arxiv: 2604.18887 · v1 · submitted 2026-04-20 · 💻 cs.RO · cs.SY· eess.SY

HALO: Hybrid Auto-encoded Locomotion with Learned Latent Dynamics, Poincar\'e Maps, and Regions of Attraction

Blake Werner , Sergio A. Esteban , Massimiliano de Sa , Max H. Cohen , Aaron D. Ames This is my paper

Pith reviewed 2026-05-10 03:45 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY

keywords reduced-order modelshybrid dynamicsPoincaré mapsautoencodersregion of attractionlegged locomotionlatent dynamicsstability analysis

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The pith

HALO learns low-dimensional latent models of legged robot dynamics that preserve stability structure and lift region-of-attraction boundaries back to the full system.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents HALO as a way to build reduced-order models for high-dimensional hybrid systems like legged robots directly from trajectory data instead of relying on hand-designed approximations. It combines an autoencoder to extract a low-dimensional latent state with a learned latent Poincaré map that describes step-to-step evolution, allowing Lyapunov stability analysis inside the latent space. The resulting region of attraction is then mapped back to the original high-dimensional robot states through the decoder. This approach aims to retain meaningful stability information that classical template models often lose while avoiding purely black-box data methods that offer no safety guarantees. If the lifting works, it would let engineers analyze and certify locomotion safety in a simpler space before applying it to the real robot.

Core claim

HALO employs an autoencoder to identify a low-dimensional latent state together with a learned latent Poincaré map that captures step-to-step locomotion dynamics. This enables Lyapunov analysis and the construction of an associated region of attraction in the latent space, both of which can be lifted back to the full-order state space through the decoder. Experiments on a simulated hopping robot and full-body humanoid locomotion demonstrate that HALO yields low-dimensional models that retain meaningful stability structure and predict full-order region-of-attraction boundaries.

What carries the argument

The hybrid autoencoder paired with a learned latent Poincaré map, which extracts a low-dimensional representation of periodic hybrid dynamics and supports lifting of Lyapunov-based stability sets via the decoder.

If this is right

Data-driven low-dimensional models can now support Lyapunov stability certificates for periodic hybrid locomotion without manual template design.
Region-of-attraction boundaries predicted in the latent space provide useful approximations for full-order robot safety margins on both hopping and humanoid platforms.
The Poincaré map in latent space captures step-to-step return dynamics well enough to enable standard discrete-time stability tools on continuous hybrid systems.
Controller synthesis or verification performed inside the latent space can be decoded and applied to the original high-dimensional robot model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the lifting property holds reliably, similar autoencoder-plus-Poincaré constructions could be tested on other hybrid periodic systems such as rhythmic manipulation or bipedal walking with contact switching.
The method implicitly suggests that latent-space Lyapunov functions might serve as practical certificates for safe reinforcement learning policies on legged robots.
One could extend the framework by adding uncertainty bounds on the decoder to produce conservative full-order attraction regions that account for reconstruction error.

Load-bearing premise

Stability and safety properties observed in the latent space transfer back to the full-order system through the decoder.

What would settle it

Compute the true region of attraction directly on the full-order simulated hopping robot dynamics and compare its boundary shape and size to the one obtained by lifting the latent-space region through the decoder; substantial mismatch would show the transfer fails.

Figures

Figures reproduced from arXiv: 2604.18887 by Aaron D. Ames, Blake Werner, Massimiliano de Sa, Max H. Cohen, Sergio A. Esteban.

**Figure 1.** Figure 1: Autoencoders enable learning of a reduced-order dynamics model in a latent space. The process of compression and decompression is performed by encoder and decoder maps. First, we define nz < nx to be the dimension of our ROM. An encoder E : R nx → R nz performs compression, and a decoder D : R nz → R nx performs decompression. Since R nz represents the hidden low-dimensional structure, we refer to R nz as … view at source ↗

**Figure 2.** Figure 2: An attractive invariant manifold for a discrete-time system (in red) can have a lower dimensional embedding (in blue). This embedding informs a choice of latent dimension and reduced-order model. Motivation from Nonlinear Systems: To begin our nonlinear analysis, recall the manifold hypothesis, which posits that data points in real-world, high-dimensional data sets are often clustered around low-dimension… view at source ↗

**Figure 3.** Figure 3: The three systems of study, their configurations, and their impacts. Thus, stability in the latent representation of the invariant manifold translates to stability in the fullorder representation of the invariant manifold. Practically—as we demonstrate below—stability on the invariant manifold can be concluded by exhibiting an asymptotic Lyapunov function for g : R nz → R nz ; such a Lyapunov function aut… view at source ↗

**Figure 4.** Figure 4: A visual guide of the losses. Loss Function: Using the insights from Section 3, we seek parameters ϕ, ψ, and ρ such that the encoder Eϕ, decoder Dψ, and latent dynamics model gρ approximate the FOM Poincare map: ´ inf ϕ,ψ,ρ X xk∈D ∥f(xk) − (Dψ ◦ gρ ◦ Eϕ)(xk)∥ 2 . (9) The encoder, decoder, and latent dynamics networks are trained jointly using a weighted sum of losses, each enforcing a different aspect of … view at source ↗

**Figure 5.** Figure 5: Per-step normalized error of the learned ROM over 3,000 test trajectories for each system. (Top) shows single-step reconstruction error, and (bottom) shows multi-step latent forward-propagation error. Solid curves denote the mean and shaded bands denote ±σ. To validate the capabilities of our autoencoder, we test both its prediction and stability estimation capabilities. To test the encoding and decoding… view at source ↗

**Figure 6.** Figure 6: Decoded visualization of the boundary of Ωz in full-order coordinates of the G1. Plotted states correspond to the center-of-mass position relative to the stance foot [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

**Figure 7.** Figure 7: Decoded visualization of the boundary of Ωz in full-order coordinates of the hopper. Plotted states correspond to the center-of-mass position relative to the hopper foot [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: A 9-step rollout with true hardware, encoded-decoded, and latent ROM trajectories. Conclusion: In this work, we proposed a method of generating reduced-order models using autoencoders to represent and study the stability of hybrid systems. Our technique enables accurate reconstruction of the original dynamical system from a lower-dimensional latent space, as well as estimation of the region of attraction … view at source ↗

read the original abstract

Reduced-order models are powerful for analyzing and controlling high-dimensional dynamical systems. Yet constructing these models for complex hybrid systems such as legged robots remains challenging. Classical approaches rely on hand-designed template models (e.g., LIP, SLIP), which, though insightful, only approximate the underlying dynamics. In contrast, data-driven methods can extract more accurate low-dimensional representations, but it remains unclear when stability and safety properties observed in the latent space meaningfully transfer back to the full-order system. To bridge this gap, we introduce HALO (Hybrid Auto-encoded Locomotion), a framework for learning latent reduced-order models of periodic hybrid dynamics directly from trajectory data. HALO employs an autoencoder to identify a low-dimensional latent state together with a learned latent Poincar\'e map that captures step-to-step locomotion dynamics. This enables Lyapunov analysis and the construction of an associated region of attraction in the latent space, both of which can be lifted back to the full-order state space through the decoder. Experiments on a simulated hopping robot and full-body humanoid locomotion demonstrate that HALO yields low-dimensional models that retain meaningful stability structure and predict full-order region-of-attraction boundaries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

HALO learns latent Poincaré maps via autoencoder and lifts Lyapunov regions through the decoder for hybrid locomotion, but the transfer's reliability is the part that still needs numbers.

read the letter

The main point is that this paper gives a concrete pipeline for turning trajectory data into a low-dimensional model that keeps the step-to-step hybrid structure and lets you compute a region of attraction that you can push back to the original robot state. The autoencoder finds the latent coordinates, a learned map handles the Poincaré return, and standard Lyapunov analysis runs in that space before the decoder lifts the set. That combination is the actual new piece; prior work either stays with hand-crafted templates or does black-box reduction without the explicit lift of stability properties.

Referee Report

0 major / 3 minor

Summary. The paper introduces HALO, a data-driven framework that uses an autoencoder to learn a low-dimensional latent state representation and a latent Poincaré map from trajectory data of periodic hybrid locomotion systems. Lyapunov analysis is performed in this latent space to construct a region of attraction (RoA), which is then lifted back to the full-order state space via the decoder. Experiments on a simulated hopping robot and a full-body humanoid are claimed to show that the learned models retain meaningful stability structure and can predict full-order RoA boundaries, offering an alternative to hand-designed reduced-order templates such as LIP or SLIP.

Significance. If the central claims hold, HALO would provide a principled way to obtain reduced-order models for hybrid legged systems that preserve stability properties under lifting, addressing a key limitation of purely data-driven approaches. The explicit use of learned Poincaré maps combined with standard Lyapunov techniques in latent space, together with the decoder-based lift, is a clear strength; the experiments on both simple hopping and complex humanoid dynamics further support generality. This could meaningfully advance analysis and control of high-dimensional robotic systems beyond classical templates, provided the transfer of stability guarantees is rigorously validated.

minor comments (3)

[Abstract] Abstract: the statement that experiments 'demonstrate retention of stability structure and predict full-order region-of-attraction boundaries' would be strengthened by explicit mention of quantitative metrics (e.g., RoA volume error, prediction accuracy), validation procedure for the decoder lift, and at least one baseline comparison even if these appear later in the manuscript.
[Method] The description of the autoencoder architecture, latent dimension choice, and training procedure should include the specific hyperparameters and loss terms used, as these are listed among the free parameters and directly affect reproducibility of the learned map and RoA.
[Experiments] Figure captions and axis labels in the results section would benefit from clearer indication of which quantities are in latent versus full-order space and how the lifted RoA boundaries are visualized.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their encouraging summary of HALO and for recognizing its potential to provide principled reduced-order models that preserve stability properties for hybrid legged systems. We appreciate the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The HALO framework learns a latent state via autoencoder and a latent Poincaré map directly from trajectory data, then performs standard Lyapunov analysis to obtain a region of attraction in latent coordinates before decoding back to full-order space. This pipeline is a conventional data-driven reduced-order modeling approach; the stability transfer is treated as an empirical property verified on hopping and humanoid examples rather than a definitional identity or fitted parameter renamed as prediction. No self-citation chains, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation appear as load-bearing steps. The derivation chain remains independent of its inputs beyond the learned mapping from data.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The framework rests on standard dynamical-systems assumptions plus learned components whose fidelity is validated only by simulation; no new physical entities are postulated.

free parameters (2)

latent dimension
Chosen to achieve dimensionality reduction while preserving dynamics; value not stated in abstract.
autoencoder architecture and training hyperparameters
Network widths, depths, and optimization settings fitted to trajectory data.

axioms (2)

domain assumption A low-dimensional latent state exists that captures the essential periodic hybrid dynamics of the locomotion system.
Invoked when the autoencoder is trained to produce a latent Poincaré map.
domain assumption Lyapunov analysis performed in latent space yields a region whose image under the decoder is meaningful for the full-order system.
Central to the claim that stability properties transfer.

pith-pipeline@v0.9.0 · 5530 in / 1448 out tokens · 34870 ms · 2026-05-10T03:45:18.226466+00:00 · methodology

discussion (0)

Reference graph

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