Journal of Advanced Transportation

Volume 2018, Article ID 2918026, 14 pages

https://doi.org/10.1155/2018/2918026

## On the Kinematics of Vehicles Relevant to Conflict Detection and Resolution

^{1}CCTAE (Center for Aeronautical and Space Science and Technology), IDMEC/LAETA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal^{2}Escola Náutica Infante D. Henrique, Av. Engenheiro Bonneville Franco, 2770–058 Paço de Arcos, Oeiras, Portugal

Correspondence should be addressed to Joaquim Guerreiro Marques; tp.aobsilu.ocincet@seuqramgmj

Received 17 April 2018; Accepted 11 July 2018; Published 19 August 2018

Academic Editor: Jose E. Naranjo

Copyright © 2018 Luís Braga Campos and Joaquim Guerreiro Marques. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper presents methods to determine the time, positions, and distance of closest approach for two vehicles following arbitrary trajectories in two or three dimensions. The distance of closest approach of two vehicles following arbitrary curved trajectories is determined by two conditions: (i) the relative velocity must be orthogonal to the relative position in order for the distance to be a nonzero extremum; (ii) the radial acceleration including centripetal terms must have a direction that increases the separation for the extremum to be a minimum. This theorem on the distance of closest approach simplifies in the case of uniform motion along rectilinear trajectories. Three examples are given: (i) the two-dimensional motion of surface vehicles changing the velocity of one of them so as to enforce a given minimum separation distance; (ii) the three-dimensional motion of two aircraft, one flying horizontally and the other climbing, changing the vertical velocity of the latter to ensure a minimum separation distance set “a priori”; (iii) the case of an aircraft flying with constant velocity in a straight line so that its closest approach to another aircraft flying in a circular holding pattern in the same plane occurs at a given time chosen “a priori”.

#### 1. Introduction

In the traffic of vehicles safety is identified with the absence of collisions or conflicts. A conflict occurs when the distance between the centroids of two vehicles is less than a safe separation distance (SSD) determined by their size. Thus (i) the absence of conflicts and (ii) the confirmation that a conflict has been resolved depend on determining the distance of closest approach (DCA) that is not less than the SSD. The conflict resolution relies (iii) on trajectory modifications that change the DCA from smaller than the SSD to larger than (or equal to) the SSD. The paper presents methods to determine the time, positions, and distance of closest approach for two vehicles following arbitrary trajectories in two or three dimensions. The two-dimensional cases include cars in road traffic, ships in sea lanes, and aircraft ground movements at an airport. The three-dimensional cases include all types of flying vehicles, like airplanes, helicopters, drones, rockets and satellites, and also submerged submarines. The differences in conflict detection and resolution (CDR) between all these types of vehicles concern the speed, size, and distances that enter as parameters in the same methods of calculation of distance and time of closest approach.

The distance and time of closest approach are essential inputs for CDR methods [1–3]. The collision risk applies to cars [4, 5], ships and submarines [6, 7], and aircraft [8, 9]. Taking as example the case of Air Traffic Management (ATM) the problem may be divided into (i) prediction of flight paths [10, 11], (ii) safety assessment [12, 13], and (iii) conflict resolution [14–16]. Collision avoidance between aircraft starts with [17] separation distances (e.g., longitudinal, lateral, and altitude) leading to high Target Level of Safety (TLS: probability of collision less than per hour) for various aircraft encounter geometries, like level crossing [18] or climb and descent [19]. The safe separation ultimately determines airspace capacity [20]. Many of these methods assume straight trajectories or approximate curved trajectories by straight segments. While this allows a continuous trajectory, the velocity becomes discontinuous in direction at the edges and the acceleration becomes singular. The purpose of the present paper is to determine the distance and time of closest approach for arbitrary curved trajectories without approximations of any kind. This may be used for CDR methods or to assess TLS.

There are a variety of CDR methods including multiagent algorithms [21–23] that apply to vehicles moving in two [1–9, 24–27] or three [10–20] dimensions. All CDR methods (i) start with the identification of a conflict, (ii) involve trajectory changes to resolve the conflict, and (iii) end with the verification that the conflict has been resolved. The safety of traffic requires that a minimum SSD be held, for example, ensuring that the “safety volumes” around two vehicles do not penetrate. If two vehicles follow two arbitrary trajectories with position vectors and , as a function of time, the modulus of the difference specifies their distance that generally varies with time: A conflict is detected if at any time the distance is less than the SSD: If a conflict is detected, then one or both trajectories must be modified to resolve the conflict. The success of the conflict resolution is checked by showing that the distance exceeds the SSD for all time:The criteria for conflict (2) or no conflict (3) are next put into a simpler form that is easier to apply.

The key concept is that of distance of closest approach between two trajectories: which occurs at the time of closest approach, when the vehicles are at position and . There is a conflict if the distance of closest approach is less than the safe separation distance (5a) and no conflict otherwise (5b): The paper presents a method to determine (i) the distance of closest approach , (ii) the time of closest approach , and (iii) the positions of the two vehicles at that time . The method applies to arbitrary trajectories: (a) curved or rectilinear; (b) with constant velocity, accelerated or decelerated motion. The method is deterministic and excludes external disturbances. The extension to include random disturbances can be made adding to the position vectors the deviations due to uncertainties or external effects and applying statistical methods.

Considering first arbitrary nonuniform motion a theorem is established (Section 2.1) specifying the conditions for minimum separation between two trajectories. In many traffic situations the future trajectories are not known, and the information available is only the current positions and velocities of two vehicles; if the motion is assumed to be uniform, simple formulas are obtained (Section 2.2) for the distance and time of closest approach. This in turn specifies the conditions for collision avoidance between two or any number of vehicles (Section 2.3). These conditions give a simple geometric interpretation of the two theorems (Section 2). The preceding theory is applied (Section 3) to three examples: (Section 3.1) two-dimensional collision avoidance between surface vehicles (ships, car, or airplanes on the ground at an airport) with constant velocity by choosing the modulus of the velocity of one of them; (Section 3.2) three-dimensional collision avoidance between two aircraft moving at constant velocity, one at constant altitude and the other climbing, by changing the vertical velocity of the latter; (Section 3.3) meeting a given time of closest approach between a vehicle in a holding pattern of uniform circular motion and another in uniform rectilinear motion. These three cases are sufficiently simple for analytical calculation (Section 3), illustrate the two theorems (Section 2), and substantiate the discussion (Section 4).

#### 2. Distance of Closest Approach between Trajectories

The minimal separation for two vehicles following arbitrary trajectories is obtained by minimizing the relative distance as a function of time. This leads to two conditions to be satisfied at the time of closest approach (Section 2.1). The simplest case is that of uniform motion from given initial positions (Section 2.2). It leads to a simple criterion for collision avoidance by maintaining a minimum separation distance (Section 2.3) and is applicable to two or more vehicles. The vehicles are represented by material vehicles placed at their centroids, and their dimensions are taken into account by setting a safe separation distance to avoid collisions.

##### 2.1. Nonuniform Motion along Curved Trajectories

Consider two vehicles moving with integrable time-dependent velocities and from initial positions and at time , so that their positions at time are given by It is necessary to find (i) the conditions for a collision, where the positions coincide with for some time(s) ; (ii) if there is no collision then the time and distance of closest approach are given by (4)(7): The relative position of the two vehicles is given at all times by:Introducing the initial relative positions (9a) at the initial time and the relative velocity (9b) at all times,the relative position at arbitrary time is given by where the change in relative position from time to time is given by (11a): and its time derivative is relative velocity (11b).

The distance between the vehicles at time is given by The first derivative of (12) with regard to time, is zero for stationary distance between the vehicles: Thus two cases may arise: (i) if the distance is zero at the stationary point(s) then there is collision between the vehicles and the relative velocity is arbitrary:(ii) if the distance is not zero then the velocity and relative position must be orthogonal: Note that the distance of closest approach is generally not the smallest distance between the paths because the two vehicles will be at these points at different times. The condition of closest approach that the relative velocity and relative position are orthogonal (Figure 1) can be explained as follows: (i) if at time then the vehicles would be moving towards each other and would be closer at some later time ; (ii) if at time then the vehicles would be moving away from each other and would have been closer for some earlier time . In either case (i) or (ii) the time would not be that of stationary distance. This proves by “reductio ad absurdum” that a stationary distance requires that the relative velocity be orthogonal to the relative position.