# Controlled Recurrence of a Biped with Torso

## TL;DR

This paper extends a symbolic reachability method to control the stability of complex hybrid systems, demonstrated on a biped robot using PD control to ensure recurrence within a specific state region.

## Contribution

It introduces an extension of the reachability method for hybrid systems with guard conditions and resets, applied to biped robot stability control.

## Key findings

- Successfully controlled a biped robot with 6 states using the method.
- Isolated a state region ensuring recurrence after footsteps.
- Demonstrated stability control with PD controllers.

## Abstract

We have recently used a symbolic reachability method for controlling the stability of special hybrid systems called 'sampled switched systems'. We show here how the method can be extended in order to control the stability of more general hybrid systems with guard conditions and state resets. We illustrate the method through the example of a biped robot with 6 state variables, using a proportional-derivative (PD) controller. More specifically, we isolate a state region R such that, starting from a state located in R just after a footstep, the PD-control makes the robot state return to R at the end of the following footstep.