A mathematical challenge...will it hit me, or not?

[chuns]

Thinking about AIS, I felt motivated to try and develop a quick and simple method of determining whether another vessel (say in a channel crossing) represented a collision risk. I certainly found a way of doing it graphically on the chart, but wondered if there was a mathematical way of doing it. Certainly my Maths ‘O’ level of 45 years ago was insufficient for the task.

So, if you know the range and bearing of a vessel, and you know its speed and COG, and, of course, know your own, is there a mathematical formula which can tell you if you will meet at the same point?

I appreciate that that probably is not enough (for example you might need to know how close it would miss) or whether it passes ahead or astern, but it might be a good starting point.

Can it be done?

 

[FlyingDutchman]

Yes.
But How???

 

[fireball]

Yes, of course it can ... otherwise AIS output wouldn't include CPA ....

 

[KenMcCulloch]

If the relative bearing of the other vessel (the angle between the other vessel and your heading) remains the same you will collide. As simple as that.

 

[BlueSkyNick]

Elementary CPA/TCPA stuff innit? Easy if you plot it on the chart or a piece of paper. Extend a line forwards from your boat and also from the other vessel so they both cross. Measure the distance from the crossing point and calculate from your speed how long it will take you to get there. Do the same for the other vessel. Then you can see the time difference and decide if its enough to avoid collision.

 

[graham]

Why make something very simple so complicated?

If the bearing remains constant or nearly constant as the range decreases theres likely to be a collision..

If the bearing is changing as you take bearings with the handbearing compass it will soon become apparent if you will pass ahead or behind the other vessel.

With a possibillity of a collision occuring I dont think its a good time to be distracted with calculations or chartwork.

 

[Cruiser2B]

The quickest and most reliable method to determine whether another vessel is a collision risk is that its bearing is steady. If you want to determine CPA, I would suggest sticking to the graphical method. A small laminated plotting board would be handy in the cockpit. It can be guesstimated - if you have a visual mind - picture the vector triangle in your mind; once you have the relative vector, it's easy to determine CPA.

 

[Amari]

I agree , simple is best. We don't even bother with a compass bearing, just use Mark one eyeball and top of stanchion rail or similar sighting point

 

[chuns]

Thanks, folks. Should have said that I was well familiar with LCBs, but I was pondering (and it is only a sort of brain exercise) whether by looking at AIS and tapping a couple of keys on a calculator I would get an instant read out.

I don't actually need this for myself, for apart from my hand bearing compass and bloodshot eyes, I also have MARPA, and I turn on the radar if crossing the shipping lanes.

So I'm looking for a theoretical solution, not a practical one.

 

[cpedw]

It's not hard, but it would be beyond my mental arithmetic. The sums needed are:
t1 = R.sin(COG+B) / (S1.sin(COG))
t2 = R.sin(B) / (S2.sin(COG))

t1 - time of vessel 1 to location where tracks cross
t2 - time of vessel 2 to location where tracks cross
R - range
B - relative bearing of 2 from 1
COG - course of 2 relative to 1
S1, S2 - speed of 1 and 2.
Use any consistent units. If speeds are knots, Range is NM then time is in hours.
The smaller the difference between t1 and t2, the scarier it is. CPA is, I think, difference between t1 and t2 times the slower speed. Easy sums to build into a plotter or similar; couldn't recommend trasnferring all those data to a calculator in a wet cockpit.

Derek

 

[johnalison]

I also use eyeball most of the time, but it has been suggested that you should take compass (or radar) bearings and log them in case you need to give evidence later. A counsel of perfection perhaps, but a bit involved for most of us.

 

[pvb]

So doesn't your AIS give CPA?

 

[Clyde_Wanderer]

Why not do like the constapated mathamatician and, "Work It Out With A Pencil"
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[shmoo]

You need a copy of this. I think it does just what you want.

Optimal Intercept Course of Vessels to a Nonzero Range
SIAM Review archive
Volume 38 , Issue 4 (December 1996)
Pages: 647 - 649
Year of Publication: 1996
ISSN:0036-1445
Authors
B. U. Nguyen
D. Nguyen
Publisher
Society for Industrial and Applied Mathematics Philadelphia, PA, USA
ABSTRACT

An exact solution, based on minimum intercept time, is proposed for the optimal course of one ship approaching another to a nonzero range.

If you need a copy and can't get it from a library PM me and I will make arrangements.

 

[SailingDoc]

I have a similar view on this - that's to say, a mathematical solution is relatively simple, and instantaneous; whereas waiting for bearings to change, or not, may well lose you precious decision-making or manoevring time.
I used a very similar formula, and entered it up on Excel (not to be confused with LIBS... /forums/images/graemlins/wink.gif ) on my PDA. Once you've used it a few times it's quick to enter your own position, speed and course, and ditto the other vessel's. This gives you the two times to the 'crossing point' - his and yours.

The problem of accurate CPA is more daunting - and knowing the CPA doesn't tell you if you're passing, say, 1/4 mile ahead or 1/4 mile astern, of large metallic sea-going object. It makes a great deal of difference to your pulse-rate, though... /forums/images/graemlins/blush.gif I've tried tackling this, with graphical representation of ships' courses, but it's not yet got beyond pre-beta testing.

Brian

 

[graham]

Not everyone will be able to tackle the math when wet cold tired or just pi$$ed off.

Theres a company recently set up that you can phone or email all the avail;able data and they calculate the CPA time to CPA etc for you.Ill dig out the details if anyone wants them.

Bit pricey at £3.99 per calculation but what price do you put on safety?

 

[BlueSkyNick]

graham, please confirm you are taking the michael now !!

BTW, if somebody is OK at maths they should be able to work out CPA/TCPA quicker than waiting for time to pass between taking handheld compass bearings.

 

[graham]

Would I joke on such a serious subject Nick ?Mistakes are only human, for less than the price of a fish supper you can have it worked out proffesionally.

Could get a bit pricy crossing busy shipping lanes they may do a better price for account holders.

Collision avoidance holding tank and anchor technical services.com

 

[Anonymous]

You've had the maths explained and it isn't mental arithmetic, it would be easy on a PDA or programmable calculator but they get wet, and small buttons with cold fingers are not good.

I think the real flaw in the thought process is to imagine that any single observation and calculation is meaningful. Vessels change COG all the time for various reasons so whether you are using AIS or MARPA for CPA/TCPA the value is the continuous re-assessment of the risk every few seconds. Quite often I have locked a few FVs on MARPA, some miles away, then had an alarm when one has altered course 180 degrees and started to steam straight for me (as only FVs can and do do!!). AIS isn't very useful with FVs though, MARPA equipped radar is a godsend.