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I have a simple system: a pendulum and two sensors (RADAR which will tell the distance $dx$ and $v_x$ and a Gyroscope, the gyroscope is giving me the information about $\omega$ - angular velocity) as it is shown below. I have to estimate the angle $\theta$ (USING THE EXTENDED KALMAN FILTER) at any given time only by using the input data I was given (see table below).

The system:

The system

Input data:

enter image description here

The data is displayed in the table as such: value, systematic error for $dx$, $v_x$, $\omega$. Since I am using sensors, an error is implied.

By far I found a formula that can estimate $\theta$ but it is not using any of the given input:

$\theta(t) = \theta(0) \cos \Big(t\sqrt{\frac{g}{l}}\Big)\ \text{where } \theta(0) = 90\ \text{degrees}$

Thanks!

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For a pendulum swinging with small oscillations (around 10 degrees), its motion is independent of all variables except the local gravitational acceleration ($g$) and its length ($l$). You have to swing the pendulum at larger amplitudes to see the effects of different starting conditions.

Here's a site with large amplitude formulas to get you started: http://hyperphysics.phy-astr.gsu.edu/hbase/pendl.html

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