[BlindMath] Understanding spoken mathematical output

[Jonathan Fine]

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