Designing Fiber-Reinforced Soft Actuators for Planar Curvilinear Shape Matching

Abstract

Fiber-reinforced soft pneumatic actuators can generate a wide variety of deformation behavior, making them popular in the field of soft robotics. Designing an actuator to meet a specified deformed shape is an important step toward the design of soft robots. We present a two-step methodology to design an actuator that matches a given planar curve on pressurization. In the first step, the curve is divided into a series of constant curvature (CC) segments that best approximate its shape. The second step involves designing a bending actuator by determining its fiber orientations for each CC segment. Further, this two-step method is extended to match two curves: a final deformed curve and an intermediate curve at a lower actuation pressure. On combining all the CC segments, the resulting actuator lies along a straight line unpressurized, and on pressurization deforms to trace the desired final curve through a preset intermediate curve. To demonstrate the method, we show different examples: an omega curve for an inchworm robot, an acronym SoRo for the Soft Robotics Journal, and a two-stage bending actuator.

Introduction

Soft robots^1–5 are increasingly becoming popular due to their (1) inherent compliance, making them safe for human interaction; (2) high power-to-weight ratio; and (3) simplicity of controls. Among pneumatically actuated soft robots, fiber-reinforced actuators are popular because of their ability to generate diverse motion patterns, produce large actuating forces, and their simplified design and fabrication.^6–16 Fiber-reinforced elastomeric enclosures (FREEs) are soft pneumatic actuators made of a hyperelastic tube of a circular cross-section with two sets of fibers wrapped in a helical shape on the external surface of the tube as shown in Figure 1. These actuators can generate a wide range of motion behavior, including extension, contraction, rotation, bending, and spiral shapes based on the angle of wrapped fibers.^6,16 There are multiple methods of analyzing the mechanics of FREEs^7,8,17 but there is a need for design methods that can guide a soft roboticist to design an actuator for any given application. This problem can have applications in designing soft continuum manipulators,¹⁴ matching the finger motions^15,16 or actuators for soft robot locomotion.

FIG. 1.

The bending behavior of FREE actuators with equal and opposite fiber angles is prompted by (a) a third strain limiting fiber running along its length. (b) Demonstration of bending in a fabricated FREE actuator. FREE, fiber reinforced elastomeric enclosure.

A FREE can be considered as a metamaterial whose deformation is dependent on global design variables such as the cylinder length, diameter, and hyperelastic material properties, and local fiber architectures that are reinforced onto the cylinder. Connolly et al.¹⁵ presented a nonlinear elasticity-based approach to analyze multisegment FREEs with varying fiber angles. They then used an optimization-based inverse design framework to determine the fiber orientations in each segment that best matches the trajectory of the deformed FREE with a required deformed shape. Their work considered various building blocks such as expanding, rotating, and bending segments and the inverse design was demonstrated to match three-dimensional (3D) trajectories to replicate the motion of the thumb and index fingers during grasping operation. Although their work is seminal in the space of fiber-reinforced actuator design, it relies on extensive computation by solving for both the global (segment length) and local design variables (fiber orientations) simultaneously. The solutions obtained are feasible for the specific problems considered, but their generalization to meeting arbitrary shapes may not be straightforward. Further, a purely computational framework provides limited design insight on the existence of a solution, or availability of alternate solutions.

In this article, we present a deterministic sequential approach toward designing multisegment fiber orientations for FREEs using an inverse design framework and nondimensional design charts. We limit our focus to match planar curves using purely bending segments with the reasoning that the insight obtained can be extended to spatial and more complex curves. In other words, given a planar curve, we design a FREE actuator that when unactuated lies along a straight line and on pressurization changes shape to closely match the shape of this curve. Our method relies on a sequential two-step approach where we first discretize a planar desired curve into a number of constant curvature (CC) segments, and then determine fiber angles for each bending segment to match the deformed shape. In the first step, the designer can choose the number of discretized segments based on the desired shape matching accuracy or ease of fabrication. After the segmentation, the next step is to design a FREE actuator for each CC segment by using bending FREEs. Bending FREEs as shown in Figure 1 are essentially FREEs with two sets of equal and opposite fibers that generate extension or increase in length on pressurization. By adding a single straight fiber¹⁸ or a strain limiting layer,¹² these extending actuators can generate bending motion where the curvature of the bend is constant at any point along its length. Therefore, by designing such a bending actuator for each CC segment that matches its required curvature at a fixed actuation pressure, we can obtain a FREE that traces the shape of the given curve.

Our sequential approach results in a rather simplified process for designing FREEs to match a single desired curve. First, the segmentation algorithm divides the desired curve into a number of CC segments and gives the desired curvature for each segment. Next, the fiber orientation of each segment is determined to match the desired curvature. Therefore, we only have one design parameter in this case, which is the fiber orientation. However, when we are required to match an intermediate curve en route the final curve, the process is slightly more complex, yet insightful. To meet the additional requirement of matching the intermediate curve, we introduce a second design parameter, the cylinder thickness. Our method throws light on the existence and feasibility of the solutions using nondimensional design charts. For feasibility, the intermediate curves must have sequentially lower curvatures in all segments than the final curve. Thus, we demonstrate how seemingly different intermediate deformed curves can be achieved by a single FREE design by just modulating actuation pressure.

In the subsequent sections, first we describe the two-step design process to design the actuator for any given curve followed by demonstrating the method using different example problems. In the Design Methodology for Single Shape Matching section, we present the algorithm to divide any given planar curve into a given number of CC segments. This is followed by a modified version of the constrained maximization algorithm⁸ that solves for the required fiber angles to generate the given curvature. This method essentially solves the inverse design problem of determining the geometry parameters of an actuator for a given curvature. Demonstration of the presented design methodology is shown in the Applications section, where different example problems are shown to explain the design process. After this, in the Shape Matching for Two Curves section, the method is extended for the case where an intermediate shape is given in addition to the final shape. A nondimensional chart-based approach to obtain a feasible solution is shown followed by an example problem going over the steps involved in the method.

Design Methodology for Single Shape Matching

In this section, we describe the two-step design method for designing a FREE to match the shape of a given curve. First, the algorithm for optimal segmentation of the curve into smooth piecewise CC segments is shown. This is followed by designing the fiber orientations of the actuator to match each CC segments.

Smooth piecewise CC segmentation

Low computational approximation of digital curves is an active area of research in the field of computer graphics and animation. For planar curves, different methods such as polygons, circular arcs, clothoids, splines, etc. have been used to obtain an approximation.¹⁹ A bending FREE on actuation deforms to take the shape of a circular arc or a CC segment. Concatenating multiple such bending FREEs along the same cylindrical tube as the core can result in an actuator that, on pressurization, deforms as a series of smooth circular arcs to best match the required curve. The smoothness here refers to the property that the tangent (or slope) at the end point of one arc coincides with the tangent at the start point of the next arc. Therefore, we are interested in approximating a curve using smooth piecewise CC segments. Toward this, we have used an algorithm that uses dynamic programming to obtain the best segmentation such that the least squares error of the Euclidean distances between the original curve and CC approximation is minimized.

Given a curve of n ordered points, we are interested in approximating this curve using m circular arcs or CC segments. The set $S$ of given n points is denoted by $S = \{(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})\}$ . To obtain the optimal approximation, we first choose a subset of $S$ containing $m + 1$ vertices $V = \{v_{1}, v_{2}, \dots, v_{m + 1}\}$ , where $v_{1} = (x_{1}, y_{1})$ and $v_{m + 1} = (x_{n}, y_{n})$ . After choosing the optimal set $V$ , a CC segment is fitted between each pair of adjacent points in $V$ .

We have used a dynamic programming algorithm that is inspired by Refs.^20,21 to obtain the optimal $V$ from a given $S$ . In this algorithm, to best fit m CC segment between the first point and last point n, first $(m - 1)$ CC segments are fitted between the kth and nth points and one segment between the first and the kth point. The kth point is chosen in such a way that the approximation error of fitting m arcs between the first point and n is minimum. For the approximation error, we use the least square measure of the distance between the given points and fitted arcs.

After determining the optimal set $V$ of $m + 1$ vertices using the dynamic programming algorithm that best segments the given curve, the next step is to fit a CC segment between each adjacent vertices. Our approach here is inspired by the CC inverse kinematics formulation developed for continuum manipulators.²²

With reference to Figure 2, given two points O and A where O is at the origin, an arc that passes through both O and A has its center on the x-axis. This arc is tangential to the y axis at the point O. From Figure 2, it can be shown that the curvature, $κ$ of the arc can be determined by using the coordinates of A by the following relation:

FIG. 2.

Fitting an arc between two points O and A.

κ = \frac{2 x}{(x^{2} + y^{2})} .

(1)

The angle $θ$ subtended by the arc at its center is given by $θ = \{\begin{matrix} {cos}^{- 1} (1 - κ x), i f y > 0 \\ 2 π - {cos}^{- 1} (1 - κ x), i f y \geq 0 \end{matrix} .$ (2)

This gives the arc parameters of the first section between the points v₁ and v₂. If the point v₁ does not coincide with the origin, we can apply a rigid body translation and rotation to the curve such that the point v₁ is at the origin and the tangent at this point is along the y axis without modifying the shape of the curve. To fit the next arc between points v₂ and v₃, first the given points on the curve are rotated about the center of the first CC segment such that the point v₂ is now at the origin and tangent at v₂ is along the y axis. Next, using the coordinates of v₃, $κ$ and $θ$ of the second CC segment can be determined by using Equations (1) and (2). Similarly, by rotating the points about the center of the second CC segment, v₃ can be moved to the origin and arc parameters of the third CC segment can be determined. By following these steps, all the m CC segments can be fitted between the $m + 1$ vertices.

Curvature matching

After obtaining the optimal CC segmentation and determining the curvatures and arc lengths of all the CC segments, the next step is to design the fiber orientations to match these desired curvatures. Toward this, a model that can solve the inverse design problem of determining the fiber orientations for given actuator deformation is desired. The deformation behavior of any FREE at any given actuation pressure can be obtained as shown in Refs.^7,8 by solving a constrained volume maximization (CVM) problem. The CVM is essentially a calculus of variations problem where the objective is to maximize the volume enclosed by the FREE subject to constraints imposed due to the inextensibility of fibers and strain energy stored in the elastomer tube. Now, we are interested in solving the inverse of this problem, where the deformed shape of the actuator is given and the FREE geometry and fiber orientations need to be solved. To achieve this, we modify the constrained maximization problem such that it can be solved to obtain the FREE geometry for any given deformation.

We consider a FREE as shown in Figure 3a of length l and radius r. Two sets of fibers are wrapped helically on the external surface of FREE at angles $α$ and $β$ . On pressurization, this FREE deforms in both the radial and longitudinal directions. The deformed length is $λ_{1} l$ , where $λ_{1}$ is the stretch ratio in longitudinal direction. Similarly, the deformed radius is $λ_{2} r$ , where $λ_{2}$ is the radial stretch ratio. Here, we have made an assumption that the FREE has a cylindrical shape with uniform radius throughout its length on deformation. This assumption is made to reduce the computational complexity of the problem.

FIG. 3.

(a) Undeformed shape of the FREE that on actuation deforms radially and longitudinally as shown in (b). (c, d) Relating the extension of the FREE to curvature in the presence of the strain limiting layer.

A FREE is an inflatable tube made of hyperelastic material that expands in all directions when pressurized. The fibers and elastomeric tube store energy on pressurization. Therefore, we can model a FREE's deformation by posing a constrained maximization problem where the objective is to maximize the enclosed volume while a strain energy constraint is imposed on both fibers and elastomer. This problem is shown next: ${Max}_{λ_{1}, λ_{2}} V = π r^{2} l λ_{1} λ_{2}^{2} .$ (3) $\begin{matrix} W_{f i b e r 1} = \frac{k}{2} Δ x_{1}^{2} \\ = \frac{k}{2} {(\sqrt{λ_{1}^{2} c os α^{2} + λ_{2}^{2} s in α^{2} {(\frac{θ + δ}{θ})}^{2}} - 1)}^{2} {(\frac{l}{cos α})}^{2} \end{matrix} .$ (4)

\begin{matrix} W_{f i b e r 2} = \frac{k}{2} Δ x_{2}^{2} \\ {= \frac{k}{2} {(\sqrt{λ_{1}^{2} c os β^{2} + λ_{2}^{2} s in β^{2} {(\frac{ϕ + δ}{ϕ})}^{2}} - 1)}^{2} (\frac{l}{cos β})}^{2} \end{matrix} .

(5)

W_{e l a s t o m e r} = V_{b} C_{10} (I_{1} - 3) .

(6)

In Equations (3)–(6), $W_{f i b e r 1}$ and $W_{f i b e r 2}$ are the strain energies stored in the fibers, k is the stiffness of the fibers, $\frac{θ}{2 π}$ and $\frac{ϕ}{2 π}$ are the number of turns the two sets of fibers make per unit length of the FREE, and $δ$ is the twist per unit length. For the elastomer, we assume a Neo-Hookean model with $C_{10}$ as the material constant. V_b is the volume of the elastomeric tube; I₁ is the first strain invariant and is given by $I_{1} = λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2}$ where $λ_{3} = 1 ∕ (λ_{1} λ_{2})$ .

In this article, we limit our focus on bending FREEs alone since they can undergo CC deformation in a given plane. These bending FREEs can be modeled as extending FREEs that bend due to a strain limiting fiber on one end, which can be modeled separately. An extending FREE has equal and opposite fiber angles such that $α = - β$ , which simplifies Equations (4) and (5) to a single constraint shown next: $W_{f i b e r} = \frac{k}{2} Δ x^{2} = \frac{k}{2} {(\sqrt{λ_{1}^{2} c os α^{2} + λ_{2}^{2} s in α^{2}} - 1)}^{2} {(\frac{l}{cos α})}^{2} .$ (7)

Necessary conditions for the maximization problem given by Equations (3), (6), and (7) are shown next: $\frac{\partial V}{\partial λ_{1}} + Λ_{1} \frac{\partial W_{f i b e r}}{\partial λ_{1}} + Λ_{2} \frac{\partial W_{e l a s t o m e r}}{\partial λ_{1}} = 0 .$ (8) $\frac{\partial V}{\partial λ_{2}} + Λ_{1} \frac{\partial W_{f i b e r}}{\partial λ_{2}} + Λ_{2} \frac{\partial W_{e l a s t o m e r}}{\partial λ_{2}} = 0 .$ (9)

In Equations (8) and (9), $Λ_{1}$ and $Λ_{2}$ are the Lagrange multipliers. By using virtual work analogy, it can be shown that $Λ_{1} = Λ_{2} = \frac{- 1}{P}$ , where P is the actuation pressure. Evaluating Equations (8) and (9) and substituting for $Λ_{1}$ and $Λ_{2}$ , we get $\begin{matrix} - C_{10} V_{b} (2 λ_{1} - \frac{2}{λ_{1}^{3} λ_{2}^{2}}) \\ - \frac{k λ_{1} l^{2} (\sqrt{λ_{1}^{2} {c os}^{2} (α) + λ_{2}^{2} {s in}^{2} (α)} - 1)}{\sqrt{λ_{1}^{2} {c os}^{2} (α) + λ_{2}^{2} {s in}^{2} (α)}} + π P r^{2} l λ_{2}^{2} = 0 . \end{matrix}$ (10)

\begin{matrix} - C_{10} V_{b} (2 λ_{2} - \frac{2}{λ_{1}^{2} λ_{2}^{3}}) \\ - \frac{k λ_{2} l^{2} {tan}^{2} (α) (\sqrt{λ_{1}^{2} {c os}^{2} (α) + λ_{2}^{2} {s in}^{2} (α)} - 1)}{\sqrt{λ_{1}^{2} {c os}^{2} (α) + λ_{2}^{2} {s in}^{2} (α)}} \\ + 2 π P r^{2} l λ_{1} λ_{2} = 0 . \end{matrix}

(11)

For given material constants, $C_{10}$ and k, and geometry parameters, r, l, V_b, and $α$ , we can solve Equations (10) and (11) to obtain $λ_{1}$ and $λ_{2}$ at any given pressure P by using any solver for a system of nonlinear equations. This is basically solving the forward analysis problem, where the geometry of actuator and material properties are known and we solve for the deformed shape. Now, we want to solve the inverse design problem where we know the material constants and geometry parameters except for the fiber angle $α$ . Instead, we are given $λ_{1}$ , so we can solve the system of two nonlinear equations given by Equations (10) and (11) for $α$ and $λ_{2}$ .

As mentioned earlier, we are limiting the design space to bending FREEs that generate CC $κ$ at any given pressure. This $κ$ can be related to the $λ_{1}$ , that is, the longitudinal stretch by modeling the effect of the strain limiting layer. As shown in Figure 3c and d, by adding a strain limiting layer in the form of a straight fiber, an extending FREE undergoes bending. The straight fiber prevents extension of the actuator along its length whereas the side of the actuator diametrically opposite to the straight fiber undergoes extension. This generates bending in the actuator, where the curvature of the bend $κ$ can be related to $λ_{1}$ by the following relation: $κ = \frac{λ_{1} - 1}{r (λ_{1} + 1)} .$ (12)

Therefore, from the segmentation algorithm, we get $κ$ that can be used to obtain $λ_{1}$ by using Equation (12). Following this, we solve for $α$ and $λ_{2}$ by using Equations (10) and (11).

Applications

In this section, we demonstrate how the design method described in the previous section can be used to design a FREE to match a given shape. All the FREE actuators designed and fabricated in this section have an inner radius of 4.75 mm and a wall thickness of 1.6 mm. The elastomer tube is made of natural latex rubber, and the fibers are made of cotton. The actuation pressure is chosen to be 0.172 MPa. For more details regarding the fabrication of FREEs, please refer to the Supplementary Data. Materials required and the steps involved in fabrication are shown in Supplementary Figures S1–S9.

Inchworm

The first example is inspired by the locomotion of an inchworm. An inchworm moves by deforming its body in a shape that looks like the Greek letter omega ( $Ω$ ).²³ By designing an actuator that matches the shape of omega on actuation, we can design a soft inchworm crawling robot.

We take a 300-mm-long curve in the shape of omega as shown in Figure 4a. Using the symmetry of this curve, we use one half of it to design a FREE that matches its shape. First, we use the segmentation algorithm described in the Design Methodology for Single Shape Matching section, to obtain CC approximation of one symmetric half of the curve. Using $m = 4$ , that is, three CC segments, generates an approximation that matches the original curve closely as shown in Figure 4b.

FIG. 4.

(a) Omega curve used for designing inchworm shape. (b) CC approximation of one symmetric half of the omega-shaped curve. (c) Design of omega-shaped FREE. (d) Prototype of Omega FREE. (e) Comparison of fabricated prototype with desired curve. CC, constant curvature. Color images are available online.

After obtaining the CC approximation, the curvature $κ$ of each section is used to calculate stretch ratio $λ_{1}$ by using Equation (12). $λ_{1}$ is then used as an input for the inverse constrained maximization formulation presented in the Design Methodology for Single Shape Matching section [Equations (10) and (11)] to solve for the fiber angle $α$ . Complete design of the omega-shaped FREE is given in Figure 4c. A fabricated prototype of this FREE actuated to 0.172 MPa and taking the shape of omega is shown in Figure 4d. This omega shape of the fabricated prototype is then overlaid on the original desired curve in Figure 4e to show the validity of the design method. A video of the fabricated prototype taking the shape of omega is shown in Supplementary Video S1.

This omega FREE is used as the body of an inchworm pipe-crawling robot as shown in Figure 5. The robot consists of two sets of grippers (G1 and G2 in Fig. 5) separated by the omega curve. The grippers are two sets of CC bending actuators used for grasping the pipe on which the robot is crawling. There are three actuation lines, one for G1, one for G2, and a third one for the body made of omega FREEs. By pressurizing and depressurizing the actuators in a specific sequence as shown in the series of images in Figure 5, the robot can generate forward motion along the pipe. Such a form of locomotion called “inching” is common in caterpillars and generates the longest step length, thereby generating the fastest gait for a given cycle frequency.²³ Supplementary Video S1 shows the inching gait of this pipe-crawling robot.

FIG. 5.

Omega-shaped actuator used for inching motion in a pipe-crawling robot.

SoRo

The second example is that of the acronym of the Soft Robotics journal “SoRo.” In this example, we have made four FREE actuators, one for each letter in the acronym. The two “o”s are trivial and can be designed with just one CC segment deforming to form a complete circle. On the other hand, “S” and “R” are generated by using Bezier splines and the target shapes are shown in Figure 6a.

FIG. 6.

(a) Target shapes to be matched for the acronym “SoRo.” (b) CC segmentation of S-shaped curve. (c) CC segmentation of R-shaped curve. (d) Design of FREE that matches S curve. (e) Design of FREE that matches R curve. (f) Fabricated prototypes of FREEs spelling out “SoRo.” (g) Comparison of fabricated prototypes with the desired curves. Color images are available online.

The curve for letter “S” is segmented by using five CC segments. The optimal segmentation using the dynamic programming algorithm is shown in Figure 6b. Using this segmentation, the arc parameters for each CC segment is determined and the corresponding stretch ratio, $λ_{1}$ is calculated, which is further used to calculate the fiber angle. The complete design for “S” using a FREE of length 300 mm is shown in Figure 6d.

Similarly, the letter “R” is designed by using five CC segments as shown in Figure 6c. Design for “R” is shown in Figure 6e. Length of the actuator in this case is 400 mm.

Fabricated prototypes of all four actuators including “S,” “R,” and two “o” are shown in Figure 6f. The “o” shape is made of a single-segment bending FREE of length 320 mm and a fiber angle of 72°. Further, the actuated shapes of fabricated prototypes are then overlaid on the desired curves in Figure 6g to show the validity of the design. Supplementary Video S1 shows the four actuators taking the shape of ‘SoRo’ upon actuation.

Shape Matching for Two Curves

In this section, we extend the design method to match not just one final target curve, but an additional intermediate curve en route the destination. The actuation pressure of the intermediate target curve is lower than the final target. Thus, by controlling the actuation pressure alone we seek to obtain two seemingly different curves. However, given a target shape, not all intermediate curves are attainable at lower pressures. Restricting our attention to bending FREEs alone, the achievable space of intermediate curves must adhere to certain constraints, which are listed next.

Each CC segment of an intermediate curve should have the same length as the corresponding segment of the target curve. This constraint is due to the presence of the strain limiting layer in bending actuators that prevent change in length on pressurization.

Each CC segment of the intermediate curve should have the same bending direction (convex or concave bend) as that of the corresponding segment in the target curve. This is because a bending actuator can bend only in one direction at any actuation pressure depending on the location of the strain limiting layer.

The intermediate curve's CC segments should have a lower curvature as compared with the corresponding segment of the target curve, because the curvature of the bending FREE increases with pressure.

First, the CC segmentation algorithm is modified such that it can solve for the simultaneous optimal segmentation of multiple curves. This is done by modifying the approximation error, which is minimized by using the dynamic programming algorithm such that it is equal to the sum of the approximation errors for both intermediate and target curves. Next, the first constraint listed earlier is enforced by using arc length parameterization and interpolation such that the set of points S are uniformly spaced along both intermediate and target curves. In addition, to account for the second and third constraints, the segmentation algorithm checks for the sign and magnitude of curvatures of the intermediate and target curve's segments such that they are of the same sign and also the curvatures of the intermediate curve's segments are smaller in magnitude as compared with those of the target curve's segments.

As an example, we consider two curves shown in Figure 7a, where one is the intermediate curve to be matched at pressure P₁ and the other is the target curve to be matched at pressure P₂ such that $P_{2} > P_{1}$ . Both the curves are 315 mm long and start from an initial straight configuration.

FIG. 7.

(a) Intermediate and target curves to be matched at different actuation pressure by the same FREE actuator. (b) CC approximation of both intermediate and target curves using three segments. Color images are available online.

The CC approximation of both the curves is shown in Figure 7b. Three segments or $m = 4$ are used for segmentation here and the arc lengths of corresponding segments in both curves are equal, but they have different curvatures. The arc length parameters and corresponding stretch ratios $λ_{1}$ for both the curves are shown in Table 1. $κ_{1}$ and $κ_{2}$ are the curvatures of the intermediate and target curves, respectively. Similarly, $λ_{1}^{1}$ and $λ_{1}^{2}$ are the corresponding longitudinal stretch ratios.

Table 1.

Arc Parameters of the Intermediate and Target Curves

Segment	Arc length	$κ_{1} (m m^{- 1})$	$κ_{2} (m m^{- 1})$	$λ_{1}^{1}$	$λ_{1}^{2}$
1	105	−0.0055	−0.0087	1.0699	1.1105
2	137	−0.0114	−0.0281	1.1448	1.3569
3	73	0.0061	0.012	1.0775	1.1524

Now, as shown in Table 1, there are two different $λ_{1}$ values for each bending segment. Therefore, solving the inverse constrained maximization formulation to determine fiber angle does not work because we have more equations than unknowns. To design an actuator then requires adding an additional design parameter besides the fiber angle. Possible choices for this include the actuator radius r, wall thickness t, or material constant $C_{10}$ . We choose wall thickness t as the second design parameter. This gives us four equations, two each for intermediate and final curves given by Equations (10) and (11) and four unknowns, namely fiber angle $α$ , thickness t, and radial stretch ratio $λ_{2}$ for both intermediate and final curves. With r fixed, t can only attain a small set of values; therefore, solving a system of four nonlinear equations does not always give a feasible solution. A nondimensional chart-based approach generated using these equations is more intuitive and practical toward finding a feasible solution.

Now, to choose the fiber angle $α$ and wall thickness t that generate the required $λ_{1}$ s at two different target curves, we use nondimensional charts that relate FREE deformation to its actuation. The nondimensional charts map the deformation of a FREE to its fiber angle; the nondimensional coefficient lumps together the actuation pressure, hyperelastic material constants, and the FREE geometry, which includes radius and thickness. These can aid in quick design and analysis of FREEs without the need to solve nonlinear equations of equilibrium. To generate nondimensional charts, we start with the following equation that can be obtained by using either the constrained maximization formulation or the virtual work method as shown in the Design Methodology for Single Shape Matching section: $P \frac{\partial V}{\partial λ_{1}} - \frac{\partial W_{e l a s t o m e r}}{\partial λ_{1}} = 0 .$ (13)

Here, we assume the fibers to be inextensible elements to simplify the overall model. We use the inextensibility constraint $(λ_{1}^{2} {cos}^{2} (α) + λ_{2}^{2} {sin}^{2} (α) - 1 = 0)$ to express $λ_{2}$ in terms of $λ_{1}$ , thereby removing one of the unknown variables. In Equation (13), the volume V can be written as $V = π r^{2} l λ_{1} λ_{2}^{2} = V_{0} λ_{1} λ_{2}^{2}$ , where V₀ is the volume enclosed by the actuator in the undeformed state. Also, substituting the expression of $W_{e l a s t o m e r}$ from Equation (8), we can write Equation (13) as $A_{c} \frac{\partial (λ_{1} λ_{2}^{2})}{\partial λ_{1}} - \frac{\partial (I_{1} - 3)}{\partial λ_{1}} = 0 .$ (14)

The quantity A_c in Equation (14) has no dimensions and is equal to $\frac{P V_{0}}{V_{b} C_{10}}$ , which can be understood as the input actuation energy normalized by the Neo-Hookean material constant times the material volume of the elastomer. We will call this quantity normalized actuation coefficient and it depends on actuation pressure P, material constant $C_{10}$ , actuator's radius r and its wall thickness t. This implies that we do not need to input each of these quantities separately in Equation (14) as they are lumped together in just one numerical value, which is that of A_c. This allows us to keep wall thickness as an unknown design parameter that we determine along with the fiber angle as shown later in this section. Now, Equation (14) is expressed in terms of only three variables, namely $λ_{1}$ , A_c, and $α$ . For all possible ranges of values for A_c and $α$ , we can solve them to obtain $λ_{1}$ . A contour plot of $λ_{1}$ for a meshgrid of A_c and $α$ is shown in Figure 8a.

FIG. 8.

(a) Contour plot of stretch ratio λ_1 versus Ac and α. Contour lines of desired λ_1 values for (b) first CC segment, (c) second CC segment, and (d) third CC segment.

We use the contour plot shown in Figure 8a to design bending segments such that it matches the desired curvatures at different pressures. For the two shape example with three CC segments shown in Figure 7, we have calculated the desired stretch ratios $λ_{1}$ for each CC segment as listed in Table 1. We can plot the Figure 8a again but now only with the contour lines corresponding to the desired $λ_{1}$ s as shown in Figure 8b–d for the three CC segments, respectively. Now, we choose $α$ and t for each segment by using these contour plots based on the process detailed next. First, we can express A_c as shown next: $A_{c} = \frac{P V_{0}}{V_{b} C_{10}} = \frac{P r^{2}}{C_{10} ({(r + t)}^{2} - r^{2})} .$ (15)

As we can see from Equation (15), A_c depends on the pressure P apart from the geometry parameters (r and t) and material constant $C_{10}$ . As previously mentioned, we choose pressures P₁ and P₂ at which the FREE deforms to match the shape of intermediate and final shapes, respectively. It can be shown by using Equation (15) that $\frac{A_{c 1}}{A_{c 2}} = \frac{P_{1}}{P_{2}}$ , where $A_{c 1}$ and $A_{c 2}$ are the values of the normalized actuation coefficient A_c for any CC segment corresponding to the intermediate and final shape, respectively. Therefore for the first CC segment, we choose $α$ by using Figure 8b such that it intersects the two contour lines at A_c values, where $\frac{A_{c 1}}{A_{c 2}} = \frac{P_{1}}{P_{2}} = 0.5$ . Here, the ratio of 0.5 is chosen arbitrarily and is up to the designer. But, because all the segments are actuated to the same pressure, $\frac{A_{c 1}}{A_{c 2}}$ has to be the same for all three segments. Therefore, only a small range of values is possible for $\frac{A_{c 1}}{A_{c 2}}$ to get a feasible design. Now, as shown in Figure 8, at α = 69.3°, we have $A_{c 1} = 0.7577$ and $A_{c 2} = 1.529$ . Using either $A_{c 1}$ or $A_{c 2}$ , we can calculate the second design parameter t by using Equation (15). For the first CC segment, we get t = 1.23 mm. Similarly, by plotting contour lines of desired $λ_{1}$ values for other two CC segments as shown in Figure 8c and d, followed by choosing $α$ such that $\frac{A_{c 1}}{A_{c 2}} = 0.5$ and then calculating t by using Equation (15), we can get the design parameters for the remaining two segments. The complete design of FREE that matches the two shape example of Figure 7 is listed in Table 2 and is also shown in Figure 9a.

FIG. 9.

(a) Design of FREE that matches two given curves. (b) FREE deforming on pressurization to trace both intermediate and target curves using FEA. (c) Prototype of FREE actuated to two different pressures to match intermediate and final curves. (d) Comparison of fabricated prototype to both intermediate and final curves. FEA, finite element analysis.

Table 2.

Design of Fiber-Reinforced Elastomeric Enclosure for Matching Two Shapes

Segment	Arc length (mm)	Fiber angle (°)	Wall thickness (t in mm)
1	−105	61.8	1.23
2	−137	85.7	1.67
3	73	69.3	2.06

To validate this design as shown in Table 2, we modeled it by using the finite element method (FEM) in the commercially available software Abaqus.¹⁶ We modeled the elastomer by using tetrahedral elements and used a Neo-Hookean material model. The fibers are modeled by using truss elements, and a tie constraint is added between the fibers and the tube. Both the intermediate and target shapes as obtained by using FEM are shown in Figure 9b. Further, a fabricated prototype of this FREE actuated to two different pressures to attain both the intermediate and final curves is shown in Figure 9c. These two shapes attained by the actuator are then overlaid on top of the desired intermediate and final curves in Figure 9d to show that the fabricated prototype matches both the desired curves. Supplementary Video S1 shows the fabricated prototype deforming in the shape of both intermediate and final curves.

Discussion

The nondimensional charts using the normalized actuation coefficient aid in the simultaneous selection of fiber angle and wall thickness for a given trajectory of curvatures or $λ_{1}$ values. For matching a trajectory of two $λ_{1}$ s in the different CC segments, we summarize the design steps as follows:

Start with the initial guess for the ratio of A_cs corresponding to the two stretch ratio $λ_{1}$ requirement for a given segment. Choose the fiber angle from the nondimensional chart that maintains this ratio and simultaneously intersects the required $λ_{1}$ s (Fig. 8b).

Keeping the ratio of A_cs fixed, choose the fiber angles for the other segments based on the required stretch ratios $λ_{1}$ s (Fig. 8c, d).

If a solution is not found, then repeat the first two steps with a different initial guess for the ratio of A_cs.

Once the A_cs are determined for all the segments at the two required $λ_{1}$ values, the wall thickness of each segment can be determined.

Although we have demonstrated this process for matching a trajectory of two curves (intermediate and final), it can be extended to matching three or more intermediate curves. For example, to match a trajectory of n curves, each will have an associated value for A_c ( $A_{c 1}$ , $A_{c 2}$ , … $A_{c n}$ ). This implies that the ratio of successive A_cs, that is, $\frac{A_{c 1}}{A_{c 2}}$ , $\frac{A_{c 2}}{A_{c 3}}$ through $\frac{A_{c (n - 1)}}{A_{c n}}$ must be maintained for all the segments. Matching all these requirements simultaneously may be quite complex with the chart-based approach, and an optimization algorithm such as the one proposed in Ref.¹⁵ may be useful, provided a solution exists.

Conclusions

Pneumatically actuated FREEs can undergo unique curvilinear deformations based on the arrangement of reinforced fibers. Although the forward analysis of predicting these deformed shapes has been extensively studied, a few efforts have focused on inverse design. In this article, we present a method for designing fiber-reinforced soft actuators to match a planar curve. This method involves two steps where the first step approximates the planar curve using CC segments. The second step is the actual inverse design process that determines the fiber angles of all the actuator segments such that they attain the desired curvatures obtained in the first step at the same actuation pressure. In this article, we propose two methods for the inverse design formulation: (1) a computational inverse design by solving the nonlinear equilibrium equations, and (2) a nondimensional design chart-based approach to match a final curve and an intermediate curve en route to the final.

Although the computational inverse formulation stems from the mechanics of FREE deformation and can handle additional complications such as fiber stiffness to make it extensible, it does not always yield a solution. The existence of a solution is better understood by using the nondimensional charts that visually span the entire feasible design space. This visualization is important to match two curves or shapes, one final and the other at an intermediate pressure, where we consider the cylinder thickness as a design parameter apart from the fiber angle. The approach critically demonstrates the difficulty in exactly matching two arbitrary curves, which makes an extension to three or more arbitrary curves extremely challenging. Although this article was limited to matching planar curves using bending FREEs alone, in future we will extend the methodology to matching spatial curves with additional building blocks such as spiral,¹⁴ rotating,⁸ and contracting FREEs. One potential way of doing this would be by using a similar two-step approach as presented in this article. In step one, the given spatial curve can be segmented by using an analogous primitive to CC arcs in 3D such as a CC and constant torsion helices. After the segmentation of the desired spatial curve to a series of smooth helices, in step two, we can design FREEs that can deform to match the curvatures and torsions of each of these segmented helices.

Footnotes

Acknowledgments

The authors would like to acknowledge Ibadat Singh Chatha and Chris Park for their assistance with the fabrication of fiber-reinforced actuators.

Author Disclosure Statement

No conflicts of interest exist.

Funding Information

This material is based upon work supported by the National Science Foundation under Grant No. CMMI-1454276.

Supplementary Material

References

Lipson

Challenges and opportunities for design, simulation, and fabrication of soft robots. Soft Robot, 2013; 1:21–27.

Kim

, Laschi

, Trimmer

. Soft robotics: a bioinspired evolution in robotics. Trends Biotechnol, 2013; 31:287–294.

Rus

, Tolley

. Design, fabrication and control of soft robots. Nature, 2015; 521:467–475.

Laschi

, Cianchetti

. Soft robotics: new perspectives for robot bodyware and control. Front Bioeng Biotechnol, 2014; 2:3.

Kovac

The bioinspiration design paradigm: a perspective for soft robotics. Soft Robot, 2013; 1:28–37.

Krishnan G. Kinematics of a new class of smart actuators for soft robots based on generalized pneumatic artificial muscles. In IEEE International Conference on Intelligent Robots and Systems, Chicago, IL, 2014, pp. 587–592.

Singh G, Krishnan G. An isoperimetric formulation to predict deformation behavior of pneumatic fiber reinforced elastomeric actuators. In 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Hamburg, Germany, 2015, pp. 1738–1743.

Singh

, Krishnan

. A constrained maximization formulation to analyze deformation of fiber reinforced elastomeric actuators. Smart Mater Struct, 2017; 26:065024.

Singh

, Xiao

, Hsiao-Wecksler

, et al. Design and analysis of coiled fiber reinforced soft pneumatic actuator. Bioinspir Biomim, 2018; 13:036010.

10.

Peel

, Jensen

, Suzumori

. Batch fabrication of fiber-reinforced elastomer prepreg. J Adv Mater, 1998; 30:3–9.

11.

Bishop-Moser

, Kota

. Design and modeling of generalized fiber-reinforced pneumatic soft actuators. IEEE Trans Robot, 2015; 31:536–545.

12.

Polygerinos

, et al. Modeling of soft fiber-reinforced bending actuators. IEEE Trans Robot, 2015; 31:778–789.

13.

Satheeshbabu S, Krishnan G. Designing systems of fiber reinforced pneumatic actuators using a pseudo-rigid body model. In 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Vancouver, Canada, 2017, pp. 1201–1206.

14.

Uppalapati

, Krishnan

Towards pneumatic spiral grippers: modeling and design considerations. Soft Robot 2018. DOI:10.1089/soro.2017.0144.

15.

Connolly

, Walsh

, Bertoldi

. Automatic design of fiber-reinforced soft actuators for trajectory matching. Proc Natl Acad Sci, 2017; 114:51–56.

16.

Connolly

, Polygerinos

, Walsh

, et al. Mechanical programming of soft actuators by varying fiber angle. Soft Robot, 2015; 2:26–32.

17.

Shan

, Philen

, Bakis

, et al. Nonlinear-elastic finite axisymmetric deformation of flexible matrix composite membranes under internal pressure and axial force. Compos Sci Technol, 2006; 66:3053–3063.

18.

Bishop-Moser J, Kota S. Towards snake-like soft robots: Design of fluidic fiber-reinforced elastomeric helical manipulators. In 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, Tokyo, Japan, 2013, pp. 5021–5026.

19.

Banchoff

, Giblin

. On the geometry of piecewise circular curves. Am Math Mon, 1994; 101:403–416.

20.

Pei

S-C

, Horng

J-H

. Optimum approximation of digital planar curves using circular arcs. Pattern Recogn, 1996; 29:383–388.

21.

Derouet-Jourdan

, Bertails-Descoubes

, Thollot

Stable inverse dynamic curves. In ACM Transactions on Graphics (TOG), Seoul, South Korea, 2010;29, p. 137.

22.

Neppalli

, Csencsits

, Jones

, et al. Closed-form inverse kinematics for continuum manipulators. Adv Robot, 2009; 23:2077–2091.

23.

Trimmer

, Lin

. Bone-free: Soft mechanics for adaptive locomotion. Integr Comp Bio, 2014; 54:1122–1135.

Supplementary Material

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