# Minimizing Energy Consumption Leads to the Emergence of Gaits in Legged Robots

arXiv: https://arxiv.org/abs/2111.01674 · HTML: https://mlsanity.com/paper/2111.01674/

## Key facts

- Cited by 76 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 42 references that are in the map.
- First posted on arXiv 25 Oct 2021.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [RMA: Rapid Motor Adaptation for Legged Robots](https://mlsanity.com/paper/2107.04034/) (2021)
- [Learning agile and dynamic motor skills for legged robots](https://mlsanity.com/paper/1901.08652/) (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)
- [Preparing for the Unknown: Learning a Universal Policy with Online System Identification](https://mlsanity.com/paper/1702.02453/) (2017)
- [Learning Symmetric and Low-energy Locomotion](https://mlsanity.com/paper/1801.08093/) (2018)
- [Sim-to-Real Transfer for Biped Locomotion](https://mlsanity.com/paper/1903.01390/) (2019)
- [Fast and Efficient Locomotion via Learned Gait Transitions](https://mlsanity.com/paper/2104.04644/) (2021)
- [Rapidly Adaptable Legged Robots via Evolutionary Meta-Learning](https://mlsanity.com/paper/2003.01239/) (2020)
- [Learning a Contact-Adaptive Controller for Robust, Efficient Legged Locomotion](https://mlsanity.com/paper/2009.10019/) (2020)

## Built on by (most cited first, 30 of 76)

- [Deep Reinforcement Learning for Robotics: A Survey of Real-World Successes](https://mlsanity.com/paper/2408.03539/) (2024)
- [Rapid Locomotion via Reinforcement Learning](https://mlsanity.com/paper/2205.02824/) (2022)
- [Extreme Parkour with Legged Robots](https://mlsanity.com/paper/2309.14341/) (2023)
- [Legged Locomotion in Challenging Terrains using Egocentric Vision](https://mlsanity.com/paper/2211.07638/) (2022)
- [Robot Parkour Learning](https://mlsanity.com/paper/2309.05665/) (2023)
- [Open-TeleVision: Teleoperation with Immersive Active Visual Feedback](https://mlsanity.com/paper/2407.01512/) (2024)
- [Expressive Whole-Body Control for Humanoid Robots](https://mlsanity.com/paper/2402.16796/) (2024)
- [Walk These Ways: Tuning Robot Control for Generalization with Multiplicity of Behavior](https://mlsanity.com/paper/2212.03238/) (2022)
- [Deep Whole-Body Control: Learning a Unified Policy for Manipulation and Locomotion](https://mlsanity.com/paper/2210.10044/) (2022)
- [Reinforcement Learning for Versatile, Dynamic, and Robust Bipedal Locomotion Control](https://mlsanity.com/paper/2401.16889/) (2024)
- [A Walk in the Park: Learning to Walk in 20 Minutes With Model-Free Reinforcement Learning](https://mlsanity.com/paper/2208.07860/) (2022)
- [Adapting Rapid Motor Adaptation for Bipedal Robots](https://mlsanity.com/paper/2205.15299/) (2022)
- [Coupling Vision and Proprioception for Navigation of Legged Robots](https://mlsanity.com/paper/2112.02094/) (2021)
- [Mobile-TeleVision: Predictive Motion Priors for Humanoid Whole-Body Control](https://mlsanity.com/paper/2412.07773/) (2024)
- [AMO: Adaptive Motion Optimization for Hyper-Dexterous Humanoid Whole-Body Control](https://mlsanity.com/paper/2505.03738/) (2025)
- [Retrospectives on the Embodied AI Workshop](https://mlsanity.com/paper/2210.06849/) (2022)
- [BeamDojo: Learning Agile Humanoid Locomotion on Sparse Footholds](https://mlsanity.com/paper/2502.10363/) (2025)
- [Learning to Jump from Pixels](https://mlsanity.com/paper/2110.15344/) (2021)
- [Legs as Manipulator: Pushing Quadrupedal Agility Beyond Locomotion](https://mlsanity.com/paper/2303.11330/) (2023)
- [Hybrid Internal Model: Learning Agile Legged Locomotion with Simulated Robot Response](https://mlsanity.com/paper/2312.11460/) (2023)
- [Robust and Versatile Bipedal Jumping Control through Reinforcement Learning](https://mlsanity.com/paper/2302.09450/) (2023)
- [CPG-RL: Learning Central Pattern Generators for Quadruped Locomotion](https://mlsanity.com/paper/2211.00458/) (2022)
- [Learning and Adapting Agile Locomotion Skills by Transferring Experience](https://mlsanity.com/paper/2304.09834/) (2023)
- [Learning Smooth Humanoid Locomotion through Lipschitz-Constrained Policies](https://mlsanity.com/paper/2410.11825/) (2024)
- [Learning Torque Control for Quadrupedal Locomotion](https://mlsanity.com/paper/2203.05194/) (2022)
- [Learning Getting-Up Policies for Real-World Humanoid Robots](https://mlsanity.com/paper/2502.12152/) (2025)
- [Learning Visual Locomotion with Cross-Modal Supervision](https://mlsanity.com/paper/2211.03785/) (2022)
- [Grow Your Limits: Continuous Improvement with Real-World RL for Robotic Locomotion](https://mlsanity.com/paper/2310.17634/) (2023)
- [ViNL: Visual Navigation and Locomotion Over Obstacles](https://mlsanity.com/paper/2210.14791/) (2022)
- [CaT: Constraints as Terminations for Legged Locomotion Reinforcement Learning](https://mlsanity.com/paper/2403.18765/) (2024)

## Abstract (arXiv)

Legged locomotion is commonly studied and expressed as a discrete set of gait patterns, like walk, trot, gallop, which are usually treated as given and pre-programmed in legged robots for efficient locomotion at different speeds. However, fixing a set of pre-programmed gaits limits the generality of locomotion. Recent animal motor studies show that these conventional gaits are only prevalent in ideal flat terrain conditions while real-world locomotion is unstructured and more like bouts of intermittent steps. What principles could lead to both structured and unstructured patterns across mammals and how to synthesize them in robots? In this work, we take an analysis-by-synthesis approach and learn to move by minimizing mechanical energy. We demonstrate that learning to minimize energy consumption plays a key role in the emergence of natural locomotion gaits at different speeds in real quadruped robots. The emergent gaits are structured in ideal terrains and look similar to that of horses and sheep. The same approach leads to unstructured gaits in rough terrains which is consistent with the findings in animal motor control. We validate our hypothesis in both simulation and real hardware across natural terrains. Videos at https://energy-locomotion.github.io