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# how we can neglect inertia of most of the robot itself when modeling robot systems [cool things]

Updated: Mar 8, 2020

So when it comes to controls in robotics such as PID Control or whatever you happen to be using. In these systems it's very important to create a model of the system so we can find things like the system's poles which tell us the response of the system under certain conditions. That means modeling physical aspects of the system like the motor, the motor's inertia, the arm connected to it, the arm's inertia and etc.

So in the case that we have a motor driving a gearbox of ratio "r," and here is a robotic arm attached at the end with inertia I_a, and the motor's rotor inertia is I_m... while we could go through the derivation it's likely just better to state this system's total inertia can be given by the following formula:

I_t = r^2*I_m+I_a

What this means is that when a gearbox is added to the end of a motor, the inertia of the motor/gearbox system increases by a property of the ratio squared. I_effective = r^2*I_m. This, of course, gets added to the arm inertia for total system inertia that must be driven by the PID control on the back-end. However, look at the formula for I_t. If "r" is very big, what happens? Let's say "r" is 100:1 which is not unreasonable for harmonic drives. Then the new motor inertia is 10000x what it originally was!

Now let's say we make the arm super light, and the inertia is very small. This means there is a point where if the gear ratio is high enough, the arm is small enough (i.e. we've designed the system well given the constrains of the gear ratio), that we can just neglect effects of the inertia of the arm in our calculations.

T_inputted (by motor) = I_t*alpha+B*omega

(Torque is equal to I-alpha+damping torque).

## extending this further... what does that mean for good robot design:

So what this also means, if we look at the leg of the Cheetah 3 robot, is if we are designing a robot arm or leg then we should place the motors as close to the "origin" or "base" of the system as possible. In the case of Cheetah 3, all the actuators for both the knee and the shoulder are packaged at the very top, there is no motor on the knee joint itself - it's driven by a chain up to the actuator on the top of the leg. What this means is the leg itself (not including the motors) is very light and has very low inertia. This type of design gives you extremely controllable, low-inertia robot mechanisms and doesn't sacrifice high amounts of torque going to the places that need it. The belt/chain system they used transfers all 230Nm of possible torque to that knee joint while still being a remote actuator.

Learning from the best in the field :)

NOTE: The diagrams and pictures in this post are NOT my own, they're taken from google images, full credit to people who created them.

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