Design of a Soft Composite Finger with Adjustable Joint Stiffness

Abstract

This article presents the design of a soft composite finger with tunable joint stiffness. The composite finger, made of two different types of silicone, uses hybrid actuation by combining tendon and pneumatic actuation schemes. Tendons control the finger shape in a prescribed direction to demonstrate discrete bending behavior due to different material moduli, similar to that of a human's finger. The pneumatic actuation changes the stiffness of joints using air chambers. The feasibility of adjustable stiffness joints is proven using both the parallel spring model and experiments that demonstrate the stiffening effect when pressurized. A set of experiments were also conducted on fingers with four different chamber shapes to observe the effect of chamber shape on stiffening and the discrete bending capability of the finger. The stiffness control can tune the structural softness of the finger, which leads to firm grasp during higher acceleration object manipulation.

Introduction

The human finger is the optimal composite structure in terms of design, compliance, and manipulation capacity.¹ One unique feature of the human finger is its intricate joint design that defines its passive compliance and the direction of motion of the finger. The human finger has both multiple degrees of freedom (DOF) and multidirectional compliance due to elastic elements (ligaments, tendons, bone cavity) in it along with an overarching joint design.^1,2 This joint design allows the human finger to conform around unknown objects, thus enhancing its adaptive capacity. Inspired by the human hand, several robotic hands have been studied previously.^3–6 Although these robotic hands have precise position control and produce strong forces at the fingertips, they have limited numbers of DOF due to simplified embedded mechanical joints. These joints have made the hands noncompliant and less adaptive to objects they manipulate. In addition, they cannot handle delicate objects such as soft body organs⁷ without advanced haptics capability.

With the advent of Soft Robotics, researchers have developed controllable structures from elastomeric materials.⁸ These structures can bend, expand, compress, and twist to achieve the desired end-effector motion. Pneumatic,⁹ tendon,¹⁰ hydraulic,¹¹ shape memory material,¹² material jamming,¹³ or electroactive polymer¹⁴ actuation schemes deform the shape of the actuator and reinforce its structural strength. Due to the inherent softness of the body material, these robots can perform a variety of tasks in unstructured environments and can interact with humans without causing harm.^15,16 Soft grippers are one of the emerging applications in Soft Robotics. They have more DOF and greater passive compliance compared with conventional rigid robotic hands,^17,18 which allow them to grab and manipulate various objects.

Despite the versatility of the soft robotic hands, they are limited by their low grasping force, which can cause objects slipping. The limitations to the soft structures can be overcome by introducing variable stiffness.¹⁹ Inspired by octopus arms,²⁰ robotics researchers have discovered actuation schemes to tune the stiffness of soft structures.^14,21,22 Shiva et al. developed a stiffness controllable robot manipulator with a hybrid and inherently antagonistic actuation scheme for potential use in minimally invasive surgery.²³ Stiffness variation is accomplished by using tendons routed inside the silicone body that are actuated using stepper motors. A similar concept was proposed by Maghooa et al.²⁴ Researchers have further demonstrated the stiffening capability of soft robots by using hybrid actuation in a bioinspired soft manipulator.²⁵ Other stiffness variation methods also implemented in soft robots include using granules that can be jammed by applying a vacuum,²⁶ varying friction between the overlapping layers of Mylar film to tune stiffness,²⁷ and thermally tunable composites that can achieve a wide range of stiffness.²⁸

In this article, we propose a potential approach to the soft composite structure with hybrid actuation for anthropomorphic soft fingers, as shown in Figure 1. Motivated by Stokes et al.,²⁹ the composite structure integrates soft silicone and relatively rigid silicone to combine the merits of both materials and enable discrete bending of the structure, which facilitates shape prediction. Compared with the previous hybrid and antagonistic actuation approach inspired by octopus musculature,^23,24 the design of the soft composite finger in this research decouples tendon and pneumatic actuation. The tendon actuation bends the finger in a defined and predictable manner,³⁰ whereas the pneumatic actuation controls the joint stiffness.

FIG. 1.

Design of soft composite finger. (a) Robotic hand made up of three soft composite fingers holding a fragile daily-life object. (b) Section view of pressurized chamber in the joint. (c) Soft composite finger similar to human finger.

The anthropomorphic finger in this research is designed to have two asymmetric soft joints, which tune the compliance of the composite structure to locally vary the stiffness properties of the finger.⁷ The joint structures are made asymmetric because the top of the joint consists of rigid silicone, whereas the bottom is made from soft silicone to enhance the discrete bending behavior (Fig. 1). At a low stiffness, the finger discretely and predictably bends to quickly adapt to various objects. With pressure, the joints are stiffened, resulting in structurally firmer fingers that can stably grasp and manipulate an object. The designed soft composite finger can be used for developing a soft gripper that can grasp complex and delicate objects and firmly hold them using pressurization for higher acceleration manipulation. Such an approach will address the challenges associated with making soft fingers more compliant, adaptive to the significant range of objects and maintain firm grasping during dynamic motions. It is noted that the entire structure remains physically soft when pressurized because the pneumatic actuation stiffens the joints only.

This article is organized as follows; it includes the design of a soft composite finger with the stiffness model to support the feasibility of pressurized joints (Design and Models section). The fabrication process is explained in the Fabrication section. Several experiments were conducted on fingers with different chamber shapes to demonstrate the stiffening capability and discrete bending of the finger (Experiments section). The Results and Discussion section compares different chamber shapes in terms of stiffening and bending behavior. Finally, the soft gripper was attached to a single-link robotic arm to experimentally validate firm grasp. The last section (Conclusion section) concludes this study by presenting future work.

Design and Models

The design of the soft composite finger is inspired by the morphology of a human finger, as shown in Figure 2a. The basic structure of the finger is composed of two joints similar to the proximal and distal interphalangeal joints. There are two thin arches connecting three distinct segments of different lengths, analogous to the proximal, middle, and distal phalanges of a human finger.³¹ In addition, there are two channels for tendon actuation on the lateral side of the finger and one channel for pneumatic actuation in the center of the finger.

FIG. 2.

Modeling of stiffening joints. (a) Composite finger showing geometrical parameters with deformed vertical ellipse chamber. Shape change due to pneumatic actuation is shown by dashed line. Shape change due to tendon actuation is shown by dotted line. (b) t_i Represents thickness of soft silicone around the joint. A_ini represents the area of silicone around the joint before pressurization. (c) x_i Represents change in lateral sides of the air chamber due to pressurization. (d) Pressure acting on rectangular cross-sectional area.

When the finger is actuated using tendons, both joints compress in the axial direction, as shown by dotted line in Figure 2a. The pneumatic actuation causes both joints to stretch in the axial direction, as shown by dashed line in Figure 2a. The stiffening capability of the finger depends on the shape of the chamber, deformation characteristics of the joints, and elastic properties of the silicones. These attributes must be considered when modeling joint and finger stiffness for a suitable design.

Joint stiffness

This section focuses on the analytical modeling of two joints to illustrate the effect of pressurizing on stiffness of the joints. Each joint has a linear spring and an air spring fixed in a parallel combination (Fig. 2a). When the tendons are pulled, the joints are compressed, so a linear spring is used to model the deformation in the joint. Joints elongate in the longitudinal direction when they are pressurized, so an air spring is used to model this behavior.^32,33 This model captures the explicit relationship between force due to pressurization and stiffness of the air spring by taking joint parameters into consideration. The stress–strain relationship for both silicones is highly nonlinear. However, nonlinearity of the materials is not accounted for in this model.

The constitutive relation for the parallel spring system in the joint can be written as: $f_{o} = (K_{i} + Δ K_{i}) (d_{i} - {(Δ d)}_{i})$ (1) $\{\begin{matrix} i = d, f o r d i s t a l j o i n t \\ i = p, f o r p r o x i m a l j o i n t \end{matrix},$

where d_i is the compression of the linear spring due to tendon force, ${(Δ d)}_{i}$ is the elongation of the air spring due to pressurization, f_o is the tendon force, K_i is the stiffness constant of the linear spring, and $Δ K_{i}$ is the additional stiffness constant of the air spring.

The pressurization of the composite finger is assumed to be a polytropic process. The pressurized air exerts a force on the air spring, ${(Δ f)}_{i}$ . This force is directly proportional to the inlet pressure caused during the process of pressurization. The force due to pressurization can be related to deformation of the air spring.³⁴ ${(Δ f)}_{i} = \frac{n P_{o} A_{o}^{2} {(Δ d)}_{i}}{V},$ (2)

where n is equal to 1 because temperature is assumed constant during pressurization, P_o is atmospheric pressure, V is volume of air in the air chamber, and A_o is cross-sectional area of the air chamber. The cross-sectional area, assumed as rectangular, plays a significant role in determining the stiffening capability of the joint (Fig. 2d).

In this article, the model is only computed for vertical ellipse chamber. We substituted expressions for A_o and V in Equation (2) to relate the deformation of air spring to the joint parameters of the finger. ${(Δ d)}_{i} = \frac{2 π b_{i} {(Δ f)}_{i}}{8 P_{o} a_{i} (h + x_{i}) - π {(Δ f)}_{i}},$ (3)

where a_i is the length of semi-major axis of the ellipse, b_i is the length of semi-minor axis of the ellipse, h is the height of finger, and x_i is the change in lateral sides of the chamber due to pressurization (Fig. 2c).

To find x_i, we have assumed that the initial volume of silicone is equal to the final volume of silicone around the joint after pressurization, as used in Ref.³⁵ $A_{i n i} h = A_{f i n} (h + x_{i}),$ (4)

A_{i n i} = π a_{i} (b_{i} + t_{i}) - π a_{i} b_{i},

A_{f i n} = π a_{i} (b_{i} + t_{i}) - π a_{i} (b_{i} + \frac{{(Δ d)}_{i}}{2}),

where b_i is the length of semi-minor axis of the ellipse, t_i is the thickness of soft silicone around the joint, $A_{i n i}$ (shown as shaded) is the initial area of silicone, and $A_{f i n}$ in is the final area of silicone around the joint after pressurization (Fig. 2b).

To determine the stiffness for each joint due to pressurization, we substituted Equation (3) into Equation (1). $K_{i} + Δ K_{i} = \frac{f_{o}}{d_{i} - \frac{2 π b_{i} {(Δ f)}_{i}}{8 P_{o} a_{i} (h + x_{i}) - π {(Δ f)}_{i}}},$ (5)

The results for the vertical ellipse chamber in Figure 3a and b show that increasing pressure at constant pulling length increases stiffness for both the distal and proximal joints, respectively.

FIG. 3.

Increasing pressure stiffens the joints. (a) Distal joint (b) proximal joint. (c) Soft composite finger with augmented structural rigidity to better constrain the object at multiple points. The inextensible tendon keeps AA′ constant, whereas B′B and CC′ increase with pressurization, which exerts a push against the object. (d) Modeling of normal force between soft fingertip and object.

Finger stiffness

The increased joint stiffness leads to the overall finger being stiffened. The finger stiffness is derived similar to the previously explained joint stiffness [Equation (5)]. $K_{f i n g e r} + Δ K_{f i n g e r} = \frac{f_{o}}{d_{f i n g e r} - Δ d_{f i n g e r}},$ (6)

Δ d_{f i n g e r} = {(Δ d)}_{p} + {(Δ d)}_{d},

where $K_{f i n g e r}$ is the stiffness constant of the finger, $Δ K_{f i n g e r}$ is the additional stiffness constant due to pressurization, $d_{f i n g e r}$ is the cumulative compression in linear spring of both joints, and $Δ d_{f i n g e r}$ is the total elongation in both air springs due to pressurization. Figure 6 shows that increasing pressure at all tendon displacements leads to stiffening of the finger.

Firm grasp

A robot hand with stiffened fingers can firmly grasp objects in dynamic situations because it has a higher structural softness, which helps it constrain objects (Fig. 3c). When the finger is pressurized and the tendon displacement length is kept constant, the joint angles increase due to the asymmetric design of the finger. Inextensible tendons constrain the lower side (AA′) and increasing pressure expands the chambers (B′ B and CC′) in the longitudinal direction (Fig. 3b). This causes the finger to apply higher normal force on the object, which can decrease the slippage. This can also be shown by using a simplified model consisting of a spring (K_s) attached at the fingertip. When the finger is pressurized, it exerts a push on the object, which increases the deformation of the finger (Δe), thus resulting in a higher normal force ( $Δ N = K_{s} Δ e$ ) on the object (Fig. 3d).

In addition, we created a model to show that increased normal force at a single point can reduce object slippage during manipulation. Initially, it is important to model the frictional force between the soft tip and the rigid object. The Coulomb model is not applicable when it comes to modeling contact forces between the soft finger and rigid objects, so a different formulation derived in Ref.³⁶ is considered for our case (Fig. 3d). $|f| \leq α N + π τ_{o} a^{2},$ (7)

where f is the friction force between the finger tip and object, α is the rate of increase of shear strength, $τ_{o}$ is the shear strength of surface force when load is zero,³⁷ a is radius of contact area, and N is the normal force.

Also, we derived the dynamics of the object to predict its behavior during manipulation.³⁸ $m_{o} ẍ_{2} + f_{s} = - f,$ (8)

where $ẍ_{2}$ is the acceleration of the object, f_s is the friction force when the object is stationary, m_o is the mass of the object, and f is the friction force between the soft fingertip and object.

Equation (8) can be substituted into Equation (7) to derive the condition for acceleration at which the object can be manipulated without slipping from the fingers.³⁶ $ẍ_{2} \leq \frac{N α + τ_{o} π a^{2} - f_{s}}{m_{o}},$ (9)

This condition validates the firm grip, as increased normal force allows for higher acceleration object manipulation while keeping material parameters constant.

Fabrication

The fabrication of the soft composite finger involves a variety of manufacturing techniques, including cutting, molding, and additive manufacturing as shown in Supplementary Video S1.³⁹ The composite finger is made of two translucent silicones: polydimethylsiloxane (PDMS) (Sylgard 184; Dow Corning, MI) and ECOFLEX 00-30 (Smooth-On, Inc., PA). These materials were chosen for their stability, strength, and ability to cohesively bond to one another. PDMS has a higher Shore A hardness and tensile strength, which enhances the strength of the overall structure of the finger. ECOFLEX has a lower Shore A hardness and tensile strength, which augments the compliant nature of the finger and helps it to adapt to different kinds of objects in unstructured environments, as shown in Table 1.

Table 1.

Materials Used in the Fabrication Process

Materials	Properties
PDMS	Shore A hardness = 50
PDMS	Tensile strength = 6.70 MPa
ECOFLEX 00-30	Shore A hardness = 30
ECOFLEX 00-30	Tensile strength = 1.40 MPa
PLA filament	Thickness = 1.75 mm
Metal rods	Diameter = 1 mm
Silicone tube	Diameter = 2 mm
Kevlar thread	Size = 46
Kevlar thread	Strength = 14 lbs

PDMS, polydimethylsiloxane; PLA, polylactic acid.

The composite finger consists of ECOFLEX encompassing a cured PDMS structure in 1:1 ECOFLEX-PDMS volumetric ratio. The PDMS and ECOFLEX are cast in 3D-printed molds made of PLA filament (polylactic acid; HATCHBOX 3D, CA). Each mold has two unique knuckles that are used to fabricate the distal and proximal joint chambers in the finger, as shown in Figure 4a.⁴⁰ The PDMS and ECOFLEX cure around the knuckles, creating indentations in the structures when removed. Also, the knuckle design allows the cured silicones to be removed from the mold without damaging the silicone tubing.

FIG. 4.

Fabrication of soft composite finger. (a) Apparatus consists of 3-D printed mold, two metal tubes, silicone tube, and two knuckles. (b) Metal tubes are inserted into the tendon channels, then silicone tubing is inserted into the pneumatic channel, and then knuckles are inserted to secure silicone tubing. (c) Filled PDMS mold. (d) Cured PDMS structure removed from the mold. (e) PDMS structure is inserted into ECOFLEX mold. (f) ECOFLEX mold is prepared as explained earlier (b). (g) Filled ECOFLEX mold. (h) Cured ECOFLEX removed from the mold. (i) Knuckles are sealed using a shallow tray filled with PDMS. (j) Soft composite finger with silicone tubing and Kevlar thread routed.

The finger fabrication process is outlined in Figure 4. The two metal rods are inserted in the base (proximal end) of the finger and extended to the tip to create channels through the structure for the tendons to be routed (Fig. 4b). Next, a silicone tube (McMASTER-Carr, IL) is sealed at one end using a silicone adhesive (Smooth-On, Inc.) to ensure that neither the PDMS nor the ECOFLEX seeps into the tubing during the curing process. Then, the silicone tubing is inserted through the base and extended to the distal knuckle (Fig. 4b). A small piece of heat shrink (McMASTER-Carr) is placed around the end of the tubing to avoid the silicone leakage. The knuckles are inserted into their designated spaces, as shown in Figure 4b, securing the silicone tubing in place. The entire prepared mold is sprayed with an Ease Release Spray (Smooth-On, Inc.) to aid the removal process.

First, a PDMS casting is made. Sylgard 184 Elastomer Base and Curing Agent are mixed in a 10 to 1 ratio. The mixture is left to settle for 15 min. Then, it is degassed thrice at −1 bar in a vacuum chamber to remove air bubbles from the mixture that could cause weakness in the structure. The mixture is poured into the prepared mold and is left to cure at room temperature for 48 h (Fig. 4c).

After the PDMS has cured, the metal rods and heat shrink are removed from the mold (Fig. 4d). The casting is carefully removed and inserted into the ECOFLEX mold (Fig. 4e). The mold is prepared again as previously explained (Fig. 4f). ECOFLEX parts A and B are mixed in a 1 to 1 ratio, followed by degassing in the vacuum chamber. The mixture is poured into the mold, which is left to cure at room temperature for 4 h (Fig. 4g).

The cured finger is removed from the mold, and the silicone tubing is trimmed at the end to remove the silicone epoxy seal allowing for pressurization of the distal air chamber. A small piece of tubing is cut out of the proximal air chamber, allowing for airflow to the chamber (Fig. 4h). The air chambers are sealed by placing the finger on top of a shallow tray filled with PDMS to create an airtight seal over the chambers, as shown in Figure 4i. This is repeated for both sides of the chambers.

The Kevlar thread (The Thread Exchange, NC) is routed through the parallel tendon channels so that the free ends of the thread exit at the base of the finger. The composite finger bends in a discrete manner when tension is applied to Kevlar threads. The end of the silicone tubing is connected to a pneumatic system to stiffen the finger (Fig. 4j). The pneumatic system consists of 3V and 6V mini air pump motors (Uxcell, New Territories, Hong Kong) used to pressurize the finger, a solenoid valve (Uxcell) to control the flow of air, and a microcontroller board (Trinket 5V; Adafruit, New York, NY) to control the timing of air pump motor and valve.

To see the effect of chamber shape on the performance of finger, different chamber shapes were fabricated in a similar way as explained earlier. The vertical ellipse (Fig. 6a) was chosen to enhance local stiffening capability and promote discrete bending. The circle (Fig. 6b) has a similar rectangular cross-sectional area as the vertical ellipse, so it is chosen to explore its effect on stiffness. The small circle (Fig. 6c) has the same volume as the vertical ellipse so it is selected to observe the effect of shape on the performance of the finger. Finally, the horizontal ellipse (Fig. 6d) is proportionally similar to the vertical ellipse but is rotated 90° to analyze the effect of orientation on the stiffening and bending of the finger.

Experiments

Stiffness experiments

The first set of experiments was conducted to investigate the effect of increasing pressure on the overall stiffness of the finger. The experiments consisted of pulling the tendons to different displacements (3, 7, and 12 mm) and measuring the resulting force needed to bend the finger using a six DOF force sensor (Nano 17; ATI Industrial Automation, NC), as shown in Figure 5a. The sensor was attached to a 3D-printed part mounted on the motorized linear platform. The data from the force sensor were recorded using a DAQ card (USB-6210; National Instruments, TX). The same procedure was also conducted at higher pressures (6.60 and 10.40 kPa) to determine stiffness values. The raw force displacement data were used to calculate the overall stiffness of the finger. This experiment was repeated for each chamber shape.

FIG. 5.

Experimental setups for soft composite finger. (a) Experimental setup for stiffness tests using force sensor and motorized linear module. (b) Soft composite finger showing discrete bending behavior.

FIG. 6.

The stiffness experiments were conducted at increasing pressure for constant tendon displacements of 3, 7, and 12 mm using a six DOF force sensor (ATI Nano 17) and a motorized linear module for various chamber shapes. (a) Vertical ellipse, (b) circle, (c) small circle, and (d) horizontal ellipse. The lines in (a) show predictions based on the model while the points are results obtained experimentally. It should be noted that the finger stiffness model works for the vertical ellipse because it satisfies the underlying assumption that the air spring is aligned with the direction of joint deformation. DOF, degrees of freedom.

Joint angle experiments

The soft composite finger shows discrete bending behavior when the Kevlar tendons are pulled; thus, the second set of experiments was performed to analyze this behavior in terms of joint angle. The finger was marked with ink before the experiment to set a datum to compare with the deformed state, as shown in Figure 5b. The first line was drawn parallel to the fixture. The second line connected the center of both joints and was parallel to middle segment. The third line connects the center of distal joint to the fingertip. The angle between the first line and the second line was called the proximal angle ( $θ_{p}$ ). The angle between the second line and the third line was denoted as the distal angle (( $θ_{d}$ ) (Fig. 5b). The Kevlar tendons routed in the finger were attached to a 3D-printed part fixed to a motorized linear mechanism, which was programmed to displace 3, 7, and 12 mm at a constant velocity. A high-definition camera (EOS Rebel SL1; Canon, Inc., NY) was used to capture photos of the soft composite finger at all displacements. This procedure was performed at 6.6 and 10.40 kPa to see the effect of pressurizing on the discrete behavior. This experiment was repeated for all chamber shapes. Finally, both the distal ( $θ_{d}$ ) and proximal ( $θ_{p}$ ) joint angles were quantified using the measuring tool in the open-source vector graphics software Inkscape, as shown in Figure 5b. This set of experiments was repeated after substantial time (∼15 min) to avoid interference of the viscoelastic nature of silicones that could significantly affect the following set of trials.

Firm grasp experiments

To validate that stiffening the joints leads to firm grasp over an object, we have designed an experiment consisting of a single-link robotic arm attached to a gripper made up of three soft fingers with the vertical ellipse joint design (Fig. 8). In this experiment, a servo motor (HS-425BB; ServoCity, KS) was attached to the gripper, which was used to actuate the tendons and control the grasping motion of the fingers. A customizable pneumatic system built in the laboratory was used to regulate the air pressure to control the stiffening effect of the fingers. During this experiment, the robotic gripper carried objects with different geometries and masses: glasses case, mouse, empty wine glass, and wine glass with weights. Each item was rotated about the robotic arm's axis for a fixed angular displacement at three different magnitudes of object acceleration. If the object did not slip during the trial, it was considered a “Pass.” Contrarily, if the object slips, the trial was considered a “Fail.” This experiment was conducted at both maximum pressure and no pressure to explicitly demonstrate the effect of stiffening on the firm grasp behavior of the fingers (Table 2).

Table 2.

Summarized Results of Firm Grasp Experiment on Various Objects

Glasses case (mass = 65 g)
	Acceleration (m/s²)
Pressure	24	28	31
No pressure	Pass	Fail	Fail
Pressure (10.40 kPa)	Pass	Pass	Pass

Mouse (mass = 80 g)
	Acceleration (m/s²)
Pressure	14	21	28
No pressure	Pass	Pass	Fail
Pressure (10.40 kPa)	Pass	Pass	Pass

Empty wine glass (mass = 90 g)
	Acceleration (m/s²)
Pressure	17	19	21
No pressure	Pass	Pass	Fail
Pressure (10.40 kPa)	Pass	Pass	Pass

Wine glass with weights (mass = 140 g)
	Acceleration (m/s²)
Pressure	17	19	21
No pressure	Pass	Fail	Fail
Pressure (10.40 kPa)	Pass	Pass	Fail

Results and Discussion

Stiffness experiments

The stiffness model is derived for the vertical ellipse chamber shape. The air spring of the vertical ellipse is well aligned with the tendon actuation direction and joint deformation, thus validating the mechanics modeling of the joint mechanism, as shown in Figure 6a. However, other chamber shapes such as the circular chambers may need multiple springs to predict the stiffness behavior due to its radial symmetry. Moreover, the horizontal ellipse may be modeled using an additional dominant air spring perpendicular to the longitudinal axis of the finger. The addition of this dominant spring is also confirmed by the results of horizontal ellipse in the joint angle experiment (see the Joint Angle Experiments section under the Results and Discussion). Therefore, this article will focus on the model of the vertical ellipse as a better design choice.

The experimental results for all the chamber shapes indicate that an increased pressure stiffens the finger at constant tendon displacements of 3, 7, and 12 mm (Fig. 6). From the results plotted, it is evident that all chamber shapes show non-zero stiffness, $K_{f i n g e r}$ , at zero pressure. Second, it is evident that increasing tendon displacement at constant pressure decreases the overall stiffness of the finger. At higher tendon displacements, both the distal and proximal chambers deform significantly due to the bending of the finger (rectangular cross-sectional area loss), so it becomes difficult for pneumatics to stiffen the finger. Although all chamber shapes demonstrate an increase in stiffness when pressure is increased, the small circle chambers show the highest stiffness compared with the other chamber shapes (Fig. 6c). The small circles have a similar volume to the vertical ellipses but have a higher stiffness likely due to their smaller semi-major axis, a_i [Equation (5)].

Despite the fact that both the vertical ellipse and the circle have similar rectangular cross-sectional area, the stiffness of the circle is less than that of the vertical ellipse (Fig. 6b). According to Equation (2), the vertical ellipses store a smaller volume of air, which leads to a higher force due to pressurization and thus higher stiffness. The results also show that changing the orientation of the vertical ellipse to horizontal ellipse decreases the overall stiffness of the finger (Fig. 6d). This could be due to the fact that the smaller cross-sectional area, compared with the vertical ellipses, decreases its stiffening capability [Equation (2)].

Joint angle experiments

To characterize the discrete bending of the finger, we performed joint angle experiments. It is evident from the results that increasing the tendon displacement increases both joint angles (distal and proximal) at fixed pressure values. This leads to the observation that the finger is able to bend in a discrete manner (Fig. 7). The magnitude of the proximal and distal angles seems to depend on the length of the longitudinal axis of the chamber. As the length of the longitudinal axis increases, the proximal and distal angle decreases for the small circle and horizontal ellipse chamber. These designs have the shortest and longest longitudinal axes, respectively. The results also show a similar behavior for vertical ellipse, circle, and small circle proving that increasing the pressure at constant tendon displacement increases the joint angle (Fig. 7a–c, e–g). The increased joint angles due to high pressure result in an increased curvature at the joints, which generates a push against the object leading to the firm grasp explained earlier in the Firm Grasp section. However, increasing the pressure at constant tendon displacement decreases joint angle for the horizontal ellipse (Fig. 7d, h). In the case of horizontal ellipse, the finger straightens rather than bending when pressurized, thus decreasing the joint angles at higher pressures. This straightening effect can be modeled using a dominant spring perpendicular to the longitudinal axis as described earlier in this section. In addition, the distal joint angle is greater than the proximal joint angle in the vertical ellipse, which shows that the finger is more compliant at the distal joint and is in line with the behavior shown by prior soft robots (Fig. 7a, e).²⁰

FIG. 7.

The joint angle experiments were conducted at constant pressures of 0, 6.60, and 10.40 kPa using a high-definition camera to capture photos, and then, open-source vector graphics software (Inkscape) was used to determine the joint angles to show the discrete bending behavior of the finger. (a) Vertical ellipse (proximal), (b) circle (proximal), (c) small circle (proximal), (d) horizontal ellipse (proximal), (e) vertical ellipse (distal), (f) circle (distal), (g) small circle (distal), and (h) horizontal ellipse (distal).

FIG. 8.

The single-link robotic arm attached to the soft gripper carried objects of various geometries and masses to validate that stiffening joints lead to firm grasp over the object. (a) Glasses case, (b) mouse, (c) empty wine glass, and (d) failed to hold the wine glass with weights without pneumatic actuation at higher object accelerations.

Firm grasp experiments

Although we have shown through model and experiments that pressurization stiffens the finger, there is a need to prove experimentally that the stiffening leads to a firm grasp over the object. To further verify the firm grasp of soft composite finger, we performed a manipulation experiment on objects with different masses and geometries (Fig. 8, Supplementary Video S2). As shown in Table 2, the grasp on all the objects with no pressure failed at high acceleration manipulation due to the highly soft joints. Although such structural compliance is necessary for soft robotic hands to conform to various object shapes, dynamic and high-speed manipulation using the hands would be limited. In the case of the proposed soft composite finger, however, pressurization stiffens the joints resulting in improved object grasping at higher acceleration. This improved grasping is attributed to the more rigid finger structures, as expected from Equation (9), which better constrain the objects.

The grasping of the glasses case (Fig. 8a) outperforms the other objects in terms of object acceleration because it is lightweight and has a rough texture, which increases the friction between the case surface and the fingers. The manipulation experiment on the mouse illustrates how the stiffened joints in the soft fingers can manipulate objects with complex geometry and smooth finish at higher object accelerations (Fig. 8b). The manipulation of the wine glasses, however, demonstrates the effect of mass on the grasping, independent of contact and grasping pose conditions (Fig. 8c, d). As predicted from Equation (9), increasing the mass of the object decreases the object acceleration achievable. Additionally, it is observed that the fingers stiffened by pressurization cannot manipulate the wine glass with weights at the highest acceleration (see Wine glass with weights in Table 2 at 21 m/s²). Therefore, it can be concluded that when the mass of the object exceeds a certain value, either the magnitude of the pressure should be increased to stably grasp the object during manipulation or it is impossible to handle it using a soft manipulator at sufficiently high acceleration.

Conclusion

In this article, we have proposed a design of a soft composite finger with two embedded soft joints inspired by the human finger. Hybrid actuation combining both tendon and pneumatic actuation systems is implemented in the finger to exhibit discrete bending behavior and stiffen the soft embedded joints, respectively. The model prediction and the experimental results suggest that anthropomorphic soft composite fingers possess stiffening capability and can discretely bend with varying curvature along their length. Also, factors such as cross-sectional area and longitudinal axis length of the joints affect the performance of finger in terms of stiffening capability and discrete bending behavior, respectively. The design of the composite finger helps it constrain the object sufficiently by enhancing stiffness using pressurization for a firm grasp during dynamic manipulation. Therefore, the soft composite finger can be used to develop a soft gripper that is capable of adapting and stably grasping items during higher object accelerations. Overall, the design presented in this article is a compromise of soft robots' flexibility and rigid robots' performance to yield more predictable behavior and firm grasping.

In future works, viscoelasticity of the materials would be accounted for in the stiffness model to better predict the dynamic performance of the finger at higher tendon displacement and pressure. Further research will focus on optimizing the design of embedded joints in the finger to achieve higher stiffness at larger tendon displacements, which will enhance the firm grasp behavior of the finger. The tip force will be measured in the stiffened composite finger to characterize the grasping behavior.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Supplementary Material

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Supplementary Material

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