2.1. Problem formulation

The original manipulator design analyzed in this work is based on the 2 DOF manipulator described in [Reference Ferrandy, Indrawanto, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena, Mihradi and Virdyawan23], which has a tubular shape with three internal chambers spaced at $120^{\circ }$ and separated by straight partition walls. It was designed to have a morphing internal structure to maximize its lateral force [Reference Garriga-Casanovas, Collison and Baena12] and is equipped with two helical windings to avoid ballooning, as well as an inextensible cable sleeve at the center, to improve its lateral force. The manipulator has a total length of 76 mm, which includes a 10 mm bottom cap and a 6 mm top cap. Figure 1 shows the original design manipulator and its cross-section dimensional specifications.

The manipulator is composed of several distinct parts, namely the main body, the fiber winding, and the central rod. The main body and central rod exhibit perfect symmetry every $120^{\circ }$ , while the fiber winding remains as the sole asymmetrical component at $120^{\circ }$ . In the original design, the fiber winding is constructed with two helical windings wound in clockwise and counterclockwise directions, a configuration commonly employed in other works [Reference Zhang, Fan, Yang, Cao and Liao24, Reference Decroly, Mertens, Lambert and Delchambre25]. While this winding shape avoids generating a twisting effect [Reference Suzumori26], it achieves symmetry only every $180^{\circ }$ .

Figure 1. Original design manipulator and cross-section dimensional specifications. $\gamma$ and $\theta$ depict the bending plane orientation angle and the bending angle of the manipulator when actuated.

In several different works, including [Reference Virdyawan, Ayatullah, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena and Indrawanto14], it has been observed that during single-chamber pressurization of the original design, the manipulator tends to skew to the side, leading to a change in its plane of bending (as can be seen in Figure 2(b)). This issue occurs in each of the three chambers, but at different pressure values and in different directions. Given the asymmetrical nature of the winding used, it is highly likely that the windings contribute to the deviation in the bending plane to some extent. The effect of winding on the single-chamber pressurization bending of 2 DOF manipulators has been investigated in [Reference Ferrandy, Indrawanto, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena, Mihradi and Virdyawan23], where it was concluded that the intersection position of the two helical windings affects the stiffness of each chamber. Building upon this insight, it is hypothesized that when the intersection of the winding is not perfectly positioned above a particular chamber, the stiffness of that chamber can increase on one side, leading to the tendency of the manipulator to skew. However, this hypothesis fails to fully explain the observed deviation, as one of the chambers in the original design is located directly below the intersection and should not experience any asymmetrical nature of the winding.

Another hypothesized source of deviation is the buckling of the partition wall. Due to the morphing inner structure feature, during single-chamber pressurization, the partition wall located directly opposite the pressurized chamber would experience a compressive load. According to Euler’s buckling theorem, this load can cause the partition wall to buckle when it reaches a critical value. This hypothesis provides an explanation for the deviation observed in chambers with symmetrical features. It would also explain the sudden decrease in bending stiffness reported in [Reference Ferrandy, Indrawanto, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena, Mihradi and Virdyawan23] and the different values of bending plane orientation angles observed in [Reference Virdyawan, Ayatullah, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena and Indrawanto14] when the manipulator is continuously pressurized in clockwise and counterclockwise orders.

In the following section, we will further investigate these two hypotheses through finite element simulations to confirm the root cause of the bending plane deviation problem.

2.2. Finite element analysis

The finite element models were developed in Abaqus/Standard (SimuliaTM; Dassault Systèmes, Vélizy-Villacoublay, France), following the procedures, configurations, and material coefficients outlined in [Reference Ferrandy, Indrawanto, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena, Mihradi and Virdyawan23], which optimize convergence and computation time. It is important to note that the models were carefully meshed using a specific model partition pattern, with meticulous attention given to the mesh number and quality. The goal was to achieve a perfectly symmetrical and uniform meshing every $120^{\circ }$ to ensure that there are no variations in stiffness caused by the mesh, which could potentially lead to a misinterpretation in the analysis.

2.2.1. Winding’s asymmetry analysis

To investigate the winding hypothesis, we simulated the original design with its two helical windings, as well as other types of windings that have been tested in [Reference Ferrandy, Indrawanto, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena, Mihradi and Virdyawan23] to preserve the bending stiffness across different chambers. These options include a circular winding, which is completely symmetrical for every angle, and a six helical winding, a winding type with three pairs of helical windings arranged to be separated $120^{\circ }$ apart, making it symmetrical for a three-chamber manipulator also spaced 120 degrees apart (see [Reference Ferrandy, Indrawanto, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena, Mihradi and Virdyawan23] for details). Each chamber was subjected to a pressure ramped up to 0.1 MPa, and the deviation was monitored through the bending plane orientation angle ( $\gamma$ ). Figure 2(a) shows the relationship between the bending plane orientation angle ( $\gamma$ ) and the bending angle ( $\theta$ ) for each chamber and different fiber winding shapes.

The simulation results indicate that the manipulator with six helical windings performed the best, deviating at higher bending angles in all chambers. In comparison, circular winding only outperformed the original double helical windings in chambers 2 and 3. These results reveal that, even with symmetrical winding shapes, the manipulator still exhibits a tendency to skew to the side. When comparing the data from symmetrical windings to the original asymmetrical winding, it becomes evident that asymmetry contributes to the deviation in the bending plane.

Firstly, looking at the chamber 2 and 3 performances, we could observe improvement from the manipulator with symmetrical windings. This could be explained by the fact that in both of these chambers, the double helical winding is asymmetric. On the other hand, at chamber 1, circular windings did not perform better than helical winding, since at this chamber, the double helical winding is symmetric, meaning the deviations here were not directly affected by the winding’s symmetry.

Secondly, we observed that at chamber 2, the manipulator with double helical winding deviates to lower value, while at chamber 3 it deviated to higher value. This direction tendency happened to be the same direction of the side where the winding’s asymmetry would create the least stiffness [Reference Ferrandy, Indrawanto, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena, Mihradi and Virdyawan23].

Figure 2. Original design simulation results. (a) Bending plane orientation angle ( $\gamma$ ) versus bending angle ( $\theta$ ) of the manipulator with different types of winding. (b) The manipulator and its cross-section during pressurization at three different pressure values given to one of the chamber (depicted with red shading).

3.1. Design requirements

The main cause of deviation has been identified as the buckling of the partition wall. Therefore, we propose several design alternatives aimed at preventing or reducing the impact of this phenomenon. For our new design requirements, we have retained the original major specifications, including three working chambers, a 2 mm working channel at the center, a morphing inner structure, and an outside diameter of 12 mm. These specifications allow for a simpler direct comparison between the original and the new designs. Additionally, the 12 mm outside diameter is in line with the requirements of minimally invasive medical procedures [Reference Runciman, Darzi and Mylonas22]. The designs should also avoid overly complicated geometry and extremely thin parts (i.e less than 0.6 mm) to ensure the feasibility of the fabrication.

3.2. Proposed designs

Before developing the new designs, we conducted further simulations on the original design’s partition wall with modified dimensions, specifically with lower aspect ratios. This modification aimed to increase the wall’s second moment of inertia, thus increase the critical buckling load. Through these initial simulations, we learned that while this approach increases the bending angle before deviation occurs, it also compromises the morphing feature of the original design, which allows the maximization of the chamber’s cross-sectional area that leads to higher lateral force. This led us to explore other alternatives by modifying the structural shape of the partition wall.

Using structural shapes with high second moment of inertia would similarly compromise the morphing design of the manipulator. Therefore, our approach shifted to exploring shapes that allow controlled deformation of the internal chambers, allowing the chamber to expand first to maximize the total surface area before the wall receives the buckling load.

Our first option involves asymmetrical walls, which due to their asymmetry, would deform in one predefined direction, intentionally allowing bending plane deviation but in predictable direction. This sub-concept could potentially provide a simple and easy-to-manufacture solution to reduce the unpredictable impact of buckling walls. Our second option features symmetrical walls, which would be more challenging to manufacture but could completely eliminate bending plane deviation as they would symmetrically deform into a more stable shape with low aspect ratio.

In this work, we propose five alternative designs, with their geometries and dimensional specifications shown in Figure 3. The first two proposed designs use the asymmetrical deforming walls: angled walls (Figure 3(a)) and zig-zag walls (Figure 3(b)), while the remaining three designs use the symmetrical deforming walls: hollow diamond walls (Figure 3(c)), hollow circle walls (Figure 3(d)), and hollow diamond-circle hybrid walls (Figure 3(e)). All the proposed designs adhere to the design requirements mentioned in Section 3.1.

Figure 3. Proposed designs’ cross-section geometry and dimensional specifications. (a) Proposed design 1: angled walls. (b) Proposed design 2: zig-zag walls. (c) Proposed design 3: hollow diamond walls. (d) Proposed design 4: hollow circle walls. (e) Proposed design 5: hollow hybrid walls.

3.3. Finite element simulations

To evaluate the performance of the proposed designs mentioned in Section 3.2, finite-element models were again developed in Abaqus/Standard (SimuliaTM; Dassault Systèmes, Vélizy-Villacoublay, France), following similar configurations as the original design’s model from Section 2.2. The models were simulated by pressurizing one of the chambers individually up to 0.1 MPa, and the performance was evaluated through bending plane orientation angle ( $\gamma$ ) and bending angle ( $\theta$ ). In all the new models, the original two-helical windings were still used, but only the partition wall directly opposite the chamber located exactly below the windings intersection was tested. This ensured that the winding’s asymmetry would not affect the deformation of the partition wall or the expansion of the pressurized chamber to cause any buckling.

Figures 4(a) and 4(b) showcase the results of the finite element simulations for the original design and five proposed designs, including the bending plane orientation angles ( $\gamma$ ) and the bending angles ( $\theta$ ).

Analyzing the cross sectional deformations of asymmetrical wall design (zig-zag wall) in Figure 4(c) shows that the wall deformed early as expected, and in one definitive direction. Furthermore, no sudden jump in bending plane orientation angle was observed, as seen in the original design (Figure 2(b)), where the manipulator would start bending in straight but suddenly deviate to the side. This proves that asymmetrical deformable walls reduce the unpredictability of the original design. Additionally, the deforming feature of the partition wall provides less resistance to the manipulator’s chamber expansion and bending, improving the manipulator’s bending angle stiffness. However, even though the bending angle orientation angle is predictable in direction, it is unstable as it continues to deviate long after the wall has fully collapsed.

For symmetrical deformable wall designs (diamond, circle, and diamond-circle hybrid wall), analyzing cross-sectional deformation and the bending angle orientation angles ( $\gamma$ ) shows that these walls collapse symmetrically, eliminating any bending plane deviation and proves to be stable up to 0.1 MPa. As the walls morph into stable shapes (Figure 4(d)), they effectively withstand the compressive load and allow for the symmetrical expansion of the pressurized chamber. Similarly, although not as much as the previous sub-concept, the symmetrical walls also provide less resistance for the chambers to expand and the manipulator to bend compared to the original design.

Figure 4. Comparison of the designs simulation results. (a) Bending plane orientation angle ( $\gamma$ ) versus bending angle ( $\theta$ ). (b) Bending angle ( $\theta$ ) versus pressure. (c) Proposed design 2 manipulator and its cross-section during pressurization. Side view of the manipulator reveals deviating but a constant bending throughout pressurization. (d) Proposed design 3 manipulator and its cross-section during pressurization.

3.4. Design selection

In order to select the best design for experimental testing, we conducted a thorough evaluation of several key parameters based on the bending plane orientation angle, bending angle, and the manufacturing process. We used a relative comparison method with a quantitative approach to assist in the design selection process.

With this method, weights were initially assigned to each of the key performance parameters, with higher values indicating greater importance. We then evaluated each of the proposed designs across all relevant parameters, assigning scores from 0 to 1 on the corresponding scale ( $N_i$ ), and calculated the weighted score or rating ( $R_i$ ) by multiplying these scores with the predetermined weights ( $R_i = N_i \times \text{weight}$ ). The overall score for each proposed design was then obtained by adding these final ratings.

Taking into account the main aim of this study, we set the parameter with highest priority as the $\gamma$ slope and $\gamma$ slope consistency. Bending stiffness was prioritized next, being a common parameter for evaluating the performance of a manipulator. Fabrication complexity and mold complexity were equally prioritized next, as both have a direct impact on the prototype quality and, hence, the final experimental performance.

To determine the corresponding scales quantitatively, we used equations (1), (2), (3), (4), and (5), which were based on the relative differences of the proposed design compared to the original design. The gamma slope and slope consistency corresponding scale ( $N_{(j,1)}$ and $N_{(j,2)}$ ) were calculated based on the simulated bending plane orientation angle gradient, while bending stiffness ( $N_{(j,3)}$ ) was based on ( $\theta$ ) versus pressure curve, ranging from $0$ to $260^\circ$ . Similarly, both fabrication ( $N_{(j,4)}$ ) and molds complexity ( $N_{(j,5)}$ ) were determined by penalizing the score for every additional step or additional custom mold parts required.

(1) \begin{equation} N_{(j,1)} = 0.5 - \frac{1}{2} \frac{\frac{1}{n} \sum _{i=1}^{n} \nabla \gamma _j - \frac{1}{n} \sum _{i=1}^{n} \nabla \gamma _0}{\frac{1}{n} \sum _{i=1}^{n} \nabla \gamma _0} \end{equation}

(2) \begin{equation} N_{(j,2)} = 0.5 - \frac{1}{2} \frac{\frac{1}{n} \sum _{i=1}^{n} \Delta (\nabla \gamma _j) - \frac{1}{n} \sum _{i=1}^{n} \Delta (\nabla \gamma _0)}{\frac{1}{n} \sum _{i=1}^{n} \Delta (\nabla \gamma _0)} \end{equation}

(3) \begin{equation} N_{(j,3)} = 0.5 + \frac{1}{2n} \sum _{i=1}^{n} \frac{\theta _{j,i} - \theta _{0,i}}{\theta _{0,i}} \end{equation}

(4) \begin{equation} N_{(j,4)} = 1 - \frac{n_{(j,s)} - n_{(0,s)}}{n_{(0,s)}} \end{equation}

(5) \begin{equation} N_{(j,5)} = 1 - \frac{n_{(j,p)} - n_{(0,p)}}{n_{(0,p)}} \end{equation}

where $\nabla \gamma _j$ and $\nabla \gamma _0$ are the slope of the bending plane orientation angles of the proposed design $j$ and the original design, respectively. $\theta _j$ and $\theta _0$ are the bending angles of the proposed design $j$ and the original design, respectively. $n_{(j,s)}$ and $n_{(j,p)}$ are the number of steps and the number of custom parts required to manufacture the proposed design $j$ , respectively. $n_{(0,s)}$ represents the original design’s number of manufacturing steps, which in this work is set to four: initial molding, fiber winding, outer layer molding, and final sealing. $n_{(0,p)}$ is the number of parts in the original design molds, which include eight parts: a one-part inner mold, a two-part initial outer mold, a two-part final outer mold, and a three-part final sealing mold.

Table I outlines our relative comparison analysis of the proposed designs. Based on this analysis, proposed design 3 and 4 have the highest calculated rating value of 0.81. In this study, we selected proposed design 3 for fabrication and experiment testing, since its cross-sectional geometry involves fewer tight angles compared to the proposed design 4, offering easier fabrication. Furthermore, the diamond shape (proposed design 3) was also expected to provide greater stability when deforming due to its geometry, which should help improve the prototype’s performance, compensating for any imperfections in the prototype.

Table I. Relative comparison table

4.1. Manufacturing process

The original design prototypes were fabricated using the process and mold sets described in [Reference Ferrandy, Indrawanto, Sugiharto, Franco, Garriga-Casanovas, Mahyuddin, Baena, Mihradi and Virdyawan23]. The resulting prototype possesses the specifications, material, and winding shape of the simulations.

On the other hand, the new design prototype was manufactured using a more complex process and special mold sets, as its design presented a set of new challenges. The main challenge arose from its intricate and relatively small geometry, which necessitated molds consisting of multiple parts that were difficult to manufacture. This complex geometry also posed a de-molding issue, as the inner mold required had a larger surface area, resulting in increased friction. Another challenge stemmed from the hollow parts of the design, which required exhaust holes to prevent the trapping of air when the partition wall collapsed, further complicating the process. Bearing all of these factors in mind, a three-part mold and another two-part mold (shown in Figure 5(b)) were used. This three-part mold was used to fabricate the main body, complete with its fiber winding, bottom cap, and exhaust holes for the partition wall at the bottom, while the additional two-part mold was used to add the final top sealing cap on the main body and was manufactured with the 3D printer.

For the three-part mold, the first main part was the smaller outer mold consisting of two constituent parts, featuring grooves in its inner surface to create a path for an exact double-helical winding and manufactured using a conventional 3D printer (Objet500, Stratasys, Ltd., Israel). The second main part was the larger outer mold, which was also manufactured using a conventional 3D printer, but contrary to its smaller counterpart, it didn’t feature any grooves and had a slightly larger inner diameter. The third main part is the inner mold, comprising a bottom mold that holds a 2 mm high-speed steel rod, three pieces of chamber shafts, and three pieces of partition wall shafts. All of the inner mold parts were manufactured using a wire EDM cutting process, allowing the fabrication of precise and low-tolerance parts.

Figure 5(a) illustrates the overall manufacturing process for the new design. The first step involved assembling the inner mold with all its constituent parts, which was then placed inside the smaller outer mold. Following a similar process to the original design, the silicone-rubber material Dragon SkinTM 10 Medium (Smooth-On, Inc., Pennsylvania, USA) was injected into the smaller outer mold, resulting in the production of the bottom cap and main body of the manipulator (as can be seen in Figure 5(a)-1).

Next, the fiber thread was manually wound around the main body, following the guide path to create perfect double-helical windings (as shown in Figure 5(a)-2). Once completed, the specimen was placed inside the larger outer mold and re-injected with silicone-rubber material to secure the windings in place (shown in Figure 5(a)-3). Due to the presence of chamber shafts in the inner mold, the resulting cast already featured three exhaust holes located at the bottom cap.

Finally, the inner mold with all its constituent parts was removed, an inextensible central rod was inserted, and the two-part mold was used to seal the top of the prototype (as can be seen in Figure 5(a)-4 and 5(a)-5). This final sealing process required meticulous attention to ensure that all of the seven chambers were separated.

Figure 5. (a) Manufacturing process of the proposed design. (b) The three-part and two part mold of the proposed design. (c) Experimental setup.

4.2. Experimental setup

The experiments were conducted by clamping the bottom cap of the prototypes upside down on a fixture (as can be seen in Figure 5(c)). The rest of the setup followed a similar configuration described in [Reference Franco and Garriga-Casanovas27], where an aurora electromagnetic 5 DoF sensor was fixed at the tip of the manipulator, and its position and rotation were measured using an electromagnetic tracking system (Aurora, NDI, Canada, 0.70 RMS).

Prior to the experiment, a calibration process was performed by actuating Chamber 1 at 0.05 MPa. A custom script was used to record the sensor’s rotation matrix before ( $\mathbf{R}_{\text{ref},1}$ ) and after actuation ( $\mathbf{R}_{\text{ref},2}$ ). The bending plane identified during the calibration was later used as the reference plane for measuring $\gamma$ .

In this work, we performed two types of pressurization for each prototype. These included individual chamber pressurization up to 0.1 MPa in each of the three chambers and two chambers vectorization. The vectorization process included pressurizing different combinations of chambers, each with various pressure ratios. After each step, both chambers were depressurized. A pneumatic actuation system, comprising proportional pressure regulators (Tecno Basic, Hoerbiger, Germany), as mentioned in [Reference Franco and Garriga-Casanovas27], was utilized to pressurize the prototypes as needed.

For each data point, a new rotation matrix was obtained 3 s after the pressurization, ensuring quasi-static condition. Performance was evaluated by calculating the bending angle ( $\theta$ ) and the bending plane orientation angle ( $\gamma$ ), using equation 6 and 7, respectively.

(6) \begin{equation} \theta = 2 \cos ^{-1}(q_{\text{ref},1} \cdot q_m) \end{equation}

(7) \begin{equation} \gamma = \cos ^{-1}\left (\frac{\hat{n}_m \cdot \hat{n}_{\text{ref}}}{\|\hat{n}_m\| \cdot \|\hat{n}_{\text{ref}}\|}\right ) \end{equation}

where $q_{\text{ref},1}$ and $q_m$ are the quaternions corresponding to $\mathbf{R}_{\text{ref},1}$ and $\mathbf{R}_m$ , respectively. Instead $\hat{n}_{\text{ref}}$ and $\hat{n}_m$ are the normal vector of the reference bending plane and measurement bending plane, respectively.

4.3. Results comparison and discussion

Figure 6(a) illustrate the performance of both prototypes during individual chamber pressurization in each of the three chambers. Figure 6(b) compares the relations between the bending plane orientation angle and bending angle of the original and new design prototypes.

Figure 6. Experimental results comparison of original and proposed design. (a) $\theta$ versus pressure for chamber 1 (top), chamber 2 (middle), and chamber 3 (bottom) (b) $\gamma$ versus $\theta$ for each chamber pressurization. (c) vectorization performance - $\gamma$ versus order of pressurized chamber.

Upon comparing the original and the new design, improvements in the bending plane angle orientation angle ( $\gamma$ ) were observed. The angles of chamber 1 and chamber 3 in the new design prototype were noticeably more stable, reaching a higher bending angle ( $\theta$ ) before $\gamma$ deviated. However, the pressurization of chamber 2 resulted in little to no improvement in terms of deviation and only reached around $110^{\circ }$ in $\gamma$ , slightly lower than expected. Additionally, at the maximum of the pressure range, the new design prototype still exhibited a slight deviation, although it only occurred at pressures exceeding 0.075 MPa, where the bending angle had almost reached $180^{\circ }$ . It is believed that this reduced improvement compared to the simulations could be attributed to the larger manufacturing errors of the new design compared to the initial prototype, since it involved additional steps and mold components.

Furthermore, when comparing the bending angle stiffness ( $\theta$ ) of both prototypes, it is evident that the new design possesses a more optimal stiffness. In all three chambers, the new design prototype managed to reach higher bending angles at lower pressure, while being surpassed by the original design at higher pressure. We believe the improvement in the experiment is less significant due to the thicker outer walls of the new design prototype as a result of imperfect molds, and the overtake was the result of the gradual decrease in stiffness that occurred in the original design due to buckling.

In conclusion, looking at the individual chamber pressurization performance, although the improvement of the new design may not be as significant as observed in simulations, it still demonstrates sufficient improvement for most practical applications.

The prototypes’ vectorization for a full $360^{\circ }$ rotation of bending planer orientation angle is depicted in Figure 6(c). It was obtained by pressurizing specific combinations of chambers at specific ratio. Comparing them with the theoretical optimal $\gamma$ based on the vectorization models described in [Reference Franco and Garriga-Casanovas27], result an MAE of $8.20 \pm 5.16$ for the original design, and $8.83 \pm 6.36$ for the proposed design prototype.

Looking at the overall vectorization error of both prototypes, it can be observed that the bending plane orientation angle of the new design is comparable to that of the original design. The new design has slightly higher deviations relative to the theoretical $\gamma$ values. This could be attributed to the higher error at ratios where one of the chambers has a significantly lower pressure compared to the other chamber. The main reason for this is likely related to the stabilizing effect of the diamond shape partition when collapsing, which helps prevent deviation during individual chamber pressurization but unintentionally pulls the bending plane towards one of the chamber’s bending planes during vectorization. Although this characteristic may not be desirable, the effect is predictable and repeatable, and therefore it can be mitigated by implementing a non-linear ratio-to- $\gamma$ control policy.

Overall, despite the new design requiring a more complex manufacturing process and exhibiting higher non-linearity during vectorization, it also demonstrates superior performance in terms of deviation resistance and bending stiffness.

5.1. Artificial neural network model

In this work, we aim to build an artificial neural network to solve the inverse kinematic problem of the manipulator, relating its angles with the respective combinations of pressure inputs in quasi-static conditions. Considering the relative simplicity of the problem, we designed a simple deep neural network architecture with three hidden layers, two inputs for the target bending angle ( $\theta$ ) and target bending plane orientation angle ( $\gamma$ ), and three outputs for the predicted pressure in each of the chambers ( $P_1$ , $P_2$ , $P_3$ ). Through k-fold validations, we empirically set the number of neurons for each of the hidden layers to be 16, 8, and 4, with the $tanh$ activation function used in each of the layers.

Each data point for the training data sets was obtained by randomly applying pressure combinations in the manipulator’s chambers, respecting the following conditions: pressure input values were in the range of 0.01 to 0.08 MPa; at least one of the chambers in each segment remained unactuated; the pressurization sequence was given in an intermittent fashion, where the manipulator was returned to the unactuated state before the next values were applied; data points were recorded after a 3-second delay to ensure a steady-state condition. In total, 600 data pairs containing the given pressure combinations and the manipulator’s angles were recorded.

5.2. ANN model training and testing

To generalize the performance of the trained model, k-fold validation with ( $k = 12$ ) was utilized. In this method, the training data set was split into 12 batches, each containing 50 different data pairs. Model training was repeated 12 times, with each iteration using eleven batches for training, and one remaining batch was used for validation. Each batch would serve as the validation set exactly once. The objective of the training was to optimize the network’s weights and biases using the adaptive moment estimation algorithm, minimizing the mean square error between the network’s predictions and the actual target data. The performance metrics from all repetition were averaged to obtain the generalized performance of the trained model. Through this process, we obtained the model validation performance with an $R^2$ score of $0.98 \pm 0.01$ and a mean absolute error (MAE) of $0.03 \pm 0.01$ .

Experimental validation was performed using the model with the closest MAE value with the average performance. Testing was conducted to predict pressure input values of 100 random combinations of $\theta$ and $\gamma$ , with a range of $10^{\circ }$ to $120^{\circ }$ for the $\theta$ . Figure 7(a) and Figure 7(c) display the final models’ actual versus target $\gamma$ and $\theta$ , respectively.

Figure 7. Neural network experimental validation results. (a) Target versus actual measured $\theta$ . (b) Target versus actual measured $\gamma$ . (c) Error heat map of $\theta$ . (d) Error heat map of $\gamma$ . Error grids were generated through linear interpolation using the recorded data points (depicted as cross markers).

In the experimental validation, the model managed to achieve $R^2$ score of 0.998 and MAE of $4.16 \pm 2.24$ for $\gamma$ prediction, and $R^2$ score of 0.95 and MAE of $3.93 \pm 3.19$ for $\theta$ prediction. The ANN model was capable of predicting the necessary combination of pressure for two chambers to achieve specific target $\theta$ and target $\gamma$ values with relatively low MAE errors. Significant improvements, up to 5 degrees, could be observed in the MAE error of $\gamma$ when we compare it to the vectorization performance of the new prototype without the ANN model (shown in Section 4.3).

To gain a better understanding of the model’s performance, we also investigated the combinations of $\theta$ and $\gamma$ where the model faced the most difficulties in making accurate predictions. Figure 7(b) and Figure 7(d) display an error heat map of target $\theta$ and $\gamma$ combinations that resulted in higher errors of actual $\gamma$ and $\theta$ .

Analyzing the heat maps, we observed that higher $\theta$ errors mostly occur around high $\theta$ targets. This is most likely caused by the increased non-linearity of the manipulator at high bending angle, making it less predictable. On the other hand, high $\gamma$ errors mostly occur at low $\theta$ targets. This was most likely caused by the limitations in the experimental setup, which faced difficulties in accurately measuring $\gamma$ when the bending angle is too low.

Nevertheless, we find that the inverse kinematics model based on ANN shows promise and demonstrates the potential of the new design. However, it also comes with some challenges that may require further improvement in future works. One of the challenges is the high number of training data points required for the model, which could potentially reduce the prototype’s life expectancy.