Variance of the position noise

[ArixII]

Hi,
I read an article in Kalman filtering. There is an example in introductory Mechanics. I just can not understand which formula for calculating the variance and covariance is used. The example is about a vehicle that is traveling straight on the road:

The postition is measured with an error of 10 feet (one standard deviation)
The command acceleration is a constant 1 foot/sec^2
The acceleration noise is 0.2 feet/sec^2 (one standard deviation)
The position is measured 10 times per second.

Since the position is proportional to 0.0005 times the acceleration, and the acceleration noise is 0.2 feet/sec^2, the variance of the position noise is (0.005)^2 * (0.2)^2 = 10^(-6)
...

[and what formula is used to calculate this? ]

[author continues]
...Similarly, since the velocity is proportional to 0.1 times the acceleration, the variance of the velocity noise is (0.1)^2 * (0.2)^2 = 4 * 10^(-4). Finally, the covariance of the position noise and velocity noise is equal to the standard deviation of the position noise times the standard deviation of the velocity noise, which can be calculated as (0.005 * 0.2) * (0.1 * 0.2) = 2 * 10^(-5).
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Well I know that:
cov(X,Y) = E[(X - mu_x) * (Y - mu_y)]
but this is not what he used, right? does somebody knows the formulas used in such calculations?

Thank you!

 

[jbsteele]

The formula used in the example is a simple calculation of the variance and covariance of two variables, X and Y. The variance of X is calculated as Var(X) = (0.005)^2 * (0.2)^2 = 10^(-6), and the variance of Y is calculated as Var(Y) = (0.1)^2 * (0.2)^2 = 4 * 10^(-4). The covariance of X and Y is then calculated as Cov(X,Y) = E[(X - mu_x) * (Y - mu_y)] = (0.005 * 0.2) * (0.1 * 0.2) = 2 * 10^(-5).

 

[thomate1]

Dear author,
Thank you for your question. The formulas used in the calculations described in the article are based on the principles of Kalman filtering, which is a mathematical technique used to estimate the state of a system based on measurements and predictions. In this case, the formulas used to calculate the variance and covariance of the position and velocity noise are based on the Kalman filter equations for a linear system. The specific equations used may vary depending on the specific implementation and assumptions made in the article, but they are based on the following general formulas:

Variance of position noise: (0.005)^2 * (0.2)^2 = 10^(-6)

Variance of velocity noise: (0.1)^2 * (0.2)^2 = 4 * 10^(-4)

Covariance of position and velocity noise: (0.005 * 0.2) * (0.1 * 0.2) = 2 * 10^(-5)

These formulas take into account the measurement and prediction errors, as well as the relationship between the position, velocity, and acceleration variables. I hope this helps to clarify the calculations used in the article. If you have any further questions, please let me know.