[BlindMath] Understanding spoken mathematical output

[kperry at blinksoft.com]

Good description.


-----Original Message-----
From: BlindMath <blindmath-bounces at nfbnet.org> On Behalf Of Jonathan Fine
via BlindMath
Sent: Saturday, April 23, 2022 1:22 PM
To: Blind Math list for those interested in mathematics
<blindmath at nfbnet.org>
Cc: Jonathan Fine <jfine2358 at gmail.com>; Pranav Lal <pranav.lal at gmail.com>
Subject: Re: [BlindMath] Understanding spoken mathematical output

Hi Pranav

You asked for help with the equation
x  =    x  sub  0  +  v  sub  0  Delta  t  +  1  over 2 a Delta  t squared
which appears on the page
https://www.kalmanfilter.net/default.aspx

I'm going to help you with the mathematical meaning of this equation. I'm a
sighted person who has taught college level math, although never to a
visually disabled person. So this is a first for me.

The Kalman Filter web page says: The Kalman Filter produces estimates of
hidden variables based on inaccurate and uncertain measurements. Also, the
Kalman Filter provides a prediction of the future system state based on past
estimations.

Back to me. A similar problem is finding the line that best fits some points
in the plane.

Another problem is finding the number that best fits a collection of
numbers. For this problem the average of the collection of numbers is often
used as the best fit.

Let's now think about a car on a straight road. At any moment of time it has
a position. If the car is moving it also has a velocity. And the car might
also have an acceleration. A positive acceleration comes from pressing the
accelerator pedal, also called stepping on the gas. A negative acceleration
is obtained by using the brakes.

For a stationary car to start moving it needs some acceleration, which gives
it a velocity. Often the acceleration is not constant. Even if you floor the
gas pedal your car won't accelerate beyond its top speed.

However, let's assume that the acceleration is constant. Let's write x(t)
for the position of the car at time t. Let's write x_0 for x(0), the
position of the car at time 0. Let's write v(t) for the velocity of the car
at time t, and v(0) for the velocity of the car at time 0. Finally, let's
write a_0 for the acceleration of the car at time 0. Because we assumed that
the acceleration is constant, the acceleration of the car at time t is equal
to a_0 for every value of t.

>From these assumptions it follows mathematically that
x(t) = x_0 + v_0 t + (a_0 / 2) t^2
is exactly true. In practice the assumptions are approximations, and the
formula is not exactly true.

The Kalman filter, as I understand it from what I've quoted, applies methods
similar to averages and least squares best fitting of a line to the problem
of predicting the location of a moving body based on limited observations,
as arise from a 1940s radar system.

I hope this helps you understand how the equation you kindly provided is
relevant to the discussion of the Kalman filter. Please ask further
questions if you wish.

By the way, when you're in a moving vehicle it is the acceleration of the
vehicle that pushes you into your seat. This is most obvious when a jet
plane is taking off, even if you can't look out of the windows.

Finally, my focus is on pure mathematics and today is the first time I've
looked at the Kalman filter.

with kind regards

Jonathan
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