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Flatness-based motion planning for a non-uniform moving cantilever Euler-Bernoulli beam with a tip-mass

Soham Chatterjee, Aman Batra, Vivek Natarajan

TL;DR

This paper addresses motion planning for a non-uniform moving cantilever Euler-Bernoulli beam with a tip-mass and a movable joint, modeled by a coupled PDE-ODE with boundary input $f(t)$ at $x=L$. It extends the generating-functions (flatness-based) framework to this PDE-ODE system by introducing two flat outputs $y_1,y_2$ in the Gevrey class $G_s[0,T]$ and their associated generating functions, yielding a convergent representation of the beam state $w$ as $w(x,t)=\sum_{k\ge0} g_k(x)y_1^{(2k)}(t)+\sum_{k\ge0} h_k(x)y_2^{(2k)}(t)$. A key result proves that, for initial and final states lying in a subspace $M$ containing all steady-states, there exists a boundary input $f(t)$ on $[0,T]$ that steers the beam from $z_0$ to $z_T$, with the commutativity ${\mathcal{L}}_1{\mathcal{L}}_2 p={\mathcal{L}}_2{\mathcal{L}}_1 p$ ensuring a well-defined flatness representation. Numerical and experimental validations demonstrate successful transfers for $T=3$ s and transfers between steady-states, confirming practical viability for precise control of spatially varying flexible structures in robotics and related applications.

Abstract

Consider a non-uniform Euler-Bernoulli beam with a tip-mass at one end and a cantilever joint at the other end. The cantilever joint is not fixed and can itself be moved along an axis perpendicular to the beam. The position of the cantilever joint is the control input to the beam. The dynamics of the beam is governed by a coupled PDE-ODE model with boundary input. On a natural state-space, there exists a unique state trajectory for this beam model for every initial state and each smooth control input which is compatible with the initial state. In this paper, we study the motion planning problem of transferring the beam from an initial state to a final state over a prescribed time interval. We address this problem by extending the generating functions approach to flatness-based control, originally proposed in the literature for motion planning of parabolic PDEs, to the beam model. We prove that such a transfer is possible if the initial and final states belong to a certain set, which also contains steady-states of the beam. We illustrate our theoretical results using simulations and experiments.

Flatness-based motion planning for a non-uniform moving cantilever Euler-Bernoulli beam with a tip-mass

TL;DR

This paper addresses motion planning for a non-uniform moving cantilever Euler-Bernoulli beam with a tip-mass and a movable joint, modeled by a coupled PDE-ODE with boundary input at . It extends the generating-functions (flatness-based) framework to this PDE-ODE system by introducing two flat outputs in the Gevrey class and their associated generating functions, yielding a convergent representation of the beam state as . A key result proves that, for initial and final states lying in a subspace containing all steady-states, there exists a boundary input on that steers the beam from to , with the commutativity ensuring a well-defined flatness representation. Numerical and experimental validations demonstrate successful transfers for s and transfers between steady-states, confirming practical viability for precise control of spatially varying flexible structures in robotics and related applications.

Abstract

Consider a non-uniform Euler-Bernoulli beam with a tip-mass at one end and a cantilever joint at the other end. The cantilever joint is not fixed and can itself be moved along an axis perpendicular to the beam. The position of the cantilever joint is the control input to the beam. The dynamics of the beam is governed by a coupled PDE-ODE model with boundary input. On a natural state-space, there exists a unique state trajectory for this beam model for every initial state and each smooth control input which is compatible with the initial state. In this paper, we study the motion planning problem of transferring the beam from an initial state to a final state over a prescribed time interval. We address this problem by extending the generating functions approach to flatness-based control, originally proposed in the literature for motion planning of parabolic PDEs, to the beam model. We prove that such a transfer is possible if the initial and final states belong to a certain set, which also contains steady-states of the beam. We illustrate our theoretical results using simulations and experiments.
Paper Structure (5 sections, 5 theorems, 35 equations)

This paper contains 5 sections, 5 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Generating functions
  4. Motion planning
  5. Numerical and experimental results

Key Result

Proposition II.2

Let $T>0$, an input $f\in C^3([0,T];{\mathbb R})$ and an initial state $z_0 = [u_0 \ \ v_0 \ \ \alpha_0 \ \ \beta_0]\in V$ with $f(0)=u_0(L)$ and $\dot f(0)=v_0(L)$ be given. There exists a unique classical solution $z\in C([0,T];Z)$ of eq:beam1-eq:beam4 on the time interval $[0,T]$ for the initial

Theorems & Definitions (13)

  • Definition II.1
  • Proposition II.2
  • proof
  • Proposition III.1
  • proof
  • Proposition III.2
  • proof
  • Proposition IV.1
  • proof
  • Remark IV.2
  • ...and 3 more