Joint position actuator question

[btaba]

Hi! I'm trying to understand the equation for qfrc_actuator for hinge/slide joints for a position joint actuator. I couldn't find a similar question, apologies if this was already asked somewhere.

Trying to write out the equation from the code:

gain is:
https://github.com/deepmind/mujoco/blob/fd1070cf039a586e0836d43baf2ff5bf4bd887b4/src/engine/engine_forward.c#L210

bias is:
https://github.com/deepmind/mujoco/blob/fd1070cf039a586e0836d43baf2ff5bf4bd887b4/src/engine/engine_forward.c#L251

actuator_length is:
https://github.com/deepmind/mujoco/blob/fd1070cf039a586e0836d43baf2ff5bf4bd887b4/src/engine/engine_core_smooth.c#L651

and $force = gain*ctrl + bias$ from here and here.

Now for a position actuator, gainprm = [kp, 0, 0] and biasprm = [0, -kp, 0] from here.

plugging these all in we get:
$force = k_p * ctrl - k_p * q * gear$

To get qfrc_actuator
moment is:
https://github.com/deepmind/mujoco/blob/fd1070cf039a586e0836d43baf2ff5bf4bd887b4/src/engine/engine_core_smooth.c#L652

qfrc_actuator is:
https://github.com/deepmind/mujoco/blob/fd1070cf039a586e0836d43baf2ff5bf4bd887b4/src/engine/engine_forward.c#L300

so $qfrc = gear * force = gear * k_p * ctrl - gear^2 * k_p * q$

What's the intuition behind the $gear^2$ term (unless I made some mistake)? I would expect the equation to look something like $gear * k_p * (ctrl - q)$

Thanks for taking a look!!!

[Balint-H]

It's to convert between ctrl's scale and the position's scale. ctrl defines the target in actuator length space, which is different from the joint position by a factor of gear. So your final expression could be: $gear * k_p * (ctrl-gear*q)$. This is same as having the $gear^2$ term for q.

[yuvaltassa]

This is a bit like the gear^2 term of reflected inertia.

The intuition is that you feel the effects of the gear twice, once on the way out, in the scaling of the angle and once again on the way in, when applying the force.

[btaba]

Thanks for the insight here!

[btaba]

(found this link to help understand reflected inertia https://physics.stackexchange.com/questions/73531/is-squared-motor-gearbox-ratio-proportional-to-inertia-ratio)

To help clarify a bit:

given $\tau_{in} = {gear} * \tau_{out}$ and $q_{out} = {gear} * q_{in}$

you are setting $\tau_{out} = {ctrl} - q_{out}$, which gives us $\tau_{in} = {gear} * ( {ctrl} - {gear} * q_{in} )$