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So we all know (and love) the typical dynamic equation of a robotic arm:

$$ M(q) \ddot{q}+C(q,\dot{q})\dot{q}+G(q)=\tau $$

Where M is the mass matrix, G is gravity and C is Coriolis and Centrifugal forces.

Coriolis and Centrifugal forces are pseudo forces that are experienced only for the frame that is attached to the body that is rotating. But my understanding is that all calculations happen relative to the base frame, which is static. The need for Coriolis and Centrifugal forces would imply that the forces are actually expressed in the frame that is attached to each link.

To put my question simply, why do we need the coriolis and centrifugal forces in the dynamic equation of a robotic arm (since everything is done relative to the initial/base frame)?

(p.s. feel free to use a single or double pendulum robotic arm example if you would prefer).

Metalzero2
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But my understanding is that all calculations happen relative to the base frame

This is incorrect. Take a look at this two-link arm manipulator. The point ($x_2,y_2$) is obviously expressed in a moving frame $y_1,x_1$. Another way to look at the problem is to use Lagrangian approach which free you from constraint forces.

enter image description here

CroCo
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  • Thanks for you answer. I think I get what you mean. But what about the case where the robot had only one link (so the above example with l1 only)? Would you not need to use coriolis and centrifugal forces? – Metalzero2 Sep 21 '19 at 11:45
  • What do you think? – CroCo Sep 21 '19 at 16:36
  • Based on the fact that in your answer you referred only to the second link because its coordinates are expressed based on the rotating frame (x1-y1), I would say no, those forces are not needed, since frame (x0-y0) is not moving. – Metalzero2 Sep 21 '19 at 18:19
  • The lagrange does not free one from constraint forces...do you mean you can use it to reduce the system and ‘hide them’ ? – DrMrstheMonarch Sep 21 '19 at 18:57
  • @DimitrisPantelis true – CroCo Sep 23 '19 at 21:52
  • @CroCo oh I see. I always thought that even if you have one mass rotating, you still need to write down Coriolis and Centrifugal forces. But that is the case only when you have a second mass that is rotating relative to another rotating frame. Thanks for the help. – Metalzero2 Sep 24 '19 at 14:40
  • @DimitrisPantelis centrifugal is not the coriolis force. – CroCo Sep 24 '19 at 15:40
  • @morbo I mean with Largrangian, we don’t know an inertial frame. – CroCo Sep 24 '19 at 15:44
  • @CroCo I don't follow you. I know that centrifugal forces and Coriolis forces are different but they both appear only in rotating frames. That is why I mentioned them together, if that was what you were pointing out. – Metalzero2 Sep 25 '19 at 11:32