# Estimate pendulum motion

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:

Input data:

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!

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.