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# Automatic design of fiber-reinforced soft actuators for trajectory matching

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 21, 2016 (received for review September 12, 2016)

## Significance

Fluid-powered elastomeric soft robots have been shown to be able to generate complex output motion using a simple control input such as pressurization of a working fluid. This capability, which mimics similar functions often found in biology, results from variations in mechanical properties of the soft robotic body that cause it to strain to different degrees when stress is applied with the fluid. In this work, we outline a mechanics- and optimization-based approach that enables the automatic selection of mechanical properties of a fiber-reinforced soft actuator to match the kinematic trajectory of the fingers or thumb during a grasping operation. This methodology can be readily extended to other applications that require mimicking or assisting biological motions.

## Abstract

Soft actuators are the components responsible for producing motion in soft robots. Although soft actuators have allowed for a variety of innovative applications, there is a need for design tools that can help to efficiently and systematically design actuators for particular functions. Mathematical modeling of soft actuators is an area that is still in its infancy but has the potential to provide quantitative insights into the response of the actuators. These insights can be used to guide actuator design, thus accelerating the design process. Here, we study fluid-powered fiber-reinforced actuators, because these have previously been shown to be capable of producing a wide range of motions. We present a design strategy that takes a kinematic trajectory as its input and uses analytical modeling based on nonlinear elasticity and optimization to identify the optimal design parameters for an actuator that will follow this trajectory upon pressurization. We experimentally verify our modeling approach, and finally we demonstrate how the strategy works, by designing actuators that replicate the motion of the index finger and thumb.

In the field of robotics, it is essential to understand how to design a robot such that it can perform a particular motion for a target application. For example, this robot could be a robot arm that moves along a certain path or a wearable robot that assists with motion of a limb. For conventional hard robots, methods have been developed to describe the forward kinematics (i.e., for given actuator inputs, what will the configuration of the robot be) and inverse kinematics (i.e., for a desired configuration of the robot, what should the actuator inputs be) (1⇓⇓–4).

Recently, there has been significant progress in the field of soft robotics, with the development of many soft grippers (5, 6), locomotion robots (7, 8), and assistive devices (9). Although their inherent compliance, easy fabrication, and ability to achieve complex output motions from simple inputs have made soft robots very popular (10, 11), there is growing recognition that the development of methods for efficiently designing actuators for particular functions is essential to the advancement of the field. To this end, some research groups have begun focusing their efforts on modeling and characterizing soft actuators (12⇓⇓⇓⇓⇓⇓⇓–20). In particular, significant progress has been made on solving the forward kinematics problem (16⇓⇓–19) and even on using dynamic modeling to perform motion planning (14). However, the practical problem of designing a soft actuator to achieve a particular motion remains an issue. Finite element (FE) analysis has previously been used as a design tool to find the optimal geometric parameters for a soft pneumatic actuator, given some design criteria (15). Although this procedure yields some nice results, only basic motions (linear or bending) were studied, because the method is computationally intensive. An alternative approach is to use analytical modeling combined with optimization to determine the properties of a soft actuator that will achieve a particular motion for some target application.

Here, we focus on fiber-reinforced actuators (17⇓⇓⇓–21), and given a particular trajectory, we find the optimal design parameters for an actuator that will replicate that trajectory upon pressurization. To achieve this goal, we first use a nonlinear elasticity approach to derive analytical models that provide a relationship between the actuator design parameters (geometry and material properties) and the actuator deformation as a function of pressure for each motion type of interest (extending, expanding, twisting, and bending). Then, we use optimization to determine properties for actuators that match the desired trajectory (Fig. 1). Whereas similar actuators were previously designed empirically (22, 23), here, we propose a robust and efficient strategy to streamline the design process. Furthermore, this strategy is not limited to the specific cases presented here (namely the trajectories of the index finger and thumb) but, rather, could be applied to produce required trajectories in a variety of soft robotic systems, such as locomotion robots, assembly line robots, or devices for pipe inspection.

## Analytical Modeling of Actuator Segments

Our approach is based on assuming a desired actuator consists of multiple segments (mimicking the links and joints of the biological digit), where each different segment undergoes some combination of axial extension, radial expansion, twisting about its axis, and bending upon pressurization. To realize actuators capable of replicating complex motions, we use segments consisting of a cylindrical elastomeric tube surrounded by fibers arranged in a helical pattern at a characteristic fiber angle *A*) (23, 24), because it has been shown that by varying the fiber angle and materials used, these segments can be easily tuned to achieve a wide range of motions (17⇓⇓⇓–21). When the elastomeric tube is of uniform stiffness, the segment undergoes some combination of axial extension, radial expansion, and twisting about its axis upon pressurization (17⇓–19). In contrast, when the tube is composed of two elastomers of different stiffness, pressurization produces a bending motion (25, 26).

Previous work has explored the design space of fiber-reinforced actuators capable of extending, expanding, and twisting using FE analysis (20) and kinematics and kinetostatics modeling (17⇓–19). Although these existing analytical models provide great insight into the behavior of fiber-reinforced actuators, they are restricted to exactly two sets of fibers (a set of fibers being fibers arranged at the same angle).

Here, we use a nonlinear elasticity approach, which facilitates modeling actuators with an arbitrary number of sets of fibers. Rather than modeling the tube and the fibers individually, we treat them as a homogeneous anisotropic material (27⇓–29). More specifically, because the fibers are located on the outside of the tube and not dispersed throughout its thickness, we model the actuator as a hollow cylinder of isotropic incompressible hyperelastic material (corresponding to the elastomer), surrounded by a thin layer of anisotropic material (corresponding to the fiber reinforcement), and impose continuity of deformation between the two layers (Fig. 2*A*). The isotropic core has initial inner radius

The inner and outer layers require different strain energy expressions, so let *SI Appendix*, Fig. S5). It is trivial to show that the strain energy density of the rod is (*SI Appendix*)

We can then use the strain energies to calculate the Cauchy stresses, which take the form

To further simplify the analytical modeling, we decouple bending from the other motions. In the following, we first introduce an analytical model describing an extending, expanding, twisting actuator and then a model for a bending actuator.

### Modeling Extension, Expansion, and Twist.

When the elastomeric part of the actuator is of uniform stiffness, we assume that the tube retains its cylindrical shape upon pressurization, and the radii become *B*). The possible extension, expansion, and twisting deformations are then described by (*SI Appendix*)

Eqs. **6****–****8** with the Cauchy stress **3** and **4** and the deformation gradient of Eq. **5** are then solved to find *SI Appendix*).

### Modeling Bending.

Because the exact solution for the finite bending of an elastic body is only possible under the assumption that the cross-sections of the cylinder remain planar upon pressurization—a condition that is severely violated by our actuator—we assume (*i*) that the radial expansion can be neglected (i.e., *ii*) vanishing stress in the radial direction (i.e., *SI Appendix*)

Because the actuator bends due to the moment created by the internal pressure acting on the actuator caps (*SI Appendix*, Fig. S9), we equate this moment*C*).

Now solving **3** and **4** (*SI Appendix*).

### Comparing Analytical and Experimental Results.

To fabricate the extending, expanding, twisting actuators, we used the elastomer Smooth-Sil 950 (_{2} = 680 kPa; Smooth-On), and for the bending actuators, we used both Smooth-Sil 950 and Dragon Skin 10 (_{2} = 85 kPa; Smooth-On). The fiber reinforcement was Kevlar, with a Young’s modulus *E* = 31,067 MPa and radius *r* = 0.0889 mm. Each actuator had an inner radius of 6.35 mm, wall thickness of 2 mm, and length of 160 mm. The effective thickness of the fiber layer (8.89 ^{-4} mm) is a fitting parameter here and was identified using the results in *SI Appendix*, Fig. S7.

For the bending model, we used an FE simulation (*SI Appendix*, Fig. S10) to determine the location of the neutral axis (*SI Appendix*, Fig. S11). However, for thicker-walled actuators, the model yielded lower than expected bend angles at any given pressure. To solve this problem, we used one FE simulation (with fiber angle *SI Appendix*). Note that because FE analysis generally provides more accurate results than our analytical bending model, an alternative solution would be to use FE simulations to build a database of simulation results for actuators with a range of different fiber angles. However, this option would be more computationally expensive, and so, although not ideal, it is preferable in our case to use just one FE simulation to identify the fitting parameters for the analytical bending model, rather than relying solely on FE.

We first consider extending actuators (with two sets of fibers, arranged symmetrically). Fig. 3*A* shows how the amount of extension undergone (illustrated by the color) depends on the fiber angle of the actuator (*A*), and the results (Fig. 3 *B* and *C*) show good agreement between the model and the experiment. Second, we consider twisting actuators, which have only one set of fibers. From Fig. 3*D*, we can see that an actuator with fiber angle around *E* and *F* shows that the analytical model accurately represents the twist per unit length undergone by an actuator with fiber angle

Finally, Fig. 3*G* illustrates the bend angle per unit length as a function of fiber angle and actuation pressure. At any given pressure, for larger fiber angles, we see less bend per unit length. Comparing analytical and experimental results for a bending actuator with fiber angles *H* and *I*), we see a good match between the model and the experiment.

## Replicating Complex Motions

We have presented two analytical models, which describe extending, expanding, twisting, and bending actuator motions. In addition to using these models to explore the actuator design space, we can use them for more complex operations, such as designing a single-input, multisegment actuator that follows a specific trajectory. In the following sections, we will demonstrate this methodology by determining the properties of multisegment actuators that can replicate finger and thumb motion.

The target motion of the actuator was determined using electromagnetic trackers that were placed on the hand at the wrist, at each joint along the finger, and at the fingertip (*SI Appendix*, Fig. S13) (22). Time series data of the coordinates of each sensor in 3D space were recorded as the hand was opened and closed. Using these data, the configuration of the fingers and thumb during a grasping motion can be obtained. Adjacent sensors are connected by links, and we use the data to calculate the length of each link and the angles between the links at each time. We smooth the data by applying a Savitzky–Golay filter. Since it will not be possible to produce an actuator that will match the finger trajectory exactly at every point, we choose just four configurations to match. (These configurations are equally spaced along the trajectory; *SI Appendix*, Figs. S14 and S15.) Because the input data represent the motion of the top of the finger, the top of the actuator we design should mimic the input motion.

The actuator will consist of multiple segments, each with a different length and fiber angle. We prescribe the number of segments and the type of each segment. (For replicating finger and thumb motion, expanding segments are not required, so each segment type is extend, bend, or twist.) The radius, wall thickness, and material parameters are the same as in the previous section. (Here, we use Dragon Skin 10 for the extending and twisting segments.) We set the maximum allowed pressure to 80 kPa. (At higher pressures, the Matlab solver fsolve is unable to solve Eqs. **6****–****8**.) Furthermore, to simplify the fabrication procedure, we impose a minimum fiber angle of *SI Appendix*). We then input all of this information, together with the models we developed in the previous sections, into a nonlinear least-squares optimization algorithm in Matlab (lsqnonlin) (Fig. 4). To find the design parameters for an actuator that will move through the given configurations (combinations of link lengths and bend angles) as it is pressurized, we minimize*c*_{1} = 100 and *c*_{2} = 1,000 are weights that balance the relative importance of the twisting, bending, and extending segments. If *i*) the fiber angle *ii*) the initial length of each of the bending and twisting segments, and (*iii*) the pressures

### Index Finger Motion.

The fiber angles and lengths required to imitate the movement of the index finger are illustrated in Fig. 5*D*. Segments 1, 3, and 6 have length 70 mm, 22 mm, and 15 mm, with fiber angles of

We fabricate the actuator as detailed in *SI Appendix*. To compare the actual performance of the actuator to the expected performance, we characterize it by taking pictures of the actuator at various different actuation pressures. We evaluate its motion by using Matlab to track points on the actuator. We see good agreement between expected and actual motion (Fig. 5 *C* and *E* and Movie S1), with some discrepancies that are most likely due to defects in the actuator fabrication (for example, segment lengths being up to 4 mm shorter than expected, due to the fibers being wound around the actuator between segments; *SI Appendix*).

### Thumb Motion.

As a second example, we consider the design of an actuator that upon pressurization, replicates the motion of a thumb. The motion of the thumb is more complex than that of the finger, because it moves out of plane. We capture this out-of-plane motion as a twisting motion. We calculate the amount of twist by fitting a plane to the twisting links at each time and then finding the angle between the normal to this plane and the normal to the initial plane.

Fig. 5*I* illustrates the fiber angles that are needed to reproduce the motion of the thumb, as predicted by the model. Segment 1 is a twisting segment of length 25 mm, with fiber angle *SI Appendix*). Fig. 5 *H* and *J* compares the input thumb kinematics and the output actuator motion (Movie S2). We see reasonable agreement between the expected and achieved motions, with discrepancies in this case most likely due to inaccuracies in measuring the actuator motion (for example, misalignment of cameras), as well as defects in actuator fabrication (such as nonuniform wall thickness).

## Conclusions

Using analytical models for fiber-reinforced actuators that extend, expand, twist, and bend, we have devised a method of designing actuators customized for a particular function. Given the kinematics of the required motion, and the number and type of segments required, the algorithm outputs the appropriate length and fiber angle of each segment, thereby providing a recipe for how the actuator should be made. The procedure is somewhat limited in its current form because it requires the user to input the type of segments required, but future versions will eliminate the need for this step, thus further automating the procedure. Future work will also focus on developing a model that combines bending with other motions, to increase the versatility of the algorithm. The design tool we have presented here has immense potential to streamline and accelerate the design of soft actuators for a particular task, eliminating much of the trial and error procedure that is currently used and broadening the scope of fiber-reinforced soft actuators.

## Acknowledgments

The authors thank Dr. J. Weaver for assistance with 3D printing and Dr. P. Polygerinos and Dr. S. Sanan for helpful discussions. This work was partially supported by National Science Foundation Grant 1317744, the Materials Research Science and Engineering Center under National Science Foundation Award DMR-1420570, the Wyss Institute, and Harvard’s Paulson School of Engineering and Applied Sciences.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: walsh{at}seas.harvard.edu or bertoldi{at}seas.harvard.edu.

Author contributions: F.C., C.J.W., and K.B. designed research; F.C. performed research; F.C. and K.B. analyzed data; and F.C., C.J.W., and K.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1615140114/-/DCSupplemental.

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