Towards Pneumatic Spiral Grippers: Modeling and Design Considerations

Abstract

There are a number of instances in nature where long and slender objects are grasped by a continuum arm spirally twirling around the object, thereby increasing the area of contact and stability between the gripper and the object. This paper investigates the design and modeling of spiral grippers using pneumatic fiber-reinforced actuators. The paper proposes two reduced order models, a pure helical model, and a spatial Cosserat rod model to capture the deformed behavior of the gripper using the mechanics of fiber-reinforced actuators in the presence of self-weight. While the former model can yield closed form expressions that aid in design, the deformation parameters deviate by greater than 40% of its length for actuators longer than 200 mm. However, the Cosserat rod model deviates by less than 8% of its length for two different prototypes validated in this work. The deformation of the gripper is then correlated to the number of spiral turns achievable about the object, which determines the quality of the grip. Together, they enable a systematic framework where the gripper parameters can be designed for a given range of object sizes to be handled. This framework is experimentally validated by successful gripping of a range of slender objects that lie between a 20 mm diameter tubelight and a 60 mm diameter PVC pipe.

Introduction

Soft robotic grippers adapt to the object being manipulated, thereby increasing the contact area, and distributing grasping forces better than a traditional gripper with localized contact regions. The inherent flexibility of soft materials enable safe gripping of flexible, brittle and other delicate objects.^1,2 An increase in contact area implies that the soft gripper must encompass, or contain the object maximally from all sides. This implies that the relative sizes of the gripper and the object must be similar. When the object is long and slender (tubelight, large wooden sticks, pipes, etc.), the size of the gripper required may be unreasonably large, or may require several grippers coordinating at different contact regions. State-of-the-art soft robots that resemble the human hand still find long and slender workpieces difficult to grasp.³

In contrast, there are a number of instances in nature where structural contact between two objects is achieved when one body spirally twines around the other, thereby maximizing the area of contact. For example, grapevine tendrils grow spirally around a long and slender tree stem⁴ for increased structural support. Elephants (Fig. 1a), snakes, and cephalopods also demonstrate spiraling to capture objects or preys. The key advantage of spiraling is that it generates a large area of contact with relatively small gripper volume. There are a number of practical applications where the object to be manipulated is long and slender. For example, underwater deep sea exploration⁵ may require grippers that spiral around objects such as corals. In agricultural applications such as in de-weeding or uprooting of a shrub, a spiral grip on the stem could result in robust manipulation (as shown in Fig. 1c). This paper explores the working design and modelling of a soft pneumatic spiral gripper (Fig. 1d).

FIG. 1.

Examples of gripping long and slender objects. (a) An elephant using its flexible trunk to grasp a log of wood (Adapted from https://thewanderlustgene.wordpress.com). (b) Rigid robot having two end grippers to grip long and slender objects; (c) spiral gripper on a four wheeled robot used to remove weeds; and (d) soft spiral gripper gripping a tube (this work).

Related work

Recently there has been a steep rise in the number of soft robotic grippers reported in literature. Some of the earliest work using soft fluidic actuators were presented by Suzumori et al.,⁶ where finger-like bending behavior is used to grip small objects including mechanical tools. More recently, Ilievski et al.⁷ developed a pneumatically actuated starfish gripper to manipulate delicate objects like an egg. Most of these grippers and their variants function by fluid induced bending^6,8 of soft stretchable chambers that are differentially constrained. Using this principle, a soft under actuated hand has been developed,³ which is capable of robust grasping of different shape and sizes of objects. The concept of granular jamming^9,10 is used to stiffen the hold of the grippers on objects of various sizes. More recently, there have been grippers that provide feedback of the object size.¹¹ With all these developments, state of the art soft grippers still find long and slender workpiece objects difficult to grasp,³ necessitating exploration of newer concepts that are biologically inspired.

In rigid robots, the concept of whole arm manipulation¹² has been demonstrated to effectively handle relatively large objects using the entire body of the manipulator. Whole arm manipulation leads to a larger contact area between the gripper and the object, thus better distributing the contact forces. Semi-soft continuum manipulators, such as the OctArm,^13,14 have demonstrated this by spirally wrapping around the object. More recently, a single actuation boa type fiber-reinforced soft actuator⁵ has been developed to grasp deep sea coral reefs of different sizes and varied fragility. The boa gripper utilizes spiral coiling, and this was shown to generate larger pulling force than a finger-like bellow counterpart. In this paper, we investigate a design framework for creating grippers that can spirally coil along long and slender workpiece objects.

Approach

This paper uses the concept of pneumatically actuated Fiber Reinforced Elastomeric Enclosures (FREEs).^15–17 FREEs are quintessential building blocks for soft robots, as they encapsulate fundamental constituents of designs in literature and nature, namely stretchable skins, fibers, muscles, and pressurized fluids. The most simplified representation of a FREE is a hollow cylinder made of stretchable elastomer material and reinforced with two families of fibers denoted by angles \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} , respectively as shown in Figure 2a. FREEs are inspired in construction and operating principle by well-known pneumatic artificial muscles or Mckibben actuators.^18,19 Based on the fiber orientations, FREEs generate several motion patterns such as axial contraction, extension, rotation, and screw motion.¹⁵ However, to generate spatial deformation, a third fiber with orientation \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\gamma$$ \end{document} that generates curvature is used. The spiral FREE gripping a cylindrical object is shown in Figure 1d.

FIG. 2.

FREEs and its design space: (a) Structure of a FREE with its deformation parameters ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1} , { \lambda _2} , \delta$$ \end{document} ); (b) FREEs undergoing extension, rotation, and contraction; and (c) Design space of FREE spanned by its fiber angles \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} . FREE, fiber reinforced elastomeric enclosure.

Preliminary investigations focused on mapping the spiral deformation to the constituent fiber orientations using kinematics alone, neglecting the effects of loading, and the strain energy stored in the elastomer.²⁰ On the other hand, high fidelity computational mechanics-based models that capture the soft interaction between pressurized chambers, grippers, and objects can be complex, offering little design insight. There have been other intermediate approaches proposed, where the FREE actuation is captured using simple axisymmetric models,^21,22 and a global reduced order model such as Cosserat rod is used to capture interaction with external forces.^23,24 In this paper, we will first extract the design insights using the idealistic helical model, and then update our understanding using the Cosserat rod theory by taking into account its self-weight due to gravity. This paper thus proposes and experimentally validates two simplified mechanics-based models to analyze and design a FREE-based spiral gripper.

The analysis methodology is used to present a design space for spiral FREEs that maps its fiber orientation, size, and geometry parameters to the range of objects that can be successfully gripped. The design space qualitatively predicts the conditions for successful gripping without accurate estimations of gripping force, its distribution, friction, etc. This design space is deemed useful in several applications,¹⁶ and this is demonstrated by an example where a spiral FREE gripper is designed to handle objects of varying diameter.

The paper is organized as follows. First, we present the materials and methods involved in fabricating and modeling a spiral FREE. This includes experiments for determining the elastomer properties, and incorporating them into Cosserat rod theory with the inclusion of gravity loading (details of the Cosserat rod theory are provided in Appendix A1). Next, we apply the methods to study gripping criterion using FREEs and map them to the fiber orientations. Then, we experimentally validate our proposed analytical methods. Following this, we present an example of a design problem. Finally, we end with conclusions and future work.

Fundamentals of Spiral FREES

Simple reduced order models indicate that two families of helically wound fibers yield a kinematically well-constrained system.²⁵ These two families of fibers span a design space denoted by angles \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} as shown in Figure 2a. The popular configuration of contracting Mckibben actuators spans just a line (shown as AF in Fig. 2c). While Mckibben actuators contract upon pressurization, FREEs can also extend, undergo axial rotation, and a combination of several motion patterns as shown in Figure 2b. It is the repository of these deformation modes that can be leveraged to design novel soft robots.

Spiral FREE design and fabrication

Two dimensional (planar) and spatial deformation modes can be achieved by the addition of a third fiber constraint to the base elastomeric structure. For example, Figure 3a–c demonstrates that the addition of a single straight fiber to an extending actuator can result in planar bending (Fig. 3b). Similarly, adding a single straight fiber to an extending-rotating actuator can exhibit spiral motion¹⁶ (Fig. 3c). We intentionally refer to the deformation as a spiral and not a helical, as the latter occurs in the ideal scenario where there is no self-weight or any other acting external force. A detailed set of possibilities may arise by adding a single fiber at any angle γ to the FREE structure as presented by Bishop-Moser and Kota.¹⁶ For ease in analysis and design, in this work we consider FREEs exclusively with a single straight third fiber ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\gamma = 0$$ \end{document} ).

FIG. 3.

Spatial deformation of FREEs using a straight fiber and modeling. Demonstration of (b) bending and (c) spiral deformation by adding a (a) third fiber to extending and extending-rotating actuators, respectively. Estimation of (d) curvature and (e) torsion from extension and rotating parameters.

The construction of the spiral FREE starts with a base layer of natural rubber latex tubing (Part No: a038132-025, Latex-Tubing.com). Fibers are then wound in a semi-automated fashion with the desired angle and orientation. Adhesive agent (Part No: 74725A12, McMaster-Carr) is applied to cement the fibers on the base latex tubing. Finally, this matrix is cured to obtain a composite structure.²¹ The actuator is tested to evaluate its material properties (more on this will be detailed in the Governing equations section). Then, a straight fiber is glued to this actuator using the adhesive agent to convert the extension and rotation into spiral motion.

Governing equations

The governing equations of FREEs are simplified by assuming cylindrical geometry for both the undeformed and deformed²² configurations. For the design space considered for the spiral gripper, this is a reasonable approximation as no substantial radial changes are observed upon actuation. The deformation parameters are represented as stretch ratios \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _2}$$ \end{document} (Fig. 2a), which are ratios of the deformed to undeformed length and diameters, respectively. Furthermore, it is assumed that the fibers are inextensible and this leads to two constraints^15,21,22 as shown in Equation (1). \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lambda _1^2 { \rm { co } } { { \rm { s } } ^2 } \alpha + \lambda _2^2 { \rm { si } } { { \rm { n } } ^2 } \alpha { \left( { \frac { { \theta + \delta } } { \theta } } \right) ^2 } = 1 \tag { 1 } \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \lambda _1^2 { \rm { co } } { { \rm { s } } ^2 } \beta + \lambda _2^2 { \rm { si } } { { \rm { n } } ^2 } \beta { \left( { \frac { { \phi + \delta } } { \phi } } \right) ^2 } = 1 \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \frac { { \Delta V } } { V } = \lambda _2^2 { \lambda _1 } - 1. \tag { 2 } \end{align*} \end{document}

In these equations, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} are the fiber angles and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta$$ \end{document} is the axial rotation of the FREE. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\phi$$ \end{document} are the number of turns due to each family of fiber on the cylinder. The volume change in the FREE can be determined based on Equation (2) where V is the initial volume. At this point, there are three unknown deformation parameters ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} , \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _2}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta$$ \end{document} ) and three equations given by Equations (1)–(2). Solving this set of nonlinear equations, we can relate the deformation parameters to an applied input volume change, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\Delta V$$ \end{document} .

In pneumatic actuation, the input is usually controlled by air pressure. We can relate applied pressure to the FREE deformation by considering the energy stored in the elastomer. We assume that the elastomer tube behaves as a Neo-Hookean or MooneyRivlin hyperelastic solid. The strain energy stored per unit volume u is expressed as a function of the first invariant (I₁) of the Cauchy-Green strain tensor, which are in turn expressed in terms of the stretch ratios as: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {I_1} = \lambda _1^2 + \lambda _2^2 + \lambda _3^2 , \tag{3} \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} u = {C_1} ( {I_1} - 3 ) + {C_2} ( {I_2} - 3 ) , \tag{4} \end{align*} \end{document}

where C₁ and C₂ are material constants. The third stretch ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _3}$$ \end{document} is the ratio of the deformed elastomer thickness to the undeformed thickness. As the material is incompressible, the total volume remains constant, which gives us the relation between the stretch ratios as: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \lambda _1}{ \lambda _2}{ \lambda _3} = 1. \tag{5} \end{align*} \end{document}

Volume of the actuator V and volume occupied by the elastomer material V_t are given by: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} V = \pi LR_i^2{ \lambda _1} \lambda _2^2. \tag{6} \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {V_t} = \pi ( R_0^2 - R_i^2 ) L. \tag{7} \end{align*} \end{document}

All the terms shown in Equations (3 )–(7) can be expressed in terms of only one deformation ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} . From principle of virtual work,²³ we equate the change in total strain energy to change in work done by pressure P. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} P \delta V = {V_t} \delta u , \tag{8} \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta V$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta u$$ \end{document} are virtual changes in the cylinder volume and energy stored in the elastomer, and are expressed in terms of the virtual change \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta { \lambda _1}$$ \end{document} . At any given pressure, by inverting the above equation we can solve for the corresponding extension \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} parameter. On solving Equation (8) one can express radial expansion ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _2}$$ \end{document} ) and the rotation ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{\delta }}$$ \end{document} ) parameter in terms of extension ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} ) from Equation (1).

In Appendix A2, we compare the model developed in this subsection with experimental results for pressure P versus extension ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} ) of the fabricated FREE (without the straight fiber). This is used to fit the material model parameters, which are found to be around 0.6 MPa for C₁ and 0 MPa for C₂.

Once we have the information of extension parameter ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} ) and rotation ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta$$ \end{document} ) parameter for different pressures, we can obtain the curvature \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\kappa$$ \end{document} and torsion \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\tau$$ \end{document} (twist per unit length) as follows (shown in Fig. 3d, e): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \kappa = \frac { 1 } { \rho } = { \frac { ( { \lambda _1 } - 1 ) } { ( { \lambda _1 } + 1 ) { r_ { act } } } } , \tag { 9 } \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \tau = \frac { \delta } { L } , \tag { 10 } \end{align*} \end{document}

Gripping Using Spiral FREEs

The efficacy of a gripper is determined by its ability to restrict relative motion between itself and the object even under the presence of external forces.²⁶ In grasping literature, the term force or form closure is commonly used to indicate the number and type of independent forces required to constrain the object.^26,27 The FREE gripper meets force or form closure conditions by spiraling about an object as shown in Figure 4.

FIG. 4.

The three possible cases of spiral gripping. (a) Overhang, perfect wrap, and curling exhibited by a spiral FREE with increase in applied pressure. Left image shows the overhang (no contact) and the right image shows curling (or twisting at a fraction of the end length). (b) Maximum gripping force (measured in grams) generated by a spiral gripper at different pressures on two different cylindrical objects with radius 20 and 15 mm.

Quality of grip: three scenarios

Consider a FREE spiral gripper undergoing several turns about the cylindrical object that it grips. The quality of grip is a measure of how uniformly the FREE conforms onto the object. To better highlight these scenarios, we experimentally evaluate the gripping force as a function of the input pressure for a FREE gripper operating on two cylindrical objects of radii 15 and 20 mm, respectively (Fig. 4b). The gripping force is the maximum axial force applied to the object to break away from the hold of the gripper, and is found by placing dead weights on the cylinder ends. We observe three scenarios.

1. Overhang: This indicates that a part of the FREE length may not contact the object as shown in Figure 4a (left insert). This is because at lower pressures, the spiral is just forming, and does not exactly conform to the object. This corresponds to the initial part of the curve in Figure 4b where the gripping force increases slowly with pressure. This scenario is disadvantageous, as it may lead to lower contact forces, and in extreme cases may fail gripping for not meeting the force closure criteria.

2. Perfect wrap: At a certain optimum input pressure the entire length of the actuator is in contact with the cylindrical object. This may correspond to the nose of the curve in Figure 4b, where there is a sudden increase in the slope of the force versus pressure curve. This is the scenario that we will strive to attain. This paper presents guidelines for attaining perfect wrap.

3. Curling: When pressure exceeds beyond the perfect wrap case, the end section of the actuator curls. At these pressures, most of the FREE length resist deformation, and maintain their spiral configuration due to contact with the cylinder. However, the contact of the FREE tip is still relatively weak, and this tip deforms further by bending. Curling may be characterized by a decrease in the slope of the force versus pressure curve at higher pressures as seen in Figure 4b. Curling is not detrimental, yet suboptimal as part of the FREE length does not involve in gripping. At higher pressures the onset of curling may lead to unwinding of the spiral as seen in Figure 4a (right insert).

For a good grip, we envision the entire actuator length involved in restricting the object from relative motion. Using the criteria for a perfect wrap, we present a method to determine range of workpiece diameters that the FREE can grip for acceptable input actuation pressures and/or acceptable strains in the elastomer. As a simplifying assumption, we will consider the deformed profile to be a pure helix,¹⁶ and then extend the design rules for more general and realistic deformation that includes the effect of gravity.

Modeling spatial deformation of spiral FREEs: helix assumption

In an ideal scenario where no external forces act, spiral FREEs deform helically with a uniform pitch and radius along its length when pressurized. In this scenario, the radius of the helix reduces and the number of turns increases as a function of applied pressure. The helix radius and the maximum number of turns may be limited by the FREE deformation: stretch ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} , and/or applied pressure P, and fiber angles \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} . In this section, without loss of generality, we consider the workpiece to be grasped as a cylinder with radius \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp}}$$ \end{document} and length \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp}}$$ \end{document} . Using simple helical equations, we formulate a design chart indicating the workpiece range that can be successfully gripped. This would assist in the design selection of fiber angles and geometry parameters (radius, etc.) of the spiral FREE actuator for a given gripping requirement.

A pure helix is formed due to the combination of a uniform curvature \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\kappa$$ \end{document} and torsion \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\tau$$ \end{document} acting on a cylinder, which for a FREE is given in Equations (9)–(10). The radius of the helix can be expressed as: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { r_ { helix } } = \frac { \kappa } { { { \kappa ^2 } + { \tau ^2 } } } . \tag { 11 } \end{align*} \end{document}

The radius of the cylindrical workpiece \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp}}$$ \end{document} gripped by the spiral FREE can be greater than or equal to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{helix}}$$ \end{document} , but corrected for the FREE radius \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{act}}$$ \end{document} and is given as: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { r_ { wp } } \ge { r_ { helix } } - { r_ { act } } = \frac { \kappa } { { { \kappa ^2 } + { \tau ^2 } } } - { r_ { act } } . \tag { 12 } \end{align*} \end{document}

The curvature \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\kappa$$ \end{document} and torsion \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\tau$$ \end{document} can be expressed as a function of axial stretch ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} as seen from Equations (9)–(10). The axial rotation parameter \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta$$ \end{document} that occurs in Equation (10) can be expressed with respect to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} by simultaneously solving Equations (1)–(2). The equations are further simplified by assuming no radial expansion ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _2} = 1$$ \end{document} ), due to one of the fiber angles being circumferential (α = 90°). With the above substitutions, the ratio of the cylinder radius that can be gripped, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp}}$$ \end{document} to the FREE radius, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{act}}$$ \end{document} can be expressed as a function of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} (one of the fiber angles) is given by: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { { r_ { wp } } } { { r_ { act } } } } \ge { \frac { { \lambda _1 } - 1 } { ( { \lambda _1 } + 1 ) \left( { \frac { 1 } { 4 } { { \left( { \sqrt 2 { { \sec } ^2 } ( \beta ) \sqrt { { { \cos } ^2 } ( \beta ) \left( { \lambda _1^2 ( - \cos ( 2 \beta ) ) - \lambda _1^2 + 2 } \right) } - 2 \tan ( \beta ) } \right) } ^2 } + { \frac { { { ( { \lambda _1 } - 1 ) } ^2 } } { { { ( { \lambda _1 } + 1 ) } ^2 } } } } \right) } } - 1. \tag { 13 } \end{align*} \end{document}

The equation reveals that the minimum workpiece radius decreases with decrease in fiber angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} and increase in axial stretch \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} (Fig. 5a).

FIG. 5.

Insight into selection of fiber angles using the helix assumption. For a given axial stretch of the spiral FREE, (a) the minimum ratio of work piece radius to actuator radius, (b) ratio of minimum length of workpiece to actuators radius, and (c) ratio of minimum length of actuator to actuators radius needed for a perfect grip for different fiber angles ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} ) with α = 90° (* when the parameters are estimated using α = 88° and β = 50°).

However, this alone does not ensure successful grip. Recent literature on continuum manipulators have shown the ability to achieve form closure by forming a spiral around an object with less than one complete turn provided there is a non-negligible friction in the grip.^28–30 Such a gripping configuration restrains the object from all sides with sufficient contact points, thus ensuring form closure. However, because the friction coefficient is unknown, or to account for the conservative scenario of very low friction coefficient, we aim for slightly greater than one complete turn around the object as the necessary and sufficient condition for gripping.

To successfully spiral around the cylinder forming one complete turn, the FREE must have a certain minimum length. If the spiral were a perfect helix then the length of the FREE for n turns around the cylinder is given by: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { l_ { act } } \ge { r_ { helix } } n2 \pi \sqrt { 1 + { \frac { { \tau ^2 } } { { \kappa ^2 } } } } , \tag { 14 } \end{align*} \end{document}

The above equation determines the minimum permissible slenderness ratio ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{act}} / {r_{act}}$$ \end{document} ) of the spiraling FREE required to successfully grip a cylindrical workpiece of radius \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp}}$$ \end{document} . This value decreases with increase in axial strain \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} and with decrease in the fiber angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} (Fig. 5b). Using this chain of thought, we can evaluate the minimum workpiece length ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp}}$$ \end{document} ) that can be gripped by the FREE from the circular helix equations as: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { { l_ { wp } } } { { r_ { act } } } } \ge 2.5 \pi \left( { { \frac { { r_ { wp } } } { { r_ { act } } } } + 1 } \right) \frac { \tau } { \kappa } . \tag { 16 } \end{align*} \end{document}

The lower limit on the workpiece length \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp}}$$ \end{document} shows a similar dependence on \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} as the FREE length \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{act}}$$ \end{document} (Fig. 5c). Furthermore, it is seen from Figure 5b, c that the minimum workpiece length is lower than the minimum actuator length.

Figures 5a–c yield insight into the selection of the fiber angle, radius, and length of the FREE actuator, and the corresponding workpiece sizes that it can grip. For example, using the curves of Figure 5c, a FREE with β = 60° and radius \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{act}} = 5 \;{ \rm{mm}}$$ \end{document} can grip a cylinder of radius \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp}}$$ \end{document} = \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$20 \;{ \rm{mm}}$$ \end{document} or more with a minimum of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$15 \%$$ \end{document} axial stretch. Even if this condition is satisfied, the grip would fail if the length of the workpiece \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp}}$$ \end{document} is lower than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$250 \;{ \rm{mm}}$$ \end{document} or the length of FREE actuator \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{act}}$$ \end{document} is lower than \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$350 \;{ \rm{mm}}$$ \end{document} . While this design space is plotted based on the assumption that the deformation behavior is an ideal helix, introducing nonidealities can alter the curves presented in Figure 5b and c. Deviation from helical deformation is usually due to body forces such as self-weight and other interaction forces. A mechanics-based model is required to consider the effect of forces.

Mechanics-based modeling of spiral FREEs

To motivate the need for a mechanics-based model that accounts for the effect of gravity loads, we compare the ideal helical deformation of the FREE¹⁶ and the exact deformation profile with the consideration of self weight under gravity (Fig. 6a). The former has a constant pitch, whereas the pitch of the latter varies with length. It can also be observed that the error between the two shapes is small for small FREE lengths,³¹ whereas for longer lengths, the mismatch is significant. In this paper, we envisage long and slender FREEs, which are greatly influenced by self-weight, thus necessitating the determination of actual deformed shapes.

FIG. 6.

Comparison between helical and exact models and perfect grasp scenario. (a) Predicted deformation profiles with and without consideration of self-weight for different FREE lengths. As FREE length increases there is a larger mismatch in the overall shape (gravity acting in positive Z direction). (b) An illustration of a perfect grasp scenario. (c) The section along the tangent at end point is inclined to the cylinder axis and (d) the cut section is an ellipse when viewed along the normal of the plane.

In this section, we model the spiral FREE using Cosserat rod theory.^{23,24,32–34} Cosserat theory has been shown to be geometrically exact in three dimensions, and has been used before to predict the deformed shape of continuum robots. An important assumption made here is that the FREE, when pressurized responds elastically to external loads, which is governed by linear stress versus strain relationship. We further simplify the model by neglecting axial and shear deformation. This is a valid assumption as the single straight fiber prevents any change in length of the actuator, and the consideration of slender FREEs justifies neglecting shear. The equations for static equilibrium for Cosserat rod mechanics, and the solution methodology are presented in Appendix A1. The model gives out the deformed position vectors ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{r}} ( s )$$ \end{document} ) and its slopes ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta ( s )$$ \end{document} or \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{ \dot r}} ( s )$$ \end{document} ) as a function of the FREE length s. Using the deformed parameters of these models, simple equations can be used to evaluate the minimum workpiece radius, length, and other relevant quantities. These are detailed below.

Determining workpiece radius for perfect wrap

We define the onset of a perfect wrap (from Fig. 6b) as the pressure at which the entire FREE length begins to contact the cylindrical work piece. At this instant the end curvature of the deformed FREE must be geometrically compatible with the curvature of the cylinder. It is seen in Figure 6c that the FREE cross-section is inclined at angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{end}}$$ \end{document} with respect to the cylinder axis. Thus, to ensure geometric compatibility, the FREE end curvature must be equal to the curvature of an ellipse that results when the cylinder is sectioned along the inclination angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{end}}$$ \end{document} . Such an ellipse is shown in Figure 6d, and its curvature \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \kappa _{ellipse}}$$ \end{document} at the point of contact with the FREE end is given by: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \kappa _ { act } } = { \kappa _ { ellipse } } = { \frac { { { \sin } ^2 } { \theta _ { end } } } { { r_ { wp } } + { r_ { act } } } } , \tag { 17 } \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp}}$$ \end{document} is the radius of the cylindrical workpiece and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \kappa _{act}}$$ \end{document} is the end curvature of the FREE. To evaluate \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \kappa _{act}}$$ \end{document} , we make an important assumption that there is no external moment acting on the FREE end, which makes its value equal to the curvature evaluated using Equation (9). The next term to be evaluated is the inclination angle of the FREE end, that is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{end}}$$ \end{document} . Here again, we make an assumption that the end inclination of the FREE upon gripping the cylinder is equal to the end inclination without gripping any object (i.e., with just self-weight acting). The value of the end inclination can be solved by applying the Cosserat beam equations in Appendix A1, and is determined by the rotation field at the end (which is detailed in Appendix A3). Thus, using Equations (17) and (9), we relate the FREE end curvature to the minimum cylinder radius that it can grip. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { r_ { wp } } = { \frac { { r_ { act } } ( { \lambda _1 } ( P ) ( { { \sin } ^2 } { \theta _ { end } } - 1 ) + 1 + { { \sin } ^2 } { \theta _ { end } } ) } { ( { \lambda _1 } ( P ) - 1 ) } } , \tag { 18 } \end{align*} \end{document}

Determining minimum workpiece length for successful grip

Equation (18) can be solved numerically to determine the minimum pressure P required such that the entire FREE length is in contact with the workpiece of radius \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp}}$$ \end{document} . However, a successful grip is realized only when the FREE is long enough to make at least one turn across the workpiece. If the spiral were a perfect helix, inclined at an angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{sp}}$$ \end{document} with respect to the cylinder axis, then the length of the FREE for n turns around the cylinder is given by: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { l_ { act } } = { \frac { 2n \pi { r_ { cyl } } } { \sin { \theta _ { sp } } } } . \tag { 19 } \end{align*} \end{document}

However, in a practical scenario, the spiral angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{sp}}$$ \end{document} is not uniform along the FREE length. It varies from a value \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{end}}$$ \end{document} at the FREE end to a larger value at the FREE base. For our purposes we will use the average of the inclination values through the length of the FREE for evaluating the length. The inclination values at different points along the length of the FREE can be obtained by evaluating the rotation field \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{R}}$$ \end{document} at these points (Appendix A1). Furthermore, to ensure one full turn we use \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$n = 1.25$$ \end{document} (if it is a perfect helix as obtained in previous section, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$n = 1$$ \end{document} assures exactly one turn) to accommodate for the deviation from perfect helix caused due to self-loading.

Similarly, the minimum length of the workpiece \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp}}$$ \end{document} that can be accommodated will depend on the minimum FREE length evaluated in Equation (19) and is given by: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { l_ { wp } } = { \frac { 2n \pi { r_ { cyl } } } { \tan { \theta _ { sp } } } } . \tag { 20 } \end{align*} \end{document}

Analysis results

Equations (18) and (20) determine the minimum workpiece radius and length that can be gripped by a FREE with the inclusion of external forces such as gravity. The proposed analysis is implemented on two FREE prototypes whose fiber angles are α = 88°, β = 50°, and 60°, respectively. Their elastomer material properties are given in Appendix A2. α = 88° was considered instead of α = 90° because of ease of fabrication. The difference between considering α = 88° and α = 90° for three design parameters (for β = 50°) leads to less than 1% variation as shown in Figure 5a–c. The length and radius of the actuators considered for the analysis are 500 and 4.7625 mm, respectively. Figure 7a shows that the minimum grippable workpiece radius reduces as actuation pressure increases. This implies that larger pressures are needed for grasping smaller objects. Furthermore, gravity has some effect (∼15%) on the minimum grippable radius especially at lower pressures.

FIG. 7.

Comparison of the minimum (a) workpiece radius and (b) minimum workpiece length required for successful grip using the helical approximation and the mechanics-based model. For the design problem 70° actuators parameters are estimated using Mooney Rivlin constants C1 = 0.6 and C2 = 0. The 70° plot is used for the length correction before fabrication.

For each pressure value, the minimum length of the FREE required to complete one turn according to Equation (19) is plotted in Figure 7b. It is seen that workpiece with smaller lengths can be gripped at higher pressures. Furthermore, the inclusion of gravity leads to a 40% increase at most in the estimation of minimum FREE length at smaller pressures. To summarize, the pure helical approximation may be good enough for estimating the workpiece radius but yields large errors in estimating the workpiece lengths. This has implications in the design, as will be seen in the next sections.

Experimental Validation

We validate our proposed analysis for estimating the exact shape of the spiral FREE and the radius of the objects that can be gripped in this section on two prototypes. Table 1 gives the parameters used for both the prototypes.

Table 1.

Parameters for Prototypes Used for Validation

Prototype	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} (degrees)	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} (degrees)	L (m)	Radius (m)
P1	88	50	0.392	4.7625e-3
P2	88	60	0.46	4.7625e-3

L, length of the actuator.

Exact shape estimation

To verify the shape predicted by the Cosserat model, the spiral actuator is fixed at one end and is actuated to different pressures in steps of 13.7895 kPa (2 psi). The shape of the spiral actuator is obtained using a Microscribe 3D digitizer as shown in Figure 8a and b.

FIG. 8.

Experimental setup and validation of exact model. (a) Setup for data collection for obtaining deformed shape of the spiral FREE using Microscribe. (b) Comparison of analytical (Cosserat model) and experimentally obtained deformation profiles (XZ and YZ views) at different pressures (axis dimensions in meters). Analytically obtained shape is in red and experimental points are in green (gravity acting in positive Z direction). Color images available online at www.liebertpub.com/soro

The experimental and analytical final shapes of the actuator at 55 kPa (8 psi) and 110 kPa (16 psi) is shown in Figure 8c–f with both XZ and YZ views. The end point of the analytical shape matches the experimental shape with an average error of 5.1% of total length for first prototype (P1) and 6.1% of total length for prototype 2 (P2). These measures are reasonable when compared to those reported in literature for similar models.²³ This validates that the Cosserat model gives a reasonable estimate for the final shape of the FREEs.

Gripping of objects of different radius

A set of 3D printed cylinders with radius 10, 12, 14, 15, 20 and 25 mm are used, and the range of pressures at which successful gripping occurs (till curling is observed) is recorded. Figure 9 shows the analytically estimated pressures from the Cosserat analysis and experimentally obtained pressure ranges for different prototypes. For prototype 2 our prediction is reasonable, whereas for prototype 1 the analysis predicts in the lower range of pressures as observed in Figure 9.

FIG. 9.

Comparison of estimated pressure for perfect grasp using the analytical mechanics-based model and experiments for different workpiece radius. The experimental estimates are given as a range of pressures between onset of perfect grasp and beginning of curling, all identified visually.

Guidelines for Design

In this section, a design framework is presented for selecting the geometry variables of a spiral FREE to grip different workpiece objects within a given radii range denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp \min }}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp \max }}$$ \end{document} , respectively, and within length range \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp \min }}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp \max }}$$ \end{document} , respectively. We need to determine the radius of the FREE \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{act}}$$ \end{document} , its length \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{act}}$$ \end{document} , and the fiber angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} that satisfactorily grip the desired workpiece sizes. For this, we first use the nondimensional curves of Figure 5, which are based on helical assumptions, and then verify the design using a more realistic Cosserat analysis framework of Appendix A1. Below, we outline with explanation the different steps in the design process.

In this example, we are interested in designing a FREE-based spiral gripper that can successfully grip a range of objects between a tubelight and a PVC pipe whose parameters are given in Table 2. In other words, any regular or irregular object whose size lies within the radius and length ranges of these two objects will be gripped.

Table 2.

Parameters of the Workpiece Objects That Are Required to Be Gripped

S. no.	Object	Radius ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wpi}}$$ \end{document} ), mm	Length ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wpi}}$$ \end{document} ), m	r_wp/r_act	l_wp/r_act
1	Tubelight	11	1.2	2.31	251.97
2	Plastic pipe	30	0.5	6.30	104

1. Step 1: Choose the radius of the FREE actuator ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{act}}$$ \end{document} ) based on ease of fabrication. This is intended as a free choice guided mostly by fabrication constraints, and may have to be smaller than the smallest workpiece radius. Here we choose \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{act}}$$ \end{document} to be 4.76 mm.

2. Step 2: Evaluate minimum and maximum workpiece radius ratios ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp \min }} / {r_{act}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp \max }} / {r_{act}}$$ \end{document} ) and length ratios ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp \min }} / {r_{act}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp \max }} / {r_{act}}$$ \end{document} ). These will then serve as inputs to Figure 5. These values are given in Table 2.

3. Step 3: Choose the appropriate fiber angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} from Figure 5a based on the required radius ratios in Table 2. This step offers some freedom, but a prudent choice of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} can be made by negotiating the trade-off between maximum stretch ratio and FREE length. The FREE is maximally stretched when gripping smaller workpiece radii \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp \min }}$$ \end{document} . Thus, a fiber angle is chosen from Figure 5a such that the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _1}$$ \end{document} corresponding to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp \min }} / {r_{act}}$$ \end{document} (which in this case is 2.31 from Table 2) is small and manageable. From Figure 5a, this may imply choosing a smaller fiber angle. However, smaller fiber angles require lower stretch ratios to accommodate larger workpiece radius \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp \max }}$$ \end{document} as seen in Figure 5a, which in turn will require larger FREE lengths to complete one full turn around the workpiece as seen in Figure 5c. Thus, the best choice for fiber angle \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} would limit the maximum stretch ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _{1 \max }}$$ \end{document} , while simultaneously maintaining sufficient minimum stretch ratio \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _{1 \min }}$$ \end{document} to grasp larger workpiece radii. In this example, we choose β = 70° as it meets both radii within acceptable stretch ratio ranges. The stretch ratio ranges [ \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _{1min}} , { \lambda _{1max}}$$ \end{document} ] are recorded for this example in Table 3.

4. Step 4: Select the FREE length. The FREE length is chosen on the consideration that at least one complete turn can be obtained while gripping a workpiece of maximum radius and length. It can then successfully grip smaller objects by wrapping with more than one turn. First, we will design the FREE length for the maximum radius requirement, and in the next step check to see if it satisfies the length requirement. The FREE length is thus determined by the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{act}} / {r_{act}}$$ \end{document} value in Figure 5c that corresponds to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _{1 \min }} = 1.17$$ \end{document} . We get \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{act}} / {r_{act}} = 76$$ \end{document} , thus the length of the FREE is 362 mm.

5. Step 5: Check if ratio of length of workpiece to actuator radius is satisfied from Figure 5b. For the range of stretch ratios \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$[ { \lambda _{1 \min }} , { \lambda _{1 \max }} ]$$ \end{document} determined from the previous steps, we obtain the values of the minimum permissible workpiece lengths from Figure 5b. The two values as seen in Table 3 are 49.7314 and 26.7280, which are both much lower than the ratios in Table 2. This implies that the FREE length chosen in step 3 can accommodate the workpiece lengths. If this condition were not satisfied, then the slenderness of the workpiece was not large enough to warrant a spiral FREE as a gripper.

6. Step 6: Exact correction. The previous design steps are based on the helical assumption. Influence of gravity changes the radius range and the workpiece lengths that can be gripped. For our example from the helical assumption we chose the actuator length to be 361 mm. We thus estimated the change in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp}}$$ \end{document} for the β = 70° FREE using the Cosserat model. The material properties used were evaluated from our previous test results ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${C_1} = 0.6 \,{ \rm{MPa}}$$ \end{document} ). The results in Figure 7a and b demonstrate a deviation in radius and length of workpiece for helical assumption and exact model. Based on this data, to accommodate for the gravity we selected the length of the actuator to be greater than 40% of length obtained by helical assumption. Here we choose it to be 500 mm.

7. Step 7: Exact design space obtained. Now the parameters of the actuator are decided, so it is fabricated and material properties are obtained (in Appendix A2). Finally the pressure versus object radius is obtained, and the pressure at which the actuator grips the objects is determined and is shown in Figure 10a.

FIG. 10.

Comparing the analytical and experimental pressures to grip the specified objects. (a) Prediction of the range of workpiece radii that can be successfully gripped by the designed prototype as a function of applied pressure. This is compared with the experimental pressure range for gripping the PVC pipe and the tubelight, as two extreme cases that define the working range for the gripper. Demonstrating the ability of the designed spiral FREE to successfully grip objects of varying diameters: (b) Tubelight, (c) a PVC tube, and (d) a workpiece object of varying diameter.

Table 3.

Strains and Minimum Workpiece Length Needed When 70° Actuator Is Selected

Fiber angle ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} )	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _{1{ \rm{min}}}}$$ \end{document}	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \lambda _{1{ \rm{max}}}}$$ \end{document}	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp}} / {r_{act}} ( { \lambda _{1 \min }} )$$ \end{document}	\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{wp}} / {r_{act}} ( { \lambda _{1 \max }} )$$ \end{document}
70	1.1696	1.3443	49.7314	26.7280

Figure 10b and c shows the gripping of the tubelight and the PVC pipe by the designed prototype. Figure 10d shows that the prototype can also grip a cylinder of varying radius that varies between the radius of tubelight (11 mm) and PVC pipe (30 mm).

Discussion

Design of spiral FREEs as grippers involve selecting the fiber angle at the local material scale and mapping it to the global deformation, which determines the size of the workpiece objects that can be successfully handled. Though the effect of gravity on the deformation pattern is considerable, FREE grippers can be successfully designed using pure helical assumptions alone, if sufficient margins are accounted for during the process, such as making the FREE length longer than needed as in Step 6. Furthermore, to accommodate the change in spiral radius of the FREE due of gravity, Steps 2 and 3 can be reworked with slightly larger workpiece radii ranges (i.e., larger \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${r_{wp \max }} - {r_{wp \min }}$$ \end{document} ). However, it is important that the workpiece to be gripped is slender, which means that its length is larger than its radius, and specifically much larger than the ratios in Figure 5b or Step 5.

Conclusions

Gripping action obtained by spirally twirling the gripper around an object can be helpful for handling long, slender, and irregular shaped objects. Although several recent publications have alluded to the usefulness of spiral configuration using fiber-reinforced pneumatic actuators in grasping underwater objects, for snake-like motion and for navigation in a pipe, there has been no formal attempt in investigating the mechanism and formulating guidelines to systematically design them. In this paper, we investigate a spiral configuration of pneumatically actuated FREEs with two families of fibers that are asymmetrically oriented.

The most significant contribution of the paper is in proposing two simplified mechanics-based models to analyze and design the FREE-based fiber gripper. The first model considers the deformed profile to be an ideal helix, while the second considers a Cosserat rod model with acting gravity forces. The Cosserat model predicts the deformation with an error less than 8% of the actuator length. The two models can help inform the design of the FREE gripper—the fiber angles, cylinder radius, and length—based on the range of workpiece objects to be manipulated. This has been presented in the paper through systematic guidelines. These reduced-order models are important because alternative models such as an extensive finite element analysis can be time consuming and numerically expensive. We have demonstrated the efficacy of the reduced-order model through experimental validation by successfully designing a FREE gripper for grasping a range of slender objects ranging from a tubelight to a larger PVC pipe.

Though successful, there are some limitations of the work presented that can be mitigated in future work. For example, the modeling and design guidelines are limited to a unique class of FREEs with two families of fibers and a single fiber. Out of these, one fiber family is circumferential (α = 90°), the other family \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} is a design variable, and the single fiber is axial running along the length of the FREE. There are several other classes of FREEs, especially with a helical single fiber yield a spiral configuration as demonstrated by a more extensive design space investigation. Furthermore, we limit our consideration to cylindrical FREEs having circular cross-section profiles, thus precluding the benefits of other cross sectional profiles as seen in Ref.³¹ Finally, we base our conclusions about the ability to grip and the design guidelines on the deformed profile by considering only the body loads due to gravity. A more thorough investigation would include estimating the exact FREE profile, considering contact forces between gripper and the object. This would better estimate the onset of grip, gripping forces, and quality of grip. However, obtaining contact forces is numerically intensive and may not be useful for design.

Footnotes

Acknowledgment

This work was supported by NSF CMMI-1454276.

Author Disclosure Statement

No competing financial interests exist.

Appendix A1. Mechanics-Based Modeling of Spiral Fiber Reinforced Elastomeric Enclosures Using Cosserat Rod Theory

In this section, we present the basic equations required to predict the deformed shape of Fiber Reinforced Elastomeric Enclosures (FREEs) under the action of external loads. These equations have been presented earlier^A1–A4 to model long and slender members routinely used in robotics.

The FREE shape is characterized by its unstretched central axis curve shown in Figure A1a. The position of any point \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s \in [ 0 , L ]$$ \end{document} on the curve is represented by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{r}} ( s ) \in {{ \rm{R}}^3}$$ \end{document} , and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{R}} ( s ) \in SO ( 3 )$$ \end{document} is the rotation matrix which converts the local coordinate frame to the global coordinate frame. The variables \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{v}} ( s )$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{u}} ( s )$$ \end{document} denote the linear and angular rates of change of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{r}} ( s )$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{R}} ( s )$$ \end{document} in the local frame, respectively. The following equations determine the final shape of the actuator:

From the force and moment balance equation and taking their derivative with respect to s, we can arrive at the following set of equations for a section of rod.^A1–A4

From Equations (A1)–(A6) we can arrive at the following system of equations where we assume no body moments ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{l}} = 0$$ \end{document} ), no shear, and no axial extension of its length. Thus by substituting \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{v}} = { [ 0 \, \, \,0 \, \, \,1 ] ^T}$$ \end{document} we obtain:

To estimate exact shape of actuator, a set of ordinary differential equations (in Eq (A7)) are solved as a boundary value problem. At \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s = 0$$ \end{document} , as the actuator is fixed to a base it provides us the initial condition on \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{r}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{R}}$$ \end{document} . Finally, at \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s = L$$ \end{document} as there are no tip loads in our model, the following holds: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{u}} ( s = L ) = {{ \bf{u}}_{ \bf{0}}}$$ \end{document} . In this work, we used Matlab bvp4c to solve the boundary value problem. \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{f}}_{ \bf{b}}}$$ \end{document} is the weight per unit length of the actuator and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{f}}_{ \bf{e}}}$$ \end{document} is the end load acting on the actuator e is the direction of gravity acting in the global direction (example e = [0 0 1] if gravity is acting in positive Z direction) and g is acceleration due to gravity.

Appendix A2. Estimation of Material Properties

To obtain a relation between actuation pressure and extension parameter, we need to estimate the Mooney Rivlin constants C₁ and C₂ of the elastomer. We take a small section of the fabricated actuator with fiber angles \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\alpha$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta$$ \end{document} , and extension of the actuator at different pressures is measured using a linear encoder. Two prototypes with the fiber angles shown in Table A1 are fabricated and the extension parameter with varying pressure is experimentally obtained and plotted in Figure A1b. The analytical estimate of the extension as a function of pressure can be found by solving Equation (8), and the values of C₁ and C₂ are varied until a good match is obtained with the experiments. These values for the constants are shown in Table A1. It is to be noted that in the pressure ranges that the experiment was conducted (less than 26 psi), the model reduces to a Neo-Hookean model.

A value of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$E = 7e5{ \rm{N / }}{{ \rm{m}}^{ \rm{2}}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$G = 4.7e5{ \rm{N / }}{{ \rm{m}}^{ \rm{2}}}$$ \end{document} (with Poisson's ratio .5) is used for this analysis. The FREE is fixed horizontally at one end and allowed to deform. The experimental deformed shape under gravity is used to fit the value of E using the Cosserat rod equations.

Appendix A3. Evaluation of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{end}}$$ \end{document}

\documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{end}}$$ \end{document} is the angle between the tangent at the end point of the spiral actuator and the central axis of the cylinder which it can grip as shown in Figure 4b. Let the cylinder axis orientation be vector \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{v}}_{ \bf{c}}}$$ \end{document} and tangent at the end point be \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{t}}_{{ \bf{end}}}}$$ \end{document} . \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _{end}}$$ \end{document} is obtained using the cross product and dot product between \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{t}}_{{ \bf{end}}}}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{v}}_{ \bf{c}}}$$ \end{document} as shown in Equation (A8).

Solving Equation (A7) provides the orientation information ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{R}}$$ \end{document} ) of the spiral actuator in the global reference frame. The Rotation matrix at any given point ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s \in [ 0 , L ]$$ \end{document} ) comprises of the normal, binormal, and tangent vector information. Therefore, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{t}}_{{ \bf{end}}}}$$ \end{document} is obtained from orientation information ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bf{R}}$$ \end{document} ) at \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$s = L$$ \end{document} .

To estimate the orientation vector, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${{ \bf{v}}_{ \bf{c}}}$$ \end{document} , singular value decomposition is used on the tangent information of the last 10% of the length of the actuator. We assume that the cylinder axis would be more aligned in direction normal to tangents of final section.

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