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

TL;DR

This work introduces a physics-aligned, interpretable reduced-order modeling framework for legged locomotion by marrying a linear autoencoder with sparse symbolic regression (SINDy) in a latent space. A multi-phase training strategy ensures coherent latent coordinates across contact transitions, enabling accurate latent dynamics that reconstruct to full-state motion. The approach yields compact, interpretable equations that capture gravity-like effects, damping, and actuation influence, and it outperforms a handcrafted aSLIP baseline in both simulation and hardware for quadruped jumping. This method promises efficient, interpretable planning and control primitives for dynamic, hybrid locomotion in real-world robots.

Abstract

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.

Figures (12)

• Figure 1: Quadruped robot with labeled joints and body frame.
• Figure 2: Methodology overview: Symbolic latent-space dynamics $\ddot{\bm{\xi}} = \bm{\Theta}(\bm{\xi}, \dot{\bm{\xi}}, \bm{\nu}) \, \bm{\Xi}$ are learned to predict the evolution of the full-state system from $(\mathbf{q}, \dot{\mathbf{q}})$.
• Figure 3: Flowchart of the pipeline. (a) The linear autoencoder is first trained on the contact phase dataset, minimizing the reconstruction loss, $\mathcal{L}_\mathrm{rec}$, depicted by the black line. Subsequently, by fixing the encoder, the decoder is fine-tuned on a dataset that includes all motion phases (see the the blue line). (b) In the third step, we train, for each motion phase separately, a symbolic dynamical model $\bm{\ddot{\xi}}_\text{pred}$ in latent space. Here, we employ the SINDy approach to regress a sparse matrix $\bm{\Xi}$ that selects and scales basis functions from the library $\bm{\Theta}(\bm{\xi}, \dot{\bm{\xi}}, \bm{\nu})$. The loss components $\mathcal{L}_{\text{SINDy}}$ and $\mathcal{L}_{\text{reg}}$ ensure accurate dynamics prediction and sparsity of the coefficients, respectively. The SINDy loss $\mathcal{L}_{\text{SINDy}}$ is calculated based on the predicted reduced-order model accelerations, $\ddot{\bm{\xi}}_\mathrm{pred}$, and the corresponding decoded configuration-space accelerations, $\ddot{\mathbf{q}}_\mathrm{pred}$.
• Figure 4: Sequential images of quadruped jumps: (a) A simulated Go1 robot performs a pronking jump. (b) A simulated A1 executes a froggy jump. (c) A real Go1 performs a pronking jump.
• Figure 5: Heatmap of the encoder weight matrix $\mathbf{W}_\mathrm{e}^\mathrm{T}$ (Go1 simulation, two latent dimensions). Brighter colors indicate larger weights; rows are 18 configuration dimensions and columns are the two latent dimensions.