# Equations of motion for a fixed-height inverted pendulum

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Hi everybody!

I'm struggling with a physics problem I though I had solved, but as it is turning out recently, I probably hadn't. The problem might actually be pretty easy, just me being unable to solve properly.

All of you are familiar with inverted pendulum. Now, imagine an inverted pendulum with a fixed pivot point that is extensible. It can stretch itself into any length. What it does, is that it stretches itself so that the height of it's end point is always at the same height above the pivot point. In order to stretch itself, it obviously has to apply some force to the ground.

You can imagine this exactly as a human when doing a step. When leaning forward, you're maintaining approximately same height above the ground. The leg touching the ground is extending according to the velocity of your body.

My question is, what are the equations of motion for this thing? I want to be able to answer questions like: if I know where the body is right now, how fast it is moving and how high above the ground it is, where exactly will it be in one second?

Thanks for your help!

## Answers and Replies

What is supplying the energy for this stretching motion? Is it some internal motor? Or is this a constraint on the system with no internal power source?

It is an "internal motor". It actually is literally a leg, that's stretching by rotating it's joints. So it's "pushing" the ground, applying some force to it, and therefore keeping the upper body at the same height.

So the vertical force must be constant and equal to the weight of the pendulum bob. The horizontal force will follow based on trigonometry.
...that it stretches itself so that the height of it's end point is always at the same height above the pivot point...
...how high above the ground it is...
That one is easy. It's a given.

Now, Assuming the "leg" is massless...

The angle of the total force will be given by the angle between the pivot point and the object's current position. You know that the vertical force is exactly adequate to maintain the pendulum bob at its current position. That should give you an equation for the horizontal force in terms of the current horizontal position.

The angle of the total force will be given by the angle between the pivot point and the object's current position. You know that the vertical force is exactly adequate to maintain the pendulum bob at its current position. That should give you an equation for the horizontal force in terms of the current horizontal position.

Thanks for your answer! This is actually correct. I even managed to figure this out myself, that's why I wrote in my first post, that I thought I had solved it. But I figured, that in reality, this is unfortunately not enough.

Assuming that the leg is massless is probably fine, the problem is that the bob is not a point. It's a whole body with mass and inertia and it's not attached to the leg at it's center of mass, it's off-centered. So if I used (which I did) what you suggested, the upper body would end up rotating. Because it's off-centered.

Now, what if I want to keep it upright? I know, that if I want to keep it upright, I have to somewhat alter the motion of the whole thing, I just have no clear idea how.

I'd be really thankful if you or anyone else could expand this answer to answer these problems. Thanks for your help!

Well, I'm trying to find the answers to the problems mentioned in my previous post, but so far I've been unsuccessful. If anyone could find them sooner than me, I'd be thankful.

In the original version of the problem you had a massless leg that could extend arbitrarily and you had an object that needed to stay at a fixed height. That's one constrained parameter in the output and one free parameter in the input. Such a problem can potentially be solved.

Now you have a version where you want to constrain the object to remain upright at a fixed height. That's two constrained parameters in the output and still only one free parameter in the input. The problem is over-constrained and cannot be solved.

So now you need some additional mechanism. Maybe two parallel legs attached to two vertically aligned pivots joined to the bob at two vertically aligned pivot points with an identical separation. But if the legs are still massless, this would not alter the solution in any way.

In the original version of the problem you had a massless leg that could extend arbitrarily and you had an object that needed to stay at a fixed height. That's one constrained parameter in the output and one free parameter in the input. Such a problem can potentially be solved.

Now you have a version where you want to constrain the object to remain upright at a fixed height. That's two constrained parameters in the output and still only one free parameter in the input. The problem is over-constrained and cannot be solved.

So now you need some additional mechanism. Maybe two parallel legs attached to two vertically aligned pivots joined to the bob at two vertically aligned pivot points with an identical separation. But if the legs are still massless, this would not alter the solution in any way.

Thanks for your answer! But, you're indeed not correct this time. It's my fault that I didn't manage to clarify my problem very well, but actually, you don't have only one free parameter in the input. You have two of them. One of them is the "well known" extension of the leg, while the other one is the rotation of the joint between the leg and the body/object attached to it. It's possible to "press the ground" by rotating that joint. It is also possible to press the ground by extending the leg.

So actually, you have two free parameters in the input and two constrained parameters in the output, so I think this is solvable (haven't solved it yet though).

Sorry, for perhaps being a little annoying, but I really want to get this solved.

Thanks for your help!