We will use the following variables and constants in the analysis.
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is the position of the cart
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is the counter-clockwise angular displacement of the pendulum from the upright position
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is the angular velocity of the pendulum
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u(t) is horizontal force applied to the cart
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L is the half-length of the pendulum
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m is the mass of the pendulum
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M is the mass of the cart
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P is the downward force exerted by the pendulum on the cart
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N is the horizontal force exerted by the pendulum on the cart
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g is the gravitational constant
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Let and .
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is the position of the center of mass of the pendulum.
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The acceleration of the center of mass is then:
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| (2.1) |
Rewrite the acceleration in terms of the cart velocity and angular velocity of pendulum.
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| (2.2) |
Apply F=ma in the horizontal (x-direction) to the pendulum.
Apply F=ma in the direction perpendicular to the pendulum.
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Apply F=ma in the horizontal direction to the cart.
Apply to the pendulum, where is the sum of all moments about the pendulum's center of mass, is the pendulum's moment of inertia, and is its angular acceleration.
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| (2.3) |
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The final nonlinear model is:
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