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Optimal Trajectory Control of Geometrically Exact Strings with Space-Time Finite Elements

Tobias Thoma, Paul Kotyczka

TL;DR

This work develops an optimal trajectory feed-forward controller for geometrically exact strings using a variational space-time formulation. By formulating a Lagrange functional and deriving variational identities, the authors obtain the optimality conditions and implement a Galerkin-in-time space-time finite element discretization, yielding an algebraic system that avoids explicit velocity discretization. A planar string with gravity demonstrates the method's ability to track a prescribed end-point trajectory $y_d(t)$ over a pre/post-actuation interval, with results indicating close tracking for sufficiently large trajectory penalties. The approach is practical for large-scale under-actuated flexible systems and can be integrated with standard FE packages, with future extensions to 3D, unstructured space-time meshes, and model predictive control.

Abstract

In this contribution, we present a variational space-time formulation which generates an optimal feed-forward controller for geometrically exact strings. More concretely, the optimization problem is solved with an indirect approach, and the space-time finite element method translates the problem to a set of algebraic equations. Thereby, only the positional field and the corresponding adjoint variable field are approximated by continuous shape functions, which makes the discretization of a velocity field unnecessary. In addition, the variational formulation can be solved using commercial or open source finite element packages. The entire approach can also be interpreted as a multiple-shooting method for solving the optimality conditions based on the semi-discrete problem. The performance of our approach is demonstrated by a numerical test.

Optimal Trajectory Control of Geometrically Exact Strings with Space-Time Finite Elements

TL;DR

This work develops an optimal trajectory feed-forward controller for geometrically exact strings using a variational space-time formulation. By formulating a Lagrange functional and deriving variational identities, the authors obtain the optimality conditions and implement a Galerkin-in-time space-time finite element discretization, yielding an algebraic system that avoids explicit velocity discretization. A planar string with gravity demonstrates the method's ability to track a prescribed end-point trajectory over a pre/post-actuation interval, with results indicating close tracking for sufficiently large trajectory penalties. The approach is practical for large-scale under-actuated flexible systems and can be integrated with standard FE packages, with future extensions to 3D, unstructured space-time meshes, and model predictive control.

Abstract

In this contribution, we present a variational space-time formulation which generates an optimal feed-forward controller for geometrically exact strings. More concretely, the optimization problem is solved with an indirect approach, and the space-time finite element method translates the problem to a set of algebraic equations. Thereby, only the positional field and the corresponding adjoint variable field are approximated by continuous shape functions, which makes the discretization of a velocity field unnecessary. In addition, the variational formulation can be solved using commercial or open source finite element packages. The entire approach can also be interpreted as a multiple-shooting method for solving the optimality conditions based on the semi-discrete problem. The performance of our approach is demonstrated by a numerical test.
Paper Structure (15 sections, 2 theorems, 30 equations, 6 figures, 1 table)

This paper contains 15 sections, 2 theorems, 30 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Geometrically exact strings
  3. Problem description
  4. Weak formulation
  5. Spatial discretization
  6. Optimal trajectory feed-forward control design
  7. Lagrange functional
  8. Variational identities
  9. Time discretization
  10. Discussion
  11. Numerical test
  12. Model scenario
  13. Details on the computational approximation process
  14. Numerical results
  15. Conclusion

Key Result

Lemma 1

Let $(r_h, u, w_h)$ be a sufficiently regular solution for Problem prob:1 minimizing eq:cost_function. Then the variational identities hold for all regular test functions $\delta r_h\in\mathcal{R}$, $\delta w_h\in\mathcal{W}$, and $\delta u\in\mathcal{U}$ with

Figures (6)

  • Figure 1: Reference and actual configuration on $\mathbb{R}^2$.
  • Figure 2: Space-time domain.
  • Figure 3: Numerical solution of $r(s,t)$ in $X_1$-direction.
  • Figure 4: Numerical solution of $u(t)$. In order to present the input force in $X_1$-direction and the one in $X_2$-direction in the same diagram, we subtract the gravitational portion.
  • Figure 5: Actual output $y(t)$ compared with $y_d(t)$. In the case of a large $\alpha$, $y(t)$ is close to $y_d(t)$ . The results in $X_2$-direction are similar and omitted for the sake of clarity.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • Remark 2
  • Remark 3