# Symbolic Learning of Interpretable Reduced-Order Models for Jumping Quadruped Robots

arXiv: https://arxiv.org/abs/2508.06538 · HTML: https://mlsanity.com/paper/2508.06538/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 4 Aug 2025.

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

- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [The dynamics of representation learning in shallow, non-linear autoencoders](https://mlsanity.com/paper/2201.02115/) (2022)
- [Scientific Exploration of Challenging Planetary Analog Environments with a Team of Legged Robots](https://mlsanity.com/paper/2307.10079/) (2023)
- [Curriculum-Based Reinforcement Learning for Quadrupedal Jumping: A Reference-free Design](https://mlsanity.com/paper/2401.16337/) (2024)
- [Physics-informed Neural Networks to Model and Control Robots: a Theoretical and Experimental Investigation](https://mlsanity.com/paper/2305.05375/) (2023)
- [Nonlinear Systems Identification Using Deep Dynamic Neural Networks](https://mlsanity.com/paper/1610.01439/) (2016)
- [Contact-timing and Trajectory Optimization for 3D Jumping on Quadruped Robots](https://mlsanity.com/paper/2110.06764/) (2021)
- [Adaptive Force-Based Control of Dynamic Legged Locomotion over Uneven Terrain](https://mlsanity.com/paper/2307.04030/) (2023)
- [Input-to-State Stable Coupled Oscillator Networks for Closed-form Model-based Control in Latent Space](https://mlsanity.com/paper/2409.08439/) (2024)
- [Legged Robots for Object Manipulation: A Review](https://mlsanity.com/paper/2303.16865/) (2023)
- [Versatile, Robust, and Explosive Locomotion with Rigid and Articulated Compliant Quadrupeds](https://mlsanity.com/paper/2504.12854/) (2025)

## Abstract (arXiv)

Reduced-order models are central to motion planning and control of quadruped robots, yet existing templates are often hand-crafted for a specific locomotion modality. This motivates the need for automatic methods that extract task-specific, interpretable low-dimensional dynamics directly from data. We propose a methodology that combines a linear autoencoder with symbolic regression to derive such models. The linear autoencoder provides a consistent latent embedding for configurations, velocities, accelerations, and inputs, enabling the sparse identification of nonlinear dynamics (SINDy) to operate in a compact, physics-aligned space. A multi-phase, hybrid-aware training scheme ensures coherent latent coordinates across contact transitions. We focus our validation on quadruped jumping-a representative, challenging, yet contained scenario in which a principled template model is especially valuable. The resulting symbolic dynamics outperform the state-of-the-art handcrafted actuated spring-loaded inverted pendulum (aSLIP) baseline in simulation and hardware across multiple robots and jumping modalities.