# Why do I need an additional moment of inertia term in the cart-pole dynamics equation?

I'm trying to understand the dynamic equations for the cart pole system according to this control tutorial from the University of Michigan, where the angular acceleration equation can be written as

$(I+ml^2)\ddot{\theta}+mglsin(\theta)+ml\ddot{x}cos(\theta)=0$

I'm having trouble understanding what forces are represented separately by the terms $ml^2\ddot{\theta}$ and $I\ddot{\theta}$. Both seem to represent a moment force exerted by the mass of the pole, so they almost seem to represent exactly the same thing. My intuition tells me that I really only need the term $ml^2\ddot{\theta}$, but the fact that there is an additional $I\ddot{\theta}$ term suggests that my intuition is not accounting for some other force present in the system.

Any help to clarify and distinguish the meaning behind these two terms would be greately appreciated!

## 1 Answer

Look up an explanation of the parallel axis theorem for the mass moment of inertia. The two terms do represent similar things, but the axis of rotation is different. The term $I \ddot\theta$ comes from summing the moments about the centroid of the link, whereas the term $m l^2 \ddot \theta$ comes from summing the horizontal forces from the free body diagram.

• I looked it up... I get the impression that $I$ is the moment of inertia of the pendulum if it were rotated only at its center of mass. Then, the other additional moment $ml^2$ comes from the fact that the pole is rotated at one of its ends (not at its center of mass). Is this correct?
– Paul
Mar 4, 2017 at 20:25
• You got it. If you read the text of the paper you reference very carefully, you'll see that the derivation is based on that. Mar 4, 2017 at 20:27
• On Stack Exchange we are are looking for comprehensive answers that provide some explanation & context. Very short answers cannot do this, so please edit your answer to explain why it is right, ideally with citations. Answers that don't include explanations may be removed. Also it is better to edit your answer to add clarification, rather than adding comments. Comments are for helping to improve questions and answers, and are distracting, so we try to keep them to a minimum. If all of the information needed to answer the question is contained within it, the comments can be tidied up (deleted). Mar 7, 2017 at 14:11