising altitude (usually above 10 km), where aircrafts often encounter turbulence, control is performed using a priori data on the probability of occurrence of turbulent zones [16]. Initially, the control system has a priori information on turbulence obtained from meteorological models, and it is specified as a priori probability of the parameter θ - the turbulence intensity. When sensors on board detect changes in air flows, the system updates the prior based on these observations using Bayes’ theorem: , (6) where - updated posterior probability of turbulence after data acquisition D; - likelihood of observed data. The aircraft’s control system, based on updated a posteriori data, can predict increased turbulence and adjust flight parameters in advance [17]. For example, if the data indicates an in-creased probability of severe turbulence ahead, the system can reduce speed or adjust altitude to mitigate the impact. This process ensures safe and stable flight, even under unexpectedly changing external conditions. Conclusion Methods of a priori statistical analysis of dis-turbed aircraft motion in turbulent environments demonstrate high efficiency in predicting trajectory deviations and flight dynamics. Stochastic models, such as the Bayesian approach, maximum likelihood method, and Monte Carlo method, allow for uncertainty and random disturbances characteristic of turbulent flows. These approaches make it possible to estimate the probability of motion deviations and improve the accuracy of forecasts under conditions of limited information. The use of a priori data and probabilistic models contributes to improving the stability and controllability of aircraft, especially when they operate in complex external conditions. However, despite significant progress, there remain challenges associated with the integration of such models into real aircraft control systems in real time. In the future, it will be necessary to improve computational methods for prompt processing of large volumes of data and adaptation of models to changing flight conditions. In addition, the development of universal approaches remains to be an important task, that will take into account the specifics of different types of aircraft and the ranges of turbulence they encounter.

About the authors

Alexander S. Ermilov

RUDN University

Email: eemilov-sasha@yandex.ru
ORCID iD: 0009-0007-4549-172X

Postgraduate student of the Department of Mechanics and Control Processes, Academy of Engineering

Moscow, Russia

Olga A. Saltykova

RUDN University

Author for correspondence.
Email: saltykova-oa@rudn.ru
ORCID iD: 0000-0002-3880-6662
SPIN-code: 3969-6707

Cand. Sc. (Physics and Mathematics), Associate Professor of the Department of Mechanics and Control Processes, Academy of Engineering

Moscow, Russia

References