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Dynamic Modeling of Branched Robots using Modular Composition

Frederico Fernandes Afonso Silva, Bruno Vilhena Adorno

TL;DR

This work introduces a modular dynamic modeling framework for branched robots with tree-like kinematics. By representing subsystems as nodes in a directed graph and propagating twists and wrenches through a wrench-interconnection matrix, the authors enable assembly of the full dynamics from partial or black-box components. A concrete instantiation using dual quaternion algebra demonstrates that the approach matches monolithic methods in accuracy while accommodating black-box subsystems and enabling model-based wrench-driven control. Numerical validations on a fixed-base BM and a holonomic MBM show high fidelity to a state-of-the-art simulator and to Featherstone’s recursive Newton-Euler, supporting practical applicability in modular robotics and control design.

Abstract

When modeling complex robot systems such as branched robots, whose kinematic structures are a tree, current techniques often require modeling the whole structure from scratch, even when partial models for the branches are available. This paper proposes a systematic modular procedure for the dynamic modeling of branched robots comprising several subsystems, each composed of an arbitrary number of rigid bodies, providing the final dynamic model by reusing previous models of each branch. Unlike previous approaches, the proposed strategy is applicable even if some subsystems are regarded as black boxes, requiring only twists and their time derivatives, and wrenches at the connection points between those subsystems. To help in the model composition, we also propose a weighted directed graph representation where the weights encode the propagation of twists and their time derivatives, and wrenches between the subsystems. A simple linear operation on the graph interconnection matrix provides the dynamics of the whole system. Numerical results using a 24-DoF fixed-base branched robot composed of eight subsystems show that the proposed formalism is as accurate as a state-of-the-art library for robotic dynamic modeling. Additional results using a 30-DoF holonomic branched mobile manipulator composed of three subsystems demonstrate the fidelity of our model to a modern robotics simulator and its capability of dealing with black box subsystems. To further illustrate how the derived dynamic model can be used in closed-loop control, we also present a simple formulation of a model-based wrench-driven pose control for branched robots.

Dynamic Modeling of Branched Robots using Modular Composition

TL;DR

This work introduces a modular dynamic modeling framework for branched robots with tree-like kinematics. By representing subsystems as nodes in a directed graph and propagating twists and wrenches through a wrench-interconnection matrix, the authors enable assembly of the full dynamics from partial or black-box components. A concrete instantiation using dual quaternion algebra demonstrates that the approach matches monolithic methods in accuracy while accommodating black-box subsystems and enabling model-based wrench-driven control. Numerical validations on a fixed-base BM and a holonomic MBM show high fidelity to a state-of-the-art simulator and to Featherstone’s recursive Newton-Euler, supporting practical applicability in modular robotics and control design.

Abstract

When modeling complex robot systems such as branched robots, whose kinematic structures are a tree, current techniques often require modeling the whole structure from scratch, even when partial models for the branches are available. This paper proposes a systematic modular procedure for the dynamic modeling of branched robots comprising several subsystems, each composed of an arbitrary number of rigid bodies, providing the final dynamic model by reusing previous models of each branch. Unlike previous approaches, the proposed strategy is applicable even if some subsystems are regarded as black boxes, requiring only twists and their time derivatives, and wrenches at the connection points between those subsystems. To help in the model composition, we also propose a weighted directed graph representation where the weights encode the propagation of twists and their time derivatives, and wrenches between the subsystems. A simple linear operation on the graph interconnection matrix provides the dynamics of the whole system. Numerical results using a 24-DoF fixed-base branched robot composed of eight subsystems show that the proposed formalism is as accurate as a state-of-the-art library for robotic dynamic modeling. Additional results using a 30-DoF holonomic branched mobile manipulator composed of three subsystems demonstrate the fidelity of our model to a modern robotics simulator and its capability of dealing with black box subsystems. To further illustrate how the derived dynamic model can be used in closed-loop control, we also present a simple formulation of a model-based wrench-driven pose control for branched robots.
Paper Structure (16 sections, 1 theorem, 31 equations, 9 figures, 4 tables, 7 algorithms)

This paper contains 16 sections, 1 theorem, 31 equations, 9 figures, 4 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Statement of contributions
  4. Abstract Model Composition
  5. Graph representation
  6. Model Composition using Dual Quaternion Algebra
  7. Computational complexity
  8. Wrench-driven End-Effector Motion Control
  9. Numerical Evaluation and Simulation Results
  10. Simulation setup
  11. Model accuracy using the BM
  12. Model accuracy using the MBM and black-box subsystems
  13. Closed-loop control of the MBM
  14. Conclusions
  15. Dual Quaternion Algebra
  16. ...and 1 more sections

Key Result

Proposition 3

Let a branched kinematic system be composed of $n$ rigid bodies divided into a set of $s$ coupled subsystems, each one containing $n_{1}$, $n_{2}$, $\ldots$, $n_{s}$ rigid bodies, respectively. Considering the proposed weighted graph representation with its corresponding adjacency matrix (eq:adjacen where $\underline{\boldsymbol{\Gamma}}_{i}\in\mathsf{W}^{n_{i}}$ is the vector of the total joint w

Figures (9)

  • Figure 1: A fixed-base $24$-DoF branched manipulator (BM) composed of eight subsystems grouped in different colored links associated with indices from 1 to 8. The root subsystem is indicated by the white circle around index 1.
  • Figure 2: Wrenches generated at the joints of the $i$th subsystem (pink region). For each subsystem, the large gray circles represent joints, solid black lines represent links, and blue crossed circles represent CoMs. The preceding subsystem $p_{i}$ is given in blue, the subsequent subsystems $j\in S_{i}$ are colored in green, and red circles on the red dashed lines, numbered from 1 to 3, indicate the connection points. Dotted arrows represent the forward propagation of twists, whereas solid arrows represent the backward propagation of wrenches.
  • Figure 3: Assembly process of a robot composed of different subsystems: (a) $3$-DoF robot given by the first subsystem shown in Fig. \ref{['fig:BM_TRO2023']} and the corresponding graph representation---the wrenches $\underline{\boldsymbol{\mathcal{W}}}_{1}\left(\underline{\boldsymbol{\boldsymbol{\Xi}}}_{1}\right)$ of the system are exclusively generated by its own twists and twist time derivatives; (b) $3$-DoF robot given by the second subsystem shown in Fig. \ref{['fig:BM_TRO2023']} and its graph representation---the wrenches $\underline{\boldsymbol{\mathcal{W}}}_{2}(\underline{\boldsymbol{\Xi}}_{2,2})$ of the system are exclusively generated by its own twists and twist derivatives; (c) $6$-DoF assembled robot given by the first two subsystems shown in Fig. \ref{['fig:BM_TRO2023']} and its graph representation. In this combined system, the wrenches of system 2 originates from its twists and twist derivatives$\underline{\boldsymbol{\Xi}}_{2,2}$ and the twist and twist derivatives$\underline{\boldsymbol{\Xi}}_{1,2}$ at the connection with system 1. Similarly, the wrenches at the joints of system 1 result from the self-motion wrenches $\underline{\boldsymbol{\mathcal{W}}}_{1}\left(\underline{\boldsymbol{\boldsymbol{\Xi}}}_{1}\right)$ of system 1 added by the wrench $\mathring{\underline{\boldsymbol{\boldsymbol{\Gamma}}}}_{2,1}$ at its connection with system 2.
  • Figure 4: Graph representation of the $24$-DoF branched robot. The colored nodes follow the color scheme adopted in Fig. \ref{['fig:BM_TRO2023']}. Only the root subsystem, represented by vertex 1, has just one incoming dashed edge, meaning that no twists and twist time derivatives are propagated from other subsystems. On the other hand, only the vertices corresponding to the leaves of the branched system (3, 4, 6, and 8) have only one incoming solid edge, meaning that no wrenches are propagated from other subsystems.
  • Figure 5: A $30$-DoF holonomic mobile branched manipulator (MBM) composed of three subsystems represented by the colored rigid bodies. The second subsystem (blue) is considered as a black box subsystem.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Example 1
  • Example 2
  • Proposition 3
  • Example 4
  • Definition 5