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

## Abstract

Developments for 3D control of the center of mass (CoM) of biped robots 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 direct transcriptions of centroidal dynamics that are delicate to handle. This paper presents an alternative that controls the CoM in 3D via an indirect transcription that is both low-dimensional and solvable fast enough for real-time control. The key to this development is the notion of boundedness condition, which quantifies the capturability of 3D CoM trajectories.

## Video

## Content

Paper | |

Slides: presentation given at NTU on 14 May 2018 | |

Posters: first part and second part | |

Source code |

## BibTeX

```
@inproceedings{caron2018icra,
title = {{Balance control using both ZMP and COM height variations: A convex boundedness approach}},
author = {Caron, St{\'e}phane and Mallein, Bastien},
booktitle = {Proceedings of the 2018 IEEE-RAS International Conference on Robotics and Automation},
url = {https://hal.archives-ouvertes.fr/hal-01590509},
year = {2018},
month = may,
organization = {IEEE},
}
```

## 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
[s0, s1]?**

That's right. The user can control the accuracy ofspatialintegration through the choice of \((s_1, \ldots, s_{N-1})\), buttimeintegration 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 for model predictive control, where precision is most important for the immediate output (the only part of the solution actually used for control).