Lagrange's equation of motion of an inverted pendulum

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＜Premise knowledge＞
　・Lagrange's equation of motion
　・Energy calculation method
　・Differential formula

■ Lagrange's equation of motion of an inverted pendulum

The following equation of motion of an inverted pendulum is derived from Lagrange's equation of motion. Lagrange's equation of motion is expressed by the energy of an object, and it is easier to obtain the equation of motion than the method of classical mechanics.



Since Lagrange's equation of motion is as follows, first find the kinetic energy and potential energy.

＜kinetic energy＞

1. kinetic energy of truck



2. Kinetic energy of an inverted pendulum



3. total kinetic energy



＜potential energy＞



＜Lagrange's equation of motion＞


The above uses derivatives of composite functions.



From the above, (6) and (8) are the equations of motion of the inverted pendulum.



＜linearized approximation＞
Since the above equation of motion contains sin and cos, it is nonlinear and difficult to handle, so it is linearized by approximation. When θ is sufficiently small, the following holds.



Therefore, formula (9) is as follows.

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