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Double pendelum

The equations of motion are calculated using Lagrangian mechanics. The controller is calculated around the linearized point where all rods are pointing straight up and the angular velocities are zero. The controller is the optimal LQR controller. The control signal u is a torque applied to the base rod around the origin.

Equation of motion

$\ \dot{\theta_1} = \omega_1$
$\ \dot{\theta_2} = \omega_2$
$\ \dot{\omega_1} = \frac{- 4 \omega_{2}^{2} \sin{\left(\theta_{1} - \theta_{2} \right)} - \left(2 \omega_{1}^{2} \sin{\left(\theta_{1} - \theta_{2} \right)} + 19.64 \sin{\left(\theta_{2} \right)}\right) \cos{\left(\theta_{1} - \theta_{2} \right)} + 19.64 \sin{\left(\theta_{1} \right)}}{2 \left(2 - \cos^{2}{\left(\theta_{1} - \theta_{2} \right)}\right)} \ \$ $\ \dot{\omega_2} = \frac{2 \omega_{1}^{2} \sin{\left(\theta_{1} - \theta_{2} \right)} + \omega_{2}^{2} \sin{\left(2 \theta_{1} - 2 \theta_{2} \right)} + 14.73 \sin{\left(\theta_{2} \right)} - 4.91 \sin{\left(2 \theta_{1} - \theta_{2} \right)}}{2 \left(2 - \cos^{2}{\left(\theta_{1} - \theta_{2} \right)}\right)} \$

The poles for the open loop system and closed loop system.

Responce of the system with controller.

Double pendelum without controller.

Pendelum with controller.

Extended Kalman filter

Kalman filter to estimate the current state.

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