Deriving commanded ZMP from DCM feedback

[wyqsnddd]

The equation above eq. (9) of the paper Stair Climbing Stabilization of the HRP-4 Humanoid Robot using Whole-body Admittance Control looks a bit confusing.

We need to assume that \xi - \xi^* = \xi^m - \xi^* to derive the equation, see details in the figure:

[stephane-caron]

That's right, basically in feedback control you assume that your state estimate is correct. Here the state is the DCM, so this assumption amounts to \xi = \xi^m.

In practice, state estimation errors are reflected into feedback errors. That's one reason why we have for instance steady-state errors. We add the integral term to compensate them as proposed in Balance control based on Capture Point error compensation for biped walking on uneven terrain.