# Multivariable integral approximation

• Jul 12th 2010, 04:19 AM
mrodgers
Multivariable integral approximation
Hello, first allow me to give some background on my problem:

I have an object being launched under propulsive forces and I need to find a way to express distance traveled in terms of time to see if it will be a safe distance away from a launcher before fins are deployed. The test data I have lasts only a total of 0.2 seconds and consists of accelerometer data associated with time steps.

The Problem:

Because I only have accelerations and the times at which they were measured I considered doing an integral approximation using either the Midpoint or Trapezoid rules with F(x)=distance. When my answer came out blatantly wrong (D=0.5*a*t^2 and the timesteps are so small that squaring them made the summed distances far too small to be correct) I realized that F(x) is really F(x,y) because both time and acceleration are changing. I have no idea how to do a multivariable integral approximation, any searching that I have done only leads me to research papers that need to be purchased. Any ideas?
• Jul 12th 2010, 05:31 AM
CaptainBlack
Quote:

Originally Posted by mrodgers
Hello, first allow me to give some background on my problem:

I have an object being launched under propulsive forces and I need to find a way to express distance traveled in terms of time to see if it will be a safe distance away from a launcher before fins are deployed. The test data I have lasts only a total of 0.2 seconds and consists of accelerometer data associated with time steps.

The Problem:

Because I only have accelerations and the times at which they were measured I considered doing an integral approximation using either the Midpoint or Trapezoid rules with F(x)=distance. When my answer came out blatantly wrong (D=0.5*a*t^2 and the timesteps are so small that squaring them made the summed distances far too small to be correct) I realized that F(x) is really F(x,y) because both time and acceleration are changing. I have no idea how to do a multivariable integral approximation, any searching that I have done only leads me to research papers that need to be purchased. Any ideas?

If you have accelerometer data why have you assumed constant acceleration?

Can we see this data?

CB
• Jul 12th 2010, 05:33 AM
Ackbeet
So, if you're trying to find position from acceleration, you're going to need to integrate twice with respect to time. You'll need an initial velocity condition, and an initial position condition. You still have the following:

$\displaystyle v(t)=\int_{t_{0}}^{t}a(t)\,dt$, and
$\displaystyle x(t)=\int_{t_{0}}^{t}v(t)\,dt.$

The formula $\displaystyle x=at^{2}/2$ only works for constant acceleration, zero initial position, and zero initial velocity, in 1-dimensional motion.

Your final formula could work like this:

$\displaystyle x(t)=\int_{t_{0}}^{t}\int_{t_{0}}^{s}a(s)\,ds\,dt.$

You have to change the inner integral's variable of integration in order not to confuse it with the outer integral's variable of integration.

Does this all make sense?