# Balance control using both ZMP and COM height variations: A convex boundedness approach

Stéphane Caron, Bastien Mallein.

## Abstract

Developments for 3D control of the center of mass (CoM) are currently located in two local minima: on the one hand, methods that allow CoM height variations but only work in the 2D sagittal plane; on the other hand, nonconvex centroidal models that are delicate to handle. This paper presents an alternative that controls the CoM in 3D by predictive control of a model with convex constraints. The key to this development is the notion of boundedness condition, which quantifies convexly the viability of CoM trajectories.

## BibTeX

@unpublished{caron2017balance,
title = {{Balance control using both ZMP and COM height variations: A convex boundedness approach}},
author = {Caron, St{\'e}phane and Mallein, Bastien},
url = {https://hal.archives-ouvertes.fr/hal-01590509},
note = {working paper or preprint},
year = {2017},
month = {September},
}


## Q & A

Thanks a lot to all the readers who took the time to ask the meaningful questions below. Feel free to write me directly if you have any other question related to this work.

It seems that dt/ds goes to infinity as s goes to zero. Does this mean that the integration accuracy of the discretization is also arbitrarily bad over [s_0, s_1]?

That's right. The user can control the accuracy of spatial integration through the choice of $$(s_1, \ldots, s_{N-1})$$, but time integration will by essence worsen close to $$s=0$$ (because the system only converges for $$t \to \infty$$). To say it the other way round, the best accuracy is obtained close to the current robot state $$(t=0)$$. This is a rather desired feature in the MPC framework, where precision is more important for immediate controls (the only part of the solution actually used for control) than at the terminal/asymptotic boundaries.

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