Frame-Coded Legged Locomotion over Noisy Terrain

Frame-Coded Legged Locomotion over Noisy Terrain

Lav R. Varshney · N/A · 2026

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Summary

Open-loop multilegged locomotion over rough terrain has been interpreted as matter transport over a noisy channel: leg-ground interactions are discrete basic active contacts, terrain deletes or perturbs those contacts, and spatial redundancy concentrates the resulting thrust and arrival time. Tha...

Abstract Summary

Open-loop multilegged locomotion over rough terrain has been interpreted as matter transport over a noisy channel: leg-ground interactions are discrete basic active contacts, terrain deletes or perturbs those contacts, and spatial redundancy concentrates the resulting thrust and arrival time. That construction is repetition-like because every module carries the same scalar locomotion task. It consequently provides neither a positive task rate nor a decoder that changes with the surviving contact set. Here we formulate locomotion instead as a quantized finite-frame expansion with erasures. A d-dimensional body-level command is mapped into N>d heterogeneous local contact commands. Rough terrain erases or corrupts frame coefficients, while a contact-gated compliant morphology physically realizes the weighted active-subframe decoder. For a linear-Gaussian model, mechanical equilibrium is exactly the posterior mean, tangent stiffness is posterior precision, and mechanical compliance is posterior covariance. Equal-norm Parseval frames are shown to be minimax optimal against one missing contact, two-contact robustness is governed by frame coherence, and a harmonic frame gives a directly realizable gait family. For independently surviving contacts of probability q, random Gaussian gait frames admit exact reconstruction at every analog dimension rate R<q, with a binomial reliability exponent, whereas recovery of arbitrary commands is impossible for R>q. Residual contact noise yields an asymptotic per-mode amplification 1/(q-R) and a vanishing mechanical stiffness margin at the threshold. An information-locomotion inequality and an exact incremental-redundancy rule direct the next gait component toward the softest task-relevant unresolved mode. The resulting analog frame-coding theorem establishes a finite relative redundancy and converse as part of a fundamental limit theory of legged locomotion.

Key Points

  • Open-loop multilegged locomotion over rough terrain has been interpreted as matter transport over...
  • That construction is repetition-like because every module carries the same scalar locomotion task
  • It consequently provides neither a positive task rate nor a decoder that changes with the survivi...
  • Here we formulate locomotion instead as a quantized finite-frame expansion with erasures
  • A d-dimensional body-level command is mapped into N>d heterogeneous local contact commands

Frame-Coded Legged Locomotion over Noisy Terrain

|Authors: Lav R. Varshney

|Venue: arXiv preprint | Year: 2026

|arXiv: 2609.10273v1

Abstract

Open-loop multilegged locomotion over rough terrain has been interpreted as matter transport over a noisy channel: leg-ground interactions are discrete basic active contacts, terrain deletes or perturbs those contacts, and spatial redundancy concentrates the resulting thrust and arrival time. That construction is repetition-like because every module carries the same scalar locomotion task. It consequently provides neither a positive task rate nor a decoder that changes with the surviving contact set. Here we formulate locomotion instead as a quantized finite-frame expansion with erasures. A d-dimensional body-level command is mapped into N>d heterogeneous local contact commands. Rough terrain erases or corrupts frame coefficients, while a contact-gated compliant morphology physically realizes the weighted active-subframe decoder. For a linear-Gaussian model, mechanical equilibrium is exactly the posterior mean, tangent stiffness is posterior precision, and mechanical compliance is posterior covariance. Equal-norm Parseval frames are shown to be minimax optimal against one missing contact, two-contact robustness is governed by frame coherence, and a harmonic frame gives a directly realizable gait family. For independently surviving contacts of probability q, random Gaussian gait frames admit exact reconstruction at every analog dimension rate R<q, with a binomial reliability exponent, whereas recovery of arbitrary commands is impossible for R>q. Residual contact noise yields an asymptotic per-mode amplification 1/(q-R) and a vanishing mechanical stiffness margin at the threshold. An information-locomotion inequality and an exact incremental-redundancy rule direct the next gait component toward the softest task-relevant unresolved mode. The resulting analog frame-coding theorem establishes a finite relative redundancy and converse as part of a fundamental limit theory of legged locomotion.

Key Contributions

  • Open-loop multilegged locomotion over rough terrain has been interpreted as matter transport over…
  • That construction is repetition-like because every module carries the same scalar locomotion task
  • It consequently provides neither a positive task rate nor a decoder that changes with the survivi…
  • Here we formulate locomotion instead as a quantized finite-frame expansion with erasures
  • A d-dimensional body-level command is mapped into N>d heterogeneous local contact commands

Topics

  • locomotion

Code & Data

No code repository linked in paper metadata.

BibTeX

@article{Varshney2026_260910273v1,
  title     = {Frame-Coded Legged Locomotion over Noisy Terrain},
  author    = {Lav R. Varshney},
  year      = {2026},
  eprint    = {2609.10273v1},
  archivePrefix = {arXiv},
  primaryClass  = {cs.IT},
  url       = {https://arxiv.org/abs/2609.10273v1}
}
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