Automatic design of fiber-reinforced soft actuators for trajectory matching

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 $\alpha$ (Fig. 2A) (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. 2A). The isotropic core has initial inner radius $R_i$ and outer radius $R_m$, and the outer anisotropic layer has initial outer radius $R_o$. The anisotropic material has a preferred direction that is determined by the initial fiber orientation $\mathrm{𝐒}=(0,\cos \alpha ,\sin \alpha )$. We define a deformation gradient $\mathrm{𝐅}$, from which we calculate the left Cauchy–Green deformation tensor $\mathrm{𝐁}={\mathrm{𝐅}\mathrm{𝐅}}^T$, the current fiber orientation $\mathrm{𝐬}=\mathrm{𝐅}\mathrm{𝐒}$, and the tensor invariants $I_1=\mathrm{tr}(\mathrm{𝐁})$ and $I_4=\mathrm{𝐬}.\mathrm{𝐬}$.

The inner and outer layers require different strain energy expressions, so let $W^{(in)}$ be the strain energy for the isotropic core and $W^{(out)}$ be the strain energy for the anisotropic outer layer. For the isotropic core, we choose a simple incompressible neo-Hookean model, so that $W^{(in)}=\mu /2(I_1-3)$, with $\mu$ denoting the initial shear modulus. For the anisotropic layer, let $W^{(out)}$ be the sum of two components, $W^{(out)}=c_1{W}^{(iso)}+c_2{W}^{(aniso)}$, where $W^{(iso)}=\mu /2(I_1-3)$ is the contribution from the isotropic elastomeric matrix, $W^{(aniso)}$ is the contribution from the fibers, and $c_1$ and $c_2$ are the corresponding volume fractions. To derive a suitable expression for $W^{(aniso)}$, we consider a small section of the helical fiber and model it as a rod subject to an axial load (SI Appendix, Fig. S5). It is trivial to show that the strain energy density of the rod is (SI Appendix)where $E$ is its Young’s modulus. By slightly modifying the strain energy, the above equations can easily be extended to account for more than one set of fibers. For example, to achieve a pure extending actuator, we might require two sets of fibers, with fiber orientations ${\mathrm{𝐬}}_{\mathrm{𝟏}}$ and ${\mathrm{𝐬}}_{\mathrm{𝟐}}$. In this case, the strain energy density iswhere $I_4={\mathrm{𝐬}}_{\mathrm{𝟏}}.{\mathrm{𝐬}}_{\mathrm{𝟏}}$ and $I_6={\mathrm{𝐬}}_{\mathrm{𝟐}}.{\mathrm{𝐬}}_{\mathrm{𝟐}}$.

We can then use the strain energies to calculate the Cauchy stresses, which take the formwhere $W_i=\frac{\partial W}{\partial I_i}$, $\mathrm{𝐈}$ is the identity matrix, and $p$ is a hydrostatic pressure (30).

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.

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 $5^{\circ }$. Also, we prescribe a maximum allowed fiber angle of ${50}^{\circ }$, because above this angle, the radial expansion of the actuators becomes significant (20). Note that for a finger, bending occurs at discrete joints, but for the actuator, it will of course take place over some finite length. To approximate the motion of the finger as closely as possible, we want this length to be as short as possible, so we impose a maximum allowed bending segment length of 30 mm (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 minimizewhere $n_{tw}$, $n_{ext}$, and $n_{bend}$ are the total numbers of twisting, extending, and bending segments, respectively, and $N$ is the number of goal configurations. The first term measures the difference between the required ($\widehat{\theta }$) and achieved ($\theta$) twist angles; the second term measures the difference between the required (${\widehat{l}}_{ext}$) and achieved ($l_{ext}$) segment lengths, and the third term measures the difference between the pressures at which the required bend angles should be achieved ($\widehat{P}$) and the pressures at which the required bend angles are actually achieved with the current set of variables ($P$). The parameters c₁ = 100 and c₂ = 1,000 are weights that balance the relative importance of the twisting, bending, and extending segments. If $f$ is not sufficiently small, the variables are updated and the optimization loop repeats. When the minimum value of $f$ is found, the optimization outputs (i) the fiber angle ${\alpha }_i$ for each segment, (ii) the initial length of each of the bending and twisting segments, and (iii) the pressures ${\widehat{P}}^{(j)}$ at which the goal configurations will occur.

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.

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.