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

Stéphane Caron and Bastien Mallein. ICRA 2018, Brisbane, Australia, May 2018.


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.



  title = {{Balance control using both ZMP and COM height variations: A convex boundedness approach}},
  author = {Caron, St{\'e}phane and Mallein, Bastien},
  booktitle = {IEEE International Conference on Robotics and Automation},
  url = {},
  year = {2018},
  month = may,


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 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 for model predictive control, where precision is most important for the immediate output (the only part of the solution actually used for control).
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