Control of an Inverted Pendulum

[kataya]

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:

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

Where $$a_{B}$$ 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

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.$$\theta''\theta$$). 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

 

[KataKoniK]

Dear Tyler,

Thank you for sharing your homework assignment with us. It seems like you have a challenging problem ahead of you. I can offer some guidance and suggestions to help you develop the equations of motion for this system.

Firstly, let's define the variables in your system. Let's call the tilt angle of the rod theta, the angular velocity of the rod omega, and the angular acceleration of the rod alpha. The length of the rod can be represented by L, the mass by m, and the moment of inertia by I. The acceleration of the palm can be represented by a, and the force applied by the palm can be represented by P.

Now, let's consider the forces acting on the rod. As you correctly identified, there are three forces: gravity, the normal force from the palm, and the force P from the palm. We can write the sum of forces in the x-direction as:

ma = Pcos(theta)

And in the y-direction as:

0 = mg - Psin(theta)

Next, let's consider the moments acting on the rod. The only moment that needs to be considered is the moment about the center of mass of the rod. We can write this as:

Ialpha = PLsin(theta)

Now, we can use the rigid body acceleration relation you mentioned in your post to relate the acceleration of the contact point (the palm) to the acceleration of the center of mass of the rod. We can write this as:

a = alpha*L

Finally, we can combine all of these equations to eliminate the forces and solve for the angular acceleration alpha. We get the following differential equation:

I(theta'') = mgLsin(theta) + PLcos(theta) - maLcos(theta)

This equation satisfies your requirement for not containing mixed order derivatives. You can also use the small angle approximation for sine and cosine to simplify this equation further. I hope this helps you in developing the equations of motion for this system. If you have any further questions, please don't hesitate to ask. Good luck with your homework!

 

[piercas]

I would first commend the student for taking on such a challenging problem and for seeking help when needed. I would also emphasize the importance of clearly defining the problem and the desired outcome, as well as making any necessary assumptions and simplifications.

In terms of the approach, I would suggest using Lagrangian mechanics to derive the equations of motion for the inverted pendulum. This method takes into account the energy and forces acting on the system and can be used to create a set of differential equations. It also allows for the inclusion of constraints, such as the restriction on vertical acceleration of the palm.

Additionally, I would recommend breaking down the problem into smaller components and solving each one individually before combining them to create the overall solution. This can help in identifying and solving any issues that may arise.

Lastly, I would suggest seeking out resources such as textbooks, online tutorials, and other examples of similar problems to gain a better understanding of the concepts involved and how to approach them. As with any scientific problem, perseverance and a willingness to learn and adapt are key to finding a solution.