Inverted Pendulum: Relate Force to Angle/Displacement?

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The discussion focuses on the dynamics of an inverted pendulum resting at an angle from the horizontal and how a dynamic force affects its movement. The force applied decreases as the angle increases, leading to a critical point where the pendulum cannot return to its resting position once it passes vertical. Participants are encouraged to share insights on relating the force's magnitude to the angle or displacement of the pendulum's end. Additionally, there is a suggestion to research the control algorithm used in Segway technology, which operates on similar principles of inverted pendulum dynamics. Understanding these relationships is essential for solving the posed problem.
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Im considering a problem were an inverted pendulum is resting against a ledge an an angle theta from the horizontal. A force is applied to the end causing it to move, This force is dynamic and decrease as the angle increases. The critical point is when the pendulum passes the vertical and cannot return to its resting point falls to the other side.

Does anyone know how i can relate the magnitude of the force to either the angle or displacement of the end of the pendulum?

suggestions welcome.
thanks
 
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I would look around the net and see if you can find any information on the control algorithm used by the Segway personal transport. That is essentially an inverted pendulum.
 

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