An Optimum Design Method of Pneu-Net Actuators for Trajectory Matching Utilizing a Bending Model and GA

This study addresses the problem in designing pneu-net actuators to achieve speciﬁc target trajectories for particular applications. The ﬁnite element analysis (FEA) method is used to study the sensitivity of 9 design parameters to bending deformation of actuators, and the results of FEA demonstrate that the width of the chamber and the width of the middle layer have great inﬂuence on the bending performance. Besides, the relations between the bending angles of actuators with diﬀerent width parameters and the pressure are always highly linear. Using FEA and the cubic polynomial ﬁtting algorithm, a uniﬁed bending model of the pneu-net actuator is established with three crucial design parameters (i.e., segment length, chamber width, and middle-layer width). An optimum design method based on the bending model and genetic algorithm (GA) is put forward to automatically adjust these three crucial parameters to realise trajectory matching. The method is developed to be an eﬀective solution for the problem in matching the target trajectory with the experimental results achieved on an actuator designed by imitating a human index ﬁnger. Furthermore, the results of the experiment also verify that the actuator can achieve stable grasping, and the proposed method has an important directive to design pneu-net actuators for their application in the ﬁelds of robotics, rescue, and detection.


Introduction
Soft actuators are widely fabricated by using compliant materials (e.g., polymers [1], elastomers [2], and hydrogels [3][4][5]) which result in a vast potential of soft actuators to handle fragile objects, resist mechanical insults (e.g., blunt impacts and compressive forces), and realise complex motions [6][7][8]. Pneumatically powered actuators aim to use new design strategies so that they can rapidly provide complex motions with simple inputs, thereby inspiring a new wave of research activities [9][10][11]. Recently, the field of soft pneumatic actuators (SPAs) has achieved a remarkable progress with the development of biomimetic materials science [12][13][14] and advanced manufacturing processes [15].
Fibre-reinforced and pneumatic network (pneu-net) actuators are the two most popular categories among the abundant designs of SPAs [7,8,16]. A fibre-reinforced actuator is mainly composed of an elastomer body with a large chamber in the centre, a strain-limiting layer at the bottom, and helically arranged fibres on the surface; these fibres protect the actuator from excessive deformation [17]. Fibre-reinforced actuators can achieve extending, twisting, expanding, and bending motions and follow a designed kinematic trajectory by changing the components and tuning oblique angle and fibre length [7,18]. A pneu-net actuator is composed of an inextensible bottom layer and an elastomer body with a network of small chambers inside; these small chambers lead to the deformation of actuators when inflated [6,8,17]. A pneu-net actuator can also realise bending and twisting motions by altering the chamber angle [8].
e field of SPAs has achieved a remarkable progress with the development of many soft grippers [19,20], locomotion robots [21,22], bionic robots [23][24][25], and biomedical devices for rehabilitation [26][27][28]. Simultaneously, the methods used for efficiently designing actuators for particular functions are essential to studying SPAs [29]. On this basis, many research groups have focused on the theoretical framework for modelling SPAs, although developing realistic and tractable models is challenging [9,11]. erefore, designing pneu-net actuators to perform a particular trajectory remains a challenge.
In this study, an efficient strategy is proposed to determine the optimal design parameters for a pneu-net actuator that can match a desired trajectory upon pressurisation. Numerous geometry models with 9 different design parameters are designed to investigate the effect of design parameters on the bending performance of pneu-net actuators in Section 2. Using the technology of FEA, we can see that the effects of these two parameters (i.e., the chamber width and the middle-layer width) on the performance of the actuator are more sensitive than those of the other parameters. Section 3 proposes a GAbased optimum design method of actuators for realising trajectory matching. In Section 4, based on the proposed method, an actuator is designed to match the trajectory of a human index finger. Section 5 presents some conclusions of this work.

FEA for Pneu-Net Actuator
Pneu-net actuators have three typical geometries: (a), (b), and (c) [17,20], as illustrated in Figure 1. Considerable deformations and stresses are excited on the top wall of geometry (a) and geometry (b) when pressurised, thereby possibly inducing failure when the top walls are pierced or the air pressure is largely increased. e main deformation zone of geometry (c) is the inside walls that may contact each other when the chambers are inflated. is form of bending motion can protect the walls from excessive deformation and thus induce the development of reliability over geometry (a) and geometry (b). us, geometry (c) is selected for intensive research in this work.
In theory, different combinations of geometrical parameters can produce actuators with different performance, which are also called design parameters. For distinguishing which design parameters have a remarkable effect on the performance of the actuator, a set of pneu-net actuators is designed with different design parameters, and each of them has one specific parameter that is different from that of the reference actuator. e length, width, and height of the actuators are fixed, that is, 85, 26, and 20 mm, respectively. e geometric structure of the reference actuator is depicted in detail in Figure 2, and the corresponding design parameters are listed in Table 1. e FEA has been widely applied as a design tool [7,17], and its practicality has been confirmed in determining the optimal design parameters. erefore, the FEA is used to analyse the influence of design parameters on the performance of the actuator. e main performance index of the actuator is the bending ability, that is, the bending angle of the actuator. e geometry models of actuators are designed in SolidWorks 2014 (Dassault Systems) and then imported into ABAQUS CAE 6.14-3 (Dassault Systems) for FEA simulations. After loading the geometry models, the end of the actuator is constrained, and pressure is applied to the walls. e material behavior is defined as hyperelastic, and the material type is isotropic. Uniaxial tensile tests are executed according to the standard ASTM D638 to determine the constitutive model parameters of the actuator's material (i.e., Dragon Skin 30). After evaluating other constitutive models, the Yeoh model is considered to be the best constitutive relation for the applications in this paper. e stress-strain curve is exhibited in Figure 3. FEA-simulated images of the bending motion of actuators are demonstrated in Figure 4. e FEA result of the reference actuator is shown in Figure 4(a). In Figure 4, it can be seen that the bending angle of the actuator f (with the middle-layer width parameter w 4 � 8 mm) has remarkably increased over that of the reference actuator under the same pressure (5 kPa). Moreover, the middle layer of each of the eight actuators is the stress concentration region. Hence, the middle-layer width parameter has a remarkable effect on the bending performance of actuators. In comparison with other actuators, the bending angle of the actuator b (with the chamber width parameter w 2 � 20 mm) is also slightly increased. us, subsequent research is conducted on the effects of the two width parameters (w 2 and w 4 ) on the bending performance of actuators. e parameter θ is used to represent the bending angle of the reference actuator, θ i represent the bending angle of M2i, and the increment of the bending angle is given as follows:

Optimization Strategy of Design Parameters
(1) e parameter R is used to represent the curvature radius of the reference actuator, R i represent the curvature radius of M2-i, and the increment of the curvature radius is given as follows: e value of bending angle θ under different pressures is shown in Figure 5(a) (i.e., the value corresponding to the longitudinal coordinate in the polygonal line Mr), and the value of curvature radius R is shown in Figure 5(c) (i.e., the value corresponding to the longitudinal coordinate in the polygonal line Mr). Figure 5(a) displays the bending angle of actuators with different values of w 2 under different pressures (i.e., 0-35 kPa). e bending angle of each of the eight actuators increases with the increase of pressure, and it also increases with the increase of w 2 under the same pressure. Figure 5(b) presents the increment of bending angle between the reference actuator and other actuators. We can see that the increments of each actuator are increased with the increase of pressure. e increments of each actuator are also increased with the increase of w 2 under the same pressure. It means that if the length of an actuator is a certain value, the actuator can change its bending angle with the same pressure by altering the design parameter w 2 .
As shown in Figure 5(c), the curvature radius of each actuator decreases with the increase of pressure. Under a certain pressure, the curvature radius decreases with the increase of w 2 . e increments of curvature radius with increasing w 2 are displayed in Figure 5(d). e gradients of increments of curvature radius become slow with the increase of pressure. e absolute values of increments of curvature radius increase with the increase of w 2 under the same pressure. It can be concluded that an actuator can also achieve a smaller curvature radius under the same pressure by increasing w 2 .
A series of actuators (M4 -1∼M4 -8) with different values of the width parameter w 4 (w 4− i � 24∼10 mm) are also designed and analysed to explore the sensitivity of the width parameter w 4 to the bending performance of the actuator.  It can be seen that the bending angle of each of the eight actuators increases with the decrease of w 4 under the same pressure in Figure 6(a). For a given actuator, the bending angle increases with the increase of pressure. Figure 6(b) presents the increment of bending angle between the reference actuator and other actuators. e increments of bending angle of each actuator are also increased with the increase of pressure. As shown in Figure 6(b), when the actuators are subjected to the same pressure, the increments of bending angle increase with the decrease of w 4 . Figure 6(c) displays that the curvature radius increases with the increase of w 4 under the same pressure. e relation between the increment of curvature radius and the increment of w 4 is displayed in Figure 6(d). e absolute value of increments of curvature radius increases with the decrease of w 4 under the same pressure. It can be concluded that an actuator can also change its bending angle and curvature radius by altering the value of w 4 .

Optimum Design Method of Pneu-Net Actuator.
e effects of the chamber width parameter and the middle-layer width parameter on the bending performance of the actuator under different pressures are discussed in detail in the above section, respectively. We can find that the relation between the increment of the chamber width and the increment of the bending angle is approximately linear under a certain pressure ( Figure 5(b)), and the relation between the increment of the middle-layer width and the increment of the bending angle is nonlinear (Figure 6(b)).
However, no matter which width parameter changes, the bending angle increases linearly with the increase of pressure (Figures 5(a) and 6(a)). Moreover, when the pressure is 0 kPa, the bending angle is 0 rad. erefore, for any width, there is only one parameter to be determined in the mathematical relationship between bending angle and pressure: the slope that changes according to the width parameter.
Combined with the linear fitting result in Figure 5(a), relative to the reference actuator, we can obtain the corresponding slopes under different increments of w 2 (Figure 7(a)), and the mathematical relation between them can be obtained by using the cubic polynomial fitting algorithm; in this case, the fitting result is given as follows: where w r2 is the value of the chamber width of the reference actuator and w 2− i is the value of the chamber width of any actuator.
Similarly, combined with the linear fitting result in Figure 6(a), the corresponding slopes under different increments of w 4 are shown in Figure 7(b), and the mathematical relation between them can also be obtained by using the cubic polynomial fitting algorithm; the fitting result is obtained in this case as follows: where w r4 is the value of the middle-layer width of the reference actuator and w 4− i is the value of the chamber width of any actuator. en, we can get the mathematical relation between the bending angle and the pressure in any width of the chamber or the middle layer: where P represents the pressure.
Combined with equations (5) and (6), because of the change in the width of the chamber and the middle layer relative to the reference actuator, a unified bending model of actuators can be established as follows:   Mathematical Problems in Engineering e actuator bending model is established according to the results of linear fitting and cubic polynomial fitting, both of which have a higher coefficient of determination (R 2 ≥ 0.99). e bending model expresses quantitatively the effect of two width parameters on the bending performance of actuators. However, the ultimate goal of adjusting the design parameters is to match the target trajectory, so we will focus on the characteristics of the end trajectory of actuators.
ree actuators with different parameters are selected in Section 2, and the pressure on each actuator is gradually increased so that the bending degree of each of them reaches the limit state (i.e., interference with itself ). e results of the three end trajectories are shown in Figure 8. Note that the three actuators that have different design parameters generate different bending angles under the same pressure. Because the three actuators are of the same length, the entire trajectories generated by them are coincident; that is, when length parameters of actuators are the same, the changes of other design parameters will not affect the trend of the end trajectory. Hence, when the operation task needs to match a trajectory of a particular path, a single pneu-net actuator needs to be made up of several interrelated segments, and each segment requires different and appropriate design parameters.   On the contrary, to achieve target trajectory matching of the pneu-net actuator, three different design parameters need to be taken into account: segment length, chamber width, and middle-layer width. Because the lengths of actuators used in the bending performance experiments are the same as that of the reference actuator (length of the reference actuator L r � 85 mm), in order to quantitatively analyse the influence of these three crucial design parameters on the bending performance, an additional comparative experiment is designed. e actuators in this experiment have only the variable-length parameter, and the other parameters are consistent with those of the reference actuator.
It can be seen from Figure 9 that the relation between the radius of the curvature and the pressure of the actuators with different length parameters is approximately the same. Hence, for actuators with different length parameters, we can use the constant characteristic of the radius of the curvature to represent the bending angle. According to equation (7), the radius of the curvature of the actuator can be expressed as When the length parameter L is changed, according to the constant characteristic of the radius of the curvature, the bending model of the actuator can be expressed as For a given target trajectory, the length parameter of the actuator is a determined parameter, i.e., a constraint condition. However, the number of segments, the length of each segment, and the corresponding two width parameters of the desired actuator need to be optimized. To tackle that problem, a multiobjective optimization algorithm based on the genetic algorithm (GA) is proposed as an automatic search optimization technology, which automatically adjusts the design parameters to achieve the optimization goal. e search of the GA for the desired actuator with better predictive capabilities is treated as a minimization problem with the mean square error (MSE) as an adaptation value, which refers to the error between the design trajectory and the target trajectory. Taking the head of the desired pneu-net actuator as the origin of the base coordinate system, the design trajectory can be calculated based on the homogeneous coordinate transformation method: x y 1 � 0 0 1 1 0 0 0 1 0 where x i � R i (P). sin Θ * P , y i � R i (P). 1 − cos Θ * P , i � 1, 2, . . . , n, n is the number of segments.
e parameters of the GA are set as follows: the size of the population is 80; the crossover rate and mutation rate are 0.8 and 0.05, respectively; the number of generations is 300; and the fitness function of the GA for the design trajectory is presented as where m is the number of comparison points selected in the target trajectory. Hence, the optimum design method of the pneu-net actuator proposed in this work consists of two steps: (1) With the results of FEA, more sensitive design parameters are determined and the bending model of the actuator is established. (2) According to the design objectives, the GA is used to optimize these three crucial design parameters (i.e., segment length, chamber width, and middle-layer width) to provide a set of optimal geometric parameters for the optimum design of the pneu-net actuator. In the next section, an example will be used to illustrate the feasibility and effectiveness of our proposed method.

Experimental Results and Discussion
To verify the effects of the optimum design method on the performance of the matching-specific target trajectory, an actuator is designed by the grasping motion of a human index finger. e index finger contains metacarpophalangeal, interphalangeal, and proximal interphalangeal joints, in which each joint generates different bending angles when completing a grasping motion. Moreover, the index finger consists of three joints connected by three links, respectively, and each segment of the actuator can be regarded as a combination of one joint and one link. To obtain the specific trajectory of the grasping motion of an index finger, one index finger is photographed to record the configurations (the length of each link and the angles between the links) and the trajectory (Figures 10(a), 10(b), and 10(d)).
On the basis of the obtained target trajectory, we need to ensure that each segment of the desired actuator can reach their respective target bending angle under the same pressure, and then the desired actuator can accurately match the target trajectory. To match the target trajectory, the GA-based optimum design method is used to design the desired actuator by adjusting the values of segment length, chamber width, and middle-layer width. e parameters of the designed actuator are illustrated in Figure 10(c): the lengths of segments 1, 2, and 3 are 24, 26, and 45 mm, respectively; segment 1 has a chamber width of 3 mm, segment 2 has a middle-layer width of 18 mm, and other parameters are the same as those of the reference actuator. Using finite element simulation, the designed actuator which consists of these three segments with these different segment lengths and design parameters can achieve different bending performances to match the configuration of the index finger.
A casting method is adopted to manufacture the designed actuator. e moulds for the designed actuator are designed in SolidWorks, and these moulds are made by 3D printing technology. First, Smooth-On Dragon Skin 30 silicone parts A and B are regularly mixed and placed in a vacuum box (Figure 11). After all the bubbles are dispelled, the silicone mixture is poured onto the moulds. As shown in  Mathematical Problems in Engineering Figure 11, the designed actuator is divided into two parts that are separately manufactured by using corresponding moulds and are bonded together when the silicone mixture was solidified.
To verify the actual performance, we capture pictures of the designed actuator under different pressures to calculate the bending angle and length of each segment (Figures 12(a)  and 12(d)). e results displayed in Figure 12(b) demonstrate that the actual bending angle of the actuator and target bending angles of the index finger are accurately matched during a grasping motion. Comparing the target trajectory with the actual trajectory of the actuator in Figure 12(c), we can see that most of the two trajectories have a good coincidence degree, but a small part of them does not coincide completely in the initial stage of the path, which is the result of the performance of the actuator itself. Each segment of the actuator has uniformly distributed chambers to drive the actuator achieve bending motion, which causes the link to get shortened under pressure. us, each segment must be designed to be slightly longer than the desired segment, thereby indicating that the actual trajectory may not exactly match each point on the target trajectory, especially without pressure. Moreover, as shown in Figure 12(c), a singlesegment actuator cannot achieve this target trajectory matching.
Furthermore, a series of grasping experiments are performed to verify that the designed actuator can simulate grasping motions of the index finger. From the experimental results displayed in Figure 13, it can be seen that the actuator can grasp different relatively small objects, such as a bottle, mould, and ping-pong ball. Hence, these experiments prove that the actuator can achieve target trajectory matching and complete the stable grasping of different objects.

Conclusion
is study has proposed a method for designing a pneunet actuator that can match the target trajectory by adjusting the design parameters. Based on the technology of FEA, we can see that the effects of the chamber width parameter and the middle-layer width parameter on the bending performance of actuators are more sensitive than those of other parameters. Besides, no matter which width parameter changes, the bending angle increases linearly with the increase of pressure. According to the relation between the bending angle and the increment of width, a bending model of the pneu-net actuator is established with the results of linear fitting and cubic polynomial fitting, and both fitting results have higher R 2 (i.e., R 2 ≥ 0.99).
Because the trajectories generated under the actuators with different design parameters are the same when the lengths of actuators are the same, we propose a segmental optimal design method for the desired actuator. To match the target trajectory, the number of segments, the length of each segment, and the corresponding two width parameters of the chamber and middle layer are optimized by using the GA. e experimental results demonstrate that the actuator which is designed on the basis of the abovementioned method can accurately match the target trajectory generated in accordance with the specific trajectory of an index finger during a grasping motion. Furthermore, the designed actuator can perform stable grasping by simulating the grasping motions of the index finger.
us, the proposed optimum design of the pneu-net actuator method can improve the performance and give a solution for designing pneu-net actuators for the target task, thereby extending their applications in the field of soft grippers, crawling robots, and bionic robots. e performance of the pneu-net actuator is slightly limited because it can only generate bending motions. us, our future work aims to combine several segments that can collectively generate bending, twisting, and extending motions in a single actuator to achieve various trajectories.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.