Airspace Encounter Model for Estimating Collision Risk
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[b]Airspace Encounter Model for Estimating Collision Risk [/b]Seminar Report Submitted in partial fulfillment of the requirements for the award of M.Tech Degree in Electrical Engineering (Guidance & Navigational Control) of the University of Kerala Presented By Ms Anumol A st 1 Semester, M.Tech, Roll No:10GNC02 Guided by Smt S. Sreeja Lecturer Dept. of EEE Department of Electrical Engineering College of Engineering, Trivandrum Thiruvananthapuram16 2010 Abstract Airspace encounter models,providing a statistical representation of geometries and air craft behavior during a close encounter, are required to estimate the safety and robustness of collision avoidance systems.Prior encounter models,developed to certify the Traffic Alert and Collision Avoidance System,have been limited in their ability to capture important charac teristics of encounters as revealed by recorded surveillance data, do not capture the current mix of aircraft types or noncoperative aircraft, and do not represent more recent airspace procedures.This papre describes a methodology for encounter model construction based on a Bayesian statistical framework connected to an extensive set of national radar data.In ad dition,this paper provides examples of using several such high fidelity models to evaluate the safety of collission avoidance systems for manned and unmanned aircraft. Airspace Encounter Model for Estimating Collision Risk.pdf (Size: 312.96 KB / Downloads: 48) Contents 1 Introduction 1 2 Encounter Model Overview 2 3 Encounter model Construction 5 4 Model Validation 7 4.1 Sampling Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 Dynamic Simulation Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 Safety Analysis 8 5.1 Estimating probability of NMAC Using a Correlated Model . . . . . . . . . . 8 5.2 Estimating probability of NMAC Using a Uncorrelated Model . . . . . . . . . 9 6 Applications 10 6.1 TCAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 6.2 Collision Avoidance for HALE Unmanned Aircraft . . . . . . . . . . . . . . . 10 6.3 Probabilistic Intruder Trajectory Estimation Model . . . . . . . . . . . . . . . 10 6.4 Additional Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 7 Conclusion 11 iii List of Figures 2.1 Model taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Processing Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Sampling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 iv Chapter 1 Introduction Statistical models of encounters are representative of what actually occurs in the airspace. Civil aviation authorities such as the Federal Aviation Administration (FAA) and Eurocon trol, require a combination of flight tests and detailed simulation studies to ensure system effectiveness and safety. Flight test can evaluate a system in actual operation. Simulation studies are required to test the robustness of the system over wide range of situations. These situations need to be generated by a statistical model of encounters. Sampling a large col lection of situations from such an encounter model and running them in simulation both with and without a collision avoidance system. It provides an estimate of the differential in collision risk. Several encounter models have been previously developed by various organizations to support the development of the Traffic Alert and Collision Avoidance System (TCAS) since the mid 1980s. These encounter models are based primarily on radar data. The encounter models developed and described in this paper extend these prior models in several important ways. This paper describes the encounter models developed based on surveillance data and outlines the general methodology for constructing encounter models. It also explains how encounter models can be used to support collision avoidance safety analysis. 1 Chapter 2 Encounter Model Overview Figure 2.1: Model taxonomy This section provides an overview of recent encounter models developed from 20062008 to evaluate TCAS and future collision avoidance systems for manned and unmanned aircraft. Aircraft encounters can be of two types: correlated or uncorrelated. The first type involves transponder equipped aircraft with at least one in contact with air traffic control (ATC). It is therefore likely that both aircraft are tracked by ATC and that at least one aircraft receives some notification about the traffic conflict and begins to take action before the involvement of a collision avoidance system. This ATC intervention often leads to a correlation between the trajectories of the two aircraft. Prior encounter models developed for TCAS analysis were of this type. The second type of encounter involves aircraft that do not receive prior ATC notification of a conflict. Such encounters include two aircraft flying under visual flight rule (VFR) without flight following services. In these encounters, the pilot must rely on visual acquisition or some other collision avoidance system at close range to detect each other and maintain separation. Such encounters are assumed to be uncorrelated because there is no coordinated intervention before the close encounter. Encounters of this type are especially important when evaluating collision avoidance systems for unmanned aircraft that must avoid non cooperative aircraft. During the development of earlier encounter models, attempts were made to manually re move the effects of TCAS so that the model would represent nominal aircraft behavior in the absence of a collision avoidance system. Recently, an automated system was developed that leverages TCAS downlink data in deciding how to best remove the effects of advisories. It 2 Airspace Encounter Model for Estimating Collision Risk was found that removing these effects from the tracks used to generate the encounter model had a negligible effect on the resultant safety metrics such as risk ratio. We also developed nine separate uncorrelated models for various categories of unconventional aircraft such as gliders and balloons, based on GPS flight recorders data. Fig1 illustrates the taxonomy of encounter models developed. Figure 2.2: Processing Sequence The general process for constructing the models from surveillance data is nearly identical for the correlated and uncorrelated models and is outlined in Fig 2. Because surveillance data can be noisy, the first step is to preprocess the raw tracks, which involves removing outliers, smoothing and interpolating. Outlier removal involves incrementally discarding individual reports until the magnitudes of dynamic variables are below set thresholds. The position and altitudes reports are smoothed using Gaussian Kernel and interpolated to 1 Hz using piecewise cubic Hermite interpolation. From the collection of high quality tracks, various features defining the dynamic char acteristics of the encounters are extracted, including airspeed, airspeed acceleration, turn rate and vertical rate. For the correlated encounter model, these features are extracted from recorded tracks from individual aircraft. Because the dynamics of these variables are depen dent upon the altitude layer L and the airspace class A, L A are included in the model. Because the correlated model needs to capture the geometry of the encounter include by ATC intervention, a collection of other features such as approach angle, hmd and vmd are extracted. To remove noise, the dynamic features are smoothed using a Gaussian Kernel. The smoothed features are then discretized into bins. From the large set of discrete, multivariate samples, we extract statistics about the joint distribution and store them as a collection of probability tables. Once the model is constructed it may be sampled as many times as necessary for anal ysis. The first step in generating an encounter is to sample from the discrete probability distribution represented by the model. This results in a series of feature bins. Uniformly sample within the bins to create a series of continuous dynamic variables that can be used in dynamic simulation. Quantitative methods are used to check the sampled features are equivalent to those observed. Department of Electrical Engineering, College of Engineering, Trivandrum 3 Airspace Encounter Model for Estimating Collision Risk Figure 2.3: Sampling Process Department of Electrical Engineering, College of Engineering, Trivandrum 4 Chapter 3 Encounter model Construction An encounter model represents the probability distribution over the initial conditions of the encounter, such as initial airspeed and turn rate, as well as the behavior of the aircraft over time. By running a large collection of encounters sampled from the encounter distribution, we can measure the effectiveness of a collision avoidance system when it intervenes. After deciding which variables to model, the next task is to choose a representation for the probability distribution over the variables. For the naturally discrete variables such as airspace class, we use a multinomial distribution. For continuous variables, such as approach angle, we discretize the variable and represent the distribution over the bins as a multinomial distribution as done in prior models. To illustrate how we model distributions, consider the altitude layer variable in the correlated model, which can take on one of ’r’ different values. We would need ’r’ different parameterθ1 ,......,θr to represent the multinomial distribution, where the variable is assigned value k with probability θk . Suppose we have a collection of counts N1 , .......Nr , whereNk is a count of the number of times an encounter occurred in layer k in our radar data. The maximum likelihood estimate forθk is r Nk / Ni (3.1) i=1 Hence if we use maximum likelihood estimate to estimate the parameters of the distri bution over altitude layer, we would produce samples from the bins with probability equal to the fraction of observed occurence in our data set. The probability tables associated with prior encounter models based on maximum likelyhood estimate. The maximum likelihood estimate is a sensible estimate of the true parameters, but there is uncertainty in this estimate. We can compute a distribution over the parameters θ given our data by applying Bayer’s rule. It can be shown that the distribution over θ, which can be thought of as a distribution over distributions, is modeled by a special kind of distribution known as Dirichlet distribution. If we start with a prior distribution represented by a Dirichlet with parameters (α1 , ....., αr ) and then observe the counts N1,......Nr, the posterior distribution is Dirichlet with parameters (α1 + N1 , ........, αr + Nr ). Sampling from the posterior Dirichlet distribution and then sampling from the multinomial distribution will result in assigning the variable to value k with probability r (αk + Nk )/ (αi + Ni ) (3.2) i=1 which is equivalent to sampling from the maximum likelihood estimate but withα1 , ........., αr added to the observed counts. Because of the wealth of radar data, the impact of adding α1 , ........., αr to the counts is fairly insignificant in practice. By this method estimate the distribution over a single variable independent of all others. A realistic encounter model is composed of many different variables, some of which may be 5 Airspace Encounter Model for Estimating Collision Risk related to each other. The relationship between variables may be represented using Bayesian network. Examples of graphical structures are illustrated in fig. Associated with each node is a conditional probability table that determines the probability distribution for the specified variable given then values of the variables that appears as parents in the network. The probability of an instantiation x (x1 , x2 , ...........xn ) of all n variable is given by the product n P (xi /πi ) (3.3) i=1 where πi is the instantiation of the parents of the ith variable andP (xi /πi ) is as given in the conditional probability tables. If the ith variable can be assigned as to ri different values andqi different instantiation for its parents, then there areri qi probability values in its associated conditional probability tables. Because the distribution must sum to 1 for a given parental instantiation, there are only (ri − 1)qi independent probabilities for the ith variable. In total, there are n (ri − 1)qi (3.4) i=1 parameters defining the conditional probability tables for a Bayesian network. Using Bayesian network to model a joint distribution over static variables such as altitude layer and the airspeed. Many of the variables in the encounter model need to change over time to capture the dynamics of real encounters. The disadvantage of this approach is that it allows single acceleration periods. Additional variables could be incorporated, increase the complexity of the model. Use a special kind of Bayesian network called dynamic Bayesian network to represent the dynamics of the variables. A dynamic Bayesian network is composed of two slices, the first slice represents the values of the variables at time t and second slice represents the values of variables at time t+1. To create an encounter trajectory sample from static Bayesian network to get the initial values of the variables and then use dynamic Bayesian network to propagate the trajectories. At the first time step, the initial values obtained by sampling the static Bayesian network are held fixed and the variables at time t+1 are sampled. The newly sampled values are then fixed and new values are sampled for the next time step. This process is repeated as long as necessary. The assumption behind this Bayesian network is that the next state of the system depends only on the current state. This is known as Markov assumption. If a system is non Markovian, then additional slices can be added to the dynamic Bayesian network or additional variables can be added to the model. Department of Electrical Engineering, College of Engineering, Trivandrum 6 Chapter 4 Model Validation The correlated and uncorrelated models, and the method for which they are sampled and used in a dynamic simulation, must be representative of the encounters and aircraft for which they are intended. It is important to have confidence that the generated samples are characteristics of the encounters and aircraft in the airspace. In addition, the dynamic simulation must provide realistic encounters. This can be quantified by comparing the encounter and aircraft characteristics that were not modeled explicitly to those that were observed. The validation steps are described as follows. 4.1 Sampling Validation The basic step in model validation is varifying the implimentation of the sampling scheme. To varify the sampling scheme, we generated a large collection of samples, collected the feature distributions as was done to compile the models,and compared them to the observed feature distributions. To validate that the correlations between variables in the generated encounters match the observed encounters, one may compute P (θ1 = θ2 /D1 D2 ), where D1 andD2 represent the data associated with the observed features and sampled features and θ1 and θ2 represent the model parameters. If the sampling was implemented correctly, this probability should be close to 1. Applying Bayes’ rule results in P (θ1 = θ2 )P (D1 , D2 /θ1 = θ2 ) P (θ1 = θ2 /D1 , D2 ) = (4.1) P (D1 , D2 ) 4.2 Dynamic Simulation Validation Comparing variables not included in the model to those extracted fro the dynamic simulation is one metric for ensuring that simulated encounters are representative of the variation in the airspace.A previous Eurocontrol correlated model explicitly captured the number of altitude crossing encounters. In our recoded radar data, 12.5 of the encounters were crossing whereas 11.42 of the simulated encounters were crossing . Similarly, we examined the rate of slow clossingencounters, or encounters being at a close range 40 s before TCA.These results suggest that the model is successful at producing encounters that are realistic and reflect what occurs in the airspace.The validity of the uncorrelated dynamic model was assessed by comparing simulated airspeed to that observed. 7 Chapter 5 Safety Analysis Encounter models are used in simulations to estimate the probability that an encounter leads to a near midair collision (NMAC), which has been defined in prior TCAS studies as being a loss of seperation 100 ft vertically and 500 ft horizontally.To estimate NMAC rate, we simply multiply the encounter rate by the probability an encounter leads to an NMAC. The encounter rate may be estimated from radar data. However, may safety studies are primarily concerned with estimating the risk ratio, which is the NMAC rate with the collision avoidance system divided by the NMAC rate without the system. To estimate the probability an encounter leads to an NMAC, we generate a large collection of encounters using an encounter model and run in simulation. Dividing the number of encounters that lead to an NMAC by the total number of encounters run in simulation provides a Monte Carlo estimate of P(nmac/enc). 5.1 Estimating probability of NMAC Using a Correlated Model The first step in generating an encounter involves sampling from the initial network, which provides the initial values of the dynamic variables, such as the vertical rate and turn rate forboth aircraft. The initial network also provides geometry of at theTCA, including the approach angle, ralative bearing,vertical miss distance, and horizontal miss distance. Using the initial values of the dynamic variables obtained from the initial network, the dynamic network propogates the trajectories by generating the values of the dynamic variables over time. We convert the sequence of turn rates, vertical rates, etc. generated by our model into trajectories in three dimensional space. We then rotate and translate the aircraft trajectories so that the geometry at TCA matches what was sampled. The initial position of the aircraft after transformation are used as the initial position when simulatng the encounters. If the collision avoidance systemdetects a threat during the course of an encounter, it will deviate at some point from the nominal path. Directly sampling from the Bayesian network and computing the average number of NMACs will provide an unbiased estimate of the probability that an encounter results inan NMAC. However,because of the rarity of NMAC events in the airspace, direct sampling from the encounter distribution will result inthe generation of relatively few NMACs. A direct sampling approach produces a colletion of N encounters z 1 , ......, z N from the probability dis tribution p(z) represented by the encounter model, permitting the following approximation: N 1 P (nmac/z i ) P (nmac/enc) = P (nmac/z)p(z)dz = (5.1) N i=1 The probabilityP (nmac/z i ) is evaluted through simulation. The approximation becomes 8 Airspace Encounter Model for Estimating Collision Risk N p(z i ) q(z) p(z) 1 P (nmac/z i ) P (nmac/enc) = P (nmac/z)p(z) dz = P (nmac/z)q(z) dz = q(z i ) q(z) q(z) N i=1 (5.2) 5.2 Estimating probability of NMAC Using a Uncorrelated Model The concept behind the uncorrelated model is that an aircraft, whose collision avoidance system is to evaluate, travels through the airspace with some large encounter cylinder fixed to its center, and the intruder aircraft is initialized on the cylinder. The appropriate dimension of the encounter cylinder depends on the aircraft dynamics and collision avoidance system. Choose the dimension of the cylinder by multiplying the magnitude of the maximum expected closure rate by the approximate time required by the collision avoidance system to detect and avoid an intruder. Simulation can focus on the behavior of intruder aircraft after penetrating the cylinder. To estimate the NMAC rate, multiply the rate at which aircraft penetrate the cylinder by the probability that an aircraft that penetrate the cylinder results in an NMAC before exiting the encounter cylinder. The rate at which aircraft penetrate the cylinder is equal to average traffic density times the mean flux of intruder aircraft through the encounter cylinder. Traffic density may be estimated from radar surveillance. Monte Carlo simulation can provide an estimate of the mean flux of intruder aircraft through the encounter cylinder. The overall uncorrelated NMAC rate, encompassing all aircraft categories, is the sum of the NMAC rates for each intruder category. Department of Electrical Engineering, College of Engineering, Trivandrum 9 Chapter 6 Applications 6.1 TCAS TCAS is mandated on longer transport aircraft in the United States and worldwide. The most recently mandated version of TCAS is 7.0. As a result of recent monitoring and mod eling efforts, several changes have been implemented to create version 7.1 of the TCAS logic.One change addresses a safety issue with preventing or delaying a reversal in the ver tical sense of the avoidance maneuver that might otherwise improve aircraft seperation in degrading conditions. 6.2 Collision Avoidance for HALE Unmanned Aircraft Collision avoidance systems for unmanned aircraft will have to demonstrate the ability to avoid noncooperative traffic as well as cooperative aircraft do not have a pilot in the cockpit to accept responsibility for visually acquiring and maintaining separation. We used the uncorrelated encounter model to evaluate a prototype collision avoidance system for a HALE unmanned aircraft. 6.3 Probabilistic Intruder Trajectory Estimation Model Accurately predicting the trajectory of an intruder aircraft is essential to a collision avoidance system. The strength of a Probabilistic approach is that it accounts for uncertainty in aircraft behavior and tends to be neither optimistic and nor overly pessimistic. 6.4 Additional Application The encounter models may also be used to develop sensor requirements for unmanned air craft.An ideal electro optical onboard sensor for detecting intruder aircraft. The encounter models can be used to evaluate the relative performance of different sensor field of view configurations in terms of probability of detection before near miss. The encounter models have also been used directly in the development of a hazard alerting systembased on target line of sight rate. 10 Chapter 7 Conclusion This paper introduces an analytic framework and methodology for encounter modeling and collision risk estimation. The methodology presented in this paper was used to create a collection of encounter models for different categories of conventional and unconventional aircraft. The encounter models are publically available to support collision avoidance system safety analysis. The analytical framework outlined in this paper serves as basis for future encounter model development and collision risk assessment. One approach that is currently being investigated, both for sense and avoid systems for unmanned aircraft and next generation TCAS, is the formulation of the collision avoidance problem as a partially observable Markov decision process(POMDP).The state dynamics of the POMDP are derived from the encounter model the observation model is derived from sensor specification. Automated POMDP solvers search for avoidance policies that minimize the expected costs of collision and maneuvering. Changes to the encounter model and sensor specifications will require resolving for the optimal avoidance policy but would require far less human effort than manually returning the collision avoidance system. 11 References [1] Mykel J. Kochenderfer., Mathew W., M. Edwards., Leo P. Espindle., James K. Kuchar., and J. Daniel Griffith., “Airspace Encounter for Estimating Collision Risk ”Journal of Guidance, Control, and Dynamics, Vol.33, No. 2,pp. 487499, MarchApril 2010. 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