2.1. Description of the module

For developing the multi-DOF soft pneumatic module, we define a simple geometry as a design requirement; in this way, the design, simulation and manufacturing process would be practical and efficient; therefore, it is an attractive alternative for use in robotics applications and soft manipulators. Likewise, it must have a wide range of mobility, as well as a length of 100–150 mm and a diameter of 40–70 mm. Thus, the proposed module is composed of four semicircular channels denoted by $Ch_1$ , $Ch_2$ , $Ch_3$ and $Ch_4$ that are controlled by independent pneumatic signals. The channels are symmetrically distributed and separated 90 $^{\circ }$ around Z axis as shown in Fig. 1(b). The design of this soft pneumatic module proposes to remove the material in the central part, allowing to optimise the channels’ design. The empty central part is sealed with covers, thus containing a volume of air inside. This strategy accordingly allows improved mobility because it reduces weight and the air contained inside works as an additional wall that amplifies mobility. Likewise, its peculiar geometry eliminates excess material in the module, reducing its size and weight and thus improving its performance. The module’s dimensions are 47 mm wide by 150 mm long, the channels’ wall thickness are 3 mm, while the wall thickness of central part has 3 mm (Fig. 1(a)). This module addresses a wide mobility range since the multi-channel structure allows performing bending motion in several directions as well elongation motion in the longitudinal axis direction. The independent activation of $Ch_1$ or $Ch_3$ produces bending that could be regarded as a rotation around $X$ -axis, while the independent activation of $Ch_2$ or $Ch_4$ produces bending motion regarded as a rotation around $Y$ -axis as shown in Fig. 1(b). Moreover, when two or three channels are activated simultaneously, bending motion is performed in different directions, as illustrated in Fig. 1(c). On the other hand, the simultaneous activation of the four channels generates elongation along the $Z$ -axis as represented in Fig. 1(b).

Figure 1. Mutli-DOF soft pneumatic module description.

The multi-channel soft pneumatic module has three DOFs: one elongation along Z-axis, represented by $l$ , and the other two degrees of freedom correspond to bending motion regarded as rotations around $X$ and $Y$ axes, represented by $\beta$ and $\delta$ , respectively, as described in Fig. 1(b).

2.2. FEM analysis

In this section, we present a FEM analysis of the multi-DOF soft module using ANSYS Workbench software. Four cases are considered: $Ch_1$ activation (Case 1), $Ch_1$ + $Ch_2$ activation (Case 2), and $Ch_1$ + $Ch_2$ + $Ch_3$ activation (Case 3); these cases are for bending motion. Finally, $Ch_1$ + $Ch_2$ + $Ch_3$ + $Ch_4$ activation (Case 4), this case generates linear displacement. However, it was performed a small linear displacement of 14.75 mm; for kinematics, we despise this motion.

The FEM analysis was performed considering the Ecoflex 00-50 as the actuator’s material, which Yeoh’s hyperelastic model was reported by ref. [Reference Kulkarni28]. On the other hand, it is considered as boundary conditions the gravity force (G) and as a restriction the actuator has a fixed support in one of its faces (Fig. 2(a)).

Figure 2. FEM analysis, boundary conditions and the graphic of bending angle – pressure relationship for $Ch_1$ , $Ch_{1}+Ch_{2}$ and $Ch_{1}+Ch_{2}+Ch_{3}$ .

For calculating the deformation angle, a coordinate system was placed on a finite element located at a point of interest in the upper part of the actuator, as shown in Fig. 2(b)). Subsequently, it is obtained its nodal coordinates from its initial state to its final state. In this way, we determine the actuator’s strain-stress, reaching angles of 124.5 $^{\circ }$ , 116.7 $^{\circ }$ and 29.36 $^{\circ }$ when activating the channels $Ch_1$ , $Ch_1$ + $Ch_2$ , $Ch_1$ + $Ch_2$ + $Ch_3$ , respectively. Finally, to determine the angle, the Eq. (13) and the coordinates obtained previously are used to calculate the deformation. For more information, see Section 3.1. Kinematic model.

2.3. Manufacturing process

This section presents the manufacturing process by using the casting technique. We used the elastomer Ecoflex 00-50 supplied by Smooth On, Inc., which has an elongation at break of 980% and a shore hardness of 00-50.

As shown in Fig. 3, we design three pairs of moulds: main moulds, channel moulds and cover moulds.

Figure 3. Description of the moulds to manufacture the soft module: (a) main moulds, (b) channel moulds and (c) cover moulds.

The main moulds shape the external geometry of the module (Fig. 3(a)). The length and width of these moulds are 152 and 50 mm, respectively. The columns in the mould on the right allow it to be assembled to the other main mould via the guideway. The guideway in both moulds avoid leaks in the casting process and non-alignment during assembling. Furthermore, the locks keep both moulds fixed, improving the assembly quality. The brackets permit an easy disassemble.

The channel mould illustrated in Fig. 3(b) creates the internal geometry of the module. The dimensions are length = 144 mm and width = 50 mm. The pillars are attached to the upper guideways of the main moulds, and the channel mould holders fit into the bottom guideways of the main moulds, preventing material leaks. The bases on channel moulds delimit the covers’ height (5 mm). Figure 3(c) presents the cover moulds. The mould on the left allows building the bottom cover. The mould on the right generates the top cover and the four air intakes. The columns in both moulds keep the module fixed, guaranteeing appropriate sealing. The assembly of the channel mould and main mould is depicted in Fig. 4(a). In this step, the elastomer is mixed and poured into the assembled moulds. The air bubbles are eliminated by using a vacuum degasser, and afterwards the elastomer is cured for 4 h. Subsequently, step (a) is repeated with the other channel and main moulds (Fig. 4(b)). Later, both parts are assembled to shape the soft module’s geometry. Again, the elastomer is cured for 4 h. In step (c), the moulds are unmounted, obtaining the module without covers. In step (d), elastomer is poured into the mould of the bottom cover. The bubbles are eliminated with the vacuum degasser, and the module generated in step (c) is assembled with this mould. Now, the elastomer is cured for 2 h. In step (e), the elastomer is poured into the mould of the top cover, generating the four air intakes and sealing the module. Here, the elastomer is cured for 2 h. Finally, in step (f), the multi-DOF soft pneumatic module is constructed.

Figure 4. Manufacturing process by casting method.

3.1. Kinematics model

Since the soft pneumatic module bends continuously, we seek to drive expressions for arc parameters ( $l(q)$ , $k(q)$ , and $\phi (q)$ ), which represent the length of the module, the inverse of the radius of curvature r and the angle out-of-plane motion of the module, respectively. In function of the joint variables $q=[l_{1}, l_{2}, l_{3}, l_{4}]^T$ which describe the lengths of four pneumatic channels as shown in Fig. 5.

Figure 5. (a) Diagram of a bending module related to the arc parameter. (b) Diagram of the distribution of the for channels in a single bending module, seen from top view.

For the kinematic analysis, we consider two bending DOF. The four independent channels can be actuated with pressurised air, where the position and orientation can be controlled by programmed pressure.

Based on the work presented in ref. [Reference Webster and Jones29], we will now describe the kinematic model for our multi-DOF soft pneumatic module related to the curvature radius. According to Eq. (1) (Fig. 5(a)), the curvature radius r measured from the module’s centre corresponds to the curvature radius for the $j-th$ channel (Fig. 5(a)):

(1) \begin{equation} r_{j} = r - bcos(\phi _j ) \end{equation}

Here, $b$ denotes the distance of a channel from the centre of the module and $\phi$ specifies the angle between the bending direction and the location of the $j-th$ channel.

Multiplied (Eq. (1)) by the arc angle $\theta$ and remembering that $l=\theta r$ and $l_{j} =\theta r_{j}$ , the result between the arc length of the module ( $l$ ) and the arc length of the $j-th$ channel ( $l_j$ ):

(2) \begin{equation} l = l_{j} + \theta b cos(\phi _j) \end{equation}

Thus, it is possible to determine $l(q)$ noting that the module’s location is related to its bending plane $\phi (q)$ with $\phi _{1}=0^{\circ }$ , $\phi _{2}= 90^{\circ }$ , $\phi _{3}= 180^{\circ }$ and $\phi _{4}=270^{\circ }$ , as shown in Fig. 5(c). Thus, $\sum _{j=1}^{4} = cos(\phi _j)=0$

(3) \begin{equation} l(q)= \frac{1}{4} \sum _{j=1}^{4} l_{j} \end{equation}

(4) \begin{equation} l(q)= \frac{1}{4} \left (l_{1}+l_{2}+l_{3}+l_{4} + \theta b \left (cos(\phi _ 1) + cos(\phi _2) + cos(\phi _3) cos(\phi _4) \right ) \right ) \end{equation}

Using Eq. (2) for the channels 1 and 3, we have $ l=l_{1}+\theta b cos(\phi _1)$ and $l=l_{3}+\theta b cos(\phi _3)$ . Equating both equations, we obtain $\theta b = \dfrac{l_{3}- l_{1}}{cos(\phi _ 1)-cos(\phi _ 3)}$ . Using the same procedure for channels 2 and 4, we compute $\phi (q)$ as shown in Eq. (5):

(5) \begin{equation} \phi (q)=tan^{-1}\left ( \frac{l_{4}-l_{2}}{l_{3}-l_{1}}\right ) \end{equation}

To obtain the module’s curvature $k(q)$ , we have $\theta =kl = \dfrac{l_{j}}{r_{j}}$ and $r_{j}= \dfrac{l_{j}}{kl}$ =, combining with (1)

(6) \begin{equation} k= \frac{l-l_{j}}{l b cos\phi _j} \end{equation}

(7) \begin{equation} k(q)=\frac{\left ( l_{1}-3l_{2}+ l_{3}+l_{4}\right )\sqrt []{\left ( l_{4}-l_{2} \right )^{2} + \left ( l_{3}-l_{1} \right )^{2}} }{b\!\left ( l_{1}+l_{2}+l_{3}+l_{4} \right ) \left ( l_{4}-l_{2} \right )} \end{equation}

On the other hand, based on the work presented in ref. [Reference Abidi, Gerboni, Brancadoro, Fras, Diodato, Cianchetti, Wurdemann, Althoefer and Menciassi30], we will now describe the kinematic model for our multi-DOF soft pneumatic module related to the curvature radius. According to the Fig. 6, the plane of the bending ( $\phi$ ) can be found by using the projection oh the position of point C on the x–y plane (Eq. (9)):

(8) \begin{equation} \theta = cos \!\left ( 1-(k \sqrt{x^2 + y^2} \right ) \end{equation}

(9) \begin{equation} \phi = tan^{-1}\left ( \frac{y}{x} \right ) \end{equation}

Figure 6. Configuration of the constant curvature parameters and their relation to the fixed coordinate system.

According to the Fig. 6 $\phi$ , can also be expressed as:

(10) \begin{equation} cos(\phi ) = \frac{OX}{OA}= \frac{x}{\sqrt{x^2 + y^2}} \end{equation}

Likewise, the author [Reference Abidi, Gerboni, Brancadoro, Fras, Diodato, Cianchetti, Wurdemann, Althoefer and Menciassi30] defines that the inverse of the radius (k) is

(11) \begin{equation} k(x,y,z)=\frac{ 2\ \sqrt []{x^{2} + y^{2}} }{x^{2} + y^{2} + z^{2}} \end{equation}

Also, recall that the arc length is l = r $\theta$ for the constant curvature (CC) assumption. If B is the centre of curvature, then from Fig. 6 the angle $\theta$ can be expressed as:

(12) \begin{equation} cos(\pi -\theta ) = \frac{BA}{BC}= \frac{\sqrt{x^2 + y^2}-r}{r} \end{equation}

Rearranging and replacing r by k of the Eq. (12), then $\theta$ can be expressed as is shown in Eq. (13):

(13) \begin{equation} \theta = cos\!\left (1- \left (k \sqrt{x^2 + y^2}\right ) \right ) \end{equation}

Since the proposed soft module generates constant curvatures with the multiple motions, the previously presented kinematic analysis was performed on the FEM analysis to determine the arc of curvature and its angle. In this way, it can be seen in Fig. 7(a), (b) and (c) that the multi-DOF soft module generates constant curvature when it activates $Ch_1$ , $Ch_1$ + $Ch_2$ , $Ch_1$ + $Ch_2$ + $Ch_3$ , respectively.

Figure 7. Distance A = B = r. (a) Constant curvature when activating $Ch_1$ ( $\theta$ =94 $^{\circ }$ ). (b) Constant curvature when activating $Ch_1$ + $Ch_2$ ( $\theta$ =145 $^{\circ }$ ). (c) Constant curvature when activating $Ch_1$ + $Ch_2$ + $Ch_3$ ( $\theta$ =103.6 $^{\circ }$ ).

3.2. Characterisation of the multi-DOF soft pneumatic module

Sensors [Reference Yap, Goh, Yeow, Lackovic and Vasic31]–[Reference Tan, Gu and Ren32] and computer vision [Reference Mosadegh, Polygerinos, Keplinger, Wennstedt, Shepherd, Gupta, Shim, Bertoldi, Walsh and Whitesides14], [Reference Polygerinos, Lyne, Wang, Nicolini, Mosadegh, Whitesides and Walsh33], [Reference Lu, Xu and Li34] are used in various characterisation methods. In this section, we present the characterisation of the multiple motion soft module through computer vision techniques. It is important to characterise the performance of soft pneumatic modules in terms of pressure, displacements (angular and linear) or the force that they can achieve to subsequently develop a type of control related to these variables.

Figure 8 shows the experiment assembly to characterise the module’s motion. The module is fixed to the bottom, two red circular marks were placed on the module, one on the top and the other on the bottom. We use the camera IDS UI-3240CP-HQ 1/ 1.8”, with a resolution of 1280 × 1024 ppi at 60 fps. The marks are using as reference points to track the module and measure the movement based on the applied pressure values. We proposed to perform six tests (T1–T6) for the characterisation of the module motion, and the six tests were averaged. Likewise, we developed a closed-loop FUZZY pressure control based on a piston, a cylinder, a worm gear and a stepper motor, as well as its user interface in LabVIEW.

Figure 8. Experiment assembly for module’s characterisation.

The presented actuator addresses a wide mobility range since bending is performed in several directions. When a single channel is activated independently, the actuator generates bending in the four axes of Cartesian plane ( $X$ , $Y$ ). When activating two channels simultaneously, the bending is achieved in four different quadrants, and when three channels are activated at the same time, the bending motion is also performed in the four quadrants of Cartesian plane but with different curvature radius. Before performing the actuator’s characterisation, three bending tests were performed to break the hysteresis [Reference Shi and Chen35].

We develop an application in Matlab to evaluate the bending angle at different pressure values. The methodology to compute the arc angle consists of capturing an image when the actuator has reached the applied pressure value. Subsequently, the image is inserted into the programme and with a tool we select the centres of the actuator’s circular marks. In this way, Matlab obtains the coordinates of the point of interest, and we compute the curve radius with the Eq. (14) and $r= \dfrac{1}{k}$ , [Reference Abidi, Gerboni, Brancadoro, Fras, Diodato, Cianchetti, Wurdemann, Althoefer and Menciassi22] for each coordinate of the mark, where $k$ is the inverse of the curvature radius $r$ and $X$ , $Y$ , and $Z$ are the position of the circle mark:

(14) \begin{equation} k(x,y,z)=\frac{ 2 \,\sqrt []{ x ^{2} + y ^{2}} }{x^{2} + y^{2} + z^{2}} \end{equation}

We evaluated the value $k$ and the coordinates of the interest points (in a pressure value) in Eq. (13) to compute $\theta$ . This process is repeated for each desired pressure value.

The bending motion characterisation is presented for the independent activation of channel 1 ( $Ch_1$ ), the simultaneous activation of channels 1 and 2 ( $Ch_1$ + $Ch_2$ ), and the synchronised activation of channels 1, 2 and 3 ( $Ch_1$ + $Ch_2$ + $Ch_3$ ). The distribution of the channels is shown in Fig. 1(b). The rest of the combination has the same bending angles but in different directions, as illustrated in Fig. 1(c). Figure 9 illustrates the mobility of the actuator in different direction according to the activation of the channel: (a) activation of a single channel, (b) synchronous activation of two channels and (c) simultaneous activation of three channels.

Figure 9. Bending motion in different directions depending on the activation of: a) one channel, b) two channels and c) three channels.

Figure 10(a) shows the graphics for the bending angle ( $^{\circ }$ ) vs pressure (kPa) related to the characterisation for $Ch_1$ , $Ch_1+Ch_2$ and $Ch_1+Ch_2+Ch_3$ . In addition, we obtained a maximum angle of $123.95^{\circ }$ at 10.19kPa when $Ch_1$ is activated independently. For the simultaneous activation of $Ch_1$ and $Ch_2$ , the resulting bending angle was $115^{\circ }$ for a pressure value of 9.32kPa; we achieved a bending angle of $28.91^{\circ }$ by applying a pressure of 9.04kPa when activating the $Ch_1+Ch_2+Ch_3$ , simultaneously.

Figure 10. Bending–angle vs pressure relationship for the activation of channels $Ch_1$ , $Ch_1$ + $Ch_2$ y $Ch_1$ + $Ch_2$ + $Ch_3$ .

Furthermore, the graphic in Fig. 10(b) shows different views of the angular and Cartesian relationship when are pressurised channels $Ch_1$ (right) and $Ch_2$ (left) independently. Likewise, Fig. 10(c) shows the angular trajectory and Cartesian relationship when we simultaneously activate channels $Ch_1$ + $Ch_2$ (right) and channels $Ch_2$ + $Ch_3$ (left).

4.1. Soft manipulator description and kinematics analysis

In order to demonstrate the impact of the proposed soft module, we developed a soft manipulator manufactured with the elastomer Ecoflex 00-50, one of the most commonly used materials in soft robotics. The proposed manipulator consists of three completely soft elements, as shown in Fig. 11, two bending modules, and a soft gripper as end effector.

Figure 11. Description of the soft manipulator. a) CAD design. b) Manufactured soft manipulator.

The length of modules one and two is 123 and 78 mm, respectively. Both modules have a width of 47 mm. The end effector consists of three fingers spaced at 120 $^{\circ }$ . Through a single channel, it can open or close the fingers. Each one is designed with four chambers with a length of 65 mm and a height of 19 mm as shown in Fig. 12(a). For the manufacturing of the soft manipulator shown in Fig. 12(b), we assembled both models and the gripper by using the bonding method [Reference Ilievski, Mazzeo, Shepherd, Chen and Whitesides36]. This method allows fusing two or more elements simply by first adding a thin layer of uncured material, similar to glue, and then finishing the curing process, the pieces accordingly stick together.

Figure 12. Morphology of the bending modules and the end effector.

The robot has two non-deformable or “death-sections” due to the manufacturing process. The eight independent channels can be actuated with compressed air, where the position and orientation can be controlled by programmed pressure.

The proposed soft manipulator is shaped by two modules which bend continuously. Previous studies [Reference Webster and Jones29, Reference Gong, Xie, Yang, Wang and Wen37] have already shown how to solve the forward transformation. The kinematic modelling describes the transformation from the channel’s length ( $l_{ij}$ ) to the coordinate of the end of the soft manipulator ( $X_n$ , $Y_n$ , $Z_n$ ), where $i$ and $j$ represent the segment and channel, respectively (see Fig. 13(a)). Furthermore, Fig. 13(b) presents the general kinematics model for a single bending module, where $r_i$ represents the curvature radius while $l_i$ is the arc length of the $i-th$ channel.

Figure 13. Kinematics modelling: (a) Diagram of transform of the $i-th$ module position parameters ( $X_i$ , $Y_i$ , $Z_i$ ) using length parameters ( $l_i$ ) via arc parameters ( $k_i$ , $\phi _i$ , $\theta _i$ ). (b) General kinematics model for a single bending module related to the curvature $k$ .

Figure 14. Sequence images of a pick and place task.

Based on the work presented in ref. [Reference Webster and Jones29], we assume: (a) the curvature rate is constant in both bending module while the non-deformable section presents a line; (b) the channels of each bending segment are parallel, and the cross sections are equal in the same module; (c) the soft manipulator is analysed without gravity forces and load.

Taking into account those assumptions, we managed to simplify the soft manipulator kinematics as independently controlled constant curves. Figure 13(a) shows the transformation between $n$ bending modules. $Q_0$ is the base static coordinate system, while $Q_{is}$ and $Q_{i\bigtriangleup }$ represent the end of a bending module and a death section in the $i-th$ module.

Considering all the modules are actuated in the same way and the kinematics model of a single bending module derived in the previous section, we can figure out the transformation in a single module. The translation of the $ith$ bending module regarded to curvature $l_n$ can be represented as a rotation around the Y-axis with curvature angle $\theta _i$ . Rotation around the Z-axis is represented by $\varphi _i$ as shown in Fig. 5. We consider that the X-axis points to the centre of the circle and that the Z-axis coincides with the tangent of the circle. Thus, multiplying the homogeneous matrix (which is constituted by a matrix of rotation and position) with the rotation matrix $R(\phi _i)$ and zero translation, we obtain the transformation matrix for one bending soft module:

(15) \begin{equation} T_{is}^{i-1\bigtriangleup }=\begin{bmatrix} c^{2} \phi _{i}c\theta _{i}+s^{2} \phi _{i}\;\;\;\; & c\phi _{i}s\phi _{i}\!\left ( c\theta _{i}-1 \right )\;\;\;\; & c\phi _{i}s\theta _{i}\;\;\;\; & rc\phi _{i}\!\left ( 1-c\theta _{i} \right ) \\[5pt] c\phi _{i}s\phi _{i}\!\left ( c\theta _{i}-1 \right )\;\;\;\; & s^{2} \phi _{i}c\theta _{i}+c^{2} \phi _{i}\;\;\;\; & s\phi _{i}s\theta _{i}\;\;\;\; & rs\phi _{i}\!\left ( 1-c\theta _{i} \right )\\[5pt] -c\phi _{i}s\theta _{i}\;\;\;\; & -s\phi _{i}s\theta _{i}\;\;\;\; & c\phi _{i}\;\;\;\; & rs\theta _{i}\\[5pt] 0\;\;\;\; & 0\;\;\;\; & 0\;\;\;\; & 1 \end{bmatrix} \end{equation}

The death module’s transformation matrix is represented by Eq. (16), where $d_{si}$ is the length of the death section of the $ith$ bending module.

(16) \begin{equation} T_{i+1\bigtriangleup }^{ is}=\begin{bmatrix} 1\;\;\;\; & 0\;\;\;\; & 0\;\;\;\; & 0\\[5pt] 0\;\;\;\; & 1\;\;\;\; & 0\;\;\;\; & 0\\[5pt] 0\;\;\;\; & 0\;\;\;\; & 1\;\;\;\; & d_{si} \\[5pt] 0\;\;\;\; & 0\;\;\;\; & 0 \;\;\;\; & 1 \end{bmatrix} \end{equation}

From the combination of $n$ bending modules and death sections, we can deduce the transformation for the soft manipulator as is described in Eq. (17):

(17) \begin{equation} T_{2\bigtriangleup }^{0} = T_{1s}^{0} \cdot T_{1\bigtriangleup }^{1s} \cdot T_{2s}^{1\bigtriangleup } \cdot T_{2\bigtriangleup }^{2s} \end{equation}

4.2. Motion sequence for the soft manipulator

We performed a motion sequence to pick an object from point A and place it at point B. The eight independent channels are controlled (in open loop) by programmed pressure signals.

The manipulation sequence starts at the rest position, as shown in Fig. 14(a). In this position, the robot is posed over the object. Thus, the soft gripper is activated and holds the object, as illustrated in Fig. 14(b). Afterwards, the orientation of the soft manipulator is controlled by pressurising one channel in each bending module, keeping the gripper activated. The activation of channel 1 ( $Ch_1$ ) of bending module 2, followed by the pressurisation of channel 4 ( $Ch_4$ ) of bending module 1, results in the manipulator being positioned over the B point, as shown in Fig. 14(c) & (d). When the manipulator is located, the gripper is depressurised and the object is placed, as illustrated in Fig. 14(e). Finally, the soft manipulator returns to its rest position (Fig. 14(f)).