Mechanism design and kinematic analysis of a robotic manipulator driven by joints with two degrees of freedom (DOF)

He Huang (Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, China)
Erbao Dong (Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, China)
Min Xu (Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, China)
Jie Yang (Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei, China)
Kin Huat Low (Nanyang Technological University, Singapore, Singapore)

Industrial Robot

ISSN: 0143-991x

Publication date: 15 January 2018

Abstract

Purpose

This paper aims to introduce a new design concept for robotic manipulator driven by the special two degrees of freedom (DOF) joints. Joint as a basic but essential component of the robotic manipulator is analysed emphatically.

Design/methodology/approach

The proposed robotic manipulator consists of several two-DOF joints and a rotary joint. Each of the two-DOF joints consists of a cylinder pairs driven by two DC motors and a universal joint (U-joint). Both kinematics of the robotic manipulator and the two-DOF joint are analysed. The influence to output ability of the joint in terms of the scale effect of the inclined plane is analysed in ADAMS simulation software. The contrast between the general and the proposed two-DOF joint is also studied. Finally, a physical prototype of the two-DOF joint is developed for experiments.

Findings

The kinematic analysis indicates that the joint can achieve omnidirectional deflection motion at a range of ±50° and the robotic manipulator can reach a similar workspace in comparison to the general robotic manipulator. Based on the kinematic analysis, two special motion modes are proposed to endow the two-DOF joint with better motion capabilities. The contrast simulation results between the general and the proposed two-DOF joints suggest that the proposed joint can perform better in the output ability. The experimental results verify the kinematic analysis and motion ability of the proposed two-DOF joint.

Originality/value

A new design concept of a robotic manipulator has been presented and verified. The complete kinematic analysis of a special two-DOF joint and a seven-DOF robotic manipulator have been resolved and verified. Compared with the general two-DOF joint, the proposed two-DOF joint can perform better in output ability.

Keywords

Citation

Huang, H., Dong, E., Xu, M., Yang, J. and Low, K. (2018), "Mechanism design and kinematic analysis of a robotic manipulator driven by joints with two degrees of freedom (DOF)", Industrial Robot, Vol. 45 No. 1, pp. 34-43. https://doi.org/10.1108/IR-07-2017-0137

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Publisher

:

Emerald Publishing Limited

Copyright © 2018, Emerald Publishing Limited


1. Introduction

No longer limited to the industrial automation (Brogardh, 2007; Dung et al., 2007), robotic manipulators have been extended to include other fields, including the service industry (Kim et al., 2009), medical field (DiMaio et al., 2011), aerospace (Jayaweera and Webb, 2010) and so on. The working environment of robotic manipulators has become more diverse and complex. Therefore, to adapt the robotic manipulators to the new environment, the performance requirements of the robotic manipulators have increased. A great deal of related research work has been conducted.

Joint as a basic but essential component of the robotic manipulator has a decisive influence on the overall performance of the robot manipulator. Therefore, developing a joint with a highly compact mechanical structure and high output ability is crucial.

Universal Robots launched a new robot called UR3 that makes table-top automation safe, easy and flexible. Torque motors and harmonic reducers were used in each of its joints to realize high compactness and high load ability. Rethink Robotics developed a high-performance collaborative Sawyer, which was equipped with series-elastic actuator (Pratt et al., 2002). YuMi, designed by ABB, was the first dual-arm robot purposed-built for application in limited space owing to the minimal and highly integrated design in its joints. Although these designs have made remarkable breakthroughs, all of these robotic manipulators were based on the single-degree of freedom (DOF) joints. Multi-DOF robotic manipulators usually need the same number of joints as the DOFs, leading to the increase in the length and inertia of the robotic manipulator. Thus, more power would be used to support the manipulator itself instead of the payload. Many gravity compensation mechanisms were developed to resist the payload on the actuators caused by the self-weight of the robot (Morita et al., 2003; Kim and Song, 2014; Arakelian, 2016). Nevertheless, these mechanisms were not practical as expected; robotic manipulators usually need several gravity compensation mechanisms, and the whole manipulator would be more complex and bulky, and sometimes the motion ability would also be restricted.

Some researchers attempted to develop a multi-DOF joint used for manipulators to cut down the number of joints and improve the integration level and output ability (Wolf et al., 2003; Memar and Esfahani, 2015). Among them, the two-DOF joint named Powerball ERB developed by SCHUNK showed the best performance. The joint integrated all main parts, and its two output axes were arranged perpendicularly. Based on this joint, they proposed an extremely compact and powerful Powerball lightweight arm LWA 4P, which could realize a weight-to-load capacity ratio of 2:1. In addition, spherical joints and actuators could also provide three-DOF motion; they can rotate around any-direction axis (Okada and Nakamura, 2005; Yan et al., 2006; Yu et al., 2009). However, due to the distribution of the three actuators, they were large in the structure dimension with low stiffness.

Ikeda and Takanashi (1987) early on proposed a novel design concept of a two-DOF joint which was composed of a passive U-joint surrounded by an angular swivel joint (Maity et al., 2009). In this way, the joint can provide a two-DOF motion and be applied in a serpentine robot proposed by Takanashi (1996). The JPL Propulsion Laboratory adopted the concept and also proposed a unique serpentine robot (Paljug et al., 1995). Salomon and Wolf (2012) developed a novel hyper-redundant elephant trunk-like robot based on this two-DOF joint. These robots were compact in mechanical structure; however, they were inappropriate to be used as robotic manipulators because the motion ranges of them were too small, and their stiffness and strength were also reduced because of the backlash introduced by the built-in structure of the U-joint. By increasing the motion range of the joint, some researchers attempted to use the joint to imitate humanoid motion (Ryew and Choi, 2001; Zhang et al., 2011). However, the two driving systems were put on the same side, which caused the whole joint to be so bulky that an extra transmission mechanism was needed to transmit one of the drive’s movements. Shammas et al. (2006) designed a three-DOF joint for a spatial hyper-redundant robot. The joint consisted of a rotary joint and angular bevel gear group instead of an angular swivel joint. This three-DOF joint allowed for a compact and high stiffness design but made the structure more complicated. The power density was also reduced by the distal DOF.

In this paper, we present a new concept of a robotic manipulator driven by joints with two-DOF. The joint itself could act as a reducer with the special series-parallel hybrid mechanical structure. Besides, the proposed mechanism consists of a special mechanical structure in which the cylinder pairs play a role as the endoskeleton, by this means stiffness of the joint is large enhanced. In addition, the proposed mechanism has two special motion modes that the two drives can always work together. This results in the improvement of the output torque over tripled as the general one under the same drive conditions. And the power density is over twice as the general one. Moreover, the proposed mechanism can be easily sealed by bellows for using in some special environment.

The rest of the paper is structured as follows. First, the mechanical structure of the robotic manipulator and two-DOF joint are analysed in Section 2. Kinematic modelling and analysis are presented in Section 3. Simulation of the two-DOF joint is described in Section 4. In Section 5, experimental results are presented and discussed. Finally, conclusions and future work are depicted in Section 6.

2. Mechanism design

2.1 Robotic manipulator design

The mechanical configuration of the robotic manipulator is presented in Figure 1(a), which is composed of three two-DOF joints, a rotary joint and a base. All these joints are connected in series. Its position control is achieved by controlling the angles of the three two-DOF joints. The two-DOF joint can realize the pitch-and-yaw DOF, but the roll DOF in the workspace is limited due to the U-joint. Thus, an additional rotary joint is mounted in the end to realize the roll DOF. Because there exists no rolling motion in the two-DOF joint, the robotic manipulator can be easily sealed by bellows without being twisted and damaged. By this means, the robotic manipulator can be applied in some special environments, such as water-proof and dust prevention environments. In addition, the motion patterns of the proposed two-DOF joint are similar to snake-like and elephant trunk-like robots. So the proposed manipulator is suitable for some special environment with limited operating space, such as in-pipe maintenance and inspection (Zhang and Yan, 2007). Moreover, through research, we also find that the motion of the proposed two-DOF joint are similar to wrist and shoulder joints of human’s arm (Ryew and Choi, 2001; Smith et al., 2012). So we can use the proposed mechanism to design dual-arm manipulator as shown in Figure 1(b).

2.2 Two-degree of freedom joint design

Five versions of the two-DOF joint have been designed as shown in Figure 2. Figure 2(a) and (b) is the earlier versions in which the U-joint is placed inside the two cylinders. However, the built-in way would increase the difficulty of assembly. In addition, the specification of the U-joint is limited by the restricted internal space of the two cylinders. The use of the larger specification of the U-joint could increase the stiffness of the two-DOF joint but make the overall structure more cumbersome; on the contrary, the use of a smaller specification of the U-joint would lower the stiffness. In Figure 2(c) and (d), we place the U-joint outside the two cylinders to increase the stiffness of the joint, but another problem is introduced because the U-joint is too long, leading to a large deflection. As there is no limit to the movement between the two cylinders along the normal direction of the inclined plane, an unavoidable backlash occurs during movement. In our latest version presented in Figure 2(e), these issues have been solved by redesigning the structure of the U-joint and changing its connection mode to the base as well as limiting the movement of the normal direction.

The 3D model of the latest two-DOF joint is shown in Figure 3. The whole mechanical structure of the joint is a series-parallel mechanical structure in which the Cylinders 1 and 2 play a role as the endoskeleton; this mechanical structure can sustain high deflection and twisting torques because the torques generated by self-weight and external payload, and by this means, payload ability of the joint is large enhanced. The joint is symmetrical about the centre of the U-joint. The U-joint is divided into three parts: two yokes and a cross. Two yokes are fixed on two bases. The U-joint is in charge of the kinematics of the joint and is used to connect the joint. Cylinders 1 and 2 are facing opposite to each other along two inclined planes. The angle of the inclined plane is ψ. Two crossed roller bearings are used to connect the two cylinders and resist the force from different directions. Another two crossed roller bearings and a rolling bearing are used between the cylinder and base. Two actuators are fixed on Bases 1 and 2 separately. Cylinders 1 and 2 are driven independently by actuators through external gears. It is necessary for the centres of the U-joint and the two cylinders to be assembled coincidently so that the joint is able to function properly.

As drawn in Figure 3(b), the relative rotation of the two actuators results in an omnidirectional deflection angle of 2ψ between the rotary centrelines of the two cylinders. As can be observed, the rotation of the two cylinders occurs only between the two inclined planes; however, bases are not rotating and acting as the support for the actuators and gears.

3. Kinematic modelling and analysis

The forward kinematics of the robotic manipulator are not straightforward, the output angles of the manipulator are not directly related to the input angles of the actuators, but to the U-joint. The directly kinematic relationship between these angles can be derived from the kinematic analysis of the two-DOF joint.

3.1 Kinematic analysis of the two-degree of freedom joint

First, we decouple the joint into two equivalent mechanisms in kinematics. Then, by using Denavit–Hartenberg (DH) method, we solve the kinematic relationship of these two mechanisms. Finally, by eliminating the intermediate variables, the direct solution can be obtained.

Two mechanisms and the corresponding kinematic models are described in Figure 4. Frames {0}, {1}, {2} and {5} are attached to Base 1, cross, Yoke 2 and Base 2, respectively. θ1 and θ2 are two rotary angles of the U-joint, and l is half length of the joint. The transformation matrix from frame {5} to frame {0} can be expressed as:

(1) T05=Dx(a)Rz(θ1)Rx(π/2)Rz(θ2)Dx(a)

Frames {0′}, {3′}, {4′} and {5′} are intermediate frames attached to Base 1, Cylinder 1, Cylinder 2 and Base 2, respectively. Frames {3} and {4} are attached to Cylinders 1 and 2, respectively. ϕ1 and ϕ2 are the input angles of the two cylinders. For the two cylinders, there only exists relative rotation with variable θ. We obtain the transformation matrix of frame {5′} with respect to frame {0′} as:

(2) 0T5=Dz(l)Rz(ϕ1)Rx(ψ)Rz(θ)Rx(ψ)Rz(ϕ2)Dz(l)

The relationship between the transformation matrix 0′T5′ and 0T5 can be expressed as:

(3) 0T5=Ry(π/2)Rz(π/2)0T5Rx(π)Ry(π/2)

By equating the corresponding items of equation (3), we eliminate the intermediate variable θ. We obtain the directly forward kinematics of the two-DOF joint as:

(4) θ1=2tan1(cψ±cψ2+sψ2sϕ12sψ2sϕ22sψsϕ1+sψsϕ2)θ2=2tan1(cψ±cψ2+sψ2cϕ22sψ2cϕ12sψcϕ1+sψcϕ2)

The directly inverse forward kinematics of the two-DOF joint is:

(5) ϕ1=2tan1(sθ1cθ2±sθ12cθ22+sθ22(1cθ1cθ2)2cot2ψsθ2(1cθ1cθ2)cotψ)ϕ2=2tan1(sθ1±sθ12+cθ12sθ22(1cθ1cθ2)2cot2ψcθ1sθ2+(1cθ1cθ2)cotψ)

Then, the workspace of the two DOF joints can be drawn as Figure 5, which is a spherical crown with the apex angle 4ψ. Due to the spherical workspace of the two-DOF joint, we can use a spherical coordinate system, which are defined in Figure 5, to represent the deflection angle α and orientation angle β (the radius of the sphere is l = 130 mm). It would be used in path planning. The transformation relations are written as:

(6) α=arccos(cosθ1cosθ2)β=arcsin(sinθ2cos2θ1sin2θ1+sin2θ2)θ1=arctan(tanαcosβ)θ2=arcsin(sinαsinβ)

Although the whole workspace of the two-DOF joint is smaller than that of the general two-DOF joint (two rotary joints connected in serial), it is sufficient to generate two-DOF motion without rolling, which is crucial for some special working environments.

Besides, by further studying the kinematic solution of the joint, two special motion modes are found. Mode 1 is when ϕ1 = ϕ2, that is, the two cylinders move in the same direction and with the same rotational angle. This results in that the joint moves in a circle, the deflection angle of the joint stays still only the orientation angle changes. Mode 2 is when ϕ1+ϕ2 = 0, the two cylinders move in the opposite direction and with the same rotational angle. In this case, the joint moves in a plane, the orientation angle of the joint stays still only the deflection angle changes. Moreover, we could use mode 1 to choose a new deflection angle, then the joint could move in a new plane by Mode 2, and vice versa. By combining these two motion modes, the motion in the whole space can be realized. These two modes would be verified in Section 5.

3.2 Kinematic analysis of the robotic manipulator

A closed-form solution for solving the inverse kinematics of the proposed manipulator is proposed. The kinematic model of the robotic manipulator can be simplified as three U-joints with a revolute joint. Hence, we simplify the robotic manipulator as seven revolute joints. As shown in Figure 6, frame {0} is the base frame, each joint frame {i} (i =1…7) is built based on the DH notation, and the DH parameters are shown in Table I. The forward kinematics of the end frame relative to the base frame can be obtained by matrix multiplication as:

(7) T07=T01T12T23T34T45T56T67=[r11r12r13pxr21r22r23pyr31r32r33pz0001]

The workspace of the robotic manipulator is calculated by Monte Carlo method, which is shown in Figure 7. The shape of the workspace is symmetrical around the x and y axes and is similar to part of a sphere.

To solve the inverse kinematics, a desired position and orientation of the end effector relative to the base frame can be expressed as equation (7).

Figure 8 shows the plane of the robotic manipulator with point A at Joint 1, point B at Joint 2, and point C at Joint 3. According to equation (6) and the law of cosine applied to triangle ABC, we obtain:

(8) cos(πα)=cosα=cosθ3cosθ4=a22+a42(x62+y62+z62)2a2a4

The origin of Frame {6} relative to base Frame {0}: 0p6 = [x6 y6 z6 1]T can be calculated from equation (7) as:

(9) [x6y6z61]=T07[00d71]=[x7d7r13y7d7r23z7d7r331]

The origin of Frame {6} relative to Frame {4} is expressed as 4p6 = [a4 0 0 1]T which can be obtained from:

(10) T24p46=(T02)1p06

Moreover, equation (10) can be calculated as:

(11) [a4cosθ3cosθ4+a2a4sinθ4a4sinθ3cosθ41]=[sinθ1cosθ2x6+sinθ2y6+cosθ1cosθ2z6sinθ1sinθ2x6cosθ2y6cosθ1sinθ2z6cosθ1x6+sinθ1z61]

We set θ1 as the redundant variable by considering that it moves most of the inertia of the manipulator and can reduce the total energy by minimizing its angle variation in a movement. Then, substituting equation (8) into equation (11), we get:

(12) θ2=2arctan(y6±y62+(cosθ1z6sinθ1x6)2(a4m+a2)2cosθ1z6sinθ1x6a4ma2)

Finally, the solutions for θ3 and θ4 can be obtained as:

(13) θ3=arctan2(sinθ2y6sinθ1cosθ2x6+cosθ1cosθ2z6a2,cosθ1x6+sinθ1z6)θ4=arcsin((sinθ1sinθ2x6cosθ2y6cosθ1sinθ2z6)/a4)

Now that θ1, θ2, θ3 and θ4 have been computed, 0T4 is known. We can rewrite equation (7) so that the right-hand side is known:

(14) T 4 5 T 5 6 T 6 7 = ( T 0 4 ) 1 T 0 7 = [ n 11 n 12 n 13 n x n 21 n 22 n 23 n y n 31 n 32 n 33 n z 0 0 0 1 ]

By equating the corresponding elements from both sides of equation (14), θ5, θ6 and θ7 can be calculated as:

(15) θ5=arctan2(n33,n13)θ6=arcsin(n23)or πarcsin(n23)θ7=arctan2(n22,n21)

Here, once that the joint position θ1 has been chosen, the remaining six joint angles can be derived analytically.

4. Simulation results

4.1 Scale effect of the inclined plane

As the cylinder pairs, especially the inclined planes, play an important role in realizing the two-DOF motion, it is necessary to analyse the scale effect caused by the inclined planes on the output of the two-DOF joint. The proposed two-DOF joint mechanism illustrated in Figure 2(b) is built in the ADAMS simulation software. The angle of the inclined planes is varying from 10 to 25° at intervals of 5°. Because in Mode 1, the joint stays still, it is not useful to find the difference. So the simulation is only implemented in Mode 2. During simulation, the speed of the two actuators is set as a constant to keep ϕ1 = –ϕ2. Because of the symmetrical structure of the two-DOF joint, we just analyse one of the two actuators.

The result is shown in Figure 9; the input torque and power required of the two actuators increase with the increasing angle ψ. The output velocity and acceleration of the joint also increased with the increasing angle ψ. Thus, we can choose the suitable angle of the inclined plane according to the different output requests.

4.2 Contrast between the general and proposed two-degree of freedom joint

The contrast simulations were conducted between the general and the proposed two-DOF joint to find which one performed better. To ensure the simulation results’ creditability, the general two-DOF joint is directly derived from the proposed two-DOF joint by changing the driven parts from the two cylinders to the U-joint; therefore, the moment of inertia and the rotary centre of these two mechanisms are still the same. Then, we analysed the input torque and power under the same output angle velocity and angle acceleration. The output angle velocity and angle acceleration were obtained when the proposed two-DOF joint moved in Mode 2. Note that the different angle ψ causes different output ability, but they have the same variation trend. So angle ψ is chose as 25° for simulation to compare the two mechanisms. The results are presented in Figure 10. Figure 10(a) and (b) is the output angular velocity and acceleration of these two mechanisms, respectively. Figure 10(c) and (d) is the input torque and power required of these two mechanisms, respectively. We observe that the proposed mechanism can save two-thirds of torque and half the power in contrast to the general mechanism with the same output ability. This also means that the proposed two-DOF joint can realize more payload ability and power density by using the same actuators as the general two-DOF joint.

5. Experimental results

We used the 3D motion capture system (MCS) made by Qualisys (Miqus M3) to study the motion of the joint. By the multiple cameras the MCS measured 3D coordinates of marker which was placed at the end of the joint. The whole experimental setup was shown in Figure 11. We used two Maxon DC motors as the actuators. Each motor was actuated by a Maxon EPOS2 Controller. Moreover, all the EPOS2 controllers were controlled through a host PC with C++.

A control block diagram shown in Figure 12 was proposed to describe the process of the control. Variables q1 and q2 were the actual input of the two motors multiplied by the total reduction ratio.

The experimental results were depicted in Figure 13. In Figure 13(a), the joint was designed to track a space curve whose projection on the XY-plane was a sine wave. In Figure 13(b), first the joint was designed to move in Mode 2 to choose the deflection angle, then the joint was moved in Mode 1 to draw circle, and repeating these movements, different circles were obtained. In Figure 13(c), the joint was first designed to move in Mode 1, and then the joint was moved in Mode 2 to draw curve in a plane, similarly, by repeating these movements, curves in different planes were obtained.

It can be observed that the simulation and experimental results are highly coincident, but with some discrepancy between them. Almost all measured curves had very little vibration in the z-axis direction in comparison to the desired curves, this was mainly caused by error or backlash during the actual machining and assembling process. We could find that the measured curves in multiple cycles were almost coincident. Both of measured and desired curves projected in XY-plane were also highly coincident. The large errors which could be avoided shown in Figure 13(c) between the measured and desired curves in XZ-plane were caused by the process of choosing the deflection angle during Mode 1. The results had verified the feasibility of the control method and shown better accuracy.

6. Conclusions

In this paper, we presented the mechanical design concept of a new seven-DOF robotic manipulator. The manipulator was composed of three two-DOF joints and a rotary joint. Through a decoupling method, the forward and inverse kinematics of the two-DOF joint were solved. Then we solved the forward and inverse kinematics of the manipulator by using a special geometry method. We obtained the workspace of the joint and the manipulator, whose shapes were similar to the general ones. Though the motion range of the joint and manipulator was smaller than the general ones, the motion patterns of the proposed two-DOF joint were similar to snake-like and elephant trunk-like robots. So the proposed manipulator was suitable for using in some special environment with limited operating space, such as in-pipe maintenance and inspection. Moreover, through some researches, we also found that the motion of the proposed two-DOF joint was similar to wrist and shoulder joints of human’s arm. Note that the motion range of joints for human’s arm was also restricted to a certain extent, but that did not matter because two arms worked collaboratively in a limited area. So we could use the proposed mechanism to design a dual-arm manipulator. The joint and manipulator were easy to seal, so it could be used in dustproof and waterproof environment. Then, we focused on analysing the motion ability and performance of two-DOF joint. The scale effect of the inclined plane caused by the varying angle was analysed by ADAMS simulation software. Moreover, contrast simulations in output ability between the general and the proposed two-DOF joints were performed. The results showed that the proposed joint could realise a better capability than the general ones. The trajectory tracking experiments conducted with 3D MCS had shown the high accuracy of the two-DOF joint. The feasibility of two-DOF joint mechanism, control method and kinematic analysis had also been verified.

As for future work, dynamic analysis and load experiments will be conducted. In addition, a prototype of the proposed robotic manipulator would also be made for manipulation experiments to verify its feasibility and practicality.

Figures

3D models of the robotic manipulators

Figure 1

3D models of the robotic manipulators

Five versions of the two-DOF joint

Figure 2

Five versions of the two-DOF joint

3D model of the two-DOF joint

Figure 3

3D model of the two-DOF joint

The two equivalent kinematic paths

Figure 4

The two equivalent kinematic paths

Workspace of the two-DOF joint

Figure 5

Workspace of the two-DOF joint

Kinematic parameters and frame assignments for the robotic manipulator

Figure 6

Kinematic parameters and frame assignments for the robotic manipulator

Workspace of the robotic manipulator

Figure 7

Workspace of the robotic manipulator

The plane of the proposed robotic manipulator

Figure 8

The plane of the proposed robotic manipulator

Scale effect caused by the varying angle of the inclined plane

Figure 9

Scale effect caused by the varying angle of the inclined plane

Contrast between the general and the proposed two-DOF joints

Figure 10

Contrast between the general and the proposed two-DOF joints

Experimental setup

Figure 11

Experimental setup

Control block diagram

Figure 12

Control block diagram

Three different types of space curves obtained in measured and desired

Figure 13

Three different types of space curves obtained in measured and desired

DH parameters of the proposed configuration

i αi−1 αi−1 di θi
1 π/2 0 0 θ1+π/2
2 π/2 0 0 θ2
3 −π/2 α2 0 θ3
4 π/2 0 0 θ4
5 −π/2 α4 0 θ5
6 π/2 0 0 θ6+π/2
7 π/2 0 d7 θ7

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Corresponding author

Erbao Dong can be contacted at: ebdong@ustc.edu.cn