# Fast Trajectory Optimization for Legged Robots using Vertex-based ZMP Constraints

arXiv: https://arxiv.org/abs/1705.10313 · HTML: https://mlsanity.com/paper/1705.10313/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2017.

## Built on by (most cited first, 5 of 5)

- [Imitation and Adaptation Based on Consistency: A Quadruped Robot Imitates Animals from Videos Using Deep Reinforcement Learning](https://mlsanity.com/paper/2203.05973/) (2022)
- [Next Steps: Learning a Disentangled Gait Representation for Versatile Quadruped Locomotion](https://mlsanity.com/paper/2112.04809/) (2021)
- [VAE-Loco: Versatile Quadruped Locomotion by Learning a Disentangled Gait Representation](https://mlsanity.com/paper/2205.01179/) (2022)
- [Isaac Sim-to-Real: Reinforcement Learning based Locomotion for Quadrupeds](https://mlsanity.com/paper/2607.18135/) (2026)
- [ADP: Adversarial Dynamics Priors for Physically Grounded Humanoid Locomotion](https://mlsanity.com/paper/2607.03454/) (2026)

## Abstract (arXiv)

This paper combines the fast Zero-Moment-Point (ZMP) approaches that work well in practice with the broader range of capabilities of a Trajectory Optimization formulation, by optimizing over body motion, footholds and Center of Pressure simultaneously. We introduce a vertex-based representation of the support-area constraint, which can treat arbitrarily oriented point-, line-, and area-contacts uniformly. This generalization allows us to create motions such quadrupedal walking, trotting, bounding, pacing, combinations and transitions between these, limping, bipedal walking and push-recovery all with the same approach. This formulation constitutes a minimal representation of the physical laws (unilateral contact forces) and kinematic restrictions (range of motion) in legged locomotion, which allows us to generate various motion in less than a second. We demonstrate the feasibility of the generated motions on a real quadruped robot.