2.1 Definitions and context

Several, mostly different definitions for air traffic conflict and conflict detection methods can be found in the literature.

According to Kelly [Reference Kelly22], a conflict can be defined either as a violation of the separation standard or the predicted violation of the separation. Separation is defined based on distance and protected zones of given shapes (oblate spheroid and cylinder). The notion of “conflict detection” and “conflict resolution” is distinguished. An alerting logic based on two protected zones is described, and the severity of the conflict is assessed based on the time to reach these ranges. Because of the different separation requirements, horizontal and vertical separation is treated separately. The proposed conflict detection algorithm is based on instantaneous state vectors and does not take into account the stochastic nature of the problem.

A review of conflict detection and resolution methods is presented by Kuchar and Yang [Reference Kuchar and Yang6], creating a taxonomy of the techniques applied. Conflict is defined as a loss of minimum separation for two or more aircraft. Detection and resolution techniques are characterised based on several characteristics: state propagation method (this paper presents a probabilistic approach), state dimensions (the paper introduces a model for the horizontal plane), detection capability, resolution capability, the type of resolution manoeuvre and the method to handle multiple conflicts (throughout the paper aircraft conflict probabilities are calculated in a pairwise manner). The paper also emphasises the importance of robustness and the need for a consistent benchmarking method.

A mathematical description of the TCAS (traffic collision avoidance system) logic is given by Tang, Zhu and Piera [Reference Tang, Zhu and Piera23], and its capabilities are extended with horizontal resolution. TCAS is a last resort airborne system to prevent mid-air collisions by notifying the pilot of the threat (traffic advisory) and suggesting coordinated resolutions of the conflict situation (resolution advisory). The effect of surrounding traffic, induced risk, and resolution generated conflicts are also analysed using a graphical modelling and analysis software. With horizontal resolution, it is possible to reduce the domino effect. Taking into account induced conflicts is beneficial for airspace stability, however, no system-wide analysis is presented in the paper.

Formal mathematical definitions of conflict and collision can be found in the paper by Mitici and Blom [Reference Mitici and Blom24], together with a comparative evaluation of collision probability estimation models. Several models are presented with their application areas and characteristics analysed in a unified mathematical framework. A survey of safety risk models and validation methods is also provided to reflect on their applicability. Presented methods may become useful in future aerospace design, especially for unmanned traffic management. None of the methods reviewed apply a method to govern global dynamics to keep conflict probability at an acceptable level and airspace complexity manageable.

2.2 Traffic collision avoidance systems

The ACAS (airborne collision avoidance system) was extended for unmanned systems (ACAS Xu) [Reference Owen, Panken, Moss, Alvarez and Leeper25] and small unmanned systems (ACAS sXu) [Reference Alvarez, Jessen, Owen, Silbermann and Wood26]. Collision detection is enabled by the surveillance and tracking module incorporating multiple surveillance inputs to determine the position of nearby aircraft, together with position uncertainty. The threat resolution module evaluates collision risk, taking into account surveillance and dynamic uncertainty and determines whether a resolution manoeuvre is necessary. Resolution manoeuvres may be horizontal, vertical or combined. Platform manoeuvrability is taken into account to ensure avoidance feasibility. The sXu variant has a different action space to reflect the capabilities of sUAS. The problem is modelled as a Markov decision process (MDP). The MDP problem is solved in real-time, coordinating with other aircraft if possible during run-time. Although multiple threats may be handled using utility decomposition and fusion, due to the MDP formulation and the “curse of dimensionality” (exponential growth of the state space with the number of agents) application in dense airspace is infeasible [Reference Li, Egorov and Kochenderfer27]. Therefore, managing airspace complexity on a higher level (middle layer of the framework proposed) is necessary and justifiable, with an underlying operation of an ACAS-like CA method, customised for UAM. It is worth noting, that the ACAS Xu operation is consistent with the model adopted in the framework investigated in the paper for lower-level collision avoidance.

In most of the algorithms, the future trajectory of the intruder aircraft is extrapolated based on a linear motion assumption [Reference Bilimoria28, Reference van den Berg, Guy, Lin and Manocha29]. In the paper by Bilimoria [Reference Bilimoria28], closed-form analytical solutions are presented based on geometric optimisation, producing trajectories with minimal deviation from nominal. Intruder trajectories are extrapolated based on current speed and heading. An instantaneous resolution manoeuvre is assumed, conflict can be resolved by changing heading only, speed only or both. It is assumed that both aircraft stay in the same horizontal plane, and no wind disturbance is taken into account. Multiple aircraft conflicts are resolved sequentially. Intruders are modelled as velocity obstacles by van den Berg et al. [Reference van den Berg, Guy, Lin and Manocha29], it is assumed that agents do not change speed for a given lookahead time, and independently decide on their velocity for the next period without communication with other agents. In the paper, sufficient conditions for reciprocal n-body collision avoidance are derived. A key drawback of the algorithm presented is the assumption of perfect sensing, not taking into account any stochastic effect. Linear motion has been extended to curved trajectories by Zhong and Xuejun [Reference Zhong and Xuejun30]. This is achieved by taking intent into account with predefined trajectory change points. To detect and resolve conflicts geometric optimisation can be used after projecting velocities to the piecewise linear trajectory. In all of the above-mentioned papers, a simple extrapolation is used for conflict detection, which may be efficient in a last resort avoidance manoeuvre, but permits no conclusions about performance, the effect on global dynamics and airspace capacity.

Conflict detection is simplified by Rebollo, Ollero and Maza [Reference Rebollo, Ollero and Maza31] by discretising the space into cubic cells and sharing residence times, thus conflicts can be identified as time overlaps. Since the space is discretised, for collision avoidance graph-based methods can be used, namely the search tree algorithm and the tabu search algorithm. The key drawback of the methods is the need for the availability of information and scalability.

For fixed-wing aircraft, a collision avoidance algorithm is given based on manoeuvre coordination in the paper by Wan, Tang and Lao [Reference Wan, Tang and Lao32]. The set of possible manoeuvres is restricted to special manoeuvre styles, uncertainty is taken into account by a conical region calculated based on error radius. Conflicts are detected using the shared information on possible trajectories. Avoidance and manoeuvre selection is such that activation can be delayed as much as possible to minimise disruptions. For a given scenario the method proposed in the paper works appropriately, however, the effects of a manoeuvre on the wider airspace are not considered, thus, it may lead to a domino effect.

A self-organising collision avoidance method is presented by Huang, Tang and Lao [Reference Huang, Tang and Lao11] for approaching drone swarms. Self-organisation is achieved by applying a combination of Reynolds rules, extended with an anti-collision force, based on the pairwise closest point of approach, and linear extrapolation. Results show that rapid reorganisation creates a domino effect when using the traditional flocking rules only. This domino effect can be successfully overcome by self-coordination, demonstrating its effectiveness. The directionality of conflict probability, however, is not taken into account, even though it could lead to more efficient swarming.

An automatic collision resolution algorithm is designed by Luongo, Di Vito and Corraro [Reference Luongo, Di Vito and Corraro33], based on an analytical solution for conflict detection. All aircraft applying the algorithm creates a self-organising system, assuming all aircraft share intent. Conflicts are resolved either by changing aircraft speed or by changing speed, flight path and track angle, respecting flight envelope. One of the requirements of the method was to solve all conflicts in a multi-aircraft scenario without creating new ones, thus induced conflicts are taken into account. Admittedly, not all conflict scenarios can be resolved. However, no method is given to identify such scenarios nor is any technique provided to avoid such scenarios or at least limit their probability.

Speed control is applied to minimise potential conflict by Constans, Fontaine and Fondacci [Reference Constans, Fontaine and Fondacci34]. The sliding horizon loop process is made up of two layers, the supervisory layer detects conflicts and calculates deconflicted arrival times using linear programming, while the localised control layer follows the set arrival times correcting for disturbances and respecting aircraft performance limits, applying predictive functional control. Conflicts are localised, they can only occur at route intersection points, therefore conflict detection is based on arrival times, and separation thus becomes time separation. Results show speed control is an effective and simple method to minimise conflict probability. The same space discretisation technique is described in the paper by Cecen and Cetek [Reference Cecen and Cetek35], focusing on en-route, level traffic. Speed control is identified as the long-term resolution method for conflicts. Airspace entry points are also considered variables of the optimisation problem, which is solved using a novel heuristic. The most important drawback of the above-described techniques is the need for a central entity for coordination.

2.3 Collision detection and risk

Analysis of collision detection methods is extended by taking into account the effect of navigation uncertainty and communication package losses in the paper by Mahjri et al. [Reference Mahjri, Dhraief, Belghith and AlMogren36] UAVs flying in a straight line are assumed to exchange position and velocity information periodically. The effect of the setting of broadcast period and lookahead time on missed and false alarms is analysed analytically and using simulation. No conflict resolution method is applied so the performance of the detection algorithm could be analysed. The results presented are averaged over all relative headings, providing no insight into the underlying dynamics of the emergence of the conflict.

Collision risk as a result of UAS intruding terminal airspace is modelled by Wang, Tan and Low [Reference Wang, Tan and Low37] to assess hazard and threat levels and prevent collisions with passenger aircraft. Based on an initial sighting of the UAS, its trajectory is propagated probabilistically to reflect on its “assumed intent”. The position and state of the passenger aircraft are prescribed, and perfectly known to ATC. Conflict probability for a given scenario is calculated by 2D Monte-Carlo simulations resulting in a framework for risk management taking the maximum speed of the UAS model as an input. Alternatively, tracking data can be incorporated into decision making when available. The study highlights the importance of stochastic analysis in risk assessment. However, in this study, only one of the participating agents exhibits stochastic behaviour. Moreover, the evaluation is based solely on simulation results, which provides no help in understanding the underlying dynamics.

Another way to detect conflicts is to predict separation violations by calculating the reachable sets of intruders, taking intruder motion uncertainty into account. In the paper by Lin and Saripalli [Reference Lin and Saripalli38] a Dubins car model is used to represent obstacle aircraft kinematics, based on current information, assuming constant speed and limited maximal turning rate the reachable set can be calculated. The x − y component and the z component of the set are calculated separately. To simplify the calculation, the reachable set is approximated by a polygon. The resolution path is generated using intermediate waypoint sampling. Although the method presented improves the success rate of collision avoidance, it does not take into account stochastic behaviour due to uncertain intent or disturbances, any relies on instantaneous information only, therefore does not reflect realistic collective dynamics.

A multi-aircraft conflict detection algorithm based on probabilistic reach sets is introduced by Yang et al. [Reference Yang, Zhang, Cai and Prandini39]. The nominal, piecewise linear trajectory of the aircraft is disturbed by a stochastic wind component, and possible realisations of the trajectory are formulated as a chance-constrained program, which is solved by the scenario approach to provide the ellipsoidal probabilistic reach sets. Conflicts are resolved as a second-order cone program. Utilising the algorithms described conflict detection can be accomplished for a given stochastic model, however, results are dependent on the given scenario and are therefore not universal.

A unified analytical framework for aircraft separation assurance and unmanned aircraft system sense-and-avoid is presented by Ramasamy, Sabatini and Gardi [Reference Ramasamy, Sabatini and Gardi40]. The paper presents mathematical models to calculate the overall uncertainty volume of the intruder aircraft, utilising unified range and bearing uncertainty descriptors, taking into account separation requirements. To determine the probability of near mid-air collisions, trajectory predictions and Monte Carlo approximations are used, based on cooperative and non-cooperative sensor selection and characteristics. Based on conflict identification the avoidance trajectory can be calculated as a result of an optimisation. For automated sensor selection, Adaptive Boolean Decision Logics are employed. Even though the paper represents a significant step towards certifiable separation assurance systems and determining UTM airspace capacity, a drawback of the methods described is that conflict probability can only be determined for a given scenario, using numerical models, therefore no general conclusions can be drawn, and the accuracy of the solutions remains uncertain.

For two common encounter geometries in a terminal manoeuvring area, a probabilistic conflict detection algorithm is formulated by Shi and Wu [Reference Shi and Wu41]. As a perturbation to the nominal trajectories, a three-dimensional Brownian motion is added and transformed to be standardised. Conflict probability is defined as the probability of the intruder entering the protection zone, which is the probability of a Brownian motion hitting a moving protection zone. Using the Bachelier-Levy theorem an analytical approximation is presented. Even though results can be used by air traffic control officers to assess the severity of a conflict for the analysed scenarios it is difficult to generalise the results of the paper.

Conflict detection for trajectory-based operations can also be found in the paper by Hao et al. [Reference Hao, Zhang, Cheng, Liu and Xing42]. Conflict probability is calculated taking into account random changes in pilot intent. The aim is to detect “whether two aircraft would violate the safe separation criteria during the look-ahead period based on the predicted aircraft trajectory”. This is done by calculating the intersections of state-time prisms in a discretised space. The uneven occurrence probability distribution within the reachable domain is considered by the truncated Brownian bridge model. Multi-aircraft conflict probability is calculated assuming independence. Other than pilot intent, no source of uncertainty is taken into account.

2.4 Directional conflict probability

None of the papers listed describe any scenario when for both the ownship and the intruder behave according to given probabilities. Also, as it can be seen from the references cited, even though conflict probabilities are calculated by many, directional analysis for further insight is rarely given. An exception is a paper and a PhD thesis published by Hernández-Romero et al. [Reference Hernández-Romero, Valenzuela and Rivas43, Reference Hernández-Romero44], which include directional analysis of the conflict probability according to the relative ground track angle of the two aircraft and the wind direction. A probabilistic approach to quantifying aircraft conflict intensity is presented, focusing on the propagation of wind forecast uncertainty using the probabilistic transformation method. The calculated conflict probability and intensity are used to improve optimal strategic deconfliction of aircraft trajectories under uncertainties derived from the ensemble weather forecast. This makes the research an important step towards minimising conflict probability taking into account weather uncertainty. Wind forecast uncertainties are relatively small compared to aircraft speeds used in the paper. Moreover, due to the small distance between the aircraft, the same wind is assumed to affect both aircraft, thus the uncertain wind vector added to the velocities is the same. In the current paper distribution parameters are not necessarily equal for both aircraft, providing a more general approach. While both papers highlight the importance of uncertainty analysis, the methodology presented here is better suited for high density airspace analysis and management, where the sources of uncertainty are more diverse.

2.5 Aim of the paper

The aim of this paper is to provide analytical formulae for geometric and speed conflict probability evaluation. The analysis is performed without constraints of any specific scenario, in free flight. Calculation results using the formulae provided can serve as measures of effectiveness of prescribed distributions for the middle layer of the proposed framework. To assess the validity of the method no underlying conflict resolution technique is applied, similarly to Mahjri et al. [Reference Mahjri, Dhraief, Belghith and AlMogren36]. Results of the analysis presented show how conflict probability is related to distribution parameters, which are to be used as decision variables in the top-level optimisation. Integrated analysis of the framework described in Section 1 is subject to future work.

3.1 Definitions

To determine the probability of a conflict two ranges are defined around the ownship, the sensing ( ${r_S}$ ) and the conflict range ( ${r_C}$ ). It is assumed that the ownship becomes aware of the intruder once it enters the sensing range. A conflict may be defined as an event when the intruder enters a range (the conflict range) with a predefined radius around the ownship, which is referred to as geometric conflict (GC) throughout the paper. For an alternative definition for conflict it may be assumed that the ownship can avoid the intruder being too close (manoeuvring, possibly in cooperation with the intruding aircraft) given enough time, therefore a conflict is defined as the event when the intruder traverses from the sensing range to the conflict range faster than a threshold time ( ${t_{{\rm{th}}}}$ ). This event is referred to as a speed conflict (SC). A conflict range value may be chosen as the NMAC (near mid-air collision) range or the well-clear range corresponding to the underlying DAA (detect and avoid) concept and method applied. As ACAS was extended to unmanned systems [Reference Owen, Panken, Moss, Alvarez and Leeper25, Reference Alvarez, Jessen, Owen, Silbermann and Wood26], it is expected that UAM vehicles would have the same capability. NMAC and well-clear ranges in practice would be defined corresponding to this system, similarly to their derivation for small UAS [Reference Weinert, Campbell, Vela, Schuldt and Kurucar45, Reference Lester and Weinert46]. The above-described concept corresponds to a simplified model of the underlying collision avoidance performance requirements, defining the relevant volumes and time thresholds. This model is appropriate for the purposes of the investigation as the point is not to evaluate the capabilities of DAA systems but to determine conflict probabilities related to an encompassing system of the proposed framework. It is expected that performance requirements for UAM DAA systems would be standardised in the future, similarly to the requirements for unmanned aircraft [47].

The conflict geometry with the ranges and angle definitions can be seen in Fig. 1. The figure shows both the ownship and the intruder in the plane in which they are flying, in a coordinate system fixed to the ownship. Note that the figures are not to scale to help understanding.

Figure 1.

The geometry of the scenario for which the conflict probabilities are calculated. The ownship and the intruder are flying in the same x − y plane. Note that the figure is not to scale to help understanding.

The angle related to the projection of the conflict range on the sensing range is denoted by $\beta $ and defined according to Equation (1). The angle at which the intruder becomes visible to the ownship is denoted by $\delta $ .

(1) \begin{align}\sin\!\left( \beta \right) = \frac{{{r_C}}}{{{r_S}}}\end{align}

The formulae for conflict probabilities can be evaluated for arbitrary sensing and conflict ranges. For the plots in this paper the sensing and conflict ranges in Equations (2)–(4) were used, based on previous analysis on the flight and communication performance of UAM vehicles [Reference Öreg, Shin and Tsourdos15, Reference Öreg, Shin and Tsourdos16].

(2) \begin{align}{r_C} = 0.135\;{\rm{[nmi]}} = 250\;{\rm{[m]}}\end{align}

(3) \begin{align}{r_S} = 2.5\;{\rm{[nmi]}} = 4630\;{\rm{[m]}}\end{align}

(4) \begin{align}\beta = \arcsin\!\left( {\frac{{{r_C}}}{{{r_S}}}} \right) \approx 3.1^{\circ} \end{align}

The speed of the ownship is denoted by ${v_{{\rm{ownship}}}}$ , and the speed of the intruder is ${v_{{\rm{intruder}}}}$ . Without loss of generality, the ownship is assumed to progress in the positive x-direction, and the angle between the intruder and ownship velocity is denoted by $\theta $ . The angle of the apparent path of the intruder is denoted by $\Omega $ .

For calculating the conflict probabilities for an intruder with perpendicular velocity it is convenient to express the ratio of speeds, which, considering the velocity triangle can be connected to the angle $\gamma $ according to Equation (5).

(5) \begin{align}r = \tan\!\left( \gamma \right) = \frac{{{v_{{\rm{ownship}}}}}}{{{v_{{\rm{intruder}}}}}}\end{align}

Given the apparent position of the intruder and the angle of the apparent trajectory, the distance to be traversed to the conflict range can be calculated according to Equation (6).

(6) \begin{align}d = \left\{ {\begin{array}{l@{\quad}l}{{r_S}\sin\!\left( {\delta + \Omega } \right)} & {}\\[5pt] { - {r_S}\sqrt {\mathop {\sin }\nolimits^2\!\left( {\delta + \Omega } \right) - \mathop {\cos }\nolimits^2 \beta } } & {{\rm{if}}\;\left( {\delta - \beta \le \frac{\pi }{2} - \Omega \le \beta + \delta \;{\rm{and}}\;\Omega \geq - \frac{\pi }{2}\;{\rm{and}}\;\delta \geq \beta - \pi } \right)}\\[5pt] {} & {\;{\rm{or}}\;\left( { - \beta \le \delta + \Omega + \frac{{3\pi }}{2} \le \beta \;{\rm{and}}\;\Omega \geq - \frac{\pi }{2}\;{\rm{and}}\;\delta \le \beta - \pi } \right)}\\[5pt] {} & {\;{\rm{or}}\;\left( { - \beta \le \delta + \Omega + \frac{{3\pi }}{2} \le \beta \;{\rm{and}}\;\Omega \lt - \frac{\pi }{2}\;{\rm{and}}\;\delta \le \pi - \beta } \right)}\\[5pt] {} & {\;{\rm{or}}\;\left( {\delta - \beta \le \frac{\pi }{2} - \Omega \le \beta + \delta \;{\rm{and}}\;\Omega \lt - \frac{\pi }{2}\;{\rm{and}}\;\delta \geq \pi - \beta } \right)}\\[5pt] \infty & {\;{\rm{else}}}\end{array}} \right.\end{align}

If the apparent trajectory is parallel to the movement of the ownship, Equation (6) can be simplified to Equation (7).

(7) \begin{align}{d_\parallel } = \left\{ {\begin{array}{l@{\quad}l}{{r_S}\cos\!(\delta ) - {r_S}\sqrt {\mathop {\cos }\nolimits^2 (\delta ) - \mathop {\cos }\nolimits^2 (\beta )} } & {{\rm{if}}\; - \beta \le \delta \le \beta }\\[5pt] { - {r_S}\cos\!(\delta ) - {r_S}\sqrt {\mathop {\cos }\nolimits^2 (\delta ) - \mathop {\cos }\nolimits^2 (\beta )} } & {{\rm{if}}\;\delta \le - \pi + \beta \;{\rm{or}}\;\pi - \beta \le \delta }\\[5pt] \infty & {{\rm{if}}\; - \pi + \beta \le \delta \le - \beta \;}\\[5pt] {} & {{\rm{or}}\;\beta \le \delta \le \pi - \beta }\end{array}} \right.\end{align}

Starting the investigations for the exponential distribution the probability distribution formulae are summarised. The probability density function and the cumulative distribution function of the exponential distribution are given by Equations (8) and (9), respectively.

(8) \begin{align}f\!\left( {x;\;\lambda } \right) = \left\{{\begin{array}{l@{\quad}l}{\lambda {e^{ - \lambda x}}} & {{\rm{if}}\;x \geq 0}\\[5pt] 0 & {{\rm{if}}\;x \lt 0}\end{array}} \right.\end{align}

(9) \begin{align}F\!\left( {X;\;\lambda } \right) = \left\{{\begin{array}{l@{\quad}l}{1 - {e^{ - \lambda X}}} & {{\rm{if}}\;X \geq 0}\\[5pt] 0 & {{\rm{if}}\;X \lt 0}\end{array}} \right.\end{align}

Considering that the range of available speeds for an aircraft is limited (minimum speed due to stall and maximum speed due to performance and structural constraints) the effect of truncating the exponential distribution has to be taken into account as well. The probability density function and the cumulative distribution function of the truncated exponential distribution with lower limit l and upper limit u are given by Equations (10) and (11).

(10) \begin{align}f'\left( {x;\;\lambda ,l,u} \right) = \left\{ {\begin{array}{l@{\quad}l}{\dfrac{{\lambda {e^{ - \lambda x}}}}{{{e^{ - \lambda l}} - {e^{ - \lambda u}}}}} & {{\rm{if}}\;x \geq 0\;{\rm{and}}\;l \le x \le u}\\[12pt] 0 & {{\rm{else}}}\end{array}} \right.\end{align}

(11) \begin{align}F'\left( {X;\;\lambda ,l,u} \right) = \left\{ {\begin{array}{l@{\quad}l}{\dfrac{{{e^{\lambda u}}\left( { - 1 + {e^{\lambda \left( {l - X} \right)}}} \right)}}{{{e^{ - \lambda l}} - {e^{ - \lambda u}}}}} & {{\rm{if}}\;X \geq 0\;{\rm{and}}\;l \le X \le u}\\[12pt] 1 & {{\rm{if}}\;X \gt u}\\[5pt] 0 & {{\rm{else}}}\end{array}} \right.\end{align}

For the ownship and the intruder, the parameters of the exponential distribution are denoted by a and b respectively, thus the cumulative distribution functions of the speeds become as given by Equations (12) and (13) for the non-truncated distributions.

(12) \begin{align}\Pr\!\left( {{v_{{\rm{ownship}}}} \le X} \right) = 1 - {e^{ - aX}}\end{align}

(13) \begin{align}\Pr\!\left( {{v_{{\rm{intruder}}}} \le Y} \right) = 1 - {e^{ - bY}}\end{align}

The formulae derived for conflict probabilities can be evaluated for arbitrary distribution parameters and speed limits. Based on previous analysis on the flight and communication performance of UAM vehicles [Reference Öreg, Shin and Tsourdos15, Reference Öreg, Shin and Tsourdos16], for the plots in this paper the parameter ranges and speed limits in Equations (14)–(16) were used for exponential distribution.

(14) \begin{align}a \in \left[ {0.0025,0.2} \right],b \in \left[ {0.0025,0.2} \right]\end{align}

(15) \begin{align}l = 15\;{\rm{[Kn]}}\end{align}

(16) \begin{align}u = 180\;{\rm{[Kn]}}\end{align}

The mean speeds for the chosen parameter range and speed limits for the truncated exponential distribution are shown in Fig. 3. Generally, higher values of the rate parameter (a or b) indicate lower mean speeds.

For the conflict probability calculations, it is necessary to calculate the sum, the difference and the ratio of exponentially distributed random variables. All related formulae can be found in Appendix A.1.

3.2 Probability of entering the conflict range for perpendicular velocities

Even though an intruder may appear from any direction on the sensing range circle as a first approximation the principal directions are analysed. Derivation for these principal directions is useful in itself for airspace structures with constrained principal directions [Reference Sunil, Hoekstra, Ellerbroek, Bussink, Nieuwenhuisen, Vidosavljevic and Kern21]. Moreover, based on these primary formulae arbitrary directions or their superposition may be analysed.

The conflict geometry for this scenario is shown in Fig. 2.

Figure 2.

The geometry of the conflict scenario for perpendicular velocities. The ownship and the intruder are flying in the same x − y plane. Note that the figure is not to scale to help understanding.

Figure 3.

Mean speed in Kn as a function of the exponential distribution parameter (a for the ownship and b for the intruder).

Assuming the velocity of the intruder is perpendicular to the velocity of the ownship, no conflict occurs if Equation (17) holds.

(17) \begin{align}\left\{ {\begin{array}{l@{\quad}l}{\gamma \lt \dfrac{\pi }{2} - \left( {\delta + \beta } \right)} & {{\rm{if}}\; - \beta \le \delta \le \dfrac{\pi }{2} - \beta }\\[12pt] {\gamma \gt \dfrac{\pi }{2} - \left( {\delta - \beta } \right)} & {{\rm{if}}\;\beta \le \delta \le \dfrac{\pi }{2} + \beta }\end{array}} \right.\end{align}

The conditions can be transformed according to Equations (18) and (19).

(18) \begin{align}\tan\!\left( \gamma \right) \lt \tan\!\left( {\frac{\pi }{2} - \left( {\delta + \beta } \right)} \right) = \cot\!\left( {\delta + \beta } \right)\end{align}

(19) \begin{align}\tan\!\left( \gamma \right) \gt \tan\!\left( {\frac{\pi }{2} - \left( {\delta - \beta } \right)} \right) = \cot\!\left( {\delta - \beta } \right)\end{align}

Therefore the probabilities that no conflict occurs become as given by Equations (20) and (21) for ranges defined in Equation (17), using the formula derived in Equation (58)

(20) \begin{align}\Pr\!\left( {r \le \cot\!\left( {\delta + \beta } \right)} \right) = \frac{{a\cot\!\left( {\delta + \beta } \right)}}{{a\cot\!\left( {\delta + \beta } \right) + b}}\end{align}

(21) \begin{align}\Pr\!\left( {r \geq \cot\!\left( {\delta - \beta } \right)} \right) = \frac{b}{{a\cot\!\left( {\delta - \beta } \right) + b}}\end{align}

Thus, for the ranges of $\delta $ where conflict is possible, defining the parameter ratio as $\rho = a/b$ , the probabilities become the following.

Case 1: $ - \beta \le \delta \le \beta $

(22) \begin{align}\mathop {\Pr }\limits_{{\rm{perp}}} \!\left( {{\rm{GC}}} \right) = \frac{b}{{a\cot\!(\beta + \delta ) + b}} = \frac{1}{{\rho \cot\!(\beta + \delta ) + 1}}\end{align}

Case 2: $\beta \le \delta \le \dfrac{\pi }{2} - \beta $

(23) \begin{align}\mathop {\Pr }\limits_{{\rm{perp}}} \!\left( {{\rm{GC}}} \right) = \frac{b}{{a\cot\!(\beta + \delta ) + b}} + \frac{b}{{a\cot\!(\beta - \delta ) - b}} = \frac{1}{{\rho \cot\!(\beta + \delta ) + 1}} + \frac{1}{{\rho \cot\!(\beta - \delta ) - 1}}\end{align}

Case 3: $\dfrac{\pi }{2} - \beta \le \delta \le \dfrac{\pi }{2} + \beta $

(24) \begin{align}\mathop {\Pr }\limits_{{\rm{perp}}}\! \left( {{\rm{GC}}} \right) = 1 - \frac{b}{{b - a\cot\!(\beta - \delta )}} = 1 - \frac{1}{{1 - \rho \cot\!(\beta - \delta )}}\end{align}

Conflict probability formulas can be evaluated for arbitrary ratio of distribution parameters ( $\rho $ ) yielding conflict probability distribution for the full circle (Figs. 4 and 5). It can be seen that for equal distribution parameters ( $\rho = 1$ ) the highest probability of a conflict occurs at head-on or perpendicular encounters. This corresponds to predicted behaviour, because with equal parameters the expected value of speed is the same, thus for an intruder arriving from a perpendicular direction the distribution is symmetric to $\pi /4$ with peaks at $\beta $ and $\pi /2 - \beta $ . By modifying the ratio of distribution parameters the distribution of conflict probability can be adjusted. For $\rho = 80$ the conflict probability for the $ - \beta $ to $\pi /2 - \beta $ region reduces almost completely, but increases significantly in the $\pi /2 - \beta $ to $\pi /2 + \beta $ range. Opposite, but symmetric behaviour can be seen for $\rho = 1/80$ .

Figure 4.

Polar plot showing the probability of geometric intrusion with perpendicular intruder velocity for exponential distribution with $\rho = 1$ ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 5.

Polar plot showing the probability of geometric intrusion with perpendicular intruder velocity for exponential distribution with $\rho = 80$ ( $a = 0.2$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

If the set of available speeds is bounded, and the velocity magnitudes assume a truncated distribution, the conflict probabilities can be calculated according to Equations (61), (62) and (63) (Appendix A.2), using the results in Equation (60).

As it can be seen in Figs. 6 and 7 the probability distribution changes significantly if the bounds considerably affect the range of available speeds ( $l = 15,\;u = 180$ ). Relaxing the bounds ( $l = 0,\;u = \infty $ ) provides the unbounded results. If the distribution is truncated, the point of maximum conflict probability shifts to $\pi /4$ for equal parameters. Not only the angle but also the magnitude of the peak probability changes. For $\rho = 80$ the peak probability decreases considerably and tilts back towards $\pi /4$ .

Figure 6.

Polar plot showing the probability of geometric intrusion with perpendicular intruder velocity for truncated exponential distribution with $\rho = 1$ ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ , $l = 15,\;u = 180$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 7.

Polar plot showing the probability of geometric intrusion with perpendicular intruder velocity for truncated exponential distribution with $\rho = 80$ ( $a = 0.2$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ , $l = 15,\;u = 180$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Since conflict probability formulae are given analytically, the sensitivity of results to parameter changes can be evaluated by calculating the partial derivatives with respect to the parameters. The analysis method is presented for geometric conflict probability with exponential distributions (given by Equations (22)–(24)), and it can be applied for all subsequent scenarios. The partial derivative with respect to the distribution parameter ratio $\left( \rho \right)$ can be calculated as in Equation (25).

(25) \begin{align}\frac{{\partial \mathop {\Pr }\limits_{{\rm{perp}}}\! \left( {{\rm{GC}}} \right)}}{{\partial \rho }} = \left\{ {\begin{array}{l@{\quad}l}{ - \dfrac{{\cot\!(\beta + \delta )}}{{{{(\rho \cot\!(\beta + \delta ) + 1)}^2}}}} & {{\rm{if}}\;\beta + \delta \geq 0\;{\rm{and}}\;\beta - \delta \geq 0}\\[15pt] { - \dfrac{{\cot\!(\beta - \delta )}}{{{{(\rho \cot\!(\beta - \delta ) - 1)}^2}}} - \dfrac{{\cot\!(\beta + \delta )}}{{{{(\rho \cot\!(\beta + \delta ) + 1)}^2}}}} & {{\rm{if}}\;\beta - \delta \le 0\;{\rm{and}}\;\beta + \delta \le \dfrac{\pi }{2}}\\[15pt] {\dfrac{{\cot\!(\beta - \delta )}}{{\rho \cot\!(\beta - \delta ) - 1}} - \dfrac{{\rho \mathop {\cot }\nolimits^2 (\beta - \delta )}}{{{{(\rho \cot\!(\beta - \delta ) - 1)}^2}}}} & {{\rm{if}}\;\beta + \delta \geq \dfrac{\pi }{2}\;{\rm{and}}\;\beta - \delta \geq - \dfrac{\pi }{2}}\end{array}} \right.\end{align}

Sensitivity as a function of heading difference can be seen in Fig. 8. Overall, the magnitude of change with parameter variation is very low. Although there are no explicit metrics for parameter sensitivity, it is still meaningful to calculate the sensitivities to parameters as they indicate how the result changes with input perturbations. Maximum locations of conflict probability and its sensitivity curve coincide. This is expected, as $\rho = 1$ represents a limiting case. For $\rho = 80$ , shown in Fig. 9, sensitivity is even lower, the peak location is at a low conflict probability angle, $\delta = \frac{\pi }{2} + \beta $ . This is because conflict at this heading difference is almost only possible at an extreme ratio, even a slight perturbation of the parameter results in a significant change in probability.

Figure 8.

Sensitivity of geometric intrusion probability to changes in parameter ratio of exponential distribution with perpendicular intruder velocity ( $\rho = 1$ , $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ ). As a function of relative azimuth angle, geometric intrusion probability is shown on the left axis, and sensitivity is shown on the right axis.

For truncated distributions, since conflict probability is not only a function of the parameter ratio, sensitivities were determined for the distribution parameters separately as shown in Figs. 10 and 11. For low parameter values, the change with parameter perturbation is significant due to the characteristics of the probability density function of the exponential distribution. As the parameter values increase the sensitivity decreases. Nevertheless, conflict probability for truncated distributions is still predominantly determined by parameter ratio, insensitivity to perturbations will be shown in Section 5.

3.3 Probability of entering the conflict range for parallel velocities

As another principal direction parallel velocities can be analysed. The conflict geometry for parallel velocities is shown in Fig. 12.

Figure 9.

Sensitivity of geometric intrusion probability to changes in parameter ratio of exponential distribution with perpendicular intruder velocity ( $\rho = 80$ , $a = 0.2$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ ). As a function of relative azimuth angle, geometric intrusion probability is shown on the left axis, and sensitivity is shown on the right axis.

Considering geometric conflicts only, if the velocities of the ownship and the intruder are not only parallel, but point in the opposite direction a conflict surely occurs ( $\Pr\!\left( {{\rm{GC}}} \right) = 1$ ) if the intruder is in the range of the projection of the conflict range on the sensing range (Equation (26)).

(26) \begin{align}\mathop {\Pr }\limits_{{\rm{parallel}}} \!\left( {{\rm{GC}}} \right) = \left\{ {\begin{array}{l@{\quad}l}1 &{{\rm{if}}\; - \beta \lt \delta \lt \beta }\\[5pt] 0 & {{\rm{if}}\;\delta \le - \beta \;{\rm{or}}\;\beta \le \delta }\end{array}} \right.\end{align}

If the velocities point in the same direction, a geometric conflict occurs only if the intruder is in the angle range determined by the projection of the conflict range, and its speed is greater than the speed of the ownship if the intruder is behind (the difference of speeds is positive ( $y - x \gt 0$ ), Equation (27), or the intruder speed is less than the ownship speed, and the intruder is ahead. Calculations are based on Equation (54)).

(27) \begin{align}\mathop {\Pr }\limits_{{\rm{parallel}}}\! \left( {{\rm{GC}}} \right) = \left\{ {\begin{array}{l@{\quad}l}{\dfrac{a}{{a + b}}} & {{\rm{if}}\;\delta \lt - \pi + \beta \;{\rm{or}}\;\pi - \beta \lt \delta }\\[12pt] {\dfrac{b}{{a + b}}} & {{\rm{if}}\; - \beta \lt \delta \lt \beta }\\[12pt] 0 & {{\rm{if}}\; - \pi + \beta \le \delta \le - \beta }\\[5pt] {} & {{\rm{or}}\;\beta \le \delta \le \pi - \beta }\end{array}} \right.\end{align}

If the distributions are truncated, the probability of a positive difference becomes different as given by Equation (28) (based on Equation (56)).

(28) \begin{align}\mathop {\Pr }\limits_{{\rm{parallel}}} \!\left( {{\rm{GC'}}} \right) = \left\{ {\begin{array}{l@{\quad}l}{\dfrac{{{e^{al}}\!\left( {a{e^{bl}} - (a + b){e^{bu}}} \right) + b{e^{u(a + b)}}}}{{(a + b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {{\rm{if}}\;\delta \lt - \pi + \beta \;{\rm{or}}\;\pi - \beta \lt \delta }\\[15pt] {\dfrac{{b{e^{bl}}\!\left( {{e^{al}} - {e^{au}}} \right) + a\!\left( {{e^{u(a + b)}} - {e^{au + bl}}} \right)}}{{(a + b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {{\rm{if}}\; - \beta \lt \delta \lt \beta }\\[15pt] 0 & {{\rm{if}}\; - \pi + \beta \le \delta \le - \beta \;}\\[5pt] {} & {{\rm{or}}\;\beta \le \delta \le \pi - \beta }\end{array}} \right.\end{align}

Having taken into account all possibilities with parallel and perpendicular intruder velocities, assuming that they have a uniform probability, the composite probability distributions weighted by arc lengths can be determined, as shown in Figs. 13 and 14. Even though the probability of head-on and rear-end collisions is much higher than that of perpendicular collisions, when weighted with the corresponding arc length it decreases below the perpendicular probability, especially in the truncated case, where the contribution of the perpendicular probability is not as significant in these directions.

Figure 10.

Sensitivity of geometric intrusion probability to changes in parameters of truncated exponential distributions with perpendicular intruder velocity ( $\rho = 1$ , $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ , $l = 15,\;u = 180$ ). As a function of relative azimuth angle, geometric intrusion probability is shown on the left axis, and sensitivity is shown on the right axis.

Figure 11.

Sensitivity of geometric intrusion probability to changes in parameters of truncated exponential distributions with perpendicular intruder velocity ( $\rho = 80$ , $a = 0.2$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ , $l = 15,\;u = 180$ ). As a function of relative azimuth angle, geometric intrusion probability is shown on the left axis, and sensitivity is shown on the right axis.

Figure 12.

The geometry of the conflict scenario for parallel velocities. The ownship and the intruder are flying in the same x − y plane. Note that the figure is not to scale to help understanding.

Figure 13.

Polar plot showing the probability of geometric intrusion with equally likely perpendicular and parallel intruder velocities for full and truncated exponential distribution with $\rho = 1$ ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \left\{ {\frac{\pi }{2},0, - \frac{\pi }{2},\pi } \right\}$ , $l = 15,\;u = 180$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

3.4 Probability of entering the conflict range for velocities with arbitrary direction

To generalise the results obtained in the previous sections the geometric conflict probability is analysed for arbitrary intruder velocity angles ( $\theta $ ) as well. The angle of the apparent path ( $\Omega $ ) can be calculated based on the relative equations of motion, according to Equation (29).

(29) \begin{align}\Omega = \left\{ {\begin{array}{l@{\quad}l}{\arctan \!\left( {\dfrac{{\tan \gamma }}{{\sin \theta }} + \cot \theta } \right)} & {{\rm{if}}\;0 \lt \theta \lt \pi }\\[15pt] {\pi + \arctan\! \left( {\dfrac{{\tan \gamma }}{{\sin \theta }} + \cot \theta } \right)}& {{\rm{if}}\; - \pi \lt \theta \lt 0}\\[15pt] {\dfrac{\pi }{2}} & {{\rm{if}}\;\theta = 0}\\[15pt] {\dfrac{\pi }{2}}& {{\rm{if}}\;\theta = \pi \;\;{\rm{and}}\;\;{v_A} \geq {v_B}}\\[15pt] { - \dfrac{\pi }{2}} & {{\rm{if}}\;\theta = \pi \;\;{\rm{and}}\;\;{v_A} \lt {v_B}}\end{array}} \right.\end{align}

In Equation (29) the angle $\theta $ is measured from the negative $x$ -axis, in the clockwise direction and $\Omega $ is measured from the negative $y$ -axis in the anticlockwise direction. Since it is easier to calculate the probability distribution of the tangent of the apparent path angle (Equation (30), based on Equation (58)), the conflict probability limits for $0 \lt \theta \lt \pi $ were transformed according to Equation (31) and (32).

(30) \begin{align}\Pr\!\left( {\tan \Omega \le Z} \right) = \left\{ {\begin{array}{l@{\quad}l}{\dfrac{{a\sin \theta \!\left( {Z - \cot \theta } \right)}}{{a\sin \theta \!\left( {Z - \cot \theta } \right) + b}}} & {{\rm{if}}\;\cot \theta \le Z}\\[15pt] 0 & {{\rm{if}}\;\cot \theta \gt Z}\end{array}} \right.\end{align}

(31) \begin{align}\left\{ {\begin{array}{l@{\quad}l}{\dfrac{\pi }{2} - \left( {\delta + \beta } \right) \le \Omega \le \dfrac{\pi }{2}} & {{\rm{if}}\; - \beta \le \delta \le \beta }\\[15pt] {\dfrac{\pi }{2} - \left( {\delta + \beta } \right) \le \Omega \le \dfrac{\pi }{2} - \left( {\delta - \beta } \right)} & {{\rm{if}}\;\beta \lt \delta \lt \pi - \beta }\\[15pt] { - \dfrac{\pi }{2} \le \Omega \le \dfrac{\pi }{2} - \left( {\delta - \beta } \right)} & {{\rm{if}}\; - \pi \le \delta \le - \pi + \beta \;{\rm{or}}\;\pi - \beta \le \delta \le \pi }\end{array}} \right.\end{align}

(32) \begin{align}\left\{ {\begin{array}{l@{\quad}l}{\cot\!\left( {\delta + \beta } \right) \le \tan \Omega } & {{\rm{if}}\; - \beta \le \delta \le \beta }\\[5pt] {\cot\!\left( {\delta + \beta } \right) \le \tan \Omega \le \cot\!\left( {\delta - \beta } \right)} & {{\rm{if}}\;\beta \lt \delta \lt \pi - \beta }\\[5pt] {\tan \Omega \le \cot\!\left( {\delta - \beta } \right)} & {{\rm{if}}\; - \pi \le \delta \le - \pi + \beta \;{\rm{or}}\;\pi - \beta \le \delta \le \pi }\end{array}} \right.\end{align}

Calculations of conflict probabilities can be found in Appendix A.3.

Figure 14.

Polar plot showing the probability of geometric intrusion with equally likely perpendicular and parallel intruder velocities for full and truncated exponential distribution with $\rho = 80$ ( $a = 0.2$ and $b = 0.0025$ , $\theta = \left\{ {\frac{\pi }{2},0, - \frac{\pi }{2},\pi } \right\}$ , $l = 15,\;u = 180$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figures 15, 16, 17 and 18 show the conflict probabilities for $\theta = \frac{\pi }{4}$ and $\theta = \frac{{3\pi }}{4}$ with full and truncated distributions. It can be seen, that having an arbitrary angle for intrusion corresponds to an angular scaling with the origin being the centre. For $\rho = 80$ and $\theta = \pi /4$ the angle for maximum conflict probability shifts from $\pi /2$ to $\pi /4$ . For truncated distributions, the same reduction and shifting behaviour is visible as in Fig. 7 for $\rho = 80$ . For $\rho = 1$ , the distribution is modified similarly as it can be seen in Fig. 6.

Figure 15.

Polar plot showing the probability of geometric intrusion for $\theta = \frac{\pi }{4}$ , exponential distribution with $\rho = 80$ ( $a = 0.2$ and $b = 0.0025$ , $\theta = \frac{\pi }{4}$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 16.

Polar plot showing the probability of geometric intrusion for $\theta = \frac{{3\pi }}{4}$ , exponential distribution with $\rho = 1$ ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{{3\pi }}{4}$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 17.

Polar plot showing the probability of geometric intrusion for $\theta = \frac{\pi }{4}$ , truncated exponential distribution with $\rho = 80$ ( $a = 0.2$ and $b = 0.0025$ , $\theta = \frac{\pi }{4}$ , $l = 15$ , $u = 180$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 18.

Polar plot showing the probability of geometric intrusion for $\theta = \frac{{3\pi }}{4}$ , truncated exponential distribution with $\rho = 1$ ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{{3\pi }}{4}$ , $l = 15$ , $u = 180$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

3.5 Probability of entering the conflict range for velocities with uniform distribution of intrusion angle

To evaluate the probability of conflicts in a realistic free flight traffic scenario, assuming a uniform distribution for intrusion velocity angles the conflict probabilities were averaged for all intrusion angles as shown in Fig. 19. If the distribution of intrusion relative headings is not uniform, but known, the same method may be applied as a weighted average, weighing the directions according to the assumed probability of intrusion with that angle. The highest probability of conflicts is for the $ - \beta $ to $\beta $ region for both the full and the truncated distribution, however, the peak value for the truncated distribution is less than half of that for the full distribution. Although for small angular deviations from the ownship heading the full distribution probability is greater, for higher angles the truncated distribution takes over.

Figure 19.

Polar plot showing the probability of geometric intrusion for full and truncated exponential distribution with $\rho = 1$ , averaged over all intrusion angles ( $a = 0.0025$ and $b = 0.0025$ , $l = 15$ , $u = 180$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

3.6 Probability of entering the conflict range faster than the threshold for perpendicular velocities

Since it is assumed that, provided enough time is available for manoeuvring, collision avoidance methods are able to handle intruders, a conflict emerges only if the traversal from the sensing range to the conflict range is faster than a certain threshold time. This is referred to as a speed conflict.

The integrals corresponding to the conflict probabilities cannot be evaluated analytically, however, lower and upper bounds can be given for them based on approximations (Equations (69)–(72), Appendix A.4).

Evaluating the speed conflict limits with a threshold time of ${t_{{\rm{th}}}} = 60{\rm{s}}$ provides the speed conflict probability distribution as it can be seen in Figs. 20 and 21. The difference between lower and upper approximations is negligible. Comparing the speed conflict with the geometric conflict probability it can be seen that specifying a time threshold corresponds to a scaling, however with the choice of this threshold time ( ${t_{{\rm{th}}}} = 60{\rm{s}}$ ) the conflict probability does not change very much for the full distribution. The selection of ${t_{{\rm{th}}}} = 60{\rm{s}}$ time threshold here is arbitrary for the plots, the formulae can be evaluated for any other parameter. Avoidance logic threshold selection for UAM vehicles in the future would probably be based on classification of airspace, type of operation, vehicle speed and category and CNS performance.

If the set of available speeds is bounded, and thus the distributions are truncated, speed conflict distributions are modified according to Equations (77) (Appendix A.5).

Conflict probabilities for full and truncated exponential distributions for intruders with perpendicular velocities can be seen in Figs. 22 and 23. For high $\rho $ the difference between upper and lower approximations is not so apparent, however, for $\rho = 1$ the effect of the speed threshold is clearly visible.

3.7 Probability of entering the conflict range faster than the threshold for parallel velocities

Speed conflict probabilities can also be calculated for intruder velocities parallel to the ownship velocity. For intruders with velocities pointing in the opposite direction the conflict probability can be calculated using the formula for the sum of exponentially distributed random variables for a full distribution is given by Equation (47), leading to Equation (33).

(33) \begin{align}\mathop {\Pr }\limits_{{\rm{parallel}}} \!\left( {{\rm{SC}}} \right) = \left\{ {\begin{array}{l@{\quad}l}{\dfrac{{{e^{\frac{{ - b\;{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}}}}\left( {b\;{d_\parallel }\left( {\delta ,\beta } \right) + {t_{{\rm{th}}}}} \right)}}{{{t_{{\rm{th}}}}}}} & {{\rm{if}}\; - \beta \lt \delta \lt \beta \;{\rm{and}}\;a = b}\\[14pt] {\dfrac{{a\;{e^{\frac{{ - b\;{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}}}} - b\;{e^{\frac{{a\;{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}}}}}}{{a - b}}} & {{\rm{if}}\; - \beta \lt \delta \lt \beta \;{\rm{and}}\;a \ne b}\\[8pt] 0 & {{\rm{if}}\;\delta \le - \beta \;{\rm{or}}\;\beta \le \delta }\end{array}} \right.\end{align}

If distributions are truncated, the sum of exponentially distributed random variables is given by Equation (51), thus speed conflict probability becomes Equation (34).

(34) \begin{align}\begin{array}{l}\mathop {\Pr }\limits_{{\rm{parallel}}} \!\left( {{\rm{SC'}}} \right) = \\[13pt] \left\{ {\begin{array}{l@{\quad}l}{\dfrac{{{e^{2au}}\left( {2al - \dfrac{{a{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}} - 1} \right)}}{{{{\left( {{e^{al}} - {e^{au}}} \right)}^2}}}\dfrac{{\exp\!\left( {2al - \dfrac{{a{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}}} \right)}}{{{{\left( {{e^{al}} - {e^{au}}} \right)}^2}}} + \dfrac{{{e^{2au}}}}{{{{\left( {{e^{al}} - {e^{au}}} \right)}^2}}}} & {{\rm{if}}\;l + u \geq \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;{\rm{and}}\;a = b\;{\rm{and}}\;}\\[14pt] {} & {2l \le \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}} \le 2u\;{\rm{and}}\; - \beta \lt \delta \lt \beta }\\[14pt] {\dfrac{1}{{{{\left( {{e^{al}} - {e^{au}}} \right)}^2}}}} & {{\rm{if}}\;l + u \le \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;{\rm{and}}\;a = b\;{\rm{and}}\;}\\[14pt] {} & {2l \le \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}} \le 2u\;{\rm{and}}\; - \beta \lt \delta \lt \beta }\\[14pt] {\dfrac{{a\exp\!\left( {al + b\!\left( {l - \dfrac{{{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}} + 2u} \right)} \right)}}{{(a - b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}} - \dfrac{{a\!\left( {{e^{au + bl}} - {e^{al + bu}} + {e^{u(a + b)}}} \right)}}{{(a - b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {}\\[14pt] { - \dfrac{{b\exp\!\left( {a\!\left( {l - \dfrac{{{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}} + 2u} \right) + bl} \right)}}{{(a - b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}} - \dfrac{{b\!\left( {{e^{au + bl}} - {e^{al + bu}} + {e^{u(a + b)}}} \right)}}{{(a - b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {{\rm{if}}\;l + u \geq \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;{\rm{and}}\;a \ne b\;{\rm{and}}\;}\\[14pt] {} & {2l \le \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}} \le 2u\;{\rm{and}}\; - \beta \lt \delta \lt \beta }\\[14pt] {\dfrac{{(a\;{e^{u(a + b)}}\left( { - \exp\!\left( {2bl - \dfrac{{b{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}}} \right)} \right)}}{{(a - b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {}\\[14pt] { + \dfrac{{b{e^{u(a + b)}}\left( {\exp\!\left( {2al - \dfrac{{a{d_\parallel }\left( {\delta ,\beta } \right)}}{{{t_{{\rm{th}}}}}}} \right) - 1} \right) + a{e^{u(a + b)}}}}{{(a - b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {{\rm{if}}\;l + u \le \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;{\rm{and}}\;a \ne b\;{\rm{and}}\;}\\[14pt] {} & {2l \le \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}} \le 2u\;{\rm{and}}\; - \beta \lt \delta \lt \beta }\\[14pt] 0 & {{\rm{if}}\;2l \gt \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;{\rm{and}}\; - \beta \lt \delta \lt \beta }\\[14pt] 1 & {{\rm{if}}\;2u \lt \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;{\rm{and}}\; - \beta \lt \delta \lt \beta }\\[14pt] 0 & {{\rm{if}}\;\delta \le - \beta \;{\rm{or}}\;\beta \le \delta }\end{array}} \right.\end{array}\end{align}

For intruders having a velocity with the same direction the probability of a speed conflict can be calculated according to Equations (35) for full and Equation (36) for truncated distributions.

(35) \begin{align}\mathop {\Pr }\limits_{{\rm{parallel}}} \!\left( {{\rm{SC}}} \right) = \left\{ {\begin{array}{l@{\quad}l}{\dfrac{{b\;{e^{\frac{{a\;{r_S}\left( {\sqrt {\mathop {\cos }\nolimits^2 (\delta ) - \mathop {\cos }\nolimits^2 (\beta )} + \cos\!(\delta )} \right)}}{{{t_{{\rm{th}}}}}}}}}}{{a + b}}} & {{\rm{if}}\;\delta \lt - \pi + \beta \;{\rm{or}}\;\pi - \beta \lt \delta }\\[15pt] {1 - \dfrac{{b\;{e^{\frac{{a\;{r_S}\left( {\sqrt {\mathop {\cos }\nolimits^2 (\delta ) - \mathop {\cos }\nolimits^2 (\beta )} + \cos\!(\delta )} \right)}}{{{t_{{\rm{th}}}}}}}}}}{{a + b}}} & {{\rm{if}}\; - \beta \lt \delta \lt \beta }\\[9pt] 0 & {{\rm{if}}\; - \pi + \beta \le \delta \le \pi - \beta }\end{array}} \right.\end{align}

(36) \begin{align}\begin{array}{l}\mathop {\Pr }\limits_{{\rm{parallel}}} \left( {{\rm{SC'}}} \right) = \\[5pt] \left\{ {\begin{array}{l@{\quad}l}0 & {{\rm{if}}\;u - l \lt \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;\;{\rm{and}}\;}\\[15pt] {} & {\left( {\delta \lt - \pi + \beta \;{\rm{or}}\;\pi - \beta \lt \delta } \right)}\\[15pt] {1 - \dfrac{{a\!\left( { - \exp\!\left( {al + bl - \dfrac{{b{r_S}\sqrt {\cos\!(2\delta ) - \cos\!(2\beta )} }}{{\sqrt 2 {t_{{\rm{th}}}}}} - \dfrac{{b{r_S}\cos\!(\delta )}}{{{t_{{\rm{th}}}}}}} \right) - {e^{au + bl}} + {e^{l(a + b)}} + {e^{u(a + b)}}} \right)}}{{(a + b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {}\\[15pt] { - \dfrac{{b\!\left( { - \exp\!\left( {u(a + b) + \dfrac{{a{r_S}\left( {\sqrt {\mathop {\cos }\nolimits^2 (\delta ) - \mathop {\cos }\nolimits^2 (\beta )} + \cos\!(\delta )} \right)}}{{{t_{{\rm{th}}}}}}} \right) - {e^{au + bl}} + {e^{l(a + b)}} + {e^{u(a + b)}}} \right)}}{{(a + b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {{\rm{if}}\;u - l \geq \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;\;{\rm{and}}\;}\\[15pt] {} & {\left( {\delta \lt - \pi + \beta \;{\rm{or}}\;\pi - \beta \lt \delta } \right)}\\[15pt] 0 & {{\rm{if}}\;u - l \lt \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;\;{\rm{and}}\;}\\[15pt] {} & { - \beta \lt \delta \lt \beta }\\[15pt] {\dfrac{{a\!\left( { - \exp\!\left( {al + bl - \dfrac{{b{r_S}\sqrt {\cos\!(2\delta ) - \cos\!(2\beta )} }}{{\sqrt 2 {t_{{\rm{th}}}}}} - \dfrac{{b{r_S}\cos\!(\delta )}}{{{t_{{\rm{th}}}}}}} \right) - {e^{au + bl}} + {e^{l(a + b)}} + {e^{u(a + b)}}} \right)}}{{(a + b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {}\\[15pt] { + \dfrac{{b\!\left( { - \exp\!\left( {u(a + b) + \dfrac{{a{r_S}\left( {\sqrt {\mathop {\cos }\nolimits^2 (\delta ) - \mathop {\cos }\nolimits^2 (\beta )} + \cos\!(\delta )} \right)}}{{{t_{{\rm{th}}}}}}} \right) - {e^{au + bl}} + {e^{l(a + b)}} + {e^{u(a + b)}}} \right)}}{{(a + b)\left( {{e^{al}} - {e^{au}}} \right)\left( {{e^{bl}} - {e^{bu}}} \right)}}} & {{\rm{if}}\;u - l \geq \dfrac{{d\left( {\delta ,\gamma } \right)}}{{{t_{{\rm{th}}}}}}\;\;{\rm{and}}\;}\\[15pt] {} & { - \beta \lt \delta \lt \beta }\\[15pt] 0 & {{\rm{if}}\; - \pi + \beta \le \delta \le \pi - \beta }\end{array}} \right.\end{array}\end{align}

3.8 Probability of entering the conflict range faster than the threshold for velocities with arbitrary direction

The speed conflict probability for intruders with arbitrary velocity directions was also analysed. Calculations were performed according to sections of the intruder velocity angle and can be found in Appendix A.6.

Figure 20.

Polar plot showing the upper and lower bounds for probability of speed intrusion, exponential distribution with $\rho = 1$ , perpendicular intruder velocity ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ , ${t_{{\rm{th}}}} = 60$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 21.

Polar plot showing the upper and lower bounds for probability of speed intrusion, exponential distribution with $\rho = 80$ , perpendicular intruder velocity ( $a = 0.2$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ , ${t_{{\rm{th}}}} = 60$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 22.

Polar plot showing the upper and lower bounds for probability of speed intrusion, full and truncated exponential distribution with $\rho = 1$ , perpendicular intruder velocity ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ , ${t_{{\rm{th}}}} = 60$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 23.

Polar plot showing the upper and lower bounds for probability of speed intrusion, full and truncated exponential distribution with $\rho = 80$ , perpendicular intruder velocity ( $a = 0.2$ and $b = 0.0025$ , $\theta = \frac{\pi }{2}$ , ${t_{{\rm{th}}}} = 60$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Assuming a truncated distribution the conflict probabilities can be calculated similarly, the range of $\theta $ is sectioned according to Equation (37).

(37) \begin{align}\begin{array}{*{20}{c}}{ - \beta \lt \delta \le \beta \;{\rm{and}}\;\theta = 0}\\[5pt] {0 \lt \theta \le 2\beta }\\[5pt] {2\beta \lt \theta \le 4\beta }\\[5pt] {4\beta \lt \theta \le \frac{\pi }{2}}\\[5pt] {\dfrac{\pi }{2} \lt \theta \le \pi }\end{array}\end{align}

Evaluating the integrals leads to the results shown in Fig. 24 for $\theta = \pi /4$ and Fig. 25 for $\theta = 3\pi /4$ . Once again the effect of changing the intruder velocity direction can be seen (central angular scaling), together with the effect of truncation, and how the difference between upper and lower approximations for truncated distributions is more conspicuous.

Figure 24.

Polar plot showing the upper and lower bounds for probability of speed intrusion, full and truncated exponential distribution with $\rho = 1$ , intruder velocity angle $\frac{\pi }{4}$ ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{\pi }{4}$ , ${t_{{\rm{th}}}} = 60$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.

Figure 25.

Polar plot showing the upper and lower bounds for probability of speed intrusion, full and truncated exponential distribution with $\rho = 1$ , intruder velocity angle $\frac{{3\pi }}{4}$ ( $a = 0.0025$ and $b = 0.0025$ , $\theta = \frac{{3\pi }}{4}$ , ${t_{{\rm{th}}}} = 60$ ). The radius coordinate represents conflict probability magnitude, while the angle coordinate shows relative azimuth.