Calculating CPA & TCPA - one for the maths gurus.

[wooslehunter]

I need some equations to calculate the closest point of approach (CPA) & time to closest point of approach (TCPA). I need to do this mathematically and can't just plot it.

Given I have the current position, course and speed of two vessels, anyone got some equations I can use?

I did derive some but they are huge and complex. There has to be a simpler way. Any ideas?

 

[sarabande]

It's simple really:

The closest point of approach between a fired projectile and a target is taken to be at the instant the projectile velocity vector (6) is orthogonal to the position error vector (13) between the projectile and target in accordance with the relationship: V.sub.p.cndot.(P.sub.P -P.sub.F)=0 where V.sub.p is the projectile velocity vector, P.sub.P is the projectile trajectory or position vector, P.sub.F is the target future position vector, .cndot. is the vector dot product and (P.sub.P -P.sub.F) is the position error vector.

/forums/images/graemlins/grin.gif

(PS> Seriously what is the environment of the problem pls ? Nautical, aeronautical, deep space....)

 

[tome]

Re: Calculating CPA & TCPA - one for the maths gurus.

First thing you need to do is to convert the positions from circular(lat,lon) to grid (Eastings & Northings). Given that both vessels will be within close proximity and that you are only really interested in relative positions, you can take some short cuts to simplify the maths

Once you have grid co-ords you can generate vectors from cog and sog. Don't have time to go into too much dtail but sure I can dig out the formulae later if yo're still stuck

 

[shmoo]

I have a paper copy of this (which is quite close to 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.

 

[shmoo]

Also slides 2 and 3 of this talk http://www.unige.ch/~gander/teaching/talk.pdf seem to offer a relatively simple geometric solution to a similar problem - close enough I think to offer insight into solving yours

 

[wooslehunter]

Thanks, clear as mud now.

This is for a nautical environment and close range so only need linear 2D trig rather than circular or 3D.

 

[shmoo]

Got it:

Go to http://www.patentstorm.us/patents/6249241-description.html

and search for:

Calculation of the CPA and TCPA is done in the following way:

and there is the formula.

 

[sarabande]

No wonder that patent agents charge so much !

 

[wooslehunter]

Thanks. They used the same method as I derived. Looks like the formulae are far simpler though so need to take a look.