Modeling of grasping force for a soft robotic gripper with variable stiffness

Mechanism and Machine Theory 128 (2018) 254–274

Contents lists available at ScienceDirect

Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Modeling of grasping force for a soft robotic gripper with variable stiffness Yin Haibin a, Kong Cheng b, Li Junfeng b,∗, Yang Guilin c a

Key Laboratory of Hubei Province for Digital Manufacture, School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China b School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China c Key Laboratory of Zhejiang Province for Robot and Intelligent Manufacturing Equipment Technology, Ningbo Institute of Material Technology and Engineering, CAS, China

a r t i c l e

i n f o

Article history: Received 18 October 2017 Revised 29 March 2018 Accepted 11 May 2018

Keywords: Soft gripper Grasping force Variable stiffness SMA Cosserat model

a b s t r a c t The purpose of this research is to present a grasping force model for a soft robotic gripper with variable stiffness. The soft robotic gripper was made of shape memory alloys (SMAs) with contraction and variable stiffness properties. A variable stiffness mechanism with embedded sets of SMA fibers was developed; however, the response characteristics of its backbone did not comply with the constant-curvature model when it was subjected to complex forces/torques, such as gravity, grasping forces and driving torques. In this case, the Cosserat theory was used to implement real-time computations of the grasping force of the soft robotic gripper that was subjected to complex forces. Finally, a series of tests were conducted on the grasping force of the soft finger and the gripper. The elicited results showed that the grasping force is related to the stiffness and to the object’s offset and friction coefficient. Moreover, experimental results showed that the grasping force of the soft robotic gripper increased by 48.7% when the Young’s modulus of the SMA-2 wires increased from 25 GPa to 48 GPa. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Conventional manipulators and end-effectors, which involve rigid components, such as linkages, gears, and motors, can exhibit precise positioning and improved mechanical performance using the control strategies and algorithms developed during the past decades [1]. However, these types of rigid robots are unsafe and exhibit poor adaptability when they interact with humans or the surrounding environment. In comparison to conventional manipulators, soft manipulators inspired by biology, such as the octopus arms and the elephant trunk, exhibit compliant and safe characteristics [2]. They can perform a variety of tasks, such as dexterous manipulation in limited space or unknown environments. Because the design principles and operation methodologies for soft robots are extremely different from conventional rigid manipulators, many different design mechanisms, fabrication and control strategies have been developed in recent years for these robots [3], including the design of soft actuators [4], fabrication of soft hands [5], and position control of soft fingers [6].

∗

Corresponding author. E-mail addresses: [email protected] (Y. Haibin), [email protected] (K. Cheng), jfl[email protected] (L. Junfeng), [email protected] (Y. Guilin). https://doi.org/10.1016/j.mechmachtheory.2018.05.005 0094-114X/© 2018 Elsevier Ltd. All rights reserved.

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

255

There is a growing trend towards the development of soft robots to serve human beings and to work with people. A soft robotic gripper is an import part of a soft robotic system that interacts with the environment and objects [7]. In some cases, the soft hands with low stiffness can effectively absorb shocks during interaction with the environment and maintain either adaptive interactions with complex shapes or safe interactions with objects that require protection, such as eggs and fruit [8]. In other cases, soft grippers with increased stiffness can pick up heavy objects, such as water cup and bottle. These advantages have attracted the attention of numerous researchers and steered efforts towards the development of soft hands with variable stiffness [9]. Common methods for grippers with tunable or variable stiffness include the hydraulic pressure, pneumatic particle jamming [10], and layer jamming [11]. However, the use of either additional pump stations, or power equipment, increases the complexity and weight of the robot system, resulting in difficulties for compact and high power-density required in autonomous mobile robots. Moreover, these variable stiffness mechanisms are coupled with the actuating process of soft robot. In addition to these methods, various decoupling strategies for obtaining both actuator and variable stiffness characteristics of soft grippers have been reported, including SMA that has actuator properties and fusible alloy (Ni-Cr wires) with tunable stiffness [12], motor that has actuator properties and magnetorheological fluids with varying stiffness characteristics [13], and a pneumatic actuator and shape memory polymer (SMP) with variable stiffness properties [14]. However, the use of these materials in variable stiffness mechanisms as low-melting alloy, fluids and low-strength polymer make these designed grippers perform the light manipulation tasks. SMAs falls into the category of smart materials and can execute direct electrothermal actuation with increased strengths. Additionally, SMA actuators possess various advantages, such as low driving voltages, biocompatibility, small size, and noiseless operations, which make them suitable for a variety of applications [15], such as soft robotics [16], robotic surgical systems [17], and grippers [18]. Among the SMA materials, the NiTi alloy has been extensively used for the design of actuators owing to its increased strain characteristics (up to 7%) [19]. Moreover, SMAs can be used to change the stiffness of structures [20]. Nevertheless, the SMA used to induce variable stiffness is not capable of producing large deformations, which are required in soft robots. To overcome these drawbacks, a new soft finger with variable stiffness was proposed [21], but its application design and grasping model remains to be investigated. In regard to kinematic and static modeling of soft robots, various methods have been developed, such as the improved Denavit-Hartenberg (D-H) approach [22], constant [23], and variable curvature [24] methods. These approximate methodologies are valid for soft robots in the absence of external loading [25]. However, the interaction of soft fingers with various objects is sensitive to not only gravity but also bending torques and varying loads. The Cosserat theory has been recently used to model kinematics and dynamics of continuum robots, and it has shown promise as a general tool used to describe the finite deformations of soft robots that withstand complex forces and torques [26]. However, the Cosserat theory was generally used in continuum robots [27] and was rarely applied to multi-finger soft hands with complex loading. The main contribution of this study is the introduction of the design and implementation of a grasping model of a new soft robotic gripper with variable stiffness. Its uniqueness lies in the structural decoupling of the actuator and the variable stiffness mechanisms of the soft robotic gripper. The actuator mechanism is realized by one thermal-induced SMA fiber with contraction in length, and the variable stiffness mechanism is realized by another thermal-induced SMA wire without change in length. In addition, a stress-induced SMA fiber with super elasticity is used as the bone structure to provide the initial stiffness and restoring force of the soft fingers. Owing to the complexity of the developed forces or torques, such as gravity, grasping force, and bending torque, the kinematics and statics of the soft robotic gripper cannot be analyzed using the conventional constant-curvature model. Based on the Cosserat theory, a grasping force model is built for the proposed soft robotic gripper with variable stiffness. Moreover, a series of experimental tests and simulations on the grasping force of the soft robotic gripper are investigated to discuss the modeling properties. The rest of this paper is organized as follows. In Section 2, a soft robot gripper is developed. A Cosserat theory-based model of the grasping force is introduced in Section 3. In Sections 4 and 5, the experimental results are compared with the simulation results for the soft finger and gripper. Finally, conclusions and comments on future work are outlined.

2. Soft robotic gripper with variable stiffness To investigate the grasping force of soft hands with variable stiffness, we designed a soft robotic gripper, as shown in Fig. 1. This soft gripper includes two fingers that are fixed using one foundation bed, where both fingers are completely identical in size and structure. The detailed configuration and partial enlargement of the soft finger are shown in Fig. 2. The SMA-3 wire (0.6 mm in diameter, NiTi alloy) with super elasticity is used as the bone structure to ensure that the soft finger can restore after bending [28], whereas four SMA-2 wires (0.5 mm in diameter, NiTi alloy) without change in length are fixed in parallel to the SMA-3 wire to realize the variable stiffness of the soft finger. The effective lengths of the fingers are referred to as the working lengths of the SMA-3 and SMA-2 wires, denoted as L3 = L2 = 100 mm. To actuate the soft finger, one SMA-1 fiber (0.15 mm in diameter, BMF150 from Toki Corporation in Japan [6]), which is used 1200 mm in length for large contraction because its strain is low, is placed into both fingers with a U shape to redouble its driving capacity, as well as to conveniently charge at one end of the fingers. As shown in Fig. 2(a), all SMA fibers were embedded in brackets to stabilize the bending deflection of the fingers and avoid kinking. These brackets were made of polylactic acid (PLA)—the same material used in the foundation bed—using a

256

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

Fig. 1. Proposed soft robotic gripper with variable stiffness.

Fingerp X

Z Bracket

Y

Base

SMA-1

SMA-3 SMA-2

(a)

X

Z Y

lSMA

d lzj (b) Fig. 2. (a) Configuration and (b) partial enlargement of the soft finger.

3D printer. The extra length of the SMA-1 fiber that protruded out of the fingers was wrapped around the 12 pulleys fixed in the foundation bed. As shown in Fig. 2(b), the parameter d is the lever of the pulling force that produces the bending torque at the fingertip. The distance between the brackets is denoted by lzj , and its value is recommended to be equal to 2.5 d [29]. lSMA is set to 4.5 mm considering the trade-off between the large space for the reduction of the thermal coupling of the SMA wires due to the thermal radiation and the compact finger.

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

257

(a)

(b) Fig. 3. (a) Strains of SMA-1 under loading 0.1 kg and (b) Young’s modulus of SMA-2.

To illustrate the thermo-mechanical properties of these SMA wires, we provide the relationship between the heating current ranging from 0 A to 0.36 A and the strains of the SMA-1 fiber under loading 0.1 kg in Fig. 3(a), as well as the relationship between the heating current ranging from 0 A to 1.70 A and the Young’s modulus E2 of the SMA-2 wire in Fig. 3(b) [30]. The Young’s modulus of SMA-2 are changed from 25 GPa to 48 GPa due to the phase transformation, so the bending stiffness of the soft finger could be adjusted and referred as EIy = E3 I3 + 4E2 I2 [31]. Choosing Poisson’s ratio as 0.3,

258

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274 Table 1 Parameters and values of the soft gripper [31]. SMA1 SMA2

SMA3

ρ A d lSMA m1 m2 Ix Iy J E G

Diameter d1 Length L1 Young’s modulus E2 Second moment of area I2 Density ρ 2 Diameter d2 Length L2 Young’s modulus E3 Second moment of area I3 Density ρ 3 Diameter d3 Length L3 Equivalent density Section area Lever of pulling force Distance between SMA-2 Mass of bracket Mass of fingertip Second moment of area in X direction Second moment of area in Y direction Polar moment of inertia Modulus of elasticity Shear modulus

0.15 mm 1200 mm 25 − 48 GPa 3.066 × 10− 3 mm−4 6.5 × 10−6 kg/mm3 0.5 mm 100 mm 56 GPa 6.359 × 10− 3 mm−4 6.5 × 10−6 kg/mm3 0.6 mm 100 mm 40 × 10−6 kg/mm3 1.0679 mm2 8 mm 4.5 mm 989.78 × 10− 6 kg 2756.18 × 10− 6 kg 39.9 mm−4 0.022 mm−4 39.9 mm−4 (E3 I3 + 4E2 I2 )/Iy E/2.6

Fig. 4. Schematic of grasping operation of soft robotic gripper.

the shear modulus of the soft finger can be described as E/2.6 [32]. The detailed mechanical parameters and values of the soft gripper are listed in Table 1. The grasping capacity of the soft robotic gripper needs to be investigated to explore its grasping behavior. 3. Model of grasping force A schematic of the grasping operation of the soft robotic gripper is shown in Fig. 4, where the gripper is grasping an object, and fingers undergo bending deflection as a result of the pulling forces (Fp ). Under the action of the bending torque Td (Td = dFp ), the fingertip will move to the required position with offset values X and Z in the X and Z directions, respectively. Considering the mechanics balance of the soft robotic gripper and an ordinary Coulomb model for friction, the grasping behavior can be described as

Fmg =

N 

(Ff j ),

j=1

Ff j =

μ j FN j ,

(1)

where Fmg is gravity of the grasped object, Ffj and FNj are the friction and normal forces between the jth fingertip and the object, respectively, and μj is the friction coefficient between the jth fingertip and the object. In addition, N is the number of fingers, and it is set to a value of two in this study. The normal force of each soft finger reflects its grasping capacity, and it is defined as the grasping force of the soft finger when the friction coefficient is set to the specific values associated to the materials used. The following section describes the formulation of grasping force using the classical Cosserat theory.

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

259

Fig. 5. Coordinate of the soft finger and arbitrary section of rod from a to s.

3.1. Cosserat theory Fig. 5 shows a Frenet–Serret coordinate frame of the soft finger, which is a one-dimensional rod. In this figure, O-XYZ is a global frame, which is stationary with respect to the base, and o-xyz is a local frame located at any point in the rod. Each point of the rod can be parameterized by its unstretched length, here represented by the variable s. The position of any point s in the global frame can be expressed as r(s), while the infinitesimal element in the local frame can be expressed as rl (s). In accordance with the Cosserat theory, a rod can be parameterized by its centerline curve in space with a threeelements vector r(s)  R3 to represent the location of a point on the rod, and a matrix R(s)  SO(3) to specify the orientation of a local frame with respect to the global coordinate frames. θ (s), a function of s, is the rotational angle from the reference coordinate X-Z to the local coordinate x-z that is attached to the rod at r(s) [26,27]. The variables vl (s) represent the linear rates of changes of the rod’s position with respect to s in the local frame, defined by vl (s) = drl / ds, in the global frame,

r  ( s ) = R ( s ) vl ( s ) ,

(2)

where the apostrophe ’ denotes a derivative with respect to s. In a similar manner, the angular rates of changes of the rod’s orientation with respect to s in the global frame is represented by

R (s ) = R(s )uˆl (s ),

(3)

where ^ indicates the conversion of the angular variables ul (s) into a skew-symmetric matrix. As shown in Fig. 5, the distributed external forces and moments applied to the arbitrary section of the rod are specified by f and τ , and the internal—or point forces—and the moments, in the global frame coordinates are denoted by n and m, respectively. In addition, ξ is the position variables in the local frame ranging from position s to a. Therefore, the force and torque equations of the static equilibrium can be described as

n ( s ) + f ( s ) = 0,

(4a)

m ( s ) + r  ( s ) × n ( s ) + τ ( s ) = 0.

(4b)

The initial linear and angular variables in the local frame are specified by v∗ (s) and u∗ (s) for the undeformed reference configuration, and the constitutive equations are thus described as

⎡

0

0

⎤

D2

0

⎦(vl (s ) − v∗ (s ) ),

0

0

D3

⎡

0

0

⎤

K2

0

⎦(ul (s ) − u∗ (s ) ),

0

K3

D1

n ( s ) = R ( s ) ⎣0

K1

m ( s ) = R ( s ) ⎣0 0

(5a)

(5b)

where D1 = D2 = GA, D3 = EA, K1 = EIx , K2 = EIy , and K3 = GJ, where E is the modulus of elasticity, G is the shear modulus, I is the second moment of area, J is the polar moment of inertia, and A is the cross-sectional area of the rod.

260

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

3.2. Grasping force In this study, the soft finger is subjected to a bending deflection in the XOZ plane because Iy is much less than Ix . Therefore, the stretch and compression of the soft finger is neglected, and the linear rates of changes of the rod’s position in the local frame is expressed as vl (s) = [0 0 1]T . Based on Eq. (2), we obtain

r (s ) = R(s )vl (s ) = [sin θ

0

cos θ ] . T

(6)

The distributed force of the soft finger is its gravity, and it is expressed as f(ξ ) = ρ Ageg , where ρ and A are the line density and cross-sectional area of the soft finger, respectively, and g is the gravitational acceleration, defined herein as eg = [0, 0, 1]T . According to Eq. (4a), we define n(s) = [C1 , C2 , C3 − ρ Ags]T . Considering the boundary condition at the fingertip n(L) = [FN , 0, μFN ]T , the coefficients are expressed as C1 = FN , C2 = 0, and C3 = μFN + ρ AgL. Hence, the point forces on the soft finger are represented by

n(s ) = [FN

0

μFN + ρ AgL − ρ Ags]T ,

(7)

Considering the distributed torque τ = 0, and substituting Eqs. (6) and (7) into Eq. (4b), we can derive the differential equation as

m  ( s ) = [0

− FN cos θ + (μFN + ρ AgL − ρ Ags ) sin θ

T

0] .

(8)

In addition, the constitutive equation [Eq. (5)] on bending is represented by



m (s ) = 0

EIy θ  (s )

T

0 ,

(9)

Hence, the kinematics and statics of the soft finger are described as

X  (s ) = sin θ ,

Z  (s ) = cos θ , −F cos θ + (μFN + ρ AgL − ρ Ags ) sin θ θ  (s ) = N , E Iy

(10)

where FN is the grasping force of any finger in this study. In order to use the Cosserat theory to compute the grasping force of the soft finger, the boundary conditions could be determined as

X ( 0 ) = 0, Z ( 0 ) = 0, θ ( 0 ) = 0, θ  ( L ) =

Td . E Iy

(11)

There is no analytical solution for the grasping force because the problem is nonlinear and may have several solutions [33]. Fig. 6 depicts the flow diagram used to calculate a solution for the grasping force and the five positions of the soft finger in the process of its deflection. As shown in Fig. 6(b), H is defined as an object offset representing the distance between the initial position of fingertip and the contact point with the grasped object along the X direction. In this study, the grasped objects are assumed as rigid cuboids, so the trajectories of the grasped objects are described as X = H. The calculation flow is explained in five steps corresponding to the five positions of the soft finger as follows: Step 1: The finger model expressed in the Eq. (10) is built and initial conditions are determined. The grasping force and driving torque are increased by 0.01 N and 1 Nmm, respectively, from their initial zero values. Under the action of the grasping force and driving torque, the fingertip’s position P(Xp , Zp ) is calculated using the finger model. Step 2: If the value Xp is ever increasing, the calculation continues. Additionally, if the value Xp is less than the object offset H, the driving torque (Td ) should be increased. Once the value of Xp becomes equal to H, the fingertip touchs the object. Step 3: To snatch up the object, the grasping force (FN ) is required to adjust to its desired value. If the grasping force is too small, the value Xp exceeds the object offset H. Otherwise, the value Xp is less than the object offset H in the calculation. Considering the trade-off between the accuracy, solvability and computational cost, the maximum permissible overshoot is set to eH = 0.1 mm, and the step sizes for the grasping force and the driving torque are given as 0.01 N and 1 Nmm, respectively. Step 4: Through the adjustment of the driving torque and grasping force, the calculation is repetitively updated. The fingertip slides along the surface of the object in the process of its interaction with the object. Step 5: The process continues until the fingertip finally detaches from the object, and the value Xp decrease progressively. Thus, the grasping force of the soft robotic gripper can be simulated and determined in accordance with this process. 4. Experimental validation of the theoretical model To validate the model, calculate the grasping force, and compare the elicited results with the experimental results presented in this section, a series of experiments were conducted on the kinematic analysis and the grasping force of the soft fingers.

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

Grasping force FN

Initial condition

Td*=Td

FN*=FN

Finger model 4

1 Fingertip position P(Xp ,Zp)

Td=Td+ Td

Driving moment Td

No

Xp everincreasing?

No

YES

261

Adjust Td

No

X 5

H

O

YES

2

5 Xp

H FN

YES

|Xp H| ≤ eH YES 3 Adjust FN FN=FN± FN

No

Ff

|Xp H| ≤ eH Z

End

(a)

4 3

Td

2

P(Xp,Zp)

1

(b)

Fig. 6. (a) Flow diagram depicting the calculation methodology for the grasping force and (b) schematic depicting the process of deflection of the soft finger.

Fp

X O

X

O

H

Z

P

A

A

Force Sensor

Td

Z

FN

P μFN

(a)

(b)

Fig. 7. Experimental scheme (a) without load and (b) with an imposed load at the soft finger.

4.1. Descriptions on experiments and setup of finger In the following experiments, the SMA-1 wire was charged to produce contraction, and both ends of the U-shaped SMA-1 wire protruding out of the finger were fixed on a force sensor (outer figures), which can measure the pulling force (Fp ) to calculate the bending torque (Td ) driving the soft finger. In addition, the Young’s modulus E2 of the SMA-2 wires was adjusted to realize the variable stiffness of the soft finger by heating the SMA-2 wires electrically [31]. The fingertip’s position marked with the symbol P was tested using graph paper. If the fingertip is not in contact with anything, the grasping force FN is set to zero in the calculation (FN = 0). In this case, only a bending torque Td is generated from the pulling wires (SMA-1) that acts on the fingertip, thereby leading to a bending deflection of the soft finger, as shown in Fig. 7(a). If the fingertip contacts with a force sensor, the grasping force

262

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

FN can be measured. In this case, both the bending torque Td and the grasping force FN act on the fingertip, which can be seen as grasping behavior of the soft finger, as shown in Fig. 7(b). In the following analyses, the experimental data presented in this paper were acquired by performing at least three times experiment iterations. 4.2. Kinematic analysis without grasping force If the fingertip is not in contact with anything, we can obtain the shape of the finger and the position of the fingertip. In this section, the Young’s modulus E2 of the SMA-2 wires is set to 25 GPa according to the stiffness characteristics at the ambient temperature [31]. Fig. 7(a) shows the experimental setup using the initial point A, which is described using the coordinate (0, 100), and the finial point P in the XOZ plane. Twelve sets of data are tested to express the fingertip’s position along the X and Z directions under a pulling force (Fp ) with 0.25 N increments, as depicted in detail in Fig. 8. Fig. 8(a) shows a kinematic analysis of the soft finger in the XOZ plane. The figure clearly shows that the bending shape of the finger is changed under the action of the bending torque, and the fingertip’s positions in the simulation (Xp , Zp ) can predict the experiment results (Xe , Ze ). Fig. 8(b) shows the fingertip’s position related to the bending torque along the X and Z directions. In order to analyze the predictive accuracy of the model, the absolute error (ep ) of the fingertip’s position estimated as the difference of the elicited simulation and experimental results is represented by

ep =



e px 2 + e pz 2 ,

(12)

where epx and epz are the errors of the fingertip’s position in the X and Z directions, and where epx = |Xp − Xe |, epz = |Zp − Ze |. Fig. 9 depicts the results of position errors expressed in Eq. (12). It clearly shows that the absolute position error increases gradually as a function of the bending torque and is mainly affected by the error along the Z direction. In addition, the trends of the errors in the X and Z directions are different. The error in the X direction first increases and then decreases from Td = 8 Nmm to Td = 16 Nmm. In contrast, the error in the Z direction exhibits an increasing trend as a function of the bending torque. This trend can be explained in Fig. 8(b). Both the simulation and experimental results show that the positions in X direction increase until Td = 16 Nmm and then decrease. The positions are almost unchanged when the torque increases from 14 to 18 Nmm, which results in a minimum error value in the X direction. However, the differences between the calculated and measured positions of the fingertip in the Z direction continually increase with the increase in the bending torque. These results show that the model and calculation method are relatively accurate in formulating the soft finger. Therefore, kinematic analysis is the basis for the investigation of grasping forces that follow. 4.3. Grasping force in the absence of variable stiffness Fig. 7(b) shows an experimental scheme of the grasping force, which can be measured using a force sensor. If the SMA-2 is not charged, the finger will perform manipulations with a stiffness that equals its initial stiffness, whereby the Young’s modulus E2 equals 25 GPa in accordance with the given parameters in Table 1. Here, the grasping force of the soft finger under the action of the bending torque is tested to validate the model and calculation, where the friction coefficient is set to μ = 0.426 according to the friction test. Fig. 10 shows the simulation and experimental results of the grasping force of the soft finger in the absence of variable stiffness. Four sets of tests with object offset H = 30 mm, 40 mm, 50 mm and 60 mm were conducted. As shown in Fig. 10(a), at the beginning of the bending deflection, there is no grasping force because the fingertip cannot establish contact with the object. This is exemplified in four cases with different H values until the bending torque reaches the respective no-load torque values of Td = 4 Nmm, 6 Nmm, 8 Nmm and 9 Nmm. The grasping force FN increases as the bending torque increases, and the fingertip slides along the surface of the object. It eventually slides away from the object at the critical bending torque, as depicted by the abrupt decrease in the grasping force at the torque values of 63 Nmm, 53 Nmm 43 Nmm and 33 Nmm. Correspondingly, the respective maximum values of grasping forces in the simulations (FN-S ) were 0.730 N, 0.576 N, 0.412 N and 0.242 N. Similarly, the maximum values of grasping forces in the conducted experiments (FN-E ) were measured to be 0.676 N, 0.571 N, 0.455 N and 0.265 N. Consequently, the error in the value of the grasping force between the simulation and experiment results is defined as eF = FN-S − FN-E . If the bending torque is relatively small, the experimental data points lie close to the positions predicted by the model; if the bending torque is close to the critical value, and the experimental data points slightly deviate from the positions predicted by the model. The analyzed results agree well with the observed error trend depicted in Fig. 10(b), where the errors lie within 0.05 N when the bending torque is less than 40 Nmm. However, the error increases significantly when the torque is larger than 40 Nmm because large deformations result from the critical bending torque of the soft finger that causes increased nonlinear responses that cannot be easily modeled. As it can be seen from Fig. 10, larger object offsets, H, lead to larger no-load torques, but produce smaller grasping forces. This is because more torque will be needed to bend the soft finger before contacting with the object when the object offset increases. Fig. 11 shows the statics responses of the grasping force FN , which are calculated using finite element methods (FEM). As shown in Fig. 11(a), a 3D model of the soft finger with specified Young’s modulus of SMA-2 (E2 = 25 GPa) is presented to cal-

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

263

(a)

(b) Fig. 8. (a) Kinematic analysis of the soft finger in the XOZ plane and (b) positions of the fingertip.

culate the grasping force FN in ABAQUS. As shown in Fig. 11(b)–(e), the grasping forces FN are calculated as 0.237 N, 0.221 N, 0.208 N and 0.162 N under the action of the bending torque Td = 20 Nmm when the object offsets are set to be 30 mm, 40 mm, 50 mm and 60 mm, respectively; while the corresponding grasping forces based on Cosserat theory are 0.232 N, 0.206 N, 0.176 N and 0.137 N, respectively. The errors between FEM-based values and Cosserat-based results of the grasping forces are 0.005 N, 0.015 N, 0.032 N and 0.025 N, respectively. Therefore, the simulation results based on FEM could be used to prove the validity of the grasping forces based on Cosserat theory. 4.4. Grasping force with variable stiffness Because the stiffness of the soft finger can be changed by adjusting the Young’s modulus E2 of the SMA-2 wires, a series of tests were conducted to measure the grasping force of the soft finger with variable stiffness by heating the SMA-2 fibers.

264

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

Fig. 9. Errors between simulation and experiment results of the fingertip’s position.

As shown in Fig. 12(a), the grasping forces of the soft finger with the object offset H = 50 mm were measured using the same method described above. In these tests, the Young’s modulus values of the SMA-2 wires were set to 25 GPa, 37 GPa and 48 GPa by heating the SMA-2 with currents 0 A, 0.73 A and 1.70 A, according to the stiffness characteristics [31]. Fig. 12(b) shows the theoretical results of the grasping force of the soft finger, which are related to both the bending torque (Td ) and the offset Z in the Z direction. The spatial mapping relationship is complex and the analysis of the results can be decoupled into torque-based and position-based results, as shown in Fig. 13. In the torque-based results, the grasping forces of the soft finger at an object offset H = 50 mm are zero until the bending torques of the soft finger at three different stiffness values were increased to 8 Nmm, 9 Nmm and 11 Nmm, respectively. At that instance, the grasping force increased linearly with the increase in bending torque. The maximum grasping forces were calculated as 0.416 N, 0.508 N and 0.594 N, for the three different Young’s modulus values of the SMA-2 wires, and the maximum grasping force values were measured to be 0.455 N, 0.549 N, and 0.656 N. The experimental and theoretical data showed the maximum grasping force increased by (0.656–0.455)/ 0.455 = 44.2% and (0.594–0.416)/0.416 = 42.8% (in accordance with simulations), owing to the increase in stiffness. In addition, the simulation results can predict the experimental data accurately with maximum absolute errors of 0.039 N, 0.041 N, and 0.062 N. The results imply that larger stiffness values of the soft finger cause larger no-load torques and model errors. Althrough the torque-based results are linear and relatively accurate, the bending torque is related to the pulling force, which results from the electrothermal contraction of the SMA-1 wire and exhibits a nonlinear response that makes the design of the controller difficult [7]. Moreover, the pulling force resulting from the contraction of the SMA-1 wire can hardly be controlled in a feedback arrangement using a force sensor because the SMA-1 wire is wrapped around pulleys, and the cavity is too small to allow the placement of a sensor. Hence, the study does not refer to the torque-based model to achieve control of the grasping force in experimental tests; however, another position-based model of grasping force is used, as presented in Fig. 13(b). This model is characterized by the increase in nonlnear grasping force, manifested at different stiffness from the same start point (no-load offset Z = 20 mm) to the maximum values (critical) attained. Comparing the data, we infer that the increase in the finger’s stiffness can improve the grasping force of the soft finger. According to the experimental data and simulations, the maximum grasping forces increased by 48.7% and 42.5% (in simulations) when Young’s modulus values E2 increased from 25 GPa to 48 GPa. Here, the simulation and experimental results of grasping forces are marked as FN-S and FN-E , respectively. And the error between the simulation and experimental results is defined as eF = FN- S − FN-E . Fig. 14 depicts the errors between simulation values and experimental results for the grasping forces. When the offset Z is less than 50 mm, the error values of the grasping forces are within the range of − 0.05 N—0.05 N. Larger errors will appear when the offset Z exceeds 55 mm and the Young’s modulus values E2 are set to 37 GPa or 48 GPa, which imply larger deflection and stiffness values for the soft finger. As a result, the position-based model can accurately predict the grasping behavior within a certain range of deflections.

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

265

(a)

(b) Fig. 10. (a) Compared results of the grasping force and (b) errors between the both results for different H values. (E2 = 25 GPa).

Both the torque-based and the position-based models reflect the same grasping behavior for the soft fingers, but the position-based model of the grasping force is more intuitive. Because the deflection of finger quantified as the offset Z can be measured on graph paper, which makes the implementation of the static grabbing of the multi-finger hand easier. Hence, the position-based model of the grasping force of the soft finger is used to investigate the grasping capacity of the soft robotic gripper.

5. Grasping force of soft robotic gripper The grasping force of a finger reflects its grasping capacity on the object, but cannot explain all the attributes of the grasping behavior. Therefore, the soft robotic gripper depicted in Fig. 1 is used to investigate its grasping behavior and grasping force response determined by gravity (Fmg ) of the object, as shown in Fig. 15. To highlight the grasping force,

266

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

(a)

(b)

(c)

(d)

(e)

Electrode clamp

Fig. 11. FEM-based Results of the grasping force for different object offset H. (a) 3D model of the soft finger; (b) H = 30 mm; (c) H = 40 mm; (d) H = 50 mm; (e) H = 60 mm.

Fig. 12. (a) Experimental scheme and (b) theoretical results (H = 50 mm).

the object offset H is set to a relatively small value, 20 mm in the grasping manipulation of the soft gripper with variable stiffness based on the analysis of the grasping force of the soft finger. 5.1. Descriptions on experiments and setup of the gripper Fig. 15(a) shows the failure and success cases in the grasping manipulation of the soft robotic gripper. In the experiments, two power sources were used for bending and changing stiffness values. The power source on the right side of the figure could change the stiffness of the soft finger. According to the stiffness characteristics, the Young’s modulus of SMA-2 wires 25 GPa, 37 GPa, and 48 GPa were achieved when the heating current of the power source on the right side of the figure for SMA-2 was adjusted to 0 A, 0.73 A, and 1.70 A [31]. And the power source on the left for SMA-1 could adjust the bending moment of the soft finger to induce deflections quantified as the offset values Z of the grasping force. A graded scale was

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

267

(a)

(b) Fig. 13. (a) Torque-based and (b) position-based results.

placed in the background of the frame and the soft robotic gripper with variable stiffness characteristics was fixed so that the offset values Z of its fingertips can also be measured. When the SMA-1 wires were charged to actuate the soft fingers with variable stiffness, the soft gripper can interact with the object. The object was printed as a hollow box using PLA, and some mass was added to the box to fill it to adjust gravity of the object during these tests. When the weight of the object keeps unchanged, both cases in Fig. 15(a) correspond to the two grasping cases with different stiffness values, namely a success case with increased stiffness and a failure case with low stiffness. When the weight of object decreases, the soft gripper with low stiffness can also grasp the object successfully. Therefore, we can conclude that the soft gripper with variable stiffness mechanism can grasp the objects with different weight. In the success cases in grasping manipulation of soft gripper with a certain stiffness, the maximum gravity (Fmg ) of the grasped object was defined as the friction force (Ff ) reflecting the grasping force (FN ) of the soft robotic gripper in the experiment.

268

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

Fig. 14. Errors between simulation and experimental results for grasping forces.

5.2. Problem on the model of grasping force Fig. 15(b) shows the experimental results of the maximum gravity (Fmg ) with different offset values Z, which are compared with the simulation results based on the flow diagram depicted in Fig. 6, and the model described by Eq. (1). In the simulation, the friction coefficient used in Eq. (1) was set to 0.426 according to the results of previous friction test on the printed parts using PLA. However, the results elicited from comparison of the experimental and simulation results of the grasping force, appear to exhibit large deviations, as shown in Fig. 15(b). The overall trend is consistent, but the experimental values are lower than the simulation values, and the maximum error is as high as 0.102 N. Poor matching between experiment and simulation might have resulted from the failure of force closure condition or an unsuitable friction coefficient used in the simulation. To investigate the reason for poor matching and improve the accuracy of the model, a series of experiments on the effects of the force closure condition and friction coefficients were conducted. 5.3. Accurate model of grasping force In the foregoing experiment of the soft gripper, a single SMA-1 fiber was symmetrically embedded in both fingers so that their bending deflections can simultaneously occur to realize symmetrical contraction of both fingers and the same offset values Z for the fingertips. The length of the SMA-1 wire used was 1200 mm to achieve adequate bending deflection and twined back and forth around the pulleys in the hollow bed to attain a compact structure. In the compact configuration, the long SMA-1 wire that is located in a small hollow bed is wrapped around eight pulleys in a plane and then around four other pulleys to change its twining directions, so the compact gripper has a complex driving system. Fig. 16(a) shows a schematic of the compact gripper, which uses one SMA wire in both fingers. For example, the numbers 1 and 2 represent both sides of the U-shaped SMA-1 wire in the left finger, and the numbers 3 and 4 represent both sides of the U-shaped SMA-1 wire in the right finger. The SMA-1 wire is then charged by a current source from the two fixed electrodes. In theory, both fingers are supposed to be simultaneously actuated to the position to attain the same offset Z and produce the same grasping force of the soft finger, thereby complying with the force closure condition. However, the noted poor matching might be explained by the failure to comply with the force closure condition, owing to the internal drag of pulleys in the compact gripper. Therefore, a smoother pulley system in the soft gripper, which requires fewer pulleys and larger space to allow the spatial arrangement of the same long wire, is developed to determine the reasoning for this finding. This kind of soft gripper with large space is called as noncompact grippers. Fig. 16(b) depicts a schematic of the noncompact gripper for the comparison of the layout of the SMA-1 wires with the compact one. In the noncompact gripper, the moveable electrodes are connected using an insulating wire with an inexten-

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

269

(1) Failure For For bending stiffness

(2) Success For For bending stiffness

(a)

(b) Fig. 15. (a) Execution of the grasping experiment and (b) results of the grasping force of the compact gripper (H = 20 mm).

sible but adjustable length to adjust the lengths of both sides of the U-shaped SMA-1 wire in each finger for symmetry purposes. In addition, to achieve smooth and simultaneous bending of the soft fingers in the noncompact gripper, we used only four pulleys with ball bearings. Fig. 17(a) shows the experimental setup of the noncompact gripper, which provides a simpler and smoother driving system for bending. If the large deviations shown in Fig. 15(b) can be reduced in the comparison results based on the noncompact gripper with smoother pulleys system, the poor matching could be explained by the failure to comply with the force closure condition, which results from the internal drag of pulleys in the compact gripper. However, the experimental results based on the noncompact gripper, shown in Fig. 17(b), are almost similar to the results elicited from the compact gripper, as shown

270

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

Fig. 16. Schematic of (a) compact and (b) noncompact grippers.

in Fig. 15(b). In a similar manner to prior findings, there is still a large deviation between the experimental and simulation results of the grasping force manifested by the maximum gravity of the grasped object. Consequently, compliance failure of the force closure condition is not the main reason for poor matching. Moreover, the manipulation of the designed soft gripper (compact gripper) meets the force closure condition. Based on these results, the unsuitable friction coefficients used in the simulations may be the reason for poor matching, because the surface of fingertip made of PLA was changed by a large number of experiments during the measurement of the grasping force of soft fingers. In both Figs. 15(b) and 17(b), the simulation values are larger than experimental values because the friction coefficient used for simulations was set to μs = 0.426, which is obtained using the freshly printed fingertip and larger than the real friction values (μr ) in the following experiments for the soft gripper. A series of future experiments on the friction coefficients are expected to investigate the accuracy of the model of the grasping force for the compact soft gripper. Fig. 18 shows a good match between the elicited experimental and simulation results of the soft gripper when the friction coefficients are set to μs = 0.335, [Fig. 18(a)]. In addition, Fig. 18(b) depicts that the experimental results (Fmg-E ) from the soft gripper with a rubbed fingertip surface can match the simulation results (Fmg-S ) considering that the friction coefficients are equal to μs = 0.426. The errors between simulation and experimental results of grasping forces of the soft robotic gripper are defined as eFmg = Fmg-S – Fmg-E . As shown in Fig. 19, the error values lie within 0.04 N, which is much less than the maximum error 0.102 N shown in Fig. 15(b). We can conclude that the soft robotic gripper with a larger friction coefficient at the contact surface has a better grasping capacity, by comparing these results for different friction coefficients. In addition, we can obtain an accurate model to simulate the grasping behavior of soft grippers when the friction coefficients for different fingertip surfaces are identified.

6. Conclusions In this study, a soft robotic gripper with variable stiffness was designed. The developed gripper includes two soft fingers, and each finger was made from bending and varying stiffness SMAs, which are characterized by structral decoulping to realize the bending deflection and variable stiffness, independently. First, the Cosserat theory was used to model the grasping force of the soft finger, which withstood complex forces/torques, such as gravity, bending torques and grasping forces. The grasping force model was nonlinear with no analytical solution. Second, a possible solution was presented to estimate the grasping force of the soft finger. The theoretical and experimental results on the soft finger show that the proposed model can quantitatively predict the kinematic and the static grasping force of the soft finger. Third, the grasping force of the soft finger could be adjusted by changing the Young’s modulus of SMA-2 fiber used in the soft finger, and the maximum grasping force increased by 48.7% (42.5% in simulations) when the Young’s modulus of SMA-2 increased from 25 GPa to 48 GPa.

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

271

Pulleys with ball bearing

Object

(a)

(b) Fig. 17. (a) Experimental setup for grasping measurements and (b) results of the grasping force of the noncompact gripper (H = 20 mm).

Finally, a series of tests on the grasping force of soft robotic grippers were conducted by heating the SMA-1 wire and the SMA-2 fibers to bend the soft fingers and change their stiffness, respectively. The results showed there were obvious deviations between the experimental results and the model predictions in the case of investigating the grasping capacity of the observed gripper. The poor matching was traced back to the choice of an unsuitable friction coefficient in the model rather than a failure of force closure condition due to the gripper construction. Consequently, the designed soft gripper was used to identify the friction coefficients for different fingertip surfaces, and we can conclude that the grasping capacity of the soft robotic gripper can be enhanced when the friction coefficient at the contact surface increased. In addition, the object offset (H) could also influence the grasping force of the soft gripper. In the future, the soft robotic gripper will be optimized and controlled to manipulate heavy objects based on the model of grasping force.

272

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

0.7

E2 = 25 GPa Simulaon E2 = 25 GPa Experiment

0.6

E2 = 37 GPa Simulaon

Fmg (N)

0.5

E2 = 37 GPa Experiment E2 = 48 GPa Simulaon

0.4

E2 = 48 GPa Experiment

0.3 0.2 0.1 0

0

2

4

6

8

10

12

14

16

8 10 ΔZ (mm)

12

14

16

ΔZ (mm)

18

(a)

0.7

E2 = 25 GPa Simulaon E2 = 25 GPa Experiment

0.6

E2 = 37 GPa Simulaon E2 = 37 GPa Experiment

Fmg (N)

0.5

E2 = 48 GPa Simulaon

0.4

E2 = 48 GPa Experiment

0.3 0.2 0.1 0

0

2

4

6

18

(b) Funding

Fig. 18. Simulation and experimental results for grasping forces of the compact gripper with (a) μs = 0.335 and (b) μs = 0.426.

This work was supported by National Natural Science Fundament of China (NSFC) under grants no. 51575409, 51475448 and 51705382.

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

(a)

(b) Fig. 19. Errors between simulation and experimental results for grasping forces of the compact gripper with (a) μs = 0.335 and (b) μs = 0.426.

273

274

Y. Haibin et al. / Mechanism and Machine Theory 128 (2018) 254–274

References [1] L. Zhou, S. Bai, A new approach to design of a lightweight anthropomorphic arm for service applications, J. Mech. Robot. 7 (3) (2015) 0310011. [2] B. Mazzolai, L. Margheri, M. Cianchetti, P. Dario, C. Laschi, Soft-robotic arm inspired by the octopus: II. From artificial requirements to innovative technological solutions, Bioinspir. Biomim. 7 (2) (2012) 025005. [3] D. Rus, M.T. Tolley, Design, fabrication and control of soft robots, Nature 521 (2015) 467. [4] L. Paez, G. Agarwal, J. Paik, Design and analysis of a soft pneumatic actuator with origami shell reinforcement, Soft. Robot. 3 (3) (2016) 109. [5] Y. She, C. Li, J. Cleary, H. Su, Design and fabrication of a soft robotic hand with embedded actuators and sensors, J. Mech. Robot. 7 (2015) 021007-2. [6] J. Li, G. Zhong, H. Yin, M. He, Y. Tan, Z. Li, Position control of a robot finger with variable stiffness actuated by shape memory alloy, in: IEEE. Int. Conf. Robot., 2017, p. 4941. [7] H.I. Kim, M.W. Han, S.H. Song, S.H. Ahn, Soft morphing hand driven by SMA tendon wire, Compos. Part. B. 105 (2016) 138. [8] R.K. Jain, S. Datta, S. Majumder, Design and control of an IPMC artificial muscle finger for micro gripper using EMG signal, Mechatronics 23 (2013) 381–394. [9] J. Shintake, B. Schubert, S. Rosset, H. Shea, D. Floreano, Variable stiffness actuator for soft robotics using dielectric elastomer and low-melting-point alloy, in: IEEE/RSJ Int. Conf. Intell. Robot. Syst., 2015, pp. 1097–1102. [10] J.R. Amend, E. Brown, N. Rodenberg, H.M. Jaeger, H. Lipson, A positive pressure universal gripper based on the jamming of granular material, IEEE. T. Robot. 28 (2) (2012) 341. [11] Y.J. Kim, S. Cheng, S. Kim, K. Iagnemma, A novel layer jamming mechanism with tunable stiffness capability for minimally invasive surgery, IEEE. T. Robot. 29 (4) (2013) 1031. [12] W. Wang, H. Rodrigue, S.H. Ahn, Smart soft composite actuator with shape retention capability using embedded fusible alloy structures, Compos. Part. B. 78 (2015) 507. [13] A. Petterssona, S. Davisb, J.O. Grayb, T. Doddc, T. Ohlsson, Design of a magnetor heological robot gripper for handling of delicate food products with varying shapes, J. Food. Eng. 98 (3) (2010) 332. [14] Y. Yang, Y. Chen, Y. Wei, Y. Li, Novel design and three-dimensional printing of variable stiffness robotic grippers, J. Mech. Robot. 8 (2016) 061010–061011. [15] M. Moallem, J. Lu, Application of shape memory alloy actuators for flexure control: theory and experiments, IEEE-ASME. T. Mech. 10 (5) (2005) 495. [16] H. Yuk, D. Kim, H. Lee, S. Jo, J.H. Shin, Shape memory alloy-based small crawling robots inspired by C. elegans, Bioinsp. Biomim. 6 (4) (2011) 046002. [17] V.R. Kode, M.C. Cavusoglu, Design and characterization of a novel hybrid actuator using shape memory alloy and DC micromotor for minimally invasive surgery applications, IEEE-ASME. T. Mech. 12 (4) (2007) 455. [18] W. Wang, H. Rodrigue, H.I. Kim, M.W. Han, S.H. Ahn, Soft composite hinge actuator and application to compliant robotic gripper, Compos Part B 98 (2016) 397. [19] M.C. Yuen, R.A. Bilodeau, R.K. Kramer, Active variable stiffness fibers for multifunctional robotic fabrics, IEEE Robot. Autom. Let. 1 (2) (2016) 708. [20] C. Liang, C.A. Rogers, Design of shape memory alloy actuators, J. Intell. Mater. Syst. Struct. 8 (4) (1997) 303. [21] J. Li, H. Yin, Y. Tan, A novel variable stiffness soft finger actuated by shape memory alloy, Int. J. Appl. Electrom. 53 (1) (2017) 727. [22] M.W. Hannan, I.D. Walker, Analysis and experiments with an elephant’s trunk robot, Adv. Robotics. 15 (8) (2001) 847. [23] W.S. Rone, P. Ben-Tzvi, Mechanics modeling of multisegment rod-driven continuum robots, J. Mech. Robot. 6 (4) (2014) 041006. [24] T. Mahl, A. Hildebrandt, O. Sawodny, A variable curvature continuum kinematics for kinematic control of the bionic handling assistant, IEEE T. Robot. 30 (4) (2014) 935. [25] D. Trivedi, A. Lotfi, C. Rahn, Geometrically exact models for soft robotic manipulators, IEEE. T. Robot. 24 (4) (2008) 773. [26] D.K. Pail, Strands: interactive simulation of thin solids using cosserat models, Comput. Graph. Forum. 21 (3) (2002) 347. [27] D.C. Rucker, R.J. Webster, Statics and dynamics of continuum robots with general tendon routing and external loading, IEEE T. Robot. 27 (6) (2011) 1033. [28] L.A. Baldelli, C. Maurini, K. Pham, A gradient approach for the macroscopic modeling of superelasticity in softening shape memeory alloys, Int. J. Solid. Struct. 52 (2015) 45–55. [29] C. Li, C.D. Rahn, Design of continuous backbone, Cable-Driven Robots, J. Mech. Design. 124 (2) (2002) 265. [30] S. Dutta, F. Ghorbel, Differential hysteresis modeling of a shape memory alloy wire actuator, IEEE/ASME Trans. Mechatron. 10 (2005) 189–197. [31] J. Li, L. Zu, G. Zhong, M. He, H. Yin, Y. Tan, Stiffness characteristics of soft finger with embedded SMA fibers, Compos. Struct. 160 (2017) 758. [32] A.J. Bryan, L.G. Ricky, T. Krishna, Three dimensional statics for continuum robotics, in: IEEE/RSJ Int. Conf. Intell. Robot. Syst., 2009, pp. 2659–2664. [33] M. Dehghani, S.A.A. Moosavian, Modeling and control of a planar continuum robot, in: IEEE-ASME. Int. Conf. Adv. Int. Mech., 2011, p. 966.