Dual Arm Steering of Deformable Linear Objects in 2-D and 3-D Environments Using Euler's Elastica Solutions

TL;DR

A novel criterion that ensures the flexible object stability during steering is introduced, which is integrated into a scheme for steering flexible linear objects in planar environments, which is lifted into a steering scheme in three-dimensional environments populated by sparsely spaced obstacles.

Abstract

This paper describes a method for steering deformable linear objects using two robot hands in environments populated by sparsely spaced obstacles. The approach involves manipulating an elastic inextensible rod by varying the gripping endpoint positions and tangents. Closed form solutions that describe the flexible linear object shape in planar environments, Euler's elastica, are described. The paper uses these solutions to formulate criteria for non self-intersection, stability and obstacle avoidance. These criteria are formulated as constraints in the flexible object six-dimensional configuration space that represents the robot gripping endpoint positions and tangents. In particular, this paper introduces a novel criterion that ensures the flexible object stability during steering. All safety criteria are integrated into a scheme for steering flexible linear objects in planar environments, which is lifted into a steering scheme in three-dimensional environments populated by sparsely spaced obstacles. Experiments with a dual-arm robot demonstrate the method.

Figures (9)

• Figure 1: A dual-arm robot has to steer a flexible linear object in a stable and non self-intersecting manner while avoiding sparsely spaced obstacles.
• Figure 2: The flexible cable elastica parameters describe its equilibrium shapes. (a) Previous work focused on flexible cable held with equal endpoint tangen-ts. (b) This paper considers the full range of elastica parameters where the flexible cable is steered with arbitrary endpoint positions and tangents.
• Figure 3: Top view of full period elastica shape with the physical flexible cable of length $L$ embedded in its periodic elastica solution of period length $\tilde{L}$. The elastica axis with angle $\phi_0$ passes through the zero curvature points and is parallel to the opposing forces of magnitude $\lambda_r$ applied at cable endpoints.
• Figure 4: (a) The elastica parameters of $\mathcal{C}_{free}$ when the flexible cable base frame is fixed at the origin, $S(0) \!=\! 0$. (b) Flexible cable distal endpoint, $(x(L),y(L),\phi(L)) \!=\! \psi(q)$ for $q \!\in\! \mathcal{C}_{free}$ with $S(0) \!=\! 0$. Purple lines on bottom of (a) are elastica shapes at which $L \!=\! \Tilde{L}$ while $s_0 \!=\! \tfrac{L}{4}$ or $s_0 \!=\! \tfrac{3L}{4}$. $\mathcal{C}_{free} \subset \mathcal{C}$ is explained in Section III.
• Figure 5: (a) The limit of stable full-period cable shapes occurs at $\mathrm{k}_c \!=\! 0.908$, which forms a figure eight. (b) The limit of non self-intersecting cable shapes occurs at $\mathrm{k}_{max} \!=\! 0.855$, at which the full-period elastica just touches itself.