Since the very begining of parallel simulation for locomotion, position-only PD control seems to be the only option.
However, due to the damping term, the target joint position must lead the actual position by a certain angle to maintain the
joint velocity, deviating from the definition of "target" joint position. To address this issue,
we introduce a velocity feedforward term where the target velocity is computed from the difference betweenlast two target joint positions.
To prevent overshoot, the velocity gain is set as the damping gain scaled by a feedforward ratio smaller than 1.
We derive a theoretical upper bound in our paper, which is further validated through experiments.
We also provide a demonstration of how velocity feedforward term affects the control behavior:
The above demonstration is conducted using PD gains corresponding to a natural frequency of 10 rad/s and a damping ratio of 1.0.
In our paper, we further show that, for low-frequency sinusoidal inputs, the induced control delay remains constant.
Since any control signal can be decomposed into a sum of sinusoidal components, this delay generalizes to arbitrary trajectories
and can be interpreted as an effective latency. Consequently, it manifests as a part of the end-to-end delay scaled by a factor of approximately 0.6,
which is an empirical observation from our real-world experiments.