1. INTRODUCTION
The predicted object CPA (Closest Point of Approach) distance DCPA and, to a lesser extent, the time interval to its occurrence TCPA are well-established approach parameters used in collision avoidance systems featuring Automatic Radar Plotting Aids (ARPAs) as well as in manual radar plots.
After the introduction of marine navigation radars for collision avoidance purposes, approach parameters of tracked objects were determined in a graphical manner by manual radar plots. At the beginning, analytical formulae for determination of motion and approach parameters and collision avoidance manoeuvres were derived in a polar coordinate system, natural for radar plots, with input values such as distances, bearings, speeds, courses and their changes.
The introduction of computer controlled ARPAs has created the need for algorithms for determination of motion and approach parameters but calculations in such systems are system-specific because they use mainly a Cartesian coordinate system. This is caused by:
- simple equations of motion in a system of Cartesian coordinates,
- simple estimation algorithms for motion parameters in digital tracking filters (because for objects travelling with constant velocities and courses their polar position changes – radial and angular velocities - are not constant but in Cartesian coordinates are constant),
- reduction of number of trigonometric and circular functions which, when used in numerical calculations, are connected with longer and less accurate calculations.
Publication of such algorithms is very rare - Jakševič (Reference Jakševič1967) and Lord (Reference Lord1968) are two of the very few that have been published - and they are without any analysis of features and special cases, which are essential in computer calculations. In Sections 2 to 4 equations for DCPA and TCPA are fully derived and analysed, including their special cases and features.
Apart from conventional ARPA displays (vectors only) there are two unconventional ones: Firstly, PADs and PPC (Predicted Areas of Danger and Predicted Points of Collision) in circles, ellipses or hexagons (Riggs, Reference Riggs1970; Reference Riggs1975; Riggs and O'Sullivan, Reference Riggs and O'Sullivan1980; Fleischer et al., Reference Fleischer, Lipsky and Tiblin1970, Cornett et al., Reference Cornett, Gentry, Seay, Tucker and Wigent1979) which are geometrical approximations of the accurate PADs (Yancey and Wood, Reference Yancey and Wood2000) which show possible course evasive manoeuvres at constant own velocity. Secondly, CTPAs (Collision Threat Parameters Areas) introduced by Lenart (Reference Lenart1983), developed by Smierzchalski (Reference Smierzchalski2000, Reference Smierzchalski2005), Szlapczynski (Reference Szlapczynski2006, Reference Szlapczynski2007a, Reference Szlapczynski2007b, Reference Szlapczynski2008a, Reference Szlapczynski2008b, Reference Szlapczynski and Smierzchalski2009) and Szlapczynski and Smierzchalski (Reference Szlapczynski and Smierzchalski2009), tested by Pedersen et al. (Reference Pedersen, Inoue and Tsugane2003) and extended for reversed solutions and other approach parameters by Lenart (Reference Lenart1999a, Reference Lenart1999b, Reference Lenart2000a, Reference Lenart2000b, Reference Lenart2010) which show possible course and/or velocity evasive manoeuvres.
In Sections 5 to 6 collision criteria are analysed, a new collision threat parameter - time to safe distance - is proposed and its application to detection of collision objects and accurate PADs is presented.
2. ASSUMPTIONS AND INPUT PARAMETERS
For the purposes of this analysis, own vessel and extraneous objects of interest are regarded as if the mass of each object was concentrated at a point. It will be assumed that all moving external objects are travelling at constant velocity and course. In the movable plane tangential to the Earth's surface Cartesian coordinates system Ox, Oy (Figure 1), with Oy pointing North, O is the present position of own vessel. It is also assumed that manual plots or the radar processing and tracking (ARPA) or AIS (Automatic Identification System) has yielded:
- the present relative position of each object of interest X, Y,
- the components of its true velocity Vtx, Vty and/or
- the components of its relative velocity Vrx, Vry.
Input parameters.
The relationship of own and an object's velocities can be described by equations
$${\rm V}_{{\rm tx}} \, = \,{\rm V}_{{\rm rx}} + {\rm V}_{\rm x} $$
$${\rm V}_{{\rm ty}} = {\rm V}_{{\rm ry}} + {\rm V}_{\rm y} $$
$${\rm V}_{\rm t} = \sqrt {{\rm V}_{{\rm tx}}^2 + {\rm V}_{\rm ty}^2} $$
$${\rm V}_{\rm r} = \sqrt {{\rm V}_{\rm rx}^2 + {\rm V}_{\rm ry}^2} $$where: Vx, Vy – own velocity components,
$${\rm V}_{\rm x} = {\rm V} \sin {\rm \psi} $$
$${\rm V}_{\rm y} = {\rm V} \cos {\rm \psi} $$
$${\rm V} = \sqrt {{\rm V}_{\rm x}^2 + {\rm V}_{\rm y}^2} $$ψ – own course (the angle measured clockwise from Oy to V).
From the above
$${\rm V}_{{\rm tx}} = {\rm V}_{{\rm rx}} + {\rm V} \sin {\rm \psi} $$
$${\rm V}_{{\rm ty}} = {\rm V}_{{\rm ry}} + {\rm V} \cos {\rm \psi} $$and
$${\rm V}_{{\rm rx}} = {\rm V}_{{\rm tx}} - {\rm V} \sin {\rm \psi} $$
$${\rm V}_{{\rm ry}} = {\rm V}_{{\rm ty}} - {\rm V} \cos {\rm \psi} $$Own and an object's motion parameters should be either ground or sea referenced and drift angle is assumed to be zero.
3. EQUATIONS OF RELATIVE MOTION
The relative position of an object, at time t, is given by
$${\rm X}\left( {\rm t} \right) = {\rm X} + {\rm V}_{{\rm rx}} \, {\rm t}$$
$${\rm Y}\left( {\rm t} \right) = {\rm Y} + {\rm V}_{{\rm ry}} \, {\rm t}$$If D(t) is the distance to an object at time t, then
$${\rm D}({\rm t}) = \sqrt {{{\rm X}^2}({\rm t}) + {{\rm Y}^2}({\rm t})} = \sqrt {{{\rm R}^2} + {\rm V}_{\rm r}^2 {{\rm t}^2} + 2({\rm X}{{\rm V}_{{\rm rx}}} + {\rm Y}{{\rm V}_{{\rm ry}}}){\rm t}} $$or after squaring both sides and rearrangements
$${\rm V}_{\rm r}^2 {{\rm t}^2} + 2({\rm X}{{\rm V}_{{\rm rx}}} + {\rm Y}{{\rm V}_{{\rm ry}}}){\rm t} + {{\rm R}^2} - {{\rm D}^2}({\rm t}) = 0$$where:
$${\rm R} = \sqrt {{\rm X}^2 + {\rm Y}^2} $$4. CPA DISTANCE AND TIME
4.1. Derivation of equations for DCPA and TCPA
In equation of relative motion Equation (14) the distance reaches a minimum (the Closest Point of Approach – CPA) when the differential
$$\displaystyle{{{\rm d}{\rm D}({\rm t})} \over {{\rm dt}}} = \displaystyle{{{\rm V}_{\rm r}^2 {\rm t} + {\rm X}{{\rm V}_{{\rm rx}}} + {\rm Y}{{\rm V}_{{\rm ry}}}} \over {\sqrt {{{\rm R}^2} + {\rm V}_{\rm r}^2 {{\rm t}^2} + 2({\rm X}{{\rm V}_{{\rm rx}}} + {\rm Y}{{\rm V}_{{\rm ry}}}){\rm t}} }} = 0$$From the above, this time t is time to achieve CPA - TCPA
$${{\rm T}_{{\rm CPA}}} = - \displaystyle{{{\rm X}{{\rm V}_{{\rm rx}}} + {\rm Y}{{\rm V}_{{\rm ry}}}} \over {{\rm V}_{\rm r}^2 }}$$and substitution of the above time for t to Equation (14) gives distance to CPA - DCPA
$${{\rm D}_{{\rm CPA}}} = \left\vert {\displaystyle{{{\rm X}{{\rm V}_{{\rm ry}}} - {\rm Y}{{\rm V}_{{\rm rx}}}} \over {{{\rm V}_{\rm r}}}}} \right\vert$$4.2 Analysis of equations for DCPA and TCPA
DCPA and TCPA are indefinable when
$${\rm V}_{{\rm rx}} = {\rm V}_{{\rm ry}} = {\rm V}_{\rm r} = 0$$In this case according to Equations (1) and (2)
$${\rm V}_{{\rm tx}} = {\rm V}_{\rm x} $$
$${\rm V}_{{\rm ty}} = {\rm V}_{\rm y} $$which means that own vessel and an object are moving with the same velocities and courses and therefore the minimal distance is the same as the present distance R
$${\rm D}_{{\rm CPA}} \left( {{\rm V}_{{\rm rx}} = {\rm V}_{{\rm ry}} = 0} \right) = {\rm R}$$and
$${\rm T}_{\rm CPA} \left( {{\rm V}_{\rm rx}} = {\rm V}_{\rm ry} = {\rm 0} \right) = {\rm 0}$$Equations (18) and (19) are based on relative velocities which, when obtained from radar tracking, are more accurate than true velocities – true velocities contain errors of own velocity.
Equation (14), after taking into consideration Equation (18), gives
$${\rm D}({\rm t}) = \sqrt {{{\rm R}^2} + {\rm V}_{\rm r}^2 {\rm t}({\rm t} - 2{{\rm T}_{{\rm CPA}}})} $$therefore for TCPA > 0
$${\rm D}\left( {{\rm t} = 0} \right) = {\rm D}\left( {{\rm t} = 2 \, {\rm T}_{{\rm CPA}}} \right) = {\rm R}$$Similarly for TCPA < 0, D(t) increases with time t, i.e., an object is moving away – CPA has taken place in the past.
After substituting t = TCPA and D(t) = DCPA into Equation (14) it yields
$${{\rm D}_{{\rm CPA}}} = \sqrt {{{\rm R}^2} - {\rm V}_{\rm r}^2 {\rm T}_{{\rm CPA}}^2 } \quad {\rm if}\quad {\rm R} \ge {{\rm V}_{\rm r}}{{\rm T}_{{\rm CPA}}}$$and
$$\left\vert {{{\rm T}_{{\rm CPA}}}} \right\vert = \displaystyle{{\sqrt {{{\rm R}^2} - {\rm D}_{{\rm CPA}}^2 } } \over {{{\rm V}_{\rm r}}}}\quad {\rm if}\quad {\rm R} \ge {{\rm D}_{{\rm CPA}}}$$The above solutions can also be obtained by solving triangle OCT in Figure 1.
For collision avoidance purposes only present and future approaches are of interest and therefore, when
$${\rm T}_{{\rm CPA}} \lt 0$$then for further calculations it should be assumed that
$${\rm D}_{{\rm CPA}} = {\rm R \, and \, T}_{{\rm CPA}} = 0$$From the above it is evident that the value of DCPA calculated from Equation (19) can be heavily corrected by the value of TCPA, therefore calculation of DCPA should be inseparably connected with calculation of TCPA.
It can be proved (Lenart, Reference Lenart2010) that the sign of formula under the modulus in Equation (19) is the sign opposite to the sign of the distance abeam Dab (Figure 1) if own course is equal to bearing to an object (when TDab > 0 and TCPA > 0). Equations for this distance are derived in Lenart (Reference Lenart2000a) - Dab > 0 means approaches on the starboard and Dab < 0 means approaches on the port side.
5. COLLISION OBJECTS
5.1. Collision criteria
Each ARPA has two settings for safe values of DCPA and TCPA preselected by an operator. It is assumed that there is a threat of collision with an object for which
$${\rm D}_{{\rm CPA}} \lt {\rm D}_{\rm S} \,{\rm and \, T}_{{\rm CPA}} \lt {\rm T}_{\rm S} $$where DS, TS are selected safe values of DCPA and TCPA respectively.
Values DCPA and TCPA yield Equations (18) and (19), taking into consideration results of analysis in Section 4.2 i.e. if Vr = 0 or TCPA < 0 we should assume
$${\rm D}_{{\rm CPA}} = {\rm R \, and \, T}_{{\rm CPA}} = 0$$5.2. Undetected dangerous object example
Assume a threat situation as illustrated in Figure 1 with an object T after passing through point A.
DS = 3 NM
TS = 10 min
X = 1 NM
Y = 2·5 NM R = 2·7 NM (Equation (16))
Vrx = −7·5 kt
Vry = −3·75 kt Vr = 8·4 kt (Equation (4))
DCPA = 1·8 NM (Equation (19))
TCPA = 14·4 min (Equation (18))
The criterion of collision threat Equation (31) classifies this object as safe (TCPA > TS) although this object is already inside the DS circle (R = 2·7 NM), therefore this criterion should be supplemented by the next condition - R < DS and Equation (31) becomes
$${\rm D}_{{\rm CPA}} \lt {\rm D}_{\rm S} \,{\rm and \, T}_{{\rm CPA}} \lt {\rm T}_{\rm S} \, {\rm or \, R} \lt {\rm D}_{\rm S} $$or we introduce a new collision threat parameter – time to safe distance TDs - and a new criterion
$${\rm D}_{{\rm CPA}} \lt {\rm D}_{\rm S} \, {\rm and \, T}_{{\rm Ds}} \lt {\rm T}_{\rm S} $$6. TIME TO SAFE DISTANCE
6.1. Derivation of equation for TDs
After substitution into Equation (15)
$${\rm t} = {\rm T}_{{\rm Ds}} \,{\rm and \, D}\left( {\rm t} \right) = {\rm D}_{\rm S} $$we get
$${\rm V}_{\rm r}^2 {{\rm T}_{{\rm Ds}}}^2 + 2({\rm X}{{\rm V}_{{\rm rx}}} + {\rm Y}{{\rm V}_{{\rm ry}}}){{\rm T}_{{\rm Ds}}} + {{\rm R}^2} - {{\rm D}_{\rm S}}^2 = 0$$and after solving this quadratic equation in TDs
$${{\rm T}_{{\rm Ds}}} = \displaystyle{{ - ({\rm X}{{\rm V}_{{\rm rx}}} + {\rm Y}{{\rm V}_{{\rm ry}}}) \pm \sqrt {{{({{\rm D}_{\rm S}}{{\rm V}_{\rm r}})}^2} - {{({\rm X}{{\rm V}_{{\rm ry}}} - {\rm Y}{{\rm V}_{{\rm rx}}})}^2}} } \over {{\rm V}_{\rm r}^2 }}$$or according to Equations (18) and (19)
$${{\rm T}_{{\rm Ds}}} = {{\rm T}_{{\rm CPA}}} \pm \displaystyle{{\sqrt {{{\rm D}_{\rm S}}^2 - {\rm D}_{{\rm CPA}}^2 } } \over {{{\rm V}_{\rm r}}}}$$where real solutions exist if
$${\rm D}_{\rm S} \, \ge \,{\rm D}_{{\rm CPA}} $$Equation (38) can also be obtained by solving triangles OCA and OCB in Figure 1.
6.2. Analysis of TDs
DCPA, TCPA and TDs are indeterminate when
$${\rm V}_{{\rm rx}} = {\rm V}_{{\rm ry}} = {\rm V}_{\rm r} = 0$$In this case for collision avoidance purposes we can assume (as in Section 4.2)
$${\rm D}_{{\rm CPA}} = {\rm R \ and \ T}_{{\rm CPA}} = 0$$and
$${\rm T}_{{\rm Ds}} = 0 \, {\rm if \, R} \lt {\rm D}_{\rm S} \, \left( {{\rm collision \, threat}} \right)$$
$${\rm T}_{{\rm Ds}} = \infty \,{\rm if \, R}\, \ge \,{\rm D}_{\rm S} \, \left( {{\rm no \, collision \, threat}} \right)$$TDs exists if
$${\rm D}_{{\rm CPA}} \, \le \,{\rm D}_{\rm S} $$when
$${\rm D}_{{\rm CPA}} \, \gt \,{\rm D}_{\rm S} $$for collision avoidance purposes we can assume
$${\rm T}_{{\rm Ds}} = \infty \,\left( {{\rm no \, collision \, threat}} \right)$$TDs, if it exists, has two values
$${\rm T}_{{\rm Ds}1} = {\rm T}_{{\rm CPA}} - \Delta {\rm T}_{{\rm Ds}} $$
$${\rm T}_{{\rm Ds}2} = {\rm T}_{{\rm CPA}} + \Delta {\rm T}_{{\rm Ds}} $$where:
$$\Delta {{\rm T}_{{\rm Ds}}} = \displaystyle{{\sqrt {{{\rm D}_{\rm S}}^2 - {\rm D}_{{\rm CPA\!\!}}{}^2 } } \over {{{\rm V}_{\rm r}}}}$$TDs1 and TDs2 are times to reach points A and B (Figure 1) respectively.
TDs (Equation (37)) is based on relative velocities – see also Section 4.2.
The condition
$$\Delta {\rm T}_{{\rm Ds}} \, \ge \,{\rm T}_{\rm S} $$or
$${\rm D}_{\rm S}{} ^2 - {\rm D}_{{\rm CPA}\!\!}{}^2 \, \ge \,\left( {{\rm V}_{\rm r} {\rm T}_{\rm S}} \right)^2 $$describes cases, when the criterion Equation (31) does not detect threat objects with R < DS that is when DS is relatively big and TDs is relatively short.
6.3. Applications of TDs
TDs can be used for detection of threat objects with the criterion Equation (34) taking into consideration special cases as in Section 6.2 and the following.
In the case when
$${\rm T}_{{\rm Ds}1} \lt 0 \, {\rm and} \, {\rm T}_{{\rm Ds}2} \lt 0$$an object is moving away with
$${\rm R} \gt {\rm D}_{\rm S} $$and this is after reaching point B and therefore
$${\rm T}_{{\rm Ds}} = \infty \left( {{\rm no \, collision \, threat}} \right)$$In the case when
$${\rm T}_{{\rm Ds}1} \ {\rm T}_{{\rm Ds}2} \, \le \,0$$an object is between points A and B and therefore
$${\rm R} \le {\rm D}_{\rm S} \,{\rm and \, T}_{{\rm Ds}} = 0 \left( {{\rm collision \, threat}} \right)$$In the case when
$${\rm T}_{{\rm Ds}1} \gt 0 \, {\rm and \, T}_{{\rm Ds}2} \gt 0$$an object is approaching with
$${\rm D}_{{\rm CPA}} \lt {\rm D}_{\rm s} $$and is not yet at point A. Thus
$${\rm T}_{{\rm Ds}} = {\rm T}_{{\rm Ds}1} $$and an object is dangerous if
$${\rm T}_{{\rm Ds}} \lt {\rm T}_{\rm S} $$This time TDs according to Equations (59) and (47) is shorter than TCPA (which can be time to collision) and therefore the criterion Equation (34) is safer than Equation (33) or TS can be shorter than used in the criterion Equation (33).
The above algorithm may seem complex because it combines calculating R, DCPA, TDs with the detection of dangerous objects. However in collision avoidance systems the detection of dangerous objects (which is made for all objects), can be much simpler than the calculation of all threat parameters (which is made for one or a few selected objects only). In fact, the criterion Equation (34) is easier to apply than Equation (33), because it mostly involves processing signs only (Section 6.3), whereas in case of the criterion Equation (33) DCPA, TCPA values have to be computed, checked and possibly replaced with the values from Equation (32).
In the undetected dangerous object example (Section 5.2) application of TDs gives:
ΔTDs = 17·1 min (Equation (49))
TDs1 = −2·7 min (Equation (47))
TDs2 = 31·5 min (Equation (48))
TDs = 0 (Equation (56))
and according to the criterion Equation (34) an object is classified as dangerous (independently of TS). See also condition Equation (50).
For the given own velocity V (real or simulated), a variable own course ψ and constant object's true velocity components Vtx, Vty, we can calculate Vrx, Vry from Equations (10) and (11).
For own courses ψ, for which (for all tracked objects)
$${\rm D}_{{\rm CPA}} \, \le \,{\rm D}_{\rm S} $$we can calculate values TDs1, TDs2 (Equations (47)-(49)) and Vx, Vy from Equations (5) and (6). By plotting positions of points
$$\left( {{\rm x} = {\rm V}_{\rm x} {\rm T}_{{\rm Ds}1}, {\rm y} = {\rm V}_{\rm y} {\rm T}_{{\rm Ds}1}} \right)$$
$$\left( {{\rm x} = {\rm V}_{\rm x} {\rm T}_{{\rm Ds}2}, {\rm y} = {\rm V}_{\rm y} {\rm T}_{{\rm Ds}2}} \right)$$we obtain boundaries of areas, for which own courses are leading to
$${\rm D}_{{\rm CPA}} \le {\rm D}_{\rm S} $$These are accurate PADs similar to Yancey and Wood (Reference Yancey and Wood2000) but obtained by a much simpler method.
Figure 2 illustrates exemplary accurate PADs plotted using parameter TDs. Points inside PADs are PPCs (Predicted Points of Collision) for which DCPA = 0. Attention should be drawn to the object at bearing 315◦, which does not have a PPC and its PAD lies entirely out of its true course line.
Accurate Predicted Areas of Danger.
7. CONCLUSIONS
It has been proved that derived equations for DCPA and TCPA have special cases and new features and collision criteria have limitations. A new collision threat parameter - time to safe distance - may have various applications. It can be applied to detection of dangerous objects, solving the problem of undetected dangerous objects, and giving safer time to dangerous situations than the time to CPA, which can be time to collision. It can be applied also to display the possible evasive manoeuvres (accurate PADs instead of their geometrical approximations).
