Re: best braking technique as one approaches red light
- From: Keith Ramsay <kramsay@xxxxxxx>
- Date: Fri, 24 Aug 2007 20:59:54 -0700
On Aug 23, 3:00 am, jeremy rutman <jeremy_spag...@xxxxxxxxxxx> wrote:
|I'm traveling at a speed V when I see a red light ahead
|at distance d with probability to change to green as function of time
p(t), which for example could be gaussian.
|I would like to find the velocity profile v(t) that maximizes my
velocity (averaged over all scenarios knowing p(t)) as I pass the
light, with the condition that I have to stop if I hit the light when
its still red.
On one set of assumptions... you're trying to maximize the
integral of v(t)p(t) subject to v being positive,
nonincreasing and the integral of v being d.
Suppose p is nondecreasing until time t0, and nonincreasing
after t0. It's an improvement, or at least doesn't make
matters worse, to replace a given v(t) with one where v(t)
is constant for t<=t0 (but with the integral of v up to t0
kept the same). In other words, get all the decelerating
done as soon as possible. Conversely for t>=t0 it's an
improvement or at least doesn't make things worse to
procrastinate on decelerating as much as possible, by
keeping v(t) constant at the highest value it can have
without increasing, until the integral reaches d: don't
break at all until you reach the light.
So break suddenly to some speed v and don't break any more
until you reach the light. The optimum v is the one where
the ratio between the probability of the light changing
in time d/v and d/v is maximized.
If there's a maximum breaking acceleration, the same basic
strategy still applies; break as quickly as possible to a
speed v, then coast until you are as close to the light as
you can be and still stop in time.
By the way, please don't do this. Many times I've wanted to
pull up to a light and come to a full stop, so I could do
some simple thing like taking off a jacket or making sure I
brought something I needed, and been foiled by someone ahead
of me trying very hard not to stop, creeping along. Or else
I'll be trying to cross the street on foot away from a light,
and there'll be this mass of traffic coasting slowly toward
the light in the next block because it's red, while the next
batch of traffic catches up with them. Just pull up to the
light and stop please.
Keith Ramsay
.
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