# Stability of Surface Contacts for Humanoid Robots: Closed-Form Formulae of the Contact Wrench Cone for Rectangular Support Areas

arXiv: https://arxiv.org/abs/1501.04719 · HTML: https://mlsanity.com/paper/1501.04719/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jan 2015.

## Built on by (most cited first, 6 of 6)

- [Agile Maneuvers in Legged Robots: a Predictive Control Approach](https://mlsanity.com/paper/2203.07554/) (2022)
- [Hand-Object Interaction Controller (HOIC): Deep Reinforcement Learning for Reconstructing Interactions with Physics](https://mlsanity.com/paper/2405.02676/) (2024)
- [Accelerating and Scaling MPC-Guided Reinforcement Learning for Humanoid Locomotion and Manipulation](https://mlsanity.com/paper/2606.05687/) (2026)
- [Toward Reliable Sim-to-Real Predictability for MoE-based Robust Quadrupedal Locomotion](https://mlsanity.com/paper/2602.00678/) (2026)
- [ODYN: An All-Shifted Non-Interior-Point Method for Quadratic Programming in Robotics and AI](https://mlsanity.com/paper/2602.16005/) (2026)
- [Autotuning Bipedal Locomotion MPC with GRFM-Net for Efficient Sim-to-Real Transfer](https://mlsanity.com/paper/2409.15710/) (2024)

## Abstract (arXiv)

Humanoid robots locomote by making and breaking contacts with their environment. A crucial problem is therefore to find precise criteria for a given contact to remain stable or to break. For rigid surface contacts, the most general criterion is the Contact Wrench Condition (CWC). To check whether a motion satisfies the CWC, existing approaches take into account a large number of individual contact forces (for instance, one at each vertex of the support polygon), which is computationally costly and prevents the use of efficient inverse-dynamics methods. Here we argue that the CWC can be explicitly computed without reference to individual contact forces, and give closed-form formulae in the case of rectangular surfaces -- which is of practical importance. It turns out that these formulae simply and naturally express three conditions: (i) Coulomb friction on the resultant force, (ii) ZMP inside the support area, and (iii) bounds on the yaw torque. Conditions (i) and (ii) are already known, but condition (iii) is, to the best of our knowledge, novel. It is also of particular interest for biped locomotion, where undesired foot yaw rotations are a known issue. We also show that our formulae yield simpler and faster computations than existing approaches for humanoid motions in single support, and demonstrate their consistency in the OpenHRP simulator.