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On robotic manipulators with time-dependent inertial parameters: From physical consistency to boundedness of the mass matrix

Tom Kaufmann, Johann Reger

TL;DR

<3-5 sentence high-level summary> This work extends robotic manipulator dynamics to accommodate time-varying inertial parameters and causative mass-distribution changes, linking physical consistency of inertia to boundedness of the mass matrix. It develops a Lagrangian-based generalized robotics equation with a regressor structure that separates pose-dependent and parameter-dependent terms, and introduces the notions of uniform physical consistency and upper boundedness for time-varying inertial parameters. The authors prove conditions under which the mass matrix M(q,Θ(t)) is uniformly positive definite and bounded, leveraging a normal Jacobian and forward-kinematic constraints akin to Gorbel’s results. These results provide a theoretical foundation for robust adaptive control of manipulators subject to load changes and internal mass redistribution, by guaranteeing a positive definite mass matrix over time.</paper_summary>

Abstract

We generalize the robotics equation describing the dynamics of an open kinematic chain to include the effect of time-dependent change of inertial parameters as well as the effects of its cause, i.e. time dependency of the distributions of mass originating from parasitic movements of mass-carrying particles. The results generate insight that allows linking the novel concepts of uniform physical consistency and upper boundedness of inertial parameters -- ruling out approaching the edge to physical inconsistency or to diverge -- with the existence of finite, positive uniform bounds of the mass matrix.

On robotic manipulators with time-dependent inertial parameters: From physical consistency to boundedness of the mass matrix

TL;DR

<3-5 sentence high-level summary> This work extends robotic manipulator dynamics to accommodate time-varying inertial parameters and causative mass-distribution changes, linking physical consistency of inertia to boundedness of the mass matrix. It develops a Lagrangian-based generalized robotics equation with a regressor structure that separates pose-dependent and parameter-dependent terms, and introduces the notions of uniform physical consistency and upper boundedness for time-varying inertial parameters. The authors prove conditions under which the mass matrix M(q,Θ(t)) is uniformly positive definite and bounded, leveraging a normal Jacobian and forward-kinematic constraints akin to Gorbel’s results. These results provide a theoretical foundation for robust adaptive control of manipulators subject to load changes and internal mass redistribution, by guaranteeing a positive definite mass matrix over time.</paper_summary>

Abstract

We generalize the robotics equation describing the dynamics of an open kinematic chain to include the effect of time-dependent change of inertial parameters as well as the effects of its cause, i.e. time dependency of the distributions of mass originating from parasitic movements of mass-carrying particles. The results generate insight that allows linking the novel concepts of uniform physical consistency and upper boundedness of inertial parameters -- ruling out approaching the edge to physical inconsistency or to diverge -- with the existence of finite, positive uniform bounds of the mass matrix.

Paper Structure

This paper contains 17 sections, 10 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Contribution and outline
  3. Notation
  4. Physical consistency of inertial parameters of rigid bodies
  5. Generalized robotics equation
  6. Model
  7. Discussion of restrictions imposed by the model
  8. Kinematics
  9. Discussion of the standing Assumption \ref{['Assumption:IndependentJacobian']}.
  10. Calculation of the energies.
  11. Lagrange formalism
  12. Structural properties
  13. Boundedness of the mass matrix
  14. Positive lower uniform bound
  15. Finite upper uniform bound
  16. ...and 2 more sections

Key Result

Theorem 1

(traversaro_identification_2016 and Theorem $3$, Eq. (16)-(17) in wensing_linear_2018) A vector $\Phi\in\mathbb{R}^{10}$ with entries according to $(eq:Phi)$ is physically consistent if and only if $f(\Phi)\succ 0$, where $f:\mathbb{R}^{10}\to\mathrm{sym}(4)$ is a function with and $h \in \mathbb{R}^3$, $I \in \mathrm{sym}(3)$ are structured as in $(eq:h)$, $(eq:I)$, respectively. The inverse tra

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Theorem 3
  • ...and 11 more