Design and Modeling of Series-Parallel Compliant Device for Reliable Assembly Under Position/Angle Deviation

: Automatic assembly using manipulator has attracted increasing attention due to low cost and high quality of assembly. As the manipulator


Introduction
In recent years, automatic assembly using manipulator has a significant impact on the cost and quality of products [1][2]. A typical shaft-hole assembly using a manipulator is shown in Fig.1. As the manipulator is entirely rigid, it often causes assembly failure and even damages the manipulator when there is a relative error on positive or angle between shaft and hole. For example, when there are position deviation and angle deviation, as shown in Fig. 2, a big contact force will occur, which can damage the shaft or hole and result in assembly failure.
Compliance is vital in assembly for protecting parts and compensating for the misalignment between them. There are two methods to increase system compliance. One is the active compliant strategy, which uses force feedback to identify the misalignment and compensates the positioning error using the feedback control. In the active compliance strategy, many control strategies have been presented to improve the assembly performance, including the stiffness control [3][4], the impedance control [5][6][7], and the force/position hybrid control [8][9][10][11][12]. Although the active compliance strategy has gained many successful applications, its complex control algorithm is often challenging to implement. Besides, it is known that a good design can achieve a good performance only using a poor control strategy. However, a poor design cannot achieve a satisfactory performance even if using an advanced control strategy. Thus, in many cases, these control methods are challenging to achieve a satisfactory performance due to a poor design. Another advisable method for assembly is the passive compliance strategy, which adds the compliance element in the manipulator from the perspective of design and hardware [13][14]. For example, various flexible grippers were designed to adapt to different objects [15][16]. Also, a specific mechanical structure was designed to produce the compliance to achieve variable contact force or reduce contact force during assembly. Chen [16] presented the remote centre compensation (RCC) structure to achieve robust assembly. Spatial remote centre compliance with an additional axial rotation was designed for accommodating the prismatic (non-axial symmetric) peg components [17]. Haskiya [18] [19] presented a chamfer-less vertical, horizontal remote centre compliance to reduce the insertion force and avoid the jamming conditions in the dynamic insertion of no-chamfer peg-in-hole assembly. Massimo [20] proposed variable remote centre compliance that could change the centre of compliance according to insertion depth. Kim [21] presented a variable passive compliance device for assembly. Kronander [22] and Liu [23] used the spring mechanism and the compliant linkage mechanism for assembly. Xing [24] used a multiple-compliant degreeof-freedoms (DOFs) mechanism (i.e., a spring) to facilitate compliant insertion in assembly. Chen [25] introduced a pneumatic mechanism to produce certain compliance. S.M [26] developed a human-robot hybrid cell for flexible assembly in manufacturing through the collaboration between a human and a robot, though most of these designs have many successful applications.
However, their compliance centre is fixed and difficult to adjust, which causes that the adaptability and versatility of the operational requirements for different stiffness are poor. In addition, they cannot assemble the shaft and hole when there is a big relative error in position or angle between shaft and hole.
Here, a series-parallel passive compliance device is designed to assemble the shaft and hole under a significant relative error on position or angle. In this mechanism, the deformation of eight elastic limbs is used to transmit the force and realize the spatial freedom of the compliance device. Unlike the traditional parallel mechanism that transmits motion through rigid hinges between the fixed and moving platforms, this mechanism is a monolithic mechanism that avoids assembly errors and the gap between the moving pairs on the dynamic characteristics. On this basis, a deformation model and a stiffness model are established to describe the compliance in different directions, and an optimization method is developed to realize the parameters of the mechanism. The shaft-hole assembly experiments are carried out, which demonstrates that the designed device has reliable assembly performance even if there is position deviation or angle deviation.

Description of design
A series-parallel compliant device is designed to realize the reliable assembly under the position deviation or the angle deviation and avoid producing a considerable contact force, as shown in Fig.3, which is used to connect the manipulator and the tool (e.g. gripper). Its core idea is that when the contact force exceeds a particular value, this device becomes compliant and can move in each direction. This guarantees that compliant assembly allows a relatively significant misalignment and produces a small force, protecting parts and manipulator. This device has two compliant components, and each component consists of the rigid frame, the four elastic limbs with a similar 'n' shape, and the square block. One end of each elastic limb is fixed on the fixed, rigid frame, and the other end is fixed on the square block placed at the centre of the component. These two compliant components are connected using a rigid block. Due to the use of elastic material, each elastic limb is equivalent to a compliant hinge (or spring), and this device is equivalent to a series-parallel compliant structure, as shown in Fig.3(b) and Fig.3(c). In this way, this device becomes compliant and can move in each direction when the contact force exceeds a particular value. As an example, when there is a considerable force vector respectively from the x, y, z-axis, the compliant component will produce deformation as shown in Fig.4(a), (b), and (c), respectively. Also, it can produce rotational deformation around the x, y, z-axis when there is a significant torque vector, as shown in Fig.4(d), (e), and (f). For this device, when there is a prominent force or moment, the deformation will be produced to effectively offset the contact force, shown in Fig.5. Thus, this new device can effectively guarantee reliable assembly under the position deviation or the angle deviation without prominent contact force. For example, as shown in Fig.6, when there is a position deviation during assembly, the big contact load will be converted to the deflection of compliant components, making the assembly process more accessible.

Analysis and optimization
This section mainly focuses on the kinematics modeling and stiffness analysis. On this basis, the structure of this designed device is optimized.

A. Kinematic analysis
Here, p r and p l is the length of A AP and AM PP , as shown in Here, s and c denote the trigonometric function sin and cos ; Here, MN is the length of connection block.
Here, p r and p l is the length of B BP and BN PP , as shown in Fig.  7.
Here, l is the length of F BP . According to the relation of these coordinates, the position TF of the end point PF can be derived:

B. Stiffness analysis
The compliant behavior of the compliant device depends on the stiffness of the elastic limb. The equivalent model of the compliance device is shown as Fig. 8. According to the classical Euler-Bernoulli beam theory, the forces and moments applied at the beam can be related to each other employing a compliance matrix [27]. Based on this theory, as shown in Fig.8(a), each elastic limb consists of two beams, and the stiffness matrix of a beam can be derived as follows.
Here, E , G , A and L denote the Young's modulus, the shear modulus, the cross-sectional area and the length of each limb, respectively; x I , y I and z I are the moments of the moments around the x-axis , y-axis and z-axis, respectively, and satisfies, Here, ie R and , ei P are the rotation matrix and the antisymmetric matrix respectively and satisfy , 1 0 0 0 0 = 0 1 0 The stiffness matrix , ei k of each elastic limb is the following, The rotation matrix ,  ( 1)) 22 Here, i is the number of elastic limbs.
Substituting Eq.(16) into Eq. (17), the compliance matrix C of the whole compliant device in the coordinate system o xyz  can be derived.
Here, 1 C is the translational compliance along x or y, 2 C is the translational compliance along z， 3 C is the rotational compliance around x or y, 4 C is the rotational compliance around z. From Eq. (18)

C. Parameter optimization
In order to guarantee the desired compliance of this device in each direction, the following objective function is constructed to optimize the geometry parameters of compliant components. , , , ,  are the desired range. The NSGA-II multi-objective optimization algorithm [28] is used to solve the above multi-objective optimization problem and the desired compliances can be achieved.

Experiment verification
In this design, the TPU material is used to manufacture the compliant component. The shrinkage, elastic modulus and Poisson's ratio of this material are 0.4% ~ 0.9%, 0.2Gpa and 0.394. According to the practical requirement, the desired compliance and movement range are set as follows, By solving the optimization problem (20) and (21), the structure parameters of the elastic limb are obtained and shown in Table I, and the practical compliant device and its shape parameters is illustrated in Fig.9. (a) (b) Fig. 9. (a) Overall dimensions; (b) the internal structure. Then, this designed compliant device is applied to connect the manipulator and the gripper (the tool). A JAKA ZU7S 6-DOF manipulator with the Advantech 610L PC and a force/Torque sensor (ATI Axia80) is used as the manipulator. The whole robot system is shown in Fig. 10.

D. Experimental verification of compliant device
The following tensile and torsional test experiments on a stretcher, as indicated in Fig.11, are conducted to test the compliant device's stiffness,  Horizontal stretching experiment: the force 20 FN  respectively along x and y is used to stretch the compliant device horizontally. The relation of force and the deformation along x and y as well as the model (19) are shown in the Fig.12;  Vertical stretching experiment: the force 20 FN  along z is used to stretch the compliant device vertically. The relation of force and the deformation along z as well as the model (19) are shown in the Fig.13;  Torsion experiment: The torque 2 T N mm  around x, y and z respectively , which is used to test the torsional deformation. The relation of torque and deformation around x, y and z as well as the model (19) are shown in the Fig.14 and 15, respectively. From Fig.12-15, the established stiffness model and the experimental results fit very well. This demonstrates the effectiveness of this model with the measured deformationforce data. The actual stiffness of compliant device along x, y, z and around x, y, z are 1

E. Assembly Experiment
Using Eq. (9), the workspace of the compliant device is calculated and shown in Fig.15. Besides, based on the practical workspace in Fig. 16, the comparison is carried out to verify the effectiveness of this compliance device. It can be seen from the figures that this design can satisfy the desired movement range in Eq. (21). Then, two kinds of assembly experiments, the shaft-hole assembly and the nut-stud assembly, are used to further verify the compliant performance of this device.
For the shaft-hole assembly experiments, four experiments, as shown in Fig.18, are conducted under different conditions: (a) verticality between shaft and hole; (b) inclination angle between shaft and hole equal to 5 ; (c) inclination angle equal to 10 ; (d) inclination angle equal to 15 . The experimental results are shown in Fig.19, from which assembly is very well as using this compliant device even if there is a big inclination angle between shaft and hole. Moreover, contact force from the shaft along x-axis is measured and shown in Fig.20. From these figures, there has a big contact force at the whole assembly process as there is without this designed compliant device and under inclination angle equal to 15 . However, as indicated in Figs. 20(b-d), there has a small contact force with short action time as using this designed compliant device. Thus, this designed compliant device can effectively reduce the contact force and its action time. Furthermore, the nut-stud assembly is used to verify the effectiveness of the designed compliant device. When it directly assembles nut and stud using the manipulator without the designed compliant device, this assembly cannot realize as the nut and stud are not aligned. As comparison, when using the manipulator with the designed compliant device, three experiments for the unaligned nutstud assembly are conducted under the inclination angle equal to 5 ,10 and 15 respectively, as shown in Fig.21. As an example, the process of unscrewing nub into stud under inclination angle equal to 15 is shown in Fig.22. From these figures, even if there is a big unaligned error, the nutstud can be well assembled with the designed compliant device. (d) Fig.22. Process of unscrewing nub into stud under inclination angle equal to 15 .

Conclusions
In conclusion, a novel compliant device with an adjusted compliant centre was briefly developed to realize the reliable assembly under the position deviation or the angle deviation and not produce a significant contact force. This designed device becomes compliant and can move in a particular direction when the contact force exceeds a particular value. Unlike the traditional parallel mechanism, it can avoid the influence of assembly errors and the gap between the moving pairs. Experiments on this device validate the effectiveness of the kinematic model and stiffness model of the compliant device and the correction of the workspace. These experiments further demonstrate that the designed device satisfies the desired design conditions. Besides, experiments on the shaft-hole assembly show a small contact force with short action time when using this designed compliant device even if there is a big inclination angle. Thus, this designed compliant device can effectively realize the reliable assembly under the position or angle deviation and avoid producing a considerable contact force. Also, even if there is a significant unaligned error, it can still effectively assemble the nut-stud with the designed compliant device.

Declarations Funding
This work was partially supported by the National Key R & R&D Program of China (2018YFB1308202), National Natural Science Foundation of China (51675539), and the Hunan Provincial Science Fund Distinguished Young Scholars under Grant 2019JJ20030.

Conflicts of interest/Competing interests
The authors declare that they have no competing interests.

Availability of data and material
Not applicable.

Code availability
Not applicable.

Authors' contributions
Du Xu provides design and experiment tests. He is a major contributor in writing the manuscript, Pro. Lu provided valuable suggestions for the revision and sorting of the whole article. All authors read and approved the final manuscript.

Ethics approval
Not applicable Consent to participate Not applicable.

Consent for publication
Not applicable. Figure 1 Shaft-hole assembly using manipulator           Tensile/torsion test.

Figure 12
Relation of force and deformation along x and y Figure 13 Relation of force and deformation along z Figure 14 Relation of torque and deformation around x and y   Shaft-hole assembly experiments: (a) verticality between shaft and hole; (b) inclination angle equal to 5º ; (c) inclination angle equal to 10º ; (d) inclination angle equal to 15º .

Figure 20
Contact force from the shaft along x-axis: (a) assembly without the compliant device under inclination angle equal to 15º ; (b) assembly with the compliant device under inclination angle equal to 5º ; (c) inclination angle equal to 10º ; (d) inclination angle equal to 15º .

Figure 22
Process of unscrewing nub into stud under inclination angle equal to 15º .