# Imitation Learning from MPC for Quadrupedal Multi-Gait Control

arXiv: https://arxiv.org/abs/2103.14331 · HTML: https://mlsanity.com/paper/2103.14331/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Placed in Imitation learning methods, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 12 of its 37 references that are in the map.
- First posted on arXiv 26 Mar 2021.

## Placed in

- Offline RL and imitation learning from data / [Imitation learning methods](https://mlsanity.com/offlinerl/imitation-learning-methods/)

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning agile and dynamic motor skills for legged robots](https://mlsanity.com/paper/1901.08652/) (2019)
- [An Algorithmic Perspective on Imitation Learning](https://mlsanity.com/paper/1811.06711/) (2018)
- [Learning by Cheating](https://mlsanity.com/paper/1912.12294/) (2019)
- [Sim-to-Real: Learning Agile Locomotion For Quadruped Robots](https://mlsanity.com/paper/1804.10332/) (2018)
- [Learning Agile Robotic Locomotion Skills by Imitating Animals](https://mlsanity.com/paper/2004.00784/) (2020)
- [Learning to Walk via Deep Reinforcement Learning](https://mlsanity.com/paper/1812.11103/) (2018)
- [Policies Modulating Trajectory Generators](https://mlsanity.com/paper/1910.02812/) (2019)
- [MPC-Net: A First Principles Guided Policy Search](https://mlsanity.com/paper/1909.05197/) (2019)
- [PLATO: Policy Learning using Adaptive Trajectory Optimization](https://mlsanity.com/paper/1603.00622/) (2016)
- [Feedback MPC for Torque-Controlled Legged Robots](https://mlsanity.com/paper/1905.06144/) (2019)
- [An Efficient Optimal Planning and Control Framework For Quadrupedal Locomotion](https://mlsanity.com/paper/1609.09861/) (2016)

## Built on by (most cited first, 18 of 18)

- [Advanced Skills by Learning Locomotion and Local Navigation End-to-End](https://mlsanity.com/paper/2209.12827/) (2022)
- [DiffuseLoco: Real-Time Legged Locomotion Control with Diffusion from Offline Datasets](https://mlsanity.com/paper/2404.19264/) (2024)
- [Learning Free Gait Transition for Quadruped Robots via Phase-Guided Controller](https://mlsanity.com/paper/2201.00206/) (2022)
- [Synthesis of Model Predictive Control and Reinforcement Learning: Survey and Classification](https://mlsanity.com/paper/2502.02133/) (2025)
- [Demonstration-Efficient Guided Policy Search via Imitation of Robust Tube MPC](https://mlsanity.com/paper/2109.09910/) (2021)
- [Adversarial Imitation Learning from Visual Observations using Latent Information](https://mlsanity.com/paper/2309.17371/) (2023)
- [Approximate non-linear model predictive control with safety-augmented neural networks](https://mlsanity.com/paper/2304.09575/) (2023)
- [Video2Reward: Generating Reward Function from Videos for Legged Robot Behavior Learning](https://mlsanity.com/paper/2412.05515/) (2024)
- [Learning Perceptual Locomotion on Uneven Terrains using Sparse Visual Observations](https://mlsanity.com/paper/2109.14026/) (2021)
- [Imitation Learning from Nonlinear MPC via the Exact Q-Loss and its Gauss-Newton Approximation](https://mlsanity.com/paper/2304.01782/) (2023)
- [DeepCPG Policies for Robot Locomotion](https://mlsanity.com/paper/2302.13191/) (2023)
- [ES-Parkour: Advanced Robot Parkour with Bio-inspired Event Camera and Spiking Neural Network](https://mlsanity.com/paper/2503.09985/) (2025)
- [GRoQ-LoCO: Generalist and Robot-agnostic Quadruped Locomotion Control using Offline Datasets](https://mlsanity.com/paper/2505.10973/) (2025)
- [Efficient Deep Learning of Robust, Adaptive Policies using Tube MPC-Guided Data Augmentation](https://mlsanity.com/paper/2303.15688/) (2023)
- [Bridging the Sim-to-real Gap: A Control Framework for Imitation Learning of Model Predictive Control](https://mlsanity.com/paper/2503.19228/) (2025)
- [Adaptive MPC-based quadrupedal robot control under periodic disturbances](https://mlsanity.com/paper/2505.12361/) (2025)
- [Unlocking the Potential of Soft Actor-Critic for Imitation Learning](https://mlsanity.com/paper/2509.24539/) (2025)
- [Bridging RL and MPC for mixed-integer optimal control with application to Formula 1 race strategies](https://mlsanity.com/paper/2604.00826/) (2026)

## Abstract (arXiv)

We present a learning algorithm for training a single policy that imitates multiple gaits of a walking robot. To achieve this, we use and extend MPC-Net, which is an Imitation Learning approach guided by Model Predictive Control (MPC). The strategy of MPC-Net differs from many other approaches since its objective is to minimize the control Hamiltonian, which derives from the principle of optimality. To represent the policies, we employ a mixture-of-experts network (MEN) and observe that the performance of a policy improves if each expert of a MEN specializes in controlling exactly one mode of a hybrid system, such as a walking robot. We introduce new loss functions for single- and multi-gait policies to achieve this kind of expert selection behavior. Moreover, we benchmark our algorithm against Behavioral Cloning and the original MPC implementation on various rough terrain scenarios. We validate our approach on hardware and show that a single learned policy can replace its teacher to control multiple gaits.