  The impressive capabilities of Large Language Models (LLMs) have led to
various efforts to enable robots to be controlled through natural language
instructions, opening exciting possibilities for human-robot interaction The
goal is for the motor-control task to be performed accurately, efficiently and
safely while also enjoying the flexibility imparted by LLMs to specify and
adjust the task through natural language. In this work, we demonstrate how a
careful layering of an LLM in combination with a Model Predictive Control (MPC)
formulation allows for accurate and flexible robotic control via natural
language while taking into consideration safety constraints. In particular, we
rely on the LLM to effectively frame constraints and objective functions as
mathematical expressions, which are later used in the motor-control module via
MPC. The transparency of the optimization formulation allows for
interpretability of the task and enables adjustments through human feedback. We
demonstrate the validity of our method through extensive experiments on
long-horizon reasoning, contact-rich, and multi-object interaction tasks. Our
evaluations show that NARRATE outperforms current existing methods on these
benchmarks and effectively transfers to the real world on two different
embodiments. Videos, Code and Prompts at narrate-mpc.github.io
