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Where these differ is how you apply the controller. $m$ and $a$ are mass and linear acceleration, respectively. Pid fajli v torque formula. So, in a way, a position controller, a velocity controller, and an acceleration (torque) controller are all different implementations of each other because each is the of the next - position is the integral of velocity, and velocity is the integral of acceleration. Torque is analogous to force for rotating systems, in that: $$ F = m a tau = I alpha $$ Where $ alpha$ is angular acceleration and $I$ is moment of inertia.

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