Tendons are a widely used actuation strategy for continuum robots that enable forces and moments to be transmitted along the robot from base-mounted actuators Most prior robots have used tendons routed in straight paths along the robot However, routing tendons through general curved paths within the robot offers potential advantages in reshaping the workspace and enabling a single section of the robot to achieve a wider variety of desired shapes In this paper, we provide a new model for the statics and dynamics of robots with general tendon routing paths that is derived by coupling the classical Cosserat-rod and Cosserat-string models This model also accounts for general external loading conditions and includes traditional axially routed tendons as a special case The advantage of the usage of this coupled model for straight-tendon robots is that it accounts for the distributed wrenches that tendons apply along the robot We show that these are necessary to consider when the robot is subjected to out-of-plane external loads Our experimental results demonstrate that the coupled model matches experimental tip positions with an error of 17% of the robot length, in a set of experiments that include both straight and nonstraight routing cases, with both point and distributed external loads

Statics and Dynamics of Continuum Robots With General Tendon Routing and External Loading

The helical coiling of plant tendrils has fascinated scientists for centuries, yet the underlying mechanism remains elusive. Moreover, despite Darwin's widely accepted interpretation of coiled tendrils as soft springs, their mechanical behavior remains unknown. Our experiments on cucumber tendrils demonstrate that tendril coiling occurs via asymmetric contraction of an internal fiber ribbon of specialized cells. Under tension, both extracted fiber ribbons and old tendrils exhibit twistless overwinding rather than unwinding, with an initially soft response followed by strong strain-stiffening at large extensions. We explain this behavior using physical models of prestrained rubber strips, geometric arguments, and mathematical models of elastic filaments. Collectively, our study illuminates the origin of tendril coiling, quantifies Darwin's original proposal, and suggests designs for biomimetic twistless springs with tunable mechanical responses.

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How the Cucumber Tendril Coils and Overwinds

We present a continuum-based discrete model for thin threads of viscous fluid by drawing upon the Rayleigh analogy to elastic rods, demonstrating canonical coiling, folding, and breakup in dynamic simulations. Our derivation emphasizes space-time symmetry, which sheds light on the role of time-parallel transport in eliminating---without approximation---all but an O(n) band of entries of the physical system's energy Hessian. The result is a fast, unified, implicit treatment of viscous threads and elastic rods that closely reproduces a variety of fascinating physical phenomena, including hysteretic transitions between coiling regimes, competition between surface tension and gravity, and the first numerical fluid-mechanical sewing machine. The novel implicit treatment also yields an order of magnitude speedup in our elastic rod dynamics.

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Discrete viscous threads

Robots consisting of several concentric, preshaped, elastic tubes can work dexterously in narrow, constrained, and/or winding spaces, as are commonly found in minimally invasive surgery. Previous models of these âactive cannulasâ assume piecewise constant precurvature of component tubes and neglect torsion in curved sections of the device. In this paper we develop a new coordinate-free energy formulation that accounts for general preshaping of an arbitrary number of component tubes, and which explicitly includes both bending and torsion throughout the device. We show that previously reported models are special cases of our formulation, and then explore in detail the implications of torsional flexibility for the special case of two tubes. Experiments demonstrate that this framework is more descriptive of physical prototype behavior than previous models1 it reduces model prediction error by 82% over the calibrated bending-only model, and 17% over the calibrated transmissional torsion model in a set of experiments.

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Equilibrium Conformations of Concentric-tube Continuum Robots

https://hal.archives-ouvertes.fr/hal-00874611/document

B-Spline interpolation of Kirchhoff-Love space rods

We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description---we represent the material frame by its angular deviation from the natural Bishop frame---as well as in the dynamical treatment---we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons.

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Stephen Robinson

Papers

Discrete elastic rods