Steering Flexible Linear Objects in Planar Environments by Two Robot Hands Using Euler's Elastica Solutions

TL;DR

The paper addresses planar steering of flexible linear objects using two robot hands by leveraging Euler's elastica as the governing equilibrium geometry under an optimal-control framework. It derives closed-form elastica solutions parameterized by $(\lambda, k, s_0)$, establishes stability and non-self-intersection criteria, and develops a quadratic-arc Bezier approximation to enable fast collision checks. A 5-D configuration-space planning scheme (with equal endpoint tangents) combines these analytic tools with A* to route cables among sparsely spaced obstacles. The approach is demonstrated through MATLAB-based representative examples and supports extensions to gravity, 3-D, and contact-rich environments; experiments validate the elastica model against static cable-tie shapes with small errors, confirming the method's accuracy and practicality.

Abstract

The manipulation of flexible objects such as cables, wires and fresh food items by robot hands forms a special challenge in robot grasp mechanics. This paper considers the steering of flexible linear objects in planar environments by two robot hands. The flexible linear object, modeled as an elastic non-stretchable rod, is manipulated by varying the gripping endpoint positions while keeping equal endpoint tangents. The flexible linear object shape has a closed form solution in terms of the grasp endpoint positions and tangents, called Euler's elastica. This paper obtains the elastica solutions under the optimal control framework, then uses the elastica solutions to obtain closed-form criteria for non self-intersection, stability and obstacle avoidance of the flexible linear object. The new tools are incorporated into a planning scheme for steering flexible linear objects in planar environments populated by sparsely spaced obstacles. The scheme is fully implemented and demonstrated with detailed examples.

Figures (21)

• Figure 1: Top view of a flexible cable steered by two robot grippers in a planar environment populated by obstacles. The cable must reach target position while avoiding self-collision and contact with obstacles (except for endpoint contacts at the start or target).
• Figure 2: Top view of a flexible cable of length $L$ embedded in its periodic elastica solution. The elastica axis with angle $\phi_0$ passes through the elastica zero curvature points and is parallel to the opposing forces of magnitude $\hbox{$\lambda$}_r$ applied at the cable endpoints.
• Figure 3: The elastica solutions are characterized by the phase parameter, $s_0$, the amplitude parameter, $A$, and the modulus parameter, $0 \!<\! \mathrm{k}\!<\! 1$, determined by $\phi(s^*)$. Note that $\hbox{\footnotesize $A$} \!=\! 2\mathrm{k} / \hbox{\scriptsize $\sqrt{\hbox{$\lambda$}}$}$, where $\hbox{\small $\lambda$}$ represents the magnitude of the force applied at the cable endpoints.
• Figure 4: Numerical solution of the non self-intersection condition, $\hbox{$\bar{x}$}_{max} \! \leq \! \hbox{$\bar{x}$}( \hbox{\scriptsize $\tilde{L}/2$})$ where $\hbox{$\bar{x}$}_{max} \!=\! \hbox{$\bar{x}$}(\bar{s}_{max})$, gives the modulus parameter $\mathrm{k}_{max} \!=\! 0.855$. The range $0 \!<\!\mathrm{k}\!<\! \mathrm{k}_{max}$ guarantees that the flexible cable is not self-intersecting.
• Figure 5: (a)-(b) Two families of energy extremal cable shapes, $(x_1(s),y_1(s))$ deter-mined by $s_0 \!=\! a_0$ and $(x_2(s),y_2(s))$ determined by $s_0 \!=\! b_0$. All cable shapes have same length of $\hbox{\scriptsize $L$} \!=\! 1$. (c) When the two families of extremal cable shapes approach the limit $a_0 \!=\! b_0 \!=\! \tfrac{\tilde{L}}{4}$, their common endpoint reaches a conjugate point along the limit cable shape.