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Control of an Inverted Pendulum

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Homework Statement



A recent homework assignment: Balancing a rod on the palm.

I need to develop the equations of motion as a function of the tilt angle for the planar case of an inverted pendulum, given an input acceleration to the palm. By accelerating the palm the pendulum (rod) can be stabilized. I eventually need to turn this into a linear differential equation which I can take the laplace transform of so the results must NOT contain mixed order derivatives. However once the equations of motion are derived correctly I don't think that I will have any trouble with the last bit.

The info on the rod is very symbolic. Length of L, mass of m, moment of inertia about its end of I

Simplifying assumptions. The palm will never accelerate vertically.

I would include the small angle assumption for sine and cosine but if need be I can linearize these via a taylor series to provide for a more general case.

Homework Equations



From rigid body dynamics, we know that:

[tex]a_{B}[/tex] = [tex]a_{A}[/tex] + [tex]\alpha[/tex] X [tex]r_{A/B}[/tex] - [tex]\omega^{2}r_{A/B}[/tex]

Where [tex]a_{B}[/tex] is the acceleration at point B on a rigid body, and alpha and omega are the angular acceleration and velocity (respectively) of the body about its center of mass.

All other equations used are sum of forces, sum of moments.

The Attempt at a Solution



First I need to make it very clear what I am looking for. I need a differential equation containing the acceleration of the contact plane/point (the palm) and derivatives of theta.

Here is a free body diagram of the system.

http://www.prism.gatech.edu/~gtg857w/S-DYN.bmp [Broken]

There are 3 forces acting on the rod: gravity and two components of a reaction force at the palm, which I have broken into a normal force and another force P.

The biggest question I have at this point is: Is the reference frame in which I drew the free body diagram non-inertial? It doesn't seem like it has to be. Imagine that the plane on which the pendulum is resting is accelerating with respect to the white space around it. Perhaps this thread should be in introductory physics....

I'm kind of thrashing around with this one right now without a good place to begin. I have tried to develop an equation using the rigid body acceleration relation and the sum of the moments about the center of mass, but I got stuck with forces I couldn't solve for and mixed order derivatives of theta ( ie.[tex]\theta''\theta[/tex]). If any point of this is not clear I will be more than happy to try and clarify, and I acknowledge that it is a rather difficult question for a general audience. I thank you in advance for considering it.

-Tyler
 
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