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Take a look at the following simple example robot arm:

Example robot arm

I want to know the torque required on that bottom motor to rotate the arm. Since it's not exactly rotating directly against gravity like the other joints I'm not sure how to analyze. Assume we know the mass of every part of the robot arm and the distances to the base.

For reference, I am planning on securing my base rotation motor's shaft through a ball bearing rotary table to the rest of the arm. I'm considering the torque required on that motor to properly pick the right component and make sure I get enough speed as well. So understanding how to analyze the forces would really help!

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    $\begingroup$ You just need to find the mass moment of inertia of the total arm around the spin axis. Since the arm moves, take worst case. It will be when the arm is fully extended in the horizontal. Use tables to find mass moment of inertia of each part. For beam around axes at its end, it is $m \frac{L^3}{3}$. Use parallel axis fo the rest. Lookup tables for these. For mass, it is just $m d^2$ where $d$ is distant from the spin axis. Once you find total $I$, then torque is just $I$ times the angular acceleration. You decide what maximum acceleration you need, and this gives you maximum torque needed. $\endgroup$ – Nasser Oct 14 '16 at 4:56
  • $\begingroup$ OK I can do that. My issue is that I need to figure out friction more than anything. It seems that since T = I*acc, even with very small torque T and high I the motor can still achieve some acc according to the equation. But I think it first has to overcome static friction before this torque is applied to rotating the arm, is this conceptually correct? $\endgroup$ – JDS Oct 14 '16 at 19:14
  • $\begingroup$ I did a similar simulation using SolidWorks: youtube.com/watch?v=crJXUlzJ918 $\endgroup$ – LCarvalho Feb 19 '18 at 19:58
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You need enough torque to overcome friction, and to accelerate the load.

If you know the friction torque ($\tau_f$), and the mass moment of inertia along the motor axis ($I$), then the minimum motor torque required is $$ \tau = \tau_f + I\alpha $$ where $\alpha$ is the required rotational acceleration.

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  • $\begingroup$ Cool and how would one approximate the frictional torque? $\endgroup$ – JDS Oct 14 '16 at 19:02
  • $\begingroup$ The motor sometimes drives a gearbox and everything is held together by a bearing. The motor, gearbox, and bearing friction should be in their respective data sheets. $\endgroup$ – Christo Oct 17 '16 at 7:08
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If you only use Newton's second law, you'll get pretty close in your analysis.

$\sum F = m a$

$\sum M = I \alpha$

Using only the first equation and the idea that a torque is a force at a distance ($\tau = F \times l$), and point masses for the payload and link masses, you will get a quick first estimate.

So choose what acceleration is acceptable to you and the length of your arm.

$\tau = m a l $

That will give you a starting point.

As you develop your design, you'll want to calculate the Inertias of each link (and the payload), as well as estimate friction from your transmissions.

The torque from the motor at the joint (on the other side of the transmission) will be a function of the efficiency of the transmission.

$\tau_{a}:$ actuator torque, $\tau_m: $ motor torque, $\eta_g:$ gears efficiency, $R_g:$ gears ratio

$\tau_a = \tau_m \eta_g R_g$

The friction in your support bearings will depend on your design.

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