# Contact-Implicit Optimization of Locomotion Trajectories for a Quadrupedal Microrobot

## TL;DR

This paper introduces a systematic framework for modeling, planning, and controlling quadrupedal microrobots, enabling the generation of feasible dynamic locomotion trajectories that are experimentally trackable, including high-speed gaits and vertical jumps.

## Contribution

It adapts a variational contact-implicit trajectory optimization method for a microrobot, demonstrating the first end-to-end planning and tracking of dynamic locomotion on such small robots.

## Key findings

- Achieved a maximum velocity of 10.87 mm/cycle, 15% faster than previous measurements.
- Successfully planned and executed a vertical jump of 9.96 mm, 78% of the robot's height.
- Demonstrated feasible whole-body locomotion plans on various surfaces and stride frequencies.

## Abstract

Planning locomotion trajectories for legged microrobots is challenging because of their complex morphology, high frequency passive dynamics, and discontinuous contact interactions with their environment. Consequently, such research is often driven by time-consuming experimental methods. As an alternative, we present a framework for systematically modeling, planning, and controlling legged microrobots. We develop a three-dimensional dynamic model of a 1.5 gram quadrupedal microrobot with complexity (e.g., number of degrees of freedom) similar to larger-scale legged robots. We then adapt a recently developed variational contact-implicit trajectory optimization method to generate feasible whole-body locomotion plans for this microrobot, and we demonstrate that these plans can be tracked with simple joint-space controllers. We plan and execute periodic gaits at multiple stride frequencies and on various surfaces. These gaits achieve high per-cycle velocities, including a maximum of 10.87 mm/cycle, which is 15% faster than previously measured velocities for this microrobot. Furthermore, we plan and execute a vertical jump of 9.96 mm, which is 78% of the microrobot's center-of-mass height. To the best of our knowledge, this is the first end-to-end demonstration of planning and tracking whole-body dynamic locomotion on a millimeter-scale legged microrobot.