The demand in air traffic has been projected to grow by two to three times by the year 2025 [1], [2]. In response to this projected growth, the Joint Planning and Development Office (JPDO) of the United States has initiated a revolutionary concept of operation, known as the Next Generation Air Transportation System (NextGen), for future air traffic operations [1]. The proposed concept of operation has identified major changes in areas such as air traffic management, air traffic surveillance, and system infrastructure. Under the NextGen, a new concept of operation allows aircraft the flexibility of changing flight routes (or flight plans) in response to changing conditions [3], [4], [5]. Furthermore, the responsibility of ensuring safe separation is delegated to individual aircraft. This new responsibility requires individual aircraft to be capable of monitoring and resolving any potential conflicts in-flight without the intervention of the air traffic controllers. In this paper, we propose a trajectory prediction and conflict detection method to facilitate air traffic management in the complex airspace of the NextGen.
Trajectory prediction methods, and the corresponding conflict detection methods, can be divided into three categories: nominal, worst case, and probabilistic [6]. The nominal trajectory prediction method projects the current state into the future along a single trajectory. This method is straightforward but does not account for deviations in aircraft trajectories due to uncertainties. Another method is to consider the worst case projection, which assumes that the aircraft will perform any of a range of maneuvers, and declare a conflict alarm if any of these maneuvers could cause a conflict. This approach is conservative and could cause high false-alarm rates in high traffic density airspace. The probabilistic method considers uncertainties (or potential variations) in the predicted aircraft trajectories. The predicted trajectories are described with probability density functions, from which a probability of conflict is computed. Hence, this approach is useful for assessing air traffic safety in operational scenarios with a high level of traffic and uncertainties.
One approach to probabilistic trajectory prediction is to propagate the state of the aircraft by using a stochastic model of the aircraft dynamics. This approach has been used in [7], which describes the aircraft dynamics with a stochastic linear difference equation from which a probabilistic trajectory is generated. Prandini et al. [8] have used a stochastic kinematic model with uncertainties represented by a two-dimensional Brownian motion for short-term trajectory prediction. The advantage of this nonintent-based approach is that it does not require knowledge of the aircraft intent or flight plan. Its limitation is that the prediction error tends to grow quadratically with time. Thus, this method is usually used for short-term trajectory prediction of up to 5 min into the future [9]. Furthermore, this approach predicts the trajectory by propagating the current state of the aircraft in a straight line and may not give accurate conflict prediction when the aircraft's trajectory consists of one or more maneuvers.
An alternative approach is to compute a nominal aircraft trajectory based on the aircraft intent or flight plan and then add a position uncertainty (or covariance) to the computed trajectory. This approach is used, for example, in the User Request Evaluation Tool (URET) [10] and the Center-TRACON Automation System (CTAS) [11]. The Performance-Based Air Traffic Management [12], currently under development, also used a conflict detection method similar to that of URET. These systems generate a probabilistic trajectory in two stages. First, a nominal trajectory is computed based on the aircraft dynamics, flight plans, and other information such as wind data [10], [13]. Then, an error covariance is added to the nominal trajectory, from which a probability of conflict is computed [14], [15]. However, this method does not account for the correlation in the cross-track error and the along-track error of a maneuvering aircraft. Furthermore, this approach requires a good knowledge of the aircraft intents. Both CTAS and URET assume that aircraft follow the advisories of air traffic control (ATC) and standard flight rules, such as instrument flight rules (IFR), standard instrument departure (SID), and standard terminal arrival routes (STAR), to predict the aircraft trajectories up to 30 min into the future. However, under the NextGen concept, aircraft intent information could dynamically change over time. The accuracy of an intent-based trajectory prediction algorithm strongly depends on the reliability of the intent information. Thus, it is necessary verify the broadcast intent of aircraft in the NextGen or to infer the intent of aircraft from radar measurements in the current ATC system.
In this paper, we propose a probabilistic trajectory prediction algorithm using a hybrid system aircraft dynamics model and verified/inferred intent information. The hybrid system models the aircraft dynamics with a number of flight modes and provides accurate short-term trajectory prediction for maneuvering or nonmaneuvering aircraft. It also allows us to compute the correlation between the cross-track error and along-track error. We utilize an intent inference algorithm to verify or infer the intent of an aircraft in real time. The inferred intent information is then used to improve the trajectory prediction in the long term. Also, in the algorithm, we propose an aircraft navigation model to describe the navigation performance of an aircraft. In the NextGen, a large variant of aircraft with very different capabilities are expected to operate in the same airspace. The navigation model enables the proposed trajectory prediction algorithm to be easily adapted for aircraft with different navigation capabilities.
Various authors have considered nominal trajectory prediction methods that incorporates aircraft intent information [16], [17], [18], [19]. This paper extends these trajectory prediction methods into a probabilistic framework by using both an aircraft dynamics model (as in [7] and [8]) for short-term prediction and an aircraft navigation model (as in [10] and [11]) for long-term prediction. Reynolds et al. [20], [21] have addressed issues of conformance monitoring in ATC and discussed how intent information could help in conformance monitoring for maneuvering aircraft. This work motivates our proposed idea of using verified intent information for accurate trajectory prediction. Yang et al. [22] proposed a Monte Carlo simulation method that incorporates aircraft intent information into probabilistic conflict detection. The method approximates the relative trajectory of a pair of aircraft with a series of straight lines. However, this could be inaccurate for turning aircraft. Furthermore, the method is not computationally efficient for three or more aircraft conflict scenarios, as it requires Monte Carlo simulations for each aircraft pair. In [23], the authors model the heading change of an aircraft with a Poisson distribution derived based on empirical data. This method does not explicitly assume any intent information.
Based on the predicted probabilistic aircraft trajectories, we then compute the probability of conflict involving two aircraft. An exact solution of this conflict probability requires computationally intensive numerical integration. Paielli et al. [14] proposed an approximate analytical solution that reduces the computation time by a few orders. However, this method assumes that the aircraft velocities are constant (in both magnitude and direction) during the encounter. We refine this algorithm and propose an approximate analytical solution to compute the conflict probability without the aforementioned assumption. It is shown that the refined algorithm significantly improves the accuracy of the approximated conflict probability, while its computational cost remains on the same order as that of Paielli's algorithm.
The rest of this paper is organized as follows. The design architecture of an intent-based probabilistic conflict detection algorithm for the NextGen is presented in Section II. We give details of the probabilistic trajectory prediction algorithm and the conflict detection algorithm in Sections III and IV, respectively. In Section V, we illustrate the proposed conflict detection algorithm with various simulation examples. Conclusions are given in Section VI.
SECTION II
OVERVIEW OF THE PROBABILISTIC CONFLICT DETECTION ALGORITHM FOR THE NextGen
The NextGen is a proposed transformation of the National Air Transportation System to significantly improve the safety, security, and capacity of air transportation operations [1]. Under the operational concept of the NextGen, aircraft are expected to have new capabilities that allow them to perform various flight management functions. Examples of these capabilities are four-dimensional (4-D) trajectory-based operations, enhanced situational awareness via data-link, and accurate positioning aided by the Global Positioning System. For more efficient use of airspace, properly equipped aircraft will be allowed to perform air management operations, such as self-separation, flight plan changes, conflict detection, and conflict resolutions. Hence, automation and/or supporting tools are needed to help in the decision-making process for both aircraft flight crew and air traffic controllers. In this section, we consider the design of an intent-based probabilistic conflict detection algorithm that would serve as this automation tool.
A. Design Architecture
The structure of the probabilistic conflict detection algorithm is given in Fig. 1. It consists of a hybrid estimation block, an intent inference block, a trajectory prediction block, and a conflict detection block.
1) Hybrid Estimation Block
To accurately model the trajectory of a maneuvering aircraft, we use a hybrid system to model the aircraft dynamics. The hybrid system consists of a continuous state that represents the aircraft's position and velocity and a discrete state which represents the aircraft flight mode. In the horizontal plane, we divide the aircraft dynamics into the following horizontal modes.
Constant Velocity (CV): The aircraft flies at constant speed and heading.
Constant Acceleration/Deceleration (CA): The aircraft increases or decreases speed while maintaining a constant heading.
Coordinated Turn (CT): The aircraft turns at a constant speed and constant turning rate.
In the vertical plane, the vertical modes are as follows.
The flight mode of an aircraft at any time is characterized by a combination of the horizontal mode and the vertical mode. For example, the flight mode of the aircraft is (CV,CH) if the aircraft is moving at constant speed and constant altitude. Thus, the set of flight modes is M = {(CV,CH),(CV,CD),(CA,CH),(CA,CD),(CT,CH), (CT,CD)}. The detailed hybrid system model of the aircraft dynamics is given in the Appendix.
We use a hybrid estimation algorithm, developed in [24], to provide the estimation of the aircraft state (position and velocity) and flight modes. A brief description of the hybrid estimation algorithm is given in Section II-B.
2) Intent Inference Block
An intent inference algorithm, developed in [17], is used to verify or infer the aircraft's intent. The algorithm considers a set of possible intents of the aircraft and infers the most likely intent based on the position, velocity, flight plan, and other relevant information. An aircraft's intent is a set of waypoints, denoted as I = {WP,WP+1,⋖,WP+Nwp}, which describes the intended flight path of the aircraft. Each waypoint WP+i is associated with a position (latitude, longitude, and altitude), a turn radius, and a controlled time of arrival (CTA). An aircraft's intent may be broadcast via onboard data-link, such as the Automatic Dependent Surveillance Broadcast (ADS-B) in the NextGen [25], [26]. In the case that broadcast information is available, we define the broadcast intent specifically as the aircraft's flight plan and the waypoints in the flight plan as trajectory change points (TCPs). The objective of the intent inference algorithm is to a) infer an aircraft's intent when no real-time flight plan information is available (e.g., nonequipped aircraft) or b) verify the reported flight plan (i.e., conformance monitoring) and infer the true intent. Thus, the intent inference algorithm could be applied to current ATC as well as the NextGen. A brief description of the intent inference algorithm is given in Section II-C.
3) Trajectory Prediction Block
In this paper, we develop a probabilistic trajectory prediction algorithm that utilizes the hybrid system aircraft dynamics model and the inferred intent to accurately describe the aircraft dynamics and to compute its future trajectory. Note that an intent describes an aircraft's future coordinates in space and time at a number of waypoints, while a probabilistic trajectory specifies the aircraft positions with the corresponding error covariance at fixed time steps. In the algorithm, we also develop an aircraft navigation model to describe the navigation capabilities of various types of aircraft. The algorithm computes the probabilistic trajectory by an analytical method that has a low computational cost. Details of the algorithm are discussed in Section III.
4) Conflict Detection Block
In the conflict detection algorithm, by extending the work in [14], we develop an analytical method to compute conflict probabilities of maneuvering or nonmaneuvering aircraft. The proposed algorithm yields accurate approximation to the conflict probabilities with computational costs several order lower than those of numerical integration methods. Details of the conflict detection algorithm are discussed in Section IV.
C. Aircraft Intent Inference
In this paper, we use the intent inference algorithm developed in our previous work [17]. As illustrated in Fig. 1, the algorithm uses the aircraft position, velocity (from hybrid estimation block), flight plan information, weather information, and static information such as special-use airspace (SUA), airports, and navigational aids (navaids). We assume the flight plan to be available from ADS-B. For an aircraft that does not broadcast its intent (i.e., flight plan), all navaids within 50 mi of the aircraft will be tested as possible intents (as in [27]). The weather region is assumed to be described by a polygonal boundary in the horizontal plane [28]. Weather information may be available, for example, from the Flight Information Service Broadcast. Other information such as SUA, temporary flight restrictions, airport information, and airspace boundaries is currently available from static databases and would be available via the Aeronautical Information Services and Geospatial Information Services in the NextGen. The various special or restricted regions, such as the SUA, are described by polygonal boundaries similar to that of the weather cell. Based on the work in [18] and [29], a finite set of intent models is considered in the algorithm. These intent models are grouped into regulation related intents and flight plan related intents. Within each group, different intent models are defined in the horizontal dimension, the vertical dimension, and the speed dimension. Examples of these intent models are given in Table 1 [17]. Each intent model Ir is associated with a set of waypoints, denoted as {WPr,WPr+1, ⋖,WPr+Nr}. We also define a unit vector er in the direction of the waypoint WPr for each intent. For example, Fig. 2 illustrates the possible intents for an aircraft whose flight plan trajectory passes through a weather cell. The intent “Go to TCP” (I1) models that the aircraft intends to follow its flight plan and fly through the weather cell. The second model, “Avoid Weather,” models that the aircraft maneuvers horizontally to avoid the weather cell. In this model, we assume that the aircraft will choose the shortest possible path in order to avoid the weather cell and then return towards the next TCP (i.e., WP2 + 1) in the original flight plan.
The intent inference algorithm first computes the likelihood function of each intent Ir and then outputs an inferred intent based on the one with the maximum likelihood. The likelihood function consists of two factors κ1 and κ2, where κ1 is a function of the estimated state of the aircraft (spatial information) and κ2 is a function of how far into the future the goals associated with each intent reside (temporal information). Hence, as first proposed in [17], the likelihood for an intent Ir at time k can be written as
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$$\Lambda(I_{r},k)=\kappa_{1}(I_{r},k)\kappa_{2}(I_{r},k).$$
In the following, we illustrate how the likelihood function is computed in the horizontal dimension. First, let us consider the case in which the aircraft is in the CV or CA flight mode. In this case, the likelihood of the intent Ir is defined to be maximum when the aircraft is heading towards the associated waypoint WPr. The likelihood of the aircraft intent based on spatial information can be represented by
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$$\kappa_{1}^{\ast}(I_{r},k)={\cal N}_{1}\left(\psi_{r}(k)-\psi_{ac}(k);0,\sigma_{h}^{2}\right)\eqno{\hbox{(1)}}$$where ψr(k) is the angle of the unit vector er (see Fig. 3); ψac(k) is the aircraft heading; and σh is a design parameter that is set to σh = 5 deg. The angles ψr(k) and ψac(k) are computed using the current state estimates of the aircraft. Since, the state estimates are noisy, the factor κ1∗(Ir,k) is filtered using a fading memory filter. Thus, for the CV or CA flight mode, we define spatial
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$$\eqalign{\kappa_{1}(I_{r},k)=&\,{1\over\phi_{k}}\sum_{l=0}^{k}f^{k-l}\kappa_{1}^{\ast}(I_{r},l)\cr&{\hbox{if flight mode}}={\hbox{CV or CA}}\cr\phi_{k}=&\,\sum_{l=0}^{k}f^{k-l},\quad 0\;< \;f\;< \;1}$$where f is the fading memory factor.
In [17], it has been shown that the spatial likelihood factor κ1(Ir,k) defined above could cause delays in detecting intent changes during aircraft maneuvers. To correct this problem, when the aircraft is in the CT flight mode, the rate of change of κ1∗(Ir,k) is used in place of (1) as follows:
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$$\kappa_{1}(I_{r},k):=\kappa_{1}^{\ast}(I_{r},k)-\kappa_{1}^{\ast}(I_{r},k-1)\quad{\hbox{if flight mode}}={\hbox{CT}}.$$Hence, the spatial likelihood factor κ1 is defined as
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$$\displaylines{\kappa_{1}(I_{r},k)\hfill\cr\hfill=\!\cases{{1\over\phi_{k}}\sum_{l=0}^{k}f^{k-l}\kappa_{1}^{\ast}(I_{r},l),&{\hbox{if flight mode}} = {\hbox{CV or CA}}\cr\kappa_{1}^{\ast}(I_{r},k) \!-\! \kappa_{1}^{\ast}(I_{r},k-1),&{\hbox{if flight mode}} = {\hbox{CT}}.}}$$
The spatial likelihood factor κ1 could give similar values for two intents Ii and Ij if their corresponding waypoints WPi and WPj are almost collinear with the aircraft's current position. To help distinguish between the two intents Ii and Ij, a likelihood factor κ2 is defined based on the time dimension as shown in the equation at the bottom of the page, where TTG(WP) denotes the time-to-go (TTG) to the waypoint WP, σreg = 900 s and σfp = 17s. Based on κ2, more weight is given to the intent whose corresponding waypoint is nearer to the aircraft's current position. Furthermore, the values of σreg and σfp are chosen based on the assumption that the pilot should follow the regulation related intent rather than the flight plan related intent when both intents are likely (based on spatial information).
The likelihood functions in the vertical dimension and speed dimension are similarly computed as above, and the interested reader is referred to [17] for details. Based on the maximum likelihood approach, the inferred intent at time k is defined as
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$$\mathhat{I}(k)=arg\max_{I_{r}}\Lambda(I_{r},k).$$For equipped aircraft, if the broadcast intent does not agree with the inferred intent, we call the aircraft “blundering” and its intent information will be treated with extra care. We intend to investigate on how to use such intent information and its effect on the accuracy of conflict prediction in future research.
SECTION III
PROBABILISTIC TRAJECTORY PREDICTION
In this section, we present the algorithm for predicting the probabilistic aircraft trajectory based on the inferred intent Î. The algorithm utilizes a hybrid system aircraft dynamics model to predict the aircraft trajectory in the short term and to propagate the correlation of the prediction errors. We also propose a navigation model to account for the deviations in aircraft trajectories about its intended flight path due to the navigation error of the aircraft. This navigation model is used with the intent information to improve the accuracy of the trajectory prediction in the long term (up to 30 min in this paper).
A. Aircraft Dynamics Model
Let the position of the aircraft in a chosen inertial reference frame be denoted as y = [ξ η h]T. We divide the aircraft dynamics into the six flight modes described in Section II-A and describe the aircraft dynamics with the following stochastic linear hybrid system [24]:
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$$x(k+1)=A_{m(k)}{x}(k)+B_{m(k)}w_{m(k)}(k)+u_{\rm nav}(k)\eqno{\hbox{(2)}}$$where 
is the continuous state vector; m is the flight mode (or discrete state); Ai, and Bi are the system matrices corresponding to flight mode m(k) = i. The process noise wi(k) is a white Gaussian sequence with covariance Qi in mode i. The white process noise models additive force disturbances due to local wind effect such as turbulence, as well as mechanical and human factors [8]. As a result, the errors of the predicted positions are correlated in time. However, the white noise model does not account for correlation effects such as that due to wind which affects two neighboring aircraft simultaneously, and we hope to address this in future research. The input unav represents the unknown control input of the aircraft's autopilot, which acts to correct any deviations of the aircraft trajectory from the intended flight path. The matrices Ai, Bi, and Qi corresponding to the six flight modes are given in the Appendix.
We consider the problem of computing the probabilistic aircraft trajectory p[y(k)] in two steps. First, we assume no information on the navigation capability of the aircraft and use only the information 
(that is, 
denotes the aircraft dynamics model in (2) with unav = 0) to predict the aircraft trajectory. We denote this trajectory as 
. Next, to consider the control input unav(k), we write, by Bayes' theorem
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$$p\left[y(k)\vert{\cal H},u_{\rm nav}(k)\right]=b_{1}p\left[u_{\rm nav}(k)\vert y(k),{\cal H}\right]p\left[y(k)\vert{\cal H}\right]\eqno{\hbox{(3)}}$$where b1 is a normalizing constant. However, it is difficult to compute 
without an extensive modeling of the aircraft's autopilot. Hence, we propose an aircraft navigation model to approximate the outcome of the autopilot as follows. We define a likelihood function
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$$L\left(y(k);{\cal H},u_{\rm nav}(k)\right):=p\left[u_{\rm nav}(k)\vert y(k),{\cal H}\right].$$We use this likelihood function to describe how the aircraft trajectory deviates from its intended flight path due to the autopilot's control unav. Specifically, the likelihood function is modeled as
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$$L\left(y(k);{\cal H},u_{\rm nav}(k)\right)=b_{2}{\cal N}_{2}\left(y(k);\bar{y}_{\cal F}(k),P_{\cal F}(k)\right)\eqno{\hbox{(4)}}$$where b2 is a constant, 
is the intended flight path of the aircraft, and 
represents the flight path deviations due to navigation errors. Note that the constant b2 is needed because, in general, a likelihood function is not a pdf. We will discuss how b2 is determined in Section III-D.
B. Trajectory Prediction Without Navigation Model
Let us first consider the trajectory prediction with 
only. The aircraft dynamics model becomes
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$$x(k+1)=A_{m(k)}x(k)+B_{m(k)}w_{m(k)}(k).\eqno{\hbox{(5)}}$$We now consider how to model the evolutions of the flight mode m(k) based on the given intent Î. The flight mode transitions are described by a set of guard conditions, as illustrated by the following example. We consider the horizontal trajectory of an aircraft following an intent Î with waypoints WP, WP + 1, etc. [see Fig. 4(a)]. Given the corresponding turn radius at each waypoint, we can compute distances such as d1∗ and d2∗ defined in Fig. 4(a). The flight mode transitions are then modeled as dependent on the distances d1(k) and d2(k). For example, in Fig. 4(b), the aircraft will be in the CT mode if a guard condition (d1(k) < d1∗)∧(d2(k) < d2∗) is true. In practice, a typical aircraft trajectory would vary randomly, such that it may start to turn at a distance, say, 
instead of d1∗. Similarly, the turn may end at the distance 
instead of d2∗. We account for this uncertainty by considering the distances d1∗ and d2∗ as random parameters in the hybrid system model of Fig. 4(b).
Hence, in general, we model a flight mode transition from m(k−1) = i to m(k) = j as determined by the following guard condition [24], [30]:
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$$L_{\theta,ij}\theta^{\ast}+L_{x,ij}x\leq 0\eqno{\hbox{(6)}}$$where 
are constant matrices. The vector θ∗ denotes a q-dimensional random vector (θ∗ = [d1∗ d2∗]T in the above example) with pdf
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$$p[\theta^{\ast}]={\cal N}_{q}(\theta^{\ast};\bar{\theta},\Sigma_{\theta})\eqno{\hbox{(7)}}$$where 
and Σθ are design parameters used to model the uncertainty in the flight mode transitions described above. We have derived the guard conditions (6) for various kinds of flight trajectories in ATC in [24].
The state pdf 
can then be computed as follows. By the total probability theorem, we can write
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$$p\left[x(k)\vert{\cal H}\right]=\sum_{i\in M}p\left[x(k)\vert m(k)=i,{\cal H}\right]p\left[m(k)=i\vert{\cal H}\right].$$We assume that the state pdf conditioned on each flight mode m(k) = i or 
is a Gaussian pdf with mean 
and covariance Pi(k), i.e.,
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$$p\left[x(k)\vert m(k)=i,{\cal H}\right]={\cal N}_{n}\left(x(k);\mathhat{x}_{i}(k),P_{i}(k)\right).$$
Initialization: At k = kc, we have 
and 
, where p[m(kc)] is the pdf of the current mode and 
is the conditional pdf of the current state. Note that both the current state and the current mode pdfs are given by the hybrid estimation algorithm (see Section II-B).
Compute Mode Transition Probability: Based on the inferred intent, we derive the guard condition (6), which models the flight mode transition as described above. Most well-known Kalman filter-based hybrid estimation algorithms, such as the interacting multiple model (IMM) algorithm [31], assume constant mode transition probabilities and are thus not applicable directly to our model here. In [32], we have shown that the mode transition probability for a mode transition governed by the guard condition in (6) is given by
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$$\eqalignno{&p\left[m(k+1)=j\vert m(k)=i,{\cal H}\right]\cr&\quad=\int\limits_{\BBR^{n}}p\left[m(k+1)=j\vert m(k)=i,x(k)=x,{\cal H}\right]\cr&\qquad\qquad\times p\left[x(k)=x\vert m(k)=i,{\cal H}\right]\,dx\cr&\quad=\Phi_{s}\left(L_{\theta,ij}\bar{\theta}+L_{x,ij}\mathhat{x}_{i}(k),L_{\theta,ij}\Sigma_{\theta}L_{\theta,ij}^{T}\right.\cr&\qquad\qquad\left.+\ L_{x,ij}P_{i}(k)L_{x,ij}^{T}\right)&\hbox{(8)}}$$where Φs(⋅) is the Gaussian cumulative density function (cdf) defined as follows.
Definition 1
Let z = [z1 z2⋖ zs]T be a multivariate Gaussian pdf with mean 
and covariance Σ. We define an s-dimensional Gaussian cdf as
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$$\Phi_{s}(\bar{z},\Sigma)\!=\!p[z\!\leq\!0]\!=\!\int\limits_{-\infty}^{0}\int\limits_{-\infty}^{0}\ldots\int\limits_{-\infty}^{0}{\cal N}_{s}(z;\bar{z},\Sigma)\,dz_{1}\,dz_{2}\ldots dz_{s}.$$Note that the first line of (8) requires us to evaluate a multivariate integral of dimension n (i.e., the dimension of the state vector x; n = 8 in this case). We have reduced the computational cost by showing that this integral is a Gaussian cdf with dimension s, where s is the number of inequalities in the guard condition (6). In our application, s is typically one or two. Hence, the computation cost is significantly reduced compared with evaluating the first line of (8) by numerical integration techniques.
Update pdfs: By the total probability theorem
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$$\displaylines{p\left[x(k\!+\!1)\vert m(k\!+\!1)\!=\!j,{\cal H}\right]\!=\!\sum_{i\in M}p\left[x(k\!+\!1)\vert m(k\!+\!1)\!=\!j,\right.\hfill\cr\hfill\left.m(k)\!=\!i,{\cal H}\right]p\left[m(k)\!=\!i\vert{\cal H}\right].\quad\hbox{(9)}}$$Equation (9) implies that we need to compute d2 pdfs 
for i,j = 1,2,⋖,d, where d is the number of flight modes (d = 6 in this paper). To reduce the computational and memory costs, we assume that each pdf 
is Gaussian and consider (9) as a weighted sum of d Gaussian pdfs. Equation (9) is then approximated by a single Gaussian pdf by moment matching. Thus, the pdf 
is approximated as [31]
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$$\displaylines{p\left[x(k+1)\vert m(k+1)=j,{\cal H}\right]\hfill\cr\hfill={\cal N}_{n}\left(x(k);\mathhat{x}_{j}(k+1),P_{j}(k+1)\right)\quad\hbox{(10)}}$$where
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$$\eqalignno{\mathhat{x}_{j}(k+1)=&\,\ A_{j}\mathhat{x}_{j}^{0}(k)&\hbox{(11)}\cr P_{j}(k+1)=&\ \,A_{j}P_{j}^{0}(k)A_{j}^{T}+B_{j}Q_{j}B_{j}^{T}&\hbox{(12)}\cr\mathhat{x}_{j}^{0}(k)=&\,\sum_{i\in M}\bar{\alpha}_{ji}(k)\mathhat{x}_{i}(k)&\hbox{(13)}\cr P_{j}^{0}(k)=&\,\sum_{i\in M}\bigg\{P_{i}(k)+\left[\mathhat{x}_{i}(k)-\mathhat{x}_{j}^{0}(k)\right]\cr&\qquad\times\left[\mathhat{x}_{i}(k)-\mathhat{x}_{j}^{0}(k)\right]^{T}\bigg\}\bar{\alpha}_{ji}(k)&\hbox{(14)}\cr\bar{\alpha}_{ji}(k)=&\,{1\over c_{j}}p\left[m(k+1)=j\vert m(k)=i,{\cal H}\right]\cr&\times p\left[m(k)=i\vert{\cal H}\right]}$$and cj is a normalizing constant. Note that the state 
is a weighted average of the states 
of all possible flight modes.
Output: The mode pdf at time k + 1 is given by
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$$\displaylines{p\left[m(k+1)=j\vert{\cal H}\right]=\sum_{i\in M}p\left[m(k+1)=j\vert m(k)=i,{\cal H}\right]\cr{\hskip170pt}\times \,p\left[m(k)=i\vert{\cal H}\right].}$$Using (10), the state pdf 
is given by
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$${\hskip -20pt}\eqalign{p\left[x(k+1)\vert{\cal H}\right]=&\,\sum_{j\in M}p\left[x(k+1)\vert m(k+1)=j,{\cal H}\right]\cr&\times p\left[m(k+1)=j\vert{\cal H}\right]\cr=&\,\sum_{j\in M}{\cal N}_{n}\left(x(k);\mathhat{x}_{j}(k+1),P_{j}(k+1)\right)\cr&\times p\left[m(k+1)=j\vert{\cal H}\right].}$$By moment matching, we approximate the pdf 
given above with a single Gaussian pdf. Hence
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$${\hskip -30pt}p\left[x(k+1)\vert{\cal H}\right]={\cal N}_{n}\left(x(k+1);\mathhat{x}(k+1),\Sigma_{x}(k+1)\right)\eqno{\hbox{(15)}}$$where
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$${\hskip -20pt}\eqalign{\mathhat{x}(k+1)=&\,\sum_{j\in M}\mathhat{x}_{j}(k+1)p\left[m(k+1)=j\vert{\cal H}\right]\cr\Sigma_{x}(k+1)=&\,\sum_{j\in M}\bigg\{P_{j}(k+1)+\left[\mathhat{x}_{j}(k+1)-\mathhat{x}(k+1)\right]\cr&\qquad\quad\times\left[\mathhat{x}_{j}(k+1)-\mathhat{x}(k+1)\right]^{T}\bigg\}\cr&\times p\left[m(k+1)=j\vert{\cal H}\right].}$$
Repeat steps 2)–4) to propagate the state pdf into the future.
The pdf of the aircraft position y(k) is then given by
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$$p\left[y(k)\vert{\cal H}\right]={\cal N}_{2}\left(y(k);\bar{y}_{\cal H}(k),P_{\cal H}(k)\right)\eqno{\hbox{(16)}}$$where
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$$\eqalign{\bar{y}_{\cal H}(k)=&\,\Upsilon\mathhat{x}(k)\quad P_{\cal H}(k)=\Upsilon\Sigma_{x}(k)\Upsilon^{T}\cr\Upsilon=&\,\left[\matrix{1&0&0&0&0&0&0&0\cr 0&0&0&1&0&0&0&0\cr 0&0&0&0&0&0&1&0\cr}\right].}$$
Note that the error covariance 
of the predicted position depends on the state covariance Σx(k), which is updated in steps 3)–4) above at each time step. Specifically, any correlation between the prediction errors (e.g., errors in the ξ-axis and the η-axis) is propagated via (12) and (14). This correlation is significant when aircraft maneuvers and during flight mode transitions.
C. Aircraft Navigation Model
In Section III-A, we have proposed the likelihood function (4) to model the navigation capability of an aircraft. We now discuss how to model the covariance 
in the likelihood function.
Consider an aircraft following an intent represented by waypoints WP and WP +1, as illustrated in Fig. 5. Let yWP and yWP1 be the coordinates of the waypoints WP and WP + 1, respectively, and TWP and TWP1 be the respective CTAs. A nominal 4-D trajectory of an aircraft can be computed by assuming that the aircraft travels at a constant velocity to reach each waypoint at the CTA. For example, the nominal 4-D trajectory up to waypoint WP is given by
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$$\bar{y}_{\cal F}(k)={y_{{\rm WP}}-y_{c}\over T_{{\rm WP}}-t_{c}}(k T_{s}-t_{c})+y_{c}$$where yc is the current aircraft position, tc = kc Ts is the current time, and Ts is the sampling period.
In practice, an aircraft trajectory could deviate from this nominal trajectory due to navigation errors which depend on the equipments onboard the aircraft. We first consider the aircraft navigation error covariance 
in a reference frame attached to the aircraft's velocity vector. The error covariance is then given by
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$$\Sigma_{\cal F}(k)=\left[\matrix{\sigma_{g}^{2}(kT_{s}-t_{c})^{2}&0&0\cr 0&\sigma_{c}^{2}&0\cr 0&0&\sigma_{h}^{2}}\right]\eqno{\hbox{(17)}}$$where σc and σh are the root mean squared (rms) cross-track position errors in the horizontal and the vertical planes, respectively, and σg is the rms along-track error growth rate. Typically, the cross-track position errors depend on the aircraft's navigation accuracy. Hence, their respective rms values are assumed to be constant and are chosen based on the navigation performance level of the aircraft. The along-track error depends on the capability of the aircraft in meeting the CTA specifications. This is illustrated in Fig. 5. We let Δ T0 denote the time tolerance associated with the CTA TWP. This tolerance specifies that the rms value of the aircraft's time of arrival at waypoint WP is TWP±Δ T0. Assuming that the aircraft travels at a constant velocity, the rms deviation of the aircraft's speed from that of the nominal could be approximated as Δ v0 = (vH0−vL0)/2, where
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$$v_{L0}={y_{{\rm WP}}-y_{c}\over T_{{\rm WP}}-t_{c}+\Delta T_{0}}\quad v_{H0}={y_{{\rm WP}}-y_{c}\over T_{{\rm WP}}-t_{c}-\Delta T_{0}}.$$The along-track rms position error is then modeled as growing at a constant rate σg = Δ v0.
Let Γ(k) be the rotational matrix from the aircraft body frame to the inertial reference frame. Then the navigation error in the inertial frame is given by
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$$P_{\cal F}(k)=\Gamma(k)\Sigma_{\cal F}(k)\Gamma(k)^{T}.\eqno{\hbox{(18)}}$$For the aircraft trajectory between waypoints WP and WP + 1, we compute nominal trajectory and the error growth rate in a similar way (with WP replaced by WP + 1 and the current position replaced by waypoint WP).
D. Trajectory Prediction With Navigation Model
We now compute the aircraft trajectory y(k) based on both the aircraft dynamics model 
and the aircraft navigation model (4). Substituting (4) and (16) into (3), we have
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$$\displaylines{p\left[y(k)\vert{\cal H},u_{\rm nav}(k)\right]=b_{1}b_{2}{\cal N}_{2}\left(y(k);\bar{y}_{\cal F}(k),P_{\cal F}(k)\right)\hfill\cr\hfill\times\,{\cal N}_{2}\left(y(k);\bar{y}_{\cal H}(k),P_{\cal H}(k)\right).\quad\hbox{(19)}}$$It can be shown that for any n-dimensional Gaussian pdfs
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$${\cal N}_{n}(y;\mu_{1},\Sigma_{1}){\cal N}_{n}(y;\mu_{2},\Sigma_{2})=\kappa{\cal N}_{n}(y;\mu_{12},\Sigma_{12})$$where κ is a constant and
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$$\eqalign{\Sigma_{12}^{-1}=&\,\Sigma_{1}^{-1}+\Sigma_{2}^{-1}\cr\mu_{12}=&\,\Sigma_{12}\left(\Sigma_{1}^{-1}\mu_{1}+\Sigma_{2}^{-1}\mu_{2}\right).}$$
Hence (19) can be written as
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$$\eqalignno{p\left[y(k)\vert{\cal H},u_{\rm nav}(k)\right]=&\,b_{1}b_{2}\kappa{\cal N}_{2}\left(y(k);\bar{y}(k),P(k)\right)\cr=&\,{\cal N}_{2}\left(y(k);\bar{y}(k),P(k)\right)&\hbox{(20)}}$$where
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$$\eqalignno{P^{-1}(k)=&\,P_{\cal H}^{-1}(k)+P_{\cal F}^{-1}(k)&\hbox{(21)}\cr\bar{y}(k)=&\,P(k)\left(P_{\cal H}^{-1}(k)\bar{y}_{\cal H}+P_{\cal F}^{-1}(k)\bar{y}_{\cal F}\right).&\hbox{(22)}}$$Note that since the total integral (i.e., the total probability) of the pdf 
must be one, we choose the constant b2 in (4) such that b1 b2κ = 1.
Example 1
Let us illustrate the probabilistic trajectory algorithm with an example. Fig. 6(a) shows the nominal flight trajectories of an aircraft with two possible intents I1 and I2. For purpose of illustration, we present the predicted probabilistic trajectories 
for both intents in Fig. 6(b). (Note that, in our proposed algorithm, we only compute the probabilistic trajectory for the inferred intent.) Fig. 7 shows the rms prediction errors of the predicted trajectory based on intent I1 in the along-track and cross-track directions. In this example, we assume that the aircraft's cross-track navigation accuracy is σc = 1 nm. We assume that the CTA tolerance at WP1 is ±15 s and the CTA tolerance at WP1 +1 is ±20 s. The along-track navigation error thus has a growth rate of σg = 8 m/s between the current aircraft position and waypoint WP1 and an error growth rate of σg = 2.8 m/s between waypoints WP1 and WP1 +1. The error covariance of the predicted trajectory P1(k) is given by (21). It can be seen that at small time, say, t≤ 5 min, the error covariance 
is small; and the error covariance of the predicted trajectory P1(k) is thus close to 
. As time increases, the error covariance 
tends to grow quadratically, and the value of the covariance error P1(k) would then approach that of 
. From (22), the mean of the predicted trajectory 
1 is a weighted sum of that computed based on aircraft dynamics 
and that computed based on the flight plan model 
. The weights of the sum depend on the relative magnitude of the inverse of the covariance errors 
and 
. Thus, at small time, the mean 1 would follow closer to the mean 
. As the covariance error of the aircraft dynamics model 
increases, the mean 1 would follow closer to the mean 
of the navigation model. Thus, the trajectory prediction algorithm automatically changes the weights of the hybrid system model and the navigation model depending on the relative magnitudes of their error covariances.
SECTION IV
PROBABILISTIC CONFLICT DETECTION
In this section, we propose an analytical method to compute the conflict probability for two aircraft whose predicted trajectories are given as in (20). By definition, a conflict (not to be confused with a collision) occurs when two aircraft come closer than a certain distance to one another. This minimum separation between aircraft is defined with a horizontal separation and a vertical separation. Currently, for en route airspace, the minimum horizontal separation is 5 nm; under the Reduced Vertical Separation Minimum standard, the minimum vertical separation is 2000 ft above the altitude of 41 000 ft and 1000 ft below that altitude for certified aircraft.
Let the predicted trajectories of aircraft A and B be
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$$\eqalign{p\left[y_{A}(k)\vert{\cal H},u_{\rm nav}\right]=&\,{\cal N}_{2}\left(y_{A}(k);\bar{y}_{A}(k),P_{A}(k)\right)\cr p\left[y_{B}(k)\vert{\cal H},u_{\rm nav}\right]=&\,{\cal N}_{2}\left(y_{B}(k);\bar{y}_{B}(k),P_{B}(k)\right).}$$Let λAB(k) = yA(k)−yB(k) be the relative distance. We assume that the trajectories of the two aircraft are independent. Then, the pdf of the relative distance λAB(k) is given by
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$$\eqalignno{p\left[\lambda_{AB}(k)\right]=&\,{\cal N}_{2}\left(\lambda_{AB}(k);\bar{\lambda}_{ab}(k),\Sigma_{ab}(k)\right)\cr\bar{\lambda}_{ab}(k)=&\,\bar{y}_{A}(k)-\bar{y}_{B}(k)\cr\Sigma_{ab}(k)=&\,P_{A}(k)+P_{B}(k).&\hbox{(23)}}$$
In the horizontal plane, a conflict occurs if the relative distance λAB is within a circle of radius ρ = 5 nm. We define a conflict zone as
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$${\cal D}=\left\{\vert\lambda\vert\;< \;\rho\vert\lambda\in\BBR^{2}\right\}.$$Our objective is to determine the probability of a conflict occurring during a period of time T = Np Ts, where Ts is the sampling period and Np is a positive integer. We define this conflict probability over the time horizon T as
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$$p[{\cal C}\vert T]=\max_{k\in[0,N_{p}]}p\left[{\cal C}(k)\right]$$where 
is the instantaneous probability of conflict at time k given by
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$$p\left[{\cal C}(k)\right]=\int\limits_{\lambda\in{\cal D}}{\cal N}_{2}\left(\lambda;\bar{\lambda}_{ab}(k),\Sigma_{ab}(k)\right)d\lambda.\eqno{\hbox{(24)}}$$
In general, there is no analytical solution to the integral in (24). Paielli and Erzberger [14] have proposed an algorithm to approximate this integral under the assumption that the aircraft velocities are constant (in both magnitude and direction) during the encounter. We propose a refinement of this algorithm such that the aforementioned assumption is not needed. In the following, we shall first present qualitatively the algorithm by Paielli and Erzberger and then present our proposed refinement of the algorithm.
Paielli and Erzberger [14] considered an aircraft encounter geometry shown in Fig. 8(a), in which both aircraft fly at constant velocities. First, the error covariance Σab(k) in (23) is assigned to one of the aircraft, called the stochastic aircraft. The other aircraft, called the reference aircraft, is assumed to have no uncertainty. The relative geometry of the encounter is then shown in Fig. 9(a). Here, the ellipse represents the error covariance Σab of the stochastic aircraft. The instantaneous probability of conflict, given by (24), is to be evaluated over the shaded circular conflict zone. Through coordinate transformation, one can make the combined covariance centered on the stochastic aircraft into a circle as shown in Fig. 9(b). Mathematically, this means that the two-dimensional Gaussian pdf 
can then be written as the product of two independent one-dimensional Gaussian pdfs 
. Note that the circular conflict zone becomes an elliptical conflict zone through the transformation. Paielli and Erzberger then approximate the instantaneous probability of conflict by integrating over the extended rectangular conflict zone, shown in Fig. 9(b), instead of over the elliptical conflict zone. The instantaneous probability of conflict at each time k is thus given by
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$$\eqalignno{p\!\left[{\cal C}(k)\vert{\cal I}_{a}^{A},{\cal I}_{b}^{B}\right]\!=\!&\,\int\limits_{\Delta_{\eta}(k)-\delta_{\eta}(k)}^{\Delta_{\eta}(k)+\delta_{\eta}(k)}\int\limits_{\!\!\!\!-\infty}\!\!^{\infty}\!{\cal N}_{1}(\lambda_{\xi};\cdot){\cal N}_{1}(\lambda_{\eta};\cdot)d\lambda_{\xi}d\lambda_{\eta}\cr\!=\!&\,\int\limits_{\Delta_{\eta}(k)-\delta_{\eta}(k)}^{\Delta_{\eta}(k)+\delta_{\eta}(k)}{\cal N}_{1}(\lambda_{\eta};\cdot)d\lambda_{\eta}\cr\!=\!&\ \Psi_{1}\left(\Delta_{\eta}(k)+\delta_{\eta}(k);\cdot\right)\cr&-\Psi_{1}\left(\Delta_{\eta}(k)-\delta_{\eta}(k);\cdot\right)&\hbox{(25)}}$$where
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$$\Psi_{1}(\delta;\cdot):=\int\limits_{-\infty}^{\delta}{\cal N}_{1}(\nu;\cdot)d\nu$$is a one-dimensional Gaussian cdf whose values are well tabulated.
The main assumption in the above algorithm is that the aircraft velocities are constant and hence the path-crossing angle is constant. This assumption may not be true in some cases, such as when one of the aircraft is turning, as illustrated in Fig. 8(b). This scenario may occur, for example, during sequencing and merging (i.e., when an aircraft is instructed to merge behind another aircraft at a given distance and time and then maintain that spacing). In this case, we could have an aircraft encounter whose relative geometry is similar to that shown in Fig. 10(a). It can be seen that the approximation by integrating over the extended conflict zone would not be accurate here. To improve the accuracy of the approximation, we note that the main idea of the algorithm in [14] is as follows. First the covariance Σab (called the combined error covariance in [14]) is diagonalized through transformation so that the two-dimensional Gaussian pdf in (24) is decoupled into a product of two one-dimensional Gaussian pdfs. Then, instead of integrating over a circular or elliptical conflict zone, the integration is carried out over an (infinite) rectangular zone. As a result, the conflict probability is approximated by an one-dimensional Gaussian cdf. Extending this idea, we found out that the integral in (24) can be better approximated by considering a finite rectangular zone as illustrated in Fig. 10(b). First, by a suitable rotational transformation, we can rotate the elliptical conflict zone 
so that its major axis or minor axis is aligned with the η-axis of the reference coordinates. We then consider a finite rectangular zone (called the modified conflict zone 1) as shown in Fig. 10(b). The instantaneous probability of conflict is then given by
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$$\eqalign{&p\left[{\cal C}(k)\vert{\cal I}_{a}^{A},{\cal I}_{b}^{B}\right]\cr&\quad=\int\limits_{\Delta_{\eta}(k)-\delta_{\eta}(k)}^{\Delta_{\eta}(k)+\delta_{\eta}(k)}\int\limits_{-\Delta_{\xi}(k)-\delta_{\xi}(k)}^{-\Delta_{\xi}(k)+\delta_{\xi}(k)}{\cal N}_{1}(\lambda_{\xi};\cdot){\cal N}_{1}(\lambda_{\eta};\cdot)d\lambda_{\xi}d\lambda_{\eta}\cr&\quad=\left[\Psi_{1}\left(\Delta_{\eta}(k)+\delta_{\eta}(k);\cdot\right)-\Psi_{1}\left(\Delta_{\eta}(k)-\delta_{\eta}(k);\cdot\right)\right]\cr&\qquad{\hskip 1.5pt}\times\left[\Psi_{1}\left(\Delta_{\xi}(k)+\delta_{\xi}(k);\cdot\right)-\Psi_{1}\left(\Delta_{\xi}(k)-\delta_{\xi}(k);\cdot\right)\right].}$$Our idea here is to use a rectangular zone to better approximate the elliptical conflict zone 
. We can further extend this idea and consider a modified zone 2, shown in Fig. 10(c), which would even better approximate the elliptical zone 
. Based on geometry, the instantaneous probability of conflict is then given by
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$$\eqalignno{&p\left[{\cal C}(k)\vert{\cal I}_{a}^{A},{\cal I}_{b}^{B}\right]\cr&\quad=\left[\Psi_{1}\left(\Delta_{\eta}(k)+\delta_{\eta}(k);\cdot\right)-\Psi_{1}\left(\Delta_{\eta}(k)-\delta_{\eta}(k);\cdot\right)\right]\cr&\qquad\times\left[\Psi_{1}\left(\Delta_{\xi}(k)+\delta_{\xi}(k);\cdot\right)-\Psi_{1}\left(\Delta_{\xi}(k)-\delta_{\xi}(k);\cdot\right)\right]\cr&\qquad-\left[\Psi_{1}\left(\Delta_{\eta}(k)+\delta_{\eta}(k);\cdot\right)-\Psi_{1}\left(\Delta_{\eta}(k)+{\delta_{\eta}(k)\over\sqrt{2}};\cdot\right)\right]\cr&\qquad\times\left[\Psi_{1}\left(\Delta_{\xi}(k)+\delta_{\xi}(k);\cdot\right)-\Psi_{1}\left(\Delta_{\xi}(k)+{\delta_{\xi}(k)\over\sqrt{2}};\cdot\right)\right].\cr&&\hbox{(26)}}$$
Example 2a
Let us consider an aircraft encounter whose relative geometry is shown in Fig. 11(a). The stochastic aircraft is considered as stationary at the origin of the reference coordinates, and the reference aircraft flies at a constant relative velocity. For simplicity, we assume in this example that the combined error covariance 
for all k. We compare three methods of computing the probability of conflict: the “exact solution,” which uses a Gaussian quadrature integration to compute (24) with a relative tolerance of 10−5; “Paielli's approximation,” which uses (25) (i.e., the algorithm in [14]); and the proposed “refined approximation,” which uses (26). The instantaneous probabilities of conflict computed at different time t = kTs are shown in Fig. 11(b). At small time t, we see that the exact instantaneous probability of conflict is very small. This can be verified from the relative geometry shown in Fig. 11(a), which shows no overlapping between the conflict zone 
and the combined error covariance at time t = 0. At time t = 28 s, the separation of the aircraft is minimum and the corresponding exact instantaneous probability of conflict is maximum. From Fig. 11(b), the probabilities of conflict over the time horizon T = [0,60] computed by the various methods are as follows:
exact solution, or (24): p[C| T] = 0.28;
Paielli's approximation, or (25): p[C| T] = 0.31;
refined approximation, or (26): p[C| T] = 0.31.
In this case, we see that both (25) and (26) give good approximations to the conflict probability p[C| T] compared with the exact solution. However, we note that the probability of conflict versus time plot of the refined approximation shows a similar behavior compared to that of the exact solution, while the probability of conflict computed by Paielli's approximation does not.
Example 2b
Next, we consider a case in which one of the aircraft is turning during the encounter. The relative geometry of such an encounter is shown in Fig. 12(a). Here, the minimum aircraft separation occurs at time t = 78 s. We can see that at small time t, the exact instantaneous probability of conflict is very small. However, during the turn, at say t = 30 s, the Paielli's approximation gives an instantaneous probability of conflict close to one. From Fig. 12(a), it can be seen that, at t = 30 s, the instantaneous relative velocity of the reference aircraft is heading towards the stochastic aircraft. The Paielli's approximation assumes that the aircraft velocity is constant during the encounter, and hence it computes a high probability of conflict. From Fig. 12(b), the probabilities of conflict over the time horizon T = [0,100] computed by the various methods are as follows:
exact solution, or (24): p[C| T] = 0.28;
Paielli's approximation, or (25): p[C| T] = 1;
refined approximation, or (26): p[C| T] = 0.31.
Thus, in this example, which illustrates an encounter with nonconstant velocities, the refined approximation (26) still gives a good approximation as in Example 2a, while the Paielli's approximation (25) gives an erroneous result. Table 2 summarizes the computation time requirements, as well as the conflict probability results, for the various methods. Note that the computational cost of the refined algorithm remains a few orders lower than those of the numerical methods.
SECTION V
ILLUSTRATIVE EXAMPLE OF AN INTENT BASED CONFLICT DETECTION
We have validated the proposed intent-based conflict detection algorithm with various air traffic scenarios, some with real air traffic data from the Enhanced Traffic Management System (ETMS). In this section, we demonstrate with an illustrative example how the intent-based conflict detection algorithm helps to correctly predict a conflict due to a change of an aircraft's intent earlier compared with a nonintent-based method.
Example 3
We consider two aircraft A and B whose planned flight paths are shown in Fig. 13(a). Based on the given flight plans, the two aircraft will have a minimum separation of about 7 nm at time t = 480 s. However, at time t = 220 s, aircraft A makes a maneuver to avoid a weather cell. The corresponding simulated trajectories of aircraft A and B are shown in Fig. 13(b). Due to the maneuver of aircraft A, the minimum separation of aircraft A and B is about 2 nm (at t = 505 s).
We compute the future aircraft trajectories and the corresponding conflict probability at each current time tc = kc Ts, where kc = 1,2,⋖ and Ts = 5 s. We consider two scenarios: a current ATC scenario and a NextGen scenario. In the current ATC scenario, we model that a typical aircraft has a cross-track navigation error of σc = 1 nm and an error growth rate of σg = 0.25 nm/min. In the NextGen scenario, we assume that σc = 0.3 nm and σg = 0.04 nm/min. In actual operations of the NextGen, different aircraft could have different navigation capabilities, and this can be modeled by using different parameters σc and σg for different aircraft.
Fig. 14 shows the inferred intent, the predicted minimum mean separation, and the predicted probability of conflict at different current times tc between 200 and 300 s. The “Avoid Weather” intent of aircraft A is inferred at time tc = 230 s, which is 10 s after the start of the weather avoidance maneuver by aircraft A. In plot (ii), the blue and red curves shows the mean minimum separations computed based on the intent-based method and the nonintent-based method, respectively. Plot (iii) shows the probabilities of conflict. We assume 5 nm radius for the conflict zone for both the current ATC scenario and the NextGen scenario for the purpose of comparison here. We expect the designated conflict zone radius to be smaller in the future, and this would lead to smaller conflict probabilities. It can be seen that the intent-based prediction method correctly predicts a conflict at tc = 230 s, while the nonintent-based method incorrectly predicts that there would be no conflict. Also, the NextGen scenario gives slightly more accurate conflict probabilities due to smaller prediction errors, as we will see in the following paragraph.
Fig. 15 shows a snapshot of the probabilistic predicted trajectories of aircraft A and B at tc = 200 s. At this time, the inferred intent of aircraft A is to follow its flight plan. A snapshot of the probabilistic predicted trajectories at tc = 250 s is shown in Fig. 16. At this time, the inferred intent of aircraft A is to avoid the weather cell. Here, the nonintent-based prediction method assumes that the aircraft would fly straight and, hence, at this instant, predicts that there would be no conflict. By using the intent information, the intent-based prediction method correctly predicts that a conflict would occur about 4.5 min later. On the other hand, our simulation shows that the nonintent-based prediction method correctly predicts a conflict only at current time tc = 460 s, or 45 s before the conflict. This example illustrates that it is important to incorporate intent information into aircraft conflict detections in order to correctly predict any potential conflict early in time. The blue ellipses represent the error covariance of the predicted positions. We can see that in the NextGen scenario, the error covariance of the trajectory prediction is much smaller because a better aircraft navigation performance level is assumed.
In the simulations, we compute the predicted trajectories yA(k) and yB(k) in steps of 10 s. The computation time for predicting both trajectories 20 min into the future, plus the computation time for the conflict probabilities, is 0.26 s. The computations are done on a Dell D410 notebook computer using MATLAB. We have also investigated the computation time for a scenario involving ten aircraft. Assuming that we need to compute the conflict probabilities for every aircraft pair (total 10 C2 = 45 pairs) at every time step, the total computation time needed for predicting conflicts up to 20 min ahead is about 4.0 s. Thus, the algorithm could possibly be implemented in real time for conflict detection involving tens or even hundreds of aircraft.
The above example has shown how intent information could be used to improve the accuracy of trajectory prediction and conflict detection. However, it is also possible that a wrong intent information could give misleading result on the conflict probability. It is thus important to ensure the reliability of the intent information. With the introduction of advanced data communication systems in the NextGen, aircraft intent would be broadcast to all users. The intent inference algorithm can be used as a conformance monitoring tool to further ensure the reliability of the reported intents. Thus, we envision that an intent-based trajectory prediction based on trusted and verified intent information would lead to better conflict detection probability in high-density air traffic operations. For current ATC operations or nonequipped aircraft in the NextGen, the proposed conflict detection algorithm could also be used since it can infer intent information from radar measurements.
In the Next Generation Air Transportation System, aircraft are expected to have better but diverse range of navigation capabilities. They would also be given more autonomy in flight plan management. We have proposed an intent-based probabilistic conflict detection algorithm that could be used as an automation tool in air traffic control for ensuring the safe separation of aircraft in the NextGen. The proposed algorithm utilizes aircraft dynamics and navigation models, as well as inferred intent information, to predict a probabilistic 4-D trajectory of the aircraft. The conflict probability is then computed analytically. Through illustrative examples, we have demonstrated that the algorithm is computationally efficient and could be implemented in real time.
In this paper, we have assumed white Gaussian noise in the aircraft dynamics model. In future research, we would like to investigate correlated noise models, which could better describe disturbances such as wind.
Let (ξ,η) be the aircraft position in a horizontal plane, h be the aircraft's altitude, and 
be the state vector. We divide the aircraft dynamics into three horizontal modes (CV, CT, and CA) and two vertical modes (CH and CD). The aircraft dynamics in each of these modes are given below. Note that the dynamics model (2) is determined by a combination of the horizontal mode and the vertical mode 
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$$\displaylines{{\hbox{CV}}\ mode:\ \left[\matrix{\xi(k+1)\cr\mathdot{\xi}(k+1)\cr\eta(k+1)\cr\mathdot{\eta}(k+1)}\right]=\left[\matrix{1&T_{s}&0&0\cr 0&1&0&0\cr 0&0&1&T_{s}\cr 0&0&0&1}\right]\left[\matrix{\xi(k)\cr\mathdot{\xi}(k)\cr\eta(k)\cr\mathdot{\eta}(k)}\right]\hfill\cr\hfill+\left[\matrix{{T_{s}^{2}\over 2}&0\cr T_{s}&0\cr 0&{T_{s}^{2}\over 2}\cr 0&T_{s}}\right]\left[\matrix{w_{\xi_{\rm CV}}(k)\cr w_{\eta_{\rm CV}}(k)}\right]}$$where 
, 
, and 
.
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$$\displaylines{{\hbox{CT}}\ mode:\left[\matrix{\xi(k+1)\cr\mathdot{\xi}(k+1)\cr\eta(k+1)\cr\mathdot{\eta}(k+1)}\right]\!\!=\!\!\left[\matrix{1\!&\!{sin(\omega T_{s})\over\omega}\!&\!0\!&\!-{1-cos(\omega T_{s})\over\omega}\cr 0\!&\!cos(\omega T_{s})\!&\!0\!&\!-sin(\omega T_{s})\cr 0\!&\!{1-cos(\omega T_{s})\over\omega}\!&\!1\!&\!{sin(\omega T_{s})\over\omega}\cr 0\!&\!sin(\omega T_{s})\!&\!0\!&\!cos(\omega T_{s})}\right]\hfill\cr\hfill\times\left[\matrix{\xi(k)\cr\mathdot{\xi}(k)\cr\eta(k)\cr\mathdot{\eta}(k)}\right]+\left[\matrix{{T_{s}^{2}\over 2}&0\cr T_{s}&0\cr 0&{T_{s}^{2}\over 2}\cr 0&T_{s}}\right]\left[\matrix{w_{\xi_{\rm CT}}(k)\cr w_{\eta_{\rm CT}}(k)}\right]}$$where ω = 1.5 deg/s is the nominal turning rate, 
, 
, and 
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$$\displaylines{{\hbox{CA}}\ mode:\ \left[\matrix{\xi(k+1)\cr\mathdot{\xi}(k+1)\cr\mathddot{\xi}(k+1)\cr\eta(k+1)\cr\mathdot{\eta}(k+1)\cr\mathddot{\eta}(k+1)}\right]=\left[\matrix{1&T_{s}&{T_{s}^{2}\over 2}&0&0&0\cr 0&1&T_{s}&0&0&0\cr 0&0&1&0&0&0\cr 0&0&0&1&T_{s}&{T_{s}^{2}\over 2}\cr 0&0&0&0&1&T_{s}\cr 0&0&0&0&0&1}\right]\hfill\cr\hfill\times\left[\matrix{\xi(k)\cr\mathdot{\xi}(k)\cr\mathddot{\xi}(k)\cr\eta(k)\cr\mathdot{\eta}(k)\cr\mathddot{\eta}(k)}\right]+\left[\matrix{{T_{s}^{2}\over 2}&0\cr T_{s}&0\cr 1&0\cr 0&{T_{s}^{2}\over 2}\cr 0&T_{s}\cr 0&1}\right]\left[\matrix{w_{\xi_{\rm CA}}(k)\cr w_{\eta_{\rm CA}}(k)}\right]}$$where 
, 
, 
is the heading of the aircraft, and 
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$${\hbox{CH}}\ mode:\quad\left[h(k+1)\right]=[1]\left[h(k)\right]+[T_{s}]w_{\rm CH}(k)$$where E[wCH2(k)] = 0.22.
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$${\hbox{CD}}\ mode:\left[\matrix{h(k+1)\cr\mathdot{h}(k+1)}\right]\!=\!\left[\matrix{1&T_{s}\cr 0&1}\right]\left[\matrix{h(k)\cr\mathdot{h}(k)}\right]\!+\!\left[\matrix{{T_{s}^{2}\over 2}\cr T_{s}}\right]w_{\rm CD}(k)$$where E[wCD2(k)] = 0.52.
is the current state estimate conditioned on flight mode i and Pi(kc) is the corresponding error covariance. The function 
n(⋅;μ,Σ) denotes an n-dimensional Gaussian pdf with mean μ and covariance Σ.
.
