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Fast and Modular Whole-Body Lagrangian Dynamics of Legged Robots with Changing Morphology

Sahand Farghdani, Omar Abdelrahman, Robin Chhabra

TL;DR

This work addresses the challenge of real-time dynamics for multi-legged robots undergoing morphological damage by introducing a singularity-free, modular dynamics framework based on Boltzmann-Hamel equations on SE(3). The main innovation is a morphology-aware mass matrix and Coriolis formulation that decouples main-body and leg dynamics and supports real-time reconfiguration when legs or links are damaged, using link and leg existence vectors. A fast modeling pipeline leverages trigonometrical multiplier functions to enable leg-morphology replication, allowing the full system to be assembled from a small set of leg modules with real-time performance. Validation through simulation and hardware on a hexapod demonstrates accurate damage adaptation and real-time operation, with the engine running roughly three times faster than real time. The approach reduces reliance on retraining for data-driven methods and avoids rigid fixed-structure analytical models, enabling robust, adaptive control and damage recovery in uncertain environments.

Abstract

Fast and modular modeling of multi-legged robots (MLRs) is essential for resilient control, particularly under significant morphological changes caused by mechanical damage. Conventional fixed-structure models, often developed with simplifying assumptions for nominal gaits, lack the flexibility to adapt to such scenarios. To address this, we propose a fast modular whole-body modeling framework using Boltzmann-Hamel equations and screw theory, in which each leg's dynamics is modeled independently and assembled based on the current robot morphology. This singularity-free, closed-form formulation enables efficient design of model-based controllers and damage identification algorithms. Its modularity allows autonomous adaptation to various damage configurations without manual re-derivation or retraining of neural networks. We validate the proposed framework using a custom simulation engine that integrates contact dynamics, a gait generator, and local leg control. Comparative simulations against hardware tests on a hexapod robot with multiple leg damage confirm the model's accuracy and adaptability. Additionally, runtime analyses reveal that the proposed model is approximately three times faster than real-time, making it suitable for real-time applications in damage identification and recovery.

Fast and Modular Whole-Body Lagrangian Dynamics of Legged Robots with Changing Morphology

TL;DR

This work addresses the challenge of real-time dynamics for multi-legged robots undergoing morphological damage by introducing a singularity-free, modular dynamics framework based on Boltzmann-Hamel equations on SE(3). The main innovation is a morphology-aware mass matrix and Coriolis formulation that decouples main-body and leg dynamics and supports real-time reconfiguration when legs or links are damaged, using link and leg existence vectors. A fast modeling pipeline leverages trigonometrical multiplier functions to enable leg-morphology replication, allowing the full system to be assembled from a small set of leg modules with real-time performance. Validation through simulation and hardware on a hexapod demonstrates accurate damage adaptation and real-time operation, with the engine running roughly three times faster than real time. The approach reduces reliance on retraining for data-driven methods and avoids rigid fixed-structure analytical models, enabling robust, adaptive control and damage recovery in uncertain environments.

Abstract

Fast and modular modeling of multi-legged robots (MLRs) is essential for resilient control, particularly under significant morphological changes caused by mechanical damage. Conventional fixed-structure models, often developed with simplifying assumptions for nominal gaits, lack the flexibility to adapt to such scenarios. To address this, we propose a fast modular whole-body modeling framework using Boltzmann-Hamel equations and screw theory, in which each leg's dynamics is modeled independently and assembled based on the current robot morphology. This singularity-free, closed-form formulation enables efficient design of model-based controllers and damage identification algorithms. Its modularity allows autonomous adaptation to various damage configurations without manual re-derivation or retraining of neural networks. We validate the proposed framework using a custom simulation engine that integrates contact dynamics, a gait generator, and local leg control. Comparative simulations against hardware tests on a hexapod robot with multiple leg damage confirm the model's accuracy and adaptability. Additionally, runtime analyses reveal that the proposed model is approximately three times faster than real-time, making it suitable for real-time applications in damage identification and recovery.

Paper Structure

This paper contains 18 sections, 8 theorems, 46 equations, 10 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Overview of Contents
  4. Related work
  5. Preliminaries and Problem Statement
  6. Special Euclidean Lie Group Notation
  7. Problem Statement
  8. Modular Dynamical Equations of MLRs
  9. Modular Mass Matrix for MLRs
  10. Modular Coriolis Matrix for MLRs
  11. Forming the Dynamical Equations of a Damaged MLR
  12. FAST Modular Dynamic Equations of Multi-Legged Robots
  13. Simulation & Experiment Results
  14. Platform Description
  15. Experiment Results
  16. ...and 3 more sections

Key Result

Theorem 1

Consider an MLR with a 6-DoF main body, whose configuration is locally parameterized by $\Phi(g_{s,b},\phi):= g_{s,b}e^{\hat{\phi}}$ (such that $\phi\in\mathbb{R}^{6}$) at $g_{s,b}$, and $N$ legs, each with $n_i$ degrees of freedom. Let $S_b$ define the relation between the local velocities $\dot\ph such that $:=\tau\in\mathbb{R}^{6+N_T}$ is the applied forces collocated with the quasi-velocities,

Figures (10)

  • Figure 1: Frames assignment for the main body and Leg $i$
  • Figure 2: Hiwonder JetHexa robot with motion-tracking LEDs
  • Figure 3: Main body orientation during the first damage scenario
  • Figure 4: Main body COM position during the first damage scenario
  • Figure 5: Snapshots of 1.625 seconds of model and robot motions for the first damage scenario
  • ...and 5 more figures

Theorems & Definitions (26)

  • Theorem 1: Boltzmann-Hamel equations of legged robots farghdani2024singularity
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Corollary 1
  • proof
  • Remark 3
  • ...and 16 more