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

## TL;DR

This paper presents a fast, unified trajectory optimization method for legged robots that uses a vertex-based ZMP support-area representation, enabling rapid generation of diverse gaits and transitions validated on a real quadruped.

## Contribution

It introduces a vertex-based support-area constraint that handles arbitrary contact orientations, enabling quick, versatile motion planning for legged robots.

## Key findings

- Generated motions in less than a second
- Successfully demonstrated on a real quadruped robot
- Supported diverse gaits and transitions

## Abstract

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.