Structural Optimization Method of a FinRay Finger for the Best Wrapping of Object

: Soft gripping, in which the gripper adapts to differently shaped objects, is in great demand for use in unknown or dynamically changing environments and is one of the main research subjects in soft robotics. Several systems have already been created, one of which is a passive shape-adaptable ﬁnger based on the FinRay effect. The geometric shape of this ﬁnger ensures that the ﬁnger wraps around the object it grips. FinRay ﬁngers have been studied in several studies, which have changed the internal structure and examined how gripping force’s dependence on ﬁnger deformation changes. So far, however, no speciﬁc way has been determined to evaluate the proposed ﬁnger regarding its ability to wrap around the object. This work comes up with a new and simple method to evaluate the ﬁnger’s wrapping around the object mathematically. Based on this evaluation method, several different patterns of the internal structure of FinRay ﬁngers were tested. The ﬁngers were ﬁrst tested in a simulation program, which simulated a steel roller indentation with a diameter of 20 mm in the middle of the ﬁnger’s contact surface. Based on the simulation results, selected types of structure were made by the Fused Filament Fabrication method from a ﬂexible ﬁlament and tested on a real test rig to verify the results of the simulation and compare it with the real behaviour. According to the methodology used, the results show that the most suitable structure of the selected tested ﬁngers from the point of view of wrapping the ﬁnger around the object is a structure without internal ﬁlling. Designers can simply use the new evaluation method to compare their designed ﬁnger variants and select the most suitable one according to the ability to wrap around the gripped object. They can also use graphs from this work’s results and determine the ﬁnger’s dimensions without internal ﬁlling according to the required forces and deﬂection.


Introduction
Adaptive gripping is one of the main areas of soft robotics applications [1]. Soft gripping, in which the gripper adapts to differently shaped objects, is in great demand in unknown or dynamically changing environments [2]. Their properties enable the gripping and manipulation of objects of various shapes, while their softness and flexibility allow them to manipulate even fragile objects. As a result, they find their application not only in industrial robotics, such as the automotive industry [3], but especially in service robotics, where they can be used, for example, in the food industry to collect fragile crops whose shapes and sizes vary [1,4], or in the home environment for handling objects of various shapes (such as keys, pens, mugs, etc.) [5,6].
The evaluation of individual finger variants is based on their behaviour under load. The finger is constrained by a fixture on one side, and the gripped object is then pressed into the finger. To find out if the finger has wrapped around the gripped object, it is necessary to know the positions of at least three points. The first point is a contact point between the gripper and the gripped object. The second one is positioned before the first one in the direction towards finger's fixed base, and the third one in the opposite direction towards finger's tip. Positions of these three points on the finger can be easily measured. The position at the beginning of the finger does not change under load due to a fixed fixture.
The maximum deflection appears at the point of contact. These three positions are on the finger's contact edge, and they are sufficient to describe the wrapping of the finger around the gripped object. Since the position at the beginning (fixed point) of the finger does not change, it is unnecessary to measure it. The positions of the remaining two points change under load and are defined by the distance by which the points move perpendicular to the contact surface in the unloaded state. Figure 1 shows these measured distances, where Ymax is the maximum deflection value and Yend is the value of the deflection at the fingertip.
Yend and Ymax values were determined at each millimeter of compression. The millimeter step is chosen because of the different internal structures; the ratio of these values may differ depending on the magnitude of the deformation. From these values, the total finger deflection coefficient Ψ is calculated according to the following formula: where ψ is the deflection coefficient, Y end i is the distance of the endpoint in the i-th millimeter of compression, Y max i is the maximum deflection of the finger in the i-th millimeter of compression, which is equal to the depth of compression of the steel roller into the finger in millimeters, n is the maximum depth of compression, which is 8 mm. The value of the maximum deflection of the finger 8 mm was determined based on the previous simulations of these sizes of fingers, in which there were failures of the simulations because of buckling in some tested structures at a greater value of deflection. There could be a different value of maximum deflection for another size, shape, or inner structure of the fingers. Yend and Ymax values were determined at each millimeter of compression. The millimeter step is chosen because of the different internal structures; the ratio of these values may differ depending on the magnitude of the deformation. From these values, the total finger deflection coefficient Ψ is calculated according to the following formula: where ψ is the deflection coefficient, Y is the distance of the endpoint in the i-th millimeter of compression, Y is the maximum deflection of the finger in the i-th millimeter of compression, which is equal to the depth of compression of the steel roller into the finger in millimeters, n is the maximum depth of compression, which is 8 mm. The value of the maximum deflection of the finger 8 mm was determined based on the previous simulations of these sizes of fingers, in which there were failures of the simulations because of buckling in some tested structures at a greater value of deflection. There could be a different value of maximum deflection for another size, shape, or inner structure of the fingers.
The smaller the deflection coefficient, the more finger's shape wraps around the manipulated object. Therefore, the most suitable type of finger is the one whose deflection coefficient Ψ is the smallest.

Tested Filling Patterns Structures
The patterns testing was based on the shape of an existing Festo finger [17], on a study of works already performed, and expert judgment. Various finger parameters were tested, such as the thickness of selected walls, the number of ribs, and the change in the geometry of the internal structure in which branching occurs, which is based on real fish fins [20]. According to aim of this paper, selected testing structures and parameters are not chosen to find the overall most optimal structure that can even exist but to compare the different structures and find the optimal by applying the new methodology.  The smaller the deflection coefficient, the more finger's shape wraps around the manipulated object. Therefore, the most suitable type of finger is the one whose deflection coefficient Ψ is the smallest.

Tested Filling Patterns Structures
The patterns testing was based on the shape of an existing Festo finger [17], on a study of works already performed, and expert judgment. Various finger parameters were tested, such as the thickness of selected walls, the number of ribs, and the change in the geometry of the internal structure in which branching occurs, which is based on real fish fins [20]. According to aim of this paper, selected testing structures and parameters are not chosen to find the overall most optimal structure that can even exist but to compare the different structures and find the optimal by applying the new methodology. Figure 2 shows the basic test structures of the fingers.
The thickness of all fingers is 15 mm, with a side length of 61 mm. The radius of the fingertip curvature is 2.5 mm, and the opening angle is 20 • (as shown in structure A). In structure A, four parameters are changing. The wall thicknesses TF and TB, the thickness of the inner ribs TR, and the number of ribs. The default values of these thicknesses is 1.6 mm, with only one selected thickness always changing and the rest remaining at the default value. The structure B has no internal filling, and values of thicknesses TF and TB are changing. These values are always equal, and, thus, both change simultaneously (the finger is always symmetrical). Structure C is an example of a branched structure where the inner rib in the middle of the jaw branches into two at an angle of 30 • . The thickness TR1 and TR2 changes. The structure D is also branched, but its branching is shifted to the left side (side to the contact surface concerning the object of manipulation) by 2 mm. The structure E is similar, but its branching is shifted by 2 mm to the opposite side. The parameters of the structures D and E change in the same way as for the structure C, and all branched structures were loaded both to the contact side and to the opposite side. All combinations of test fingers are shown in the Table 1, with some parameters listed referring to the figure above. The values of the different thicknesses of the parameters are graduated in 0.4 mm increments, based on the printer nozzle diameter on which the selected fingers is printed.
Appl. Sci. 2021, 11, x FOR PROOFREADING 4 of 1 structure E is similar, but its branching is shifted by 2 mm to the opposite side . The pa  rameters of the structures D and E change in the same way as for the structure C, and a  branched structures were loaded both to the contact side and to the opposite side. A  combinations of test fingers are shown in the Table 1, with some parameters listed refer ring to the figure above. The values of the different thicknesses of the parameters are grad uated in 0.4 mm increments, based on the printer nozzle diameter on which the selected fingers is printed.

Loading of the Testing Finger
A steel roller with a diameter of 20 mm and a length of 25 mm was pressed into the finger. It acts perpendicular to the contact surface in the middle of the finger, which, according to Festo [17], is the most suitable area for gripping handling objects. In the testing of the finger, this grip area is located 25 mm from its tip, as shown in the Figure 3. ppl. Sci. 2021, 11, x FOR PROOFREADING

Loading of the Testing Finger
A steel roller with a diameter of 20 mm and a length of 25 mm was pr finger. It acts perpendicular to the contact surface in the middle of the fing cording to Festo [17], is the most suitable area for gripping handling objects of the finger, this grip area is located 25 mm from its tip, as shown in the Fi Three ways of loading the finger were simulated. In the first method, pressed into the finger to a depth of 8 mm. At each millimeter of compressio deflection values (point of contact of the finger with the cylinder) and de fingertip were measured. These values were used to calculate the deflect according to which the most suitable structure was selected from all tested In the second test, the steel roller was pressed this time not to the req but by force of 5 N, while the deformation of the entire contact surface of measured. This force value was chosen with respect to the structures use the number of variations of the fingers, which would reach irreversible defo case occurred with a single variation of the finger, namely the structure B fo TF = TB 1.6 mm, where the simulation stopped with the message that the so in the buckling or at the limit point, i.e., the displacements increase at consta test is done only to prove that the simulations at a constant load of 5 N are n the selection of the finger shape based on its suitability from the point of view the fingers around the object, as examined by some previously performed w The third method is only valid for selected types of fingers when comp tests. With this method, the roller load force is gradually increased until th at the point of contact by 8 mm. The deflection of the point of contact is me play a graph of the dependence of this deflection on the applied forces.

Basic Simulation Settings
All finger simulations were performed as nonlinear simulations in S 2019 [21] software. Due to the finger's constant cross-section across its en Simplification was used to simplify the computational model and thus sho putational time. The finger material was set as hyperelastic Mooney Rivlin m values from the tensile curve were used to set its parameters, as described in chapter. The Roller material is steel. The finger is rigidly attached to the edg Three ways of loading the finger were simulated. In the first method, the roller was pressed into the finger to a depth of 8 mm. At each millimeter of compression, the maximal deflection values (point of contact of the finger with the cylinder) and deflection of the fingertip were measured. These values were used to calculate the deflection coefficient according to which the most suitable structure was selected from all tested.
In the second test, the steel roller was pressed this time not to the required distance but by force of 5 N, while the deformation of the entire contact surface of the finger was measured. This force value was chosen with respect to the structures used to minimize the number of variations of the fingers, which would reach irreversible deformation. This case occurred with a single variation of the finger, namely the structure B for the thickness TF = TB 1.6 mm, where the simulation stopped with the message that the solution may be in the buckling or at the limit point, i.e., the displacements increase at constant forces. This test is done only to prove that the simulations at a constant load of 5 N are not suitable for the selection of the finger shape based on its suitability from the point of view of wrapping the fingers around the object, as examined by some previously performed works [4,13].
The third method is only valid for selected types of fingers when compared with real tests. With this method, the roller load force is gradually increased until the finger bends at the point of contact by 8 mm. The deflection of the point of contact is measured to display a graph of the dependence of this deflection on the applied forces.

Basic Simulation Settings
All finger simulations were performed as nonlinear simulations in SOLIDWORKS 2019 [21] software. Due to the finger's constant cross-section across its entire width, 2D Simplification was used to simplify the computational model and thus shorten the computational time. The finger material was set as hyperelastic Mooney Rivlin model, and the values from the tensile curve were used to set its parameters, as described in the following chapter. The Roller material is steel. The finger is rigidly attached to the edges on one side. A No Penetration contact bond is set between the edges of the Steel Roller and the Finger. Automatic Solver Selection is set for the analysis. Figure 4 shows the finger attachment and the points for the measurement.  For the simulation to determine the deflection coefficient, the Stell roller is set move in the positive Y-axis direction by a distance of 8 mm. X-axis movement is prohi ited. The deflections of the points Ymax and Yend are measured at each mm of the ste roller movement.
For the simulation applied a force of 5 N is applied to the Steel roller in the positiv Y-axis direction, where the Steel roller is allowed to move. In the X-axis, the movement the roller is again prohibited. In this simulation, the entire contact edge deflection is mea ured, which is further shown in the graphs.
When simulating selected types of internal finger structures for comparison with re measurements, the force is gradually increased (for each finger separately) until the Yma is deflected by approximately 8 mm.

Finger Material
Fingers and samples are printed from Fiberflex TPU 30D [22]. It is a flexible materi whose hardness value is according to the Shore 30D scale and is currently one of the sof est materials for FFF printers on the market [23]. Printing took place on the Original Prus i3 MK3S printer [24] with a modified Flexion extruder [25] for printing flexible material The G-code for the printer was generated using PrusaSlicer v2.3 software [26]. The bas printing parameters are shown in the Table 2. Due to the fact that the tested material TPU 30D is not in the standard offer of mat rials for simulation, the possibility of software to set the properties of this material b uploading the Stress-stretch ratio curve from a tensile test to the simulation software wa used. Samples, according to ISO 527, were printed for the tensile test [27]. The dimension For the simulation to determine the deflection coefficient, the Stell roller is set to move in the positive Y-axis direction by a distance of 8 mm. X-axis movement is prohibited. The deflections of the points Ymax and Yend are measured at each mm of the steel roller movement.
For the simulation applied a force of 5 N is applied to the Steel roller in the positive Y-axis direction, where the Steel roller is allowed to move. In the X-axis, the movement of the roller is again prohibited. In this simulation, the entire contact edge deflection is measured, which is further shown in the graphs.
When simulating selected types of internal finger structures for comparison with real measurements, the force is gradually increased (for each finger separately) until the Ymax is deflected by approximately 8 mm.

Finger Material
Fingers and samples are printed from Fiberflex TPU 30D [22]. It is a flexible material whose hardness value is according to the Shore 30D scale and is currently one of the softest materials for FFF printers on the market [23]. Printing took place on the Original Prusa i3 MK3S printer [24] with a modified Flexion extruder [25] for printing flexible materials. The G-code for the printer was generated using PrusaSlicer v2.3 software [26]. The basic printing parameters are shown in the Table 2.

Material Settings for Simulation
Due to the fact that the tested material TPU 30D is not in the standard offer of materials for simulation, the possibility of software to set the properties of this material by uploading the Stress-stretch ratio curve from a tensile test to the simulation software was used. Samples, according to ISO 527, were printed for the tensile test [27]. The dimensions of the sample are shown in Figure 5.  The samples were tested for tensile test on a Testometric M500/50 CT machine [28]. The test specimens failed to break even at the maximum unloading of the machine. According to the filament manufacturer catalogue [22], the material can stretch by up to 900%. Figure 6 shows its extension. On the left is the sample before the test, and on the right during the test. For the required purpose, the sample did not need to break, because the elastic deformation took place to a strain of about 25%, and then plastic deformations began to appear. For the simulation, it was necessary to obtain the stress dependence on the stretch ratio from the tensile curve, which is calculated according to the following formula: where λ is the stretch ratio, is the final length, and is the initial length. The Figure 7 shows the stress dependence of the stretch ratio for the TPU 30D material. The samples were tested for tensile test on a Testometric M500/50 CT machine [28]. The test specimens failed to break even at the maximum unloading of the machine. According to the filament manufacturer catalogue [22], the material can stretch by up to 900%. Figure 6 shows its extension. On the left is the sample before the test, and on the right during the test.  The samples were tested for tensile test on a Testometric M500/50 CT machine [28]. The test specimens failed to break even at the maximum unloading of the machine. According to the filament manufacturer catalogue [22], the material can stretch by up to 900%. Figure 6 shows its extension. On the left is the sample before the test, and on the right during the test. For the required purpose, the sample did not need to break, because the elastic deformation took place to a strain of about 25%, and then plastic deformations began to appear. For the simulation, it was necessary to obtain the stress dependence on the stretch ratio from the tensile curve, which is calculated according to the following formula: where λ is the stretch ratio, is the final length, and is the initial length. The Figure 7 shows the stress dependence of the stretch ratio for the TPU 30D material. For the required purpose, the sample did not need to break, because the elastic deformation took place to a strain of about 25%, and then plastic deformations began to appear. For the simulation, it was necessary to obtain the stress dependence on the stretch ratio from the tensile curve, which is calculated according to the following formula: where λ is the stretch ratio, l is the final length, and L is the initial length. The Figure 7 shows the stress dependence of the stretch ratio for the TPU 30D material.

Real Testing
Since real fingers are most often made using 3D printing [4,13,16,18], a test is performed to compare simulated material behaviour concerning real printed fingers. According to the simulations results, the most suitable types of fingers were printed and tested on a test device. An industrial robot ABB IRB 1600 [29] with ABB Small force sensor [30] was used for testing. This assembly makes it possible to measure the force magnitude up to 495 N with an accuracy of 0.11 N, while the positioning accuracy is 0.02 mm [29,30]. The setup of the real experiment is shown in the Figure 8.

Real Testing
Since real fingers are most often made using 3D printing [4,13,16,18], a test is performed to compare simulated material behaviour concerning real printed fingers. According to the simulations results, the most suitable types of fingers were printed and tested on a test device. An industrial robot ABB IRB 1600 [29] with ABB Small force sensor [30] was used for testing. This assembly makes it possible to measure the force magnitude up to 495 N with an accuracy of 0.11 N, while the positioning accuracy is 0.02 mm [29,30]. The setup of the real experiment is shown in the Figure 8. The test finger is firmly attached to a stand that is screwed to the worktop. The robot moves the roller to the required position 2 mm above the contact surface. It is then moved at a speed of 0.1 mm/s perpendicular to the contact surface to a depth of 10 mm (the finger deflection is thus 8 mm). This testing measures the dependence of the deformation on the loading force. In the same way, a simulation is created for selected fingers for mutual comparison. The results of this test assess the difference between the simulated and real behaviour of the fingers. On the other hand, the results can help designers design selected fingers according to the required force and deflection.

Real Testing
Since real fingers are most often made using 3D printing [4,13,16,18], a test is performed to compare simulated material behaviour concerning real printed fingers. According to the simulations results, the most suitable types of fingers were printed and tested on a test device. An industrial robot ABB IRB 1600 [29] with ABB Small force sensor [30] was used for testing. This assembly makes it possible to measure the force magnitude up to 495 N with an accuracy of 0.11 N, while the positioning accuracy is 0.02 mm [29,30]. The setup of the real experiment is shown in the Figure 8. The test finger is firmly attached to a stand that is screwed to the worktop. The robot moves the roller to the required position 2 mm above the contact surface. It is then moved at a speed of 0.1 mm/s perpendicular to the contact surface to a depth of 10 mm (the finger deflection is thus 8 mm). This testing measures the dependence of the deformation on the loading force. In the same way, a simulation is created for selected fingers for mutual comparison. The results of this test assess the difference between the simulated and real behaviour of the fingers. On the other hand, the results can help designers design selected fingers according to the required force and deflection. The test finger is firmly attached to a stand that is screwed to the worktop. The robot moves the roller to the required position 2 mm above the contact surface. It is then moved at a speed of 0.1 mm/s perpendicular to the contact surface to a depth of 10 mm (the finger deflection is thus 8 mm). This testing measures the dependence of the deformation on the loading force. In the same way, a simulation is created for selected fingers for mutual comparison. The results of this test assess the difference between the simulated and real behaviour of the fingers. On the other hand, the results can help designers design selected fingers according to the required force and deflection.

Demonstration of Grasping Various Objects
Several selected variants of fingers (based on simulations) were used to grip selected objects to prove the fingers suitability for gripping objects of various shapes. Selected items are a lighter, a cube (which will be gripped first by its side and then by one edge) with a side size of 20 mm, an egg (which will be gripped first of its longest sides, the second time around its circumference) and a car bulb, which will be grasped by its metal sleeve. The items are shown in Figure 9.

Demonstration of Grasping Various Objects
Several selected variants of fingers (based on simulations) were used to grip selected objects to prove the fingers suitability for gripping objects of various shapes. Selected items are a lighter, a cube (which will be gripped first by its side and then by one edge) with a side size of 20 mm, an egg (which will be gripped first of its longest sides, the second time around its circumference) and a car bulb, which will be grasped by its metal sleeve. The items are shown in Figure 9.

Results
This chapter describes the results obtained during the simulations of individual variants of structures to determine the deflection coefficient and thus evaluate the most suitable structure from the point of view of wrapping around the object. Furthermore, the simulation results of individual variations of structures under load with a constant force of 5 N are presented. Some selected types of fingers are also printed and tested on a real device while also compared with simulations to verify the simulated finger behaviour to the real one. Finally, a demonstration of grasping various objects with selected fingers is showed.

Simulations
In the first set of simulations, all finger variations were tested for deflection coefficient. Out of all the tested structures, the smallest deflection coefficients were measured for structure B, i.e., a finger without an internal structure. The best result was obtained on the variation of this structure with a wall thickness of 2.8 mm, whose deflection coefficient is equal to 0.27. The second best result was achieved on structure A with the number of ribs 6, namely, the value 0.42. Of the branched structures, structures E performed best when loaded towards the contact side with the same thicknesses TR1 = TR2 = 0.8 mm, for which the value of the deflection coefficient is 0.44. The load towards the opposite side at the same wall thicknesses TR1 = TR2 = 0.8 mm, for which the value of the deflection coefficient is 0.46. Based on these results, for which printing and comparison with real testing were selected, fingers from the structure B at wall thicknesses of 1.6 mm, 2.8 mm, and 4 mm, fingers from the structure A with the number of ribs six, and the two branched structures E, which achieved the best deflection values coefficients. Table 3 lists all the tested fingers and their calculated deflection coefficients. The table also contains the ranking, representing the order of the most advantageous fingers in terms of deflection coefficient. If more than one tested finger has the same deflection coefficient value, they have the same ranking value.

Results
This chapter describes the results obtained during the simulations of individual variants of structures to determine the deflection coefficient and thus evaluate the most suitable structure from the point of view of wrapping around the object. Furthermore, the simulation results of individual variations of structures under load with a constant force of 5 N are presented. Some selected types of fingers are also printed and tested on a real device while also compared with simulations to verify the simulated finger behaviour to the real one. Finally, a demonstration of grasping various objects with selected fingers is showed.

Simulations
In the first set of simulations, all finger variations were tested for deflection coefficient. Out of all the tested structures, the smallest deflection coefficients were measured for structure B, i.e., a finger without an internal structure. The best result was obtained on the variation of this structure with a wall thickness of 2.8 mm, whose deflection coefficient is equal to 0.27. The second best result was achieved on structure A with the number of ribs 6, namely, the value 0.42. Of the branched structures, structures E performed best when loaded towards the contact side with the same thicknesses TR1 = TR2 = 0.8 mm, for which the value of the deflection coefficient is 0.44. The load towards the opposite side at the same wall thicknesses TR1 = TR2 = 0.8 mm, for which the value of the deflection coefficient is 0.46. Based on these results, for which printing and comparison with real testing were selected, fingers from the structure B at wall thicknesses of 1.6 mm, 2.8 mm, and 4 mm, fingers from the structure A with the number of ribs six, and the two branched structures E, which achieved the best deflection values coefficients. Table 3 lists all the tested fingers and their calculated deflection coefficients. The table also contains the ranking, representing the order of the most advantageous fingers in terms of deflection coefficient. If more than one tested finger has the same deflection coefficient value, they have the same ranking value.
When simulating structures for which their parameters changed, i.e., structures A and B, the deformation was dependent on the changed parameter. The lower the value of this parameter, the greater the deformation, as shown in the Figure 10, where the deformation's dependence on the distance of the contact surface from the fixed region for structure B under load 5 N is shown.
Due to the deformation's dependence on the given parameters' size, only graphs of the largest and lowest deformations for each structure and its changed parameter are used to compare all structures in Figure 11. Of the branched structures, only the structures are shown that achieved the greatest and the smallest deformation.  When simulating structures for which their parameters changed, i.e., structures A and B, the deformation was dependent on the changed parameter. The lower the value of this parameter, the greater the deformation, as shown in the Figure 10, where the deformation's dependence on the distance of the contact surface from the fixed region for structure B under load 5 N is shown. Due to the deformation's dependence on the given parameters' size, only graphs of the largest and lowest deformations for each structure and its changed parameter are used to compare all structures in Figure 11. Of the branched structures, only the structures are shown that achieved the greatest and the smallest deformation.  When simulating structures for which their parameters changed, i.e., structures A and B, the deformation was dependent on the changed parameter. The lower the value of this parameter, the greater the deformation, as shown in the Figure 10, where the deformation's dependence on the distance of the contact surface from the fixed region for structure B under load 5 N is shown. Due to the deformation's dependence on the given parameters' size, only graphs of the largest and lowest deformations for each structure and its changed parameter are used to compare all structures in Figure 11. Of the branched structures, only the structures are shown that achieved the greatest and the smallest deformation. From the figure, it is evident that the finger with one rib reached the greatest deformation. Nevertheless, from the deflection coefficient point of view, this finger is only in the 10th place. The finger of structure B with a wall thickness of 4 mm has reached three, From the figure, it is evident that the finger with one rib reached the greatest deformation. Nevertheless, from the deflection coefficient point of view, this finger is only in the 10th place. The finger of structure B with a wall thickness of 4 mm has reached three, the smallest deformations, yet according to the deflection coefficient 6, it is the best of all. From these results, it is clear that the amount of deflection at constant load is not a suitable parameter for assessing the fingers from the point of view of their wrapping around the gripped object.
Due to the fact that for branched structures, not only one parameter changed, but also their shape, the individual results are shown separately in Figure 12.
Appl. Sci. 2021, 11, x FOR PROOFREADING 12 of 18 the smallest deformations, yet according to the deflection coefficient 6, it is the best of all. From these results, it is clear that the amount of deflection at constant load is not a suitable parameter for assessing the fingers from the point of view of their wrapping around the gripped object. Due to the fact that for branched structures, not only one parameter changed, but also their shape, the individual results are shown separately in Figure 12.

Real Testing
For real testing, fingers were selected from the group with the lowest deflection coefficient, i.e., fingers without internal structure (structure B). A total of three fingers were tested, with a minimum wall thickness of 1.6 mm, a maximum wall thickness of 4 mm and a mean value of 2.8 mm. Furthermore, a sample with six ribs, branched structure E at the same thickness was tested, both under load from the contact surface and under load to the opposite side.
During real testing, the dependence of the deformation at the point of contact on the applied force was determined. Figure 13 shows this dependence of all selected structures when measured, while the values measured on a real device are compared with simulations. Figure 13. Dependence of loading force on deformation for finger without filling for simulated and real tests. Figure 13 shows that the simulations are very close to the results of real testing. The simulated behaviour of the material can therefore be considered to correspond to the real

Real Testing
For real testing, fingers were selected from the group with the lowest deflection coefficient, i.e., fingers without internal structure (structure B). A total of three fingers were tested, with a minimum wall thickness of 1.6 mm, a maximum wall thickness of 4 mm and a mean value of 2.8 mm. Furthermore, a sample with six ribs, branched structure E at the same thickness was tested, both under load from the contact surface and under load to the opposite side.
During real testing, the dependence of the deformation at the point of contact on the applied force was determined. Figure 13 shows this dependence of all selected structures when measured, while the values measured on a real device are compared with simulations.
Appl. Sci. 2021, 11, x FOR PROOFREADING 12 of 18 the smallest deformations, yet according to the deflection coefficient 6, it is the best of all. From these results, it is clear that the amount of deflection at constant load is not a suitable parameter for assessing the fingers from the point of view of their wrapping around the gripped object. Due to the fact that for branched structures, not only one parameter changed, but also their shape, the individual results are shown separately in Figure 12.

Real Testing
For real testing, fingers were selected from the group with the lowest deflection coefficient, i.e., fingers without internal structure (structure B). A total of three fingers were tested, with a minimum wall thickness of 1.6 mm, a maximum wall thickness of 4 mm and a mean value of 2.8 mm. Furthermore, a sample with six ribs, branched structure E at the same thickness was tested, both under load from the contact surface and under load to the opposite side.
During real testing, the dependence of the deformation at the point of contact on the applied force was determined. Figure 13 shows this dependence of all selected structures when measured, while the values measured on a real device are compared with simulations. Figure 13. Dependence of loading force on deformation for finger without filling for simulated and real tests. Figure 13 shows that the simulations are very close to the results of real testing. The simulated behaviour of the material can therefore be considered to correspond to the real Figure 13. Dependence of loading force on deformation for finger without filling for simulated and real tests. Figure 13 shows that the simulations are very close to the results of real testing. The simulated behaviour of the material can therefore be considered to correspond to the real behaviour. When the wall thickness of structure B (parameters TF and TB) increases, so do the force required for the given deflection. Figure 14 shows the deformed finger in the simulation compared to the real testing with the structure B and the wall thickness of 1.6, 2.8, and 4 mm.
Appl. Sci. 2021, 11, x FOR PROOFREADING 13 of 18 behaviour. When the wall thickness of structure B (parameters TF and TB) increases, so do the force required for the given deflection. Figure 14 shows the deformed finger in the simulation compared to the real testing with the structure B and the wall thickness of 1.6, 2.8, and 4 mm.  Figure 15 shows the deformed finger in the simulation in comparison with the rea testing with the structure E mark 5 and 11, and the structure A with the number of ribs six. Figure 15. Comparison of the simulation with the printed tested finger, left: simulation, right: real test; top: structure E mark EO0808, middle: structure E mark EC0808, bottom: structure A with the number of ribs 6.
For possible application, it is appropriate to point out the hysteresis of flexible mate rials. For the printed finger, the hysteresis gave a different dependence of deformations on the applied force depending on whether the object was pushed towards the finger (loaded) or was moved away from the finger (unloading). This hysteresis also varies ac cording to the number of load cycles. A total of 500 load cycles were used for this testing Figure 16 shows the cyclic hysteresis with different loading and unloading values and for  behaviour. When the wall thickness of structure B (parameters TF and TB) increases, s do the force required for the given deflection. Figure 14 shows the deformed finger in the simulation compared to the real testin with the structure B and the wall thickness of 1.6, 2.8, and 4 mm.  Figure 15 shows the deformed finger in the simulation in comparison with the rea testing with the structure E mark 5 and 11, and the structure A with the number of rib six. Figure 15. Comparison of the simulation with the printed tested finger, left: simulation, right: rea test; top: structure E mark EO0808, middle: structure E mark EC0808, bottom: structure A with th number of ribs 6.
For possible application, it is appropriate to point out the hysteresis of flexible mate rials. For the printed finger, the hysteresis gave a different dependence of deformation on the applied force depending on whether the object was pushed towards the finge (loaded) or was moved away from the finger (unloading). This hysteresis also varies ac cording to the number of load cycles. A total of 500 load cycles were used for this testing Figure 16 shows the cyclic hysteresis with different loading and unloading values and fo Figure 15. Comparison of the simulation with the printed tested finger, left: simulation, right: real test; top: structure E mark EO0808, middle: structure E mark EC0808, bottom: structure A with the number of ribs 6.
For possible application, it is appropriate to point out the hysteresis of flexible materials. For the printed finger, the hysteresis gave a different dependence of deformations on the applied force depending on whether the object was pushed towards the finger (loaded) or was moved away from the finger (unloading). This hysteresis also varies according to the number of load cycles. A total of 500 load cycles were used for this testing. Figure 16 shows the cyclic hysteresis with different loading and unloading values and for different cycles for a finger with structure B with a wall thickness of 2.8 mm. The results show the deformation dependence at the place of loading on the loaded force according to the number of load cycles. As the number of cycles increases, the amount of force required to induce a certain amount of deformation decreases. Figure 17 shows the results from real testing of structures B and structure A with six ribs. The number of load cycles of each finger tested is 500.

Demonstration of Grasping Various Objects
For grasping demonstration, fingers were selected from the group with the lowest deflection coefficient, i.e., fingers without internal structure (Structure B with a wall thickness of 1.6 mm and 4 mm) and for comparison the structure A with six ribs. The Figure 18 demonstrates their grip with selected objects. The results show the deformation dependence at the place of loading on the loaded force according to the number of load cycles. As the number of cycles increases, the amount of force required to induce a certain amount of deformation decreases. Figure 17 shows the results from real testing of structures B and structure A with six ribs. The number of load cycles of each finger tested is 500. The results show the deformation dependence at the place of loading on the loaded force according to the number of load cycles. As the number of cycles increases, the amount of force required to induce a certain amount of deformation decreases. Figure 17 shows the results from real testing of structures B and structure A with six ribs. The number of load cycles of each finger tested is 500.

Demonstration of Grasping Various Objects
For grasping demonstration, fingers were selected from the group with the lowest deflection coefficient, i.e., fingers without internal structure (Structure B with a wall thickness of 1.6 mm and 4 mm) and for comparison the structure A with six ribs. The Figure 18 demonstrates their grip with selected objects.

Demonstration of Grasping Various Objects
For grasping demonstration, fingers were selected from the group with the lowest deflection coefficient, i.e., fingers without internal structure (Structure B with a wall thickness of 1.6 mm and 4 mm) and for comparison the structure A with six ribs. The

Discussion
This work comes up with a new and simple method of evaluating the finger wrapping around the object mathematically. Several types of internal structures have been designed, in which their parameters, such as the thickness of the selected walls or the number of internal ribs have changed. To evaluate the wrapping of the finger around the object, a method was determined that measures the ratio between the deflection of the endpoint and the maximum deflection of the finger. Tensile test data was used to set up the material for simulation analysis properly. The tested material is TPU 30D. It is a flexible filament designed for printing on Fused Filament Fabrication printers. The pressed object is a steel roller with a diameter of 20 mm.
Three tasks were chosen for the simulation. In the first simulation, the roller is pressed into the finger 8 mm to the depth. Based on the results of this simulation, the deflection coefficients are calculated. In the second simulation, the roller is pressed into the finger with a force of 5 N. The simulation illustrates and points out that from the point of view of wrapping the finger around the object, the magnitude of deformation after loading with a certain force is not very appropriate, as examined by some previously performed works [4,13]. The third method is done only for few selected types of fingers to compare them with real tests. With this method, the roller load force is gradually increased until the finger bends at the point of contact by 8 mm. The deflection of the point of contact is measured to display a graph of this deflection's dependence on the applied forces. Graphs showing the hysteresis of the printed fingers of structures B and structure A with six ribs were also measured and shown for designers possible needs. Selected finger variations were chosen to demonstrate their ability to grasp various objects. All the selected structures were able to grasp all the selected objects and their contact sides wrapped around their shape appropriately.
From the analyses results, it is evident that the greatest wrapping of the fingers around the object occurs in the types of fingers without an internal structure (structure B). For all structures, the dependence between individual parameters (for example, the thickness of selected walls or the number of inner ribs) on the deflection at a constant force was observed. The simulations were verified on selected types of fingers printed on a 3D printer and tested on a real test device.
A comparison of the calculated deflection coefficients from the first set of simulations and deformation caused by constant force of 5 N from the second set of tests revealed that the amount of deflection at constant load is not a suitable parameter for judging fingers view of their wrapping around the object.
The work aimed not to find the most suitable structure that can exist or include all possible structure variants. The aim was to find a methodology for comparing different structures with each other according to their ability to wrap around a wrapped object. This methodology can be used to test other variants of fingers, changes in their other parameters, sizes, and shapes, etc.
This work aims to serve the designers in comparing the FinRay finger structures to find the maximum finger wrap around the gripped objects. For the selected structures in this paper, the work shows graphs of deformation dependence on the gripping force for individual parameters of structures (for example, wall thickness), which could be useful for the designers of the fingers.