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Stability of Cantilever-like Structures with Applications to Soft Robot Arms

Siva Prasad Chakri Dhanakoti

TL;DR

The paper addresses the stability of cantilever-like elastic rods with intrinsic curvature under end loads by extending the Jacobi condition to fixed-free boundary conditions within a Kirchhoff rod model. It combines a variational formulation with Euler-parameter/G Hamiltonian dynamics, deriving the second variation $\delta^2 J$ and the Jacobi operator $\mathcal{S}$, and uses a Matrix Riccati approach via $\mathbf{Q} + \mathbf{W}' = (\mathbf{C}+\mathbf{W}) \mathbf{P}^{-1} (\mathbf{C}^{T}+\mathbf{W})$ to certify positive definiteness. Stability is assessed by solving an 8-dimensional Jacobi system and constructing a 3×3 stability matrix from IVP solutions; a vanishing determinant indicates a conjugate point and instability. Numerical experiments reveal how intrinsic curvature and auxiliary factors such as a load arm and torsion induce snap-back and hysteresis, providing design insights for soft robotic arms and novel mechanisms.

Abstract

The application of variational principles for analyzing problems in the physical sciences is widespread. Cantilever-like problems, where one end is fixed and the other end is free, have received less attention in terms of their stability despite their prevalence. In this article, we establish stability conditions for these problems by examining the second variation of the energy functional through the generalized Jacobi condition. This requires computing conjugate points determined by solving a set of initial value problems from the linearized equilibrium equations. We apply these conditions to investigate the nonlinear stability of intrinsically curved elastic cantilevers subject to an end load. The rod deformations are modelled using Kirchhoff rod theory. The role of intrinsic curvature in inducing complex nonlinear phenomena, such as snap-back instability, is particularly emphasized. The numerical examples highlight its dependence on the system parameters. These examples illustrate potential applications in the design of flexible soft robot arms and innovative mechanisms.

Stability of Cantilever-like Structures with Applications to Soft Robot Arms

TL;DR

The paper addresses the stability of cantilever-like elastic rods with intrinsic curvature under end loads by extending the Jacobi condition to fixed-free boundary conditions within a Kirchhoff rod model. It combines a variational formulation with Euler-parameter/G Hamiltonian dynamics, deriving the second variation and the Jacobi operator , and uses a Matrix Riccati approach via to certify positive definiteness. Stability is assessed by solving an 8-dimensional Jacobi system and constructing a 3×3 stability matrix from IVP solutions; a vanishing determinant indicates a conjugate point and instability. Numerical experiments reveal how intrinsic curvature and auxiliary factors such as a load arm and torsion induce snap-back and hysteresis, providing design insights for soft robotic arms and novel mechanisms.

Abstract

The application of variational principles for analyzing problems in the physical sciences is widespread. Cantilever-like problems, where one end is fixed and the other end is free, have received less attention in terms of their stability despite their prevalence. In this article, we establish stability conditions for these problems by examining the second variation of the energy functional through the generalized Jacobi condition. This requires computing conjugate points determined by solving a set of initial value problems from the linearized equilibrium equations. We apply these conditions to investigate the nonlinear stability of intrinsically curved elastic cantilevers subject to an end load. The rod deformations are modelled using Kirchhoff rod theory. The role of intrinsic curvature in inducing complex nonlinear phenomena, such as snap-back instability, is particularly emphasized. The numerical examples highlight its dependence on the system parameters. These examples illustrate potential applications in the design of flexible soft robot arms and innovative mechanisms.
Paper Structure (13 sections, 2 theorems, 59 equations, 14 figures)

This paper contains 13 sections, 2 theorems, 59 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Calculus of Variations Problem
  3. Fixed-Fixed Ends
  4. Fixed-Free Ends
  5. Necessary condition
  6. Sufficient condition
  7. Application to Cantilever structures
  8. Kinematics and Equilibrium
  9. Stability Analysis
  10. Numerical Examples
  11. Intrinsically Planar Curvatures
  12. Effect of Torsion Component or Load Arm
  13. Summary and Discussion

Key Result

Theorem 1

If the matrix $\mathbf{P}$ is strictly positive definite, and the interval $[0,l]$ contains no point conjugate to $l$, then the second variation quadratic functional $\delta^{2}J(\boldsymbol \zeta_{o})[\mathbf{h}]$ is positive for all $\mathbf{h}(s)$ satisfying the boundary conditions eqn:Linearized

Figures (14)

  • Figure 1: The broken accessory extremal $\gamma(s)$ satisfying the boundary conditions \ref{['eqn:Linearized_BCs']}
  • Figure 2: Schematic showing an elastic rod with an external tip load acting through a massless rigid arm (in blue). The arm is assumed to be fixed to the tip of the elastic rod.
  • Figure 3: The schematic of the tip-loaded cantilever setup. One end of the rod is fixed to the quasi-statically rotating shaft, while the other end is attached to a load through a massless rigid lever arm (indicated in blue).
  • Figure 4: (a)(Top) The bifurcation diagram depicting the twist moment at the clamped end $m_{3}(0)$ as $\theta$ is varied. The equilibria corresponding to $\theta=0$ and $\theta=\pi/2$ are chosen for stability analysis. (Bottom) The conjugate point computations for equilibria with labels $1$ and $2$. The determinant never vanishes, indicating the absence of conjugate points, and therefore, both equilibria are stable. (b) The tip trace of the cantilever as $\theta$ is varied and the centrelines for the equilibria with labels $1$ and $2$. The tip load is represented by a solid dot, with its direction indicated by arrows.
  • Figure 5: (a) (Top) The bifurcation diagram depicting the twist moment at the clamped end $m_{3}(0)$ as $\theta$ is varied. The equilibria corresponding to $\theta=0$ and $\theta=\pi/2$ are selected for stability analysis. The plot has two folds and three equilibria exist for $\theta=0$. (Bottom) The conjugate point computations for the selected configurations. The determinant corresponding to the equilibrium $2$ vanishes indicating the presence of a conjugate point and is unstable. The remaining equilibria have no conjugate points and are stable. (b) The tip trace during the control maneuver and the rod centrelines of the selected equilibria. The tip load is represented by a solid dot, with its direction indicated by arrows. The tip corresponding to the equilibria lying between the folds is indicated by the black dotted line. The equilibria labelled $1$ and $3$ are mirror images about the $\mathbf{e}_{2} - \mathbf{e}_{3}$ plane and are nearly identical, which explains why the curves corresponding to conjugate tests in (a) bottom coincide.
  • ...and 9 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof