1. Introduction
Polishing refers to as the process of smoothing or giving gloss to the surface of a processed product. It is mainly used for trimming and finishing various products such as metal, plastic, and glass. To ensure accurate and precise work, polishing tasks require skilled workers [1]. Polishing tasks are repetitive and require only a small number of skilled workers due to the high intensity of the work. Additionally, it is difficult for even skilled workers to work for long periods of time [2]. Recently, polishing tasks using robots become important, and research has been conducted to apply a constant force by providing polishing tools with compliance similar to that of workers [3]. In order to increase the precision of polishing, research has been conducted on position/force control of robots based on force sensors [4] and force control considering the rigidity of the robot [5]. In addition, research was also conducted on simultaneous position/force control using AI to create machining paths [6]. In previous studies, the position/force control was performed by using a 6-axis compliance device [7-10].
In this paper, a polishing tool mechanism with 3-axis compliance is presented, which consists of 2-axis rotational and 1-axis linear compliances in series. First, the stiffness of the 2-axis rotational compliance mechanism is derived in an analytical manner. Second, the stiffness matrix becomes diagonal by
changing design parameters. Finally, the prototype of the polishing tool with 3-axis compliance is designed by simply adding linear springs to the 2-axis rotational compliance mechanism in series.
2. Stiffness Analysis of a 2-axis Rotational Compliance Mechanism
This chapter presents the stiffness analysis of a 2-axis rotational compliance mechanism. First, the working direction of the polishing tool is defined as shown in Fig. 1.
Fig. 1 Definition of polishing tasks
Fig. 2 Conceptual design of a 2-axis rotational compliance mechanism with four cantilever beams and one flexure universal joint
The direction along the z-axis is the direction in which the polishing tool applies force to the workpiece, and the directions about the xand y-axes are the directions of rotational compliance for the polishing tool to be in close contact with the workpiece. The polishing press force is defined as , and the reaction moments are defined as , . The 2-axis rotational compliance mechanism is shown in Fig. 2. Additionally, the frame is for . Fig. 3 [11] is modeled as Euler-Bernoulli beams with the frame as the origin. The compliance matrix of a cantilever beam can be obtained by Eq. (1).
\(F _ { k i } = \left[ \begin{array} { c c c c c c } { \frac { L ^ { 3 } } { 3 E I _ { y } } } & { 0 } & { 0 } & { 0 } & { \frac { L ^ { 2 } } { 2 E I _ { y } } } & { 0 } \\ { 0 } & { \frac { L ^ { 3 } } { 3 E I _ { x } } } & { 0 } & { - \frac { L ^ { 2 } } { 2 E I _ { x } } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { \frac { L } { A E } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - \frac { L ^ { 2 } } { 2 E I _ { x } } } & { 0 } & { \frac { L } { E I _ { x } } } & { 0 } & { 0 } \\ { \frac { L ^ { 2 } } { 2 E I _ { y } } } & { 0 } & { 0 } & { 0 } & { \frac { L } { E I _ { y } } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { \frac { L } { G I _ { p } } } \end{array} \right]\)
where and denote link area and length; and are the modulus of elasticity and shear modulus; and , and are the area moments of inertia about the x- and y-axes, and the polar area moment of inertia, respectively.
Fig. 3 Elastic model of link
The structure of four cantilever beams in Fig. 2 has linear compliance along the z-axis as well as rotational compliance, according to Eq. (1). By employing a flexure universal joint at the center, the linear compliance along the z-axis can be almost eliminated. The resulting structure has only 2-axis rotational compliance about the x- and y-axes.
Quantities specified in a local frame are combined with others in another reference frame by introducing × rigid body transformations.
\({ } ^ { G } E _ { L } = \left[ \begin{array} { c c } { { } ^ { G } R _ { L } { } ^ { G } \hat { p } _ { L } { } ^ { G } R _ { L } } \\ { 0 } & { { } ^ { G } R _ { L } } \end{array} \right]\)
where is the rotation matrix in frame of frame , is the vector in frame from origin to origin expressed as a × skew-symmetric matrix.
Referring to Fig. 4(a), the compliance matrix for the cantilever beam in frame can be expressed in frame by the following transformation.
\({ } ^ { B } F _ { 1 i } = { } ^ { B } E _ { B i } { } ^ { B i } F _ { 1 i } { } ^ { B } E _ { B i } ^ { T } \quad \text { for } i = 1,2,3,4\)
Since the four cantilever beams are connected in parallel to the moving platform, the stiffness matrix by four cantilever beams, is obtained by
\({ } ^ { B } F _ { B } ^ { - 1 } = \sum _ { i = 1 } ^ { 4 } { } ^ { B } F _ { 1 i } ^ { - 1 }\)
The one rotational axis of a flexure universal joint can be modeled as two small cantilever beams as shown in Fig. 4(b). The compliance matrix for the small beam in frame can be expressed in frame by the following transformation.
\({ } ^ { B } F _ { 2 j } = { } ^ { B } E _ { B 5 j } { } ^ { B 5 j } F _ { 2 j } { } ^ { B } E _ { B 5 j } ^ { T } \quad \text { for } j = 1,2\)
Since two small beams are connected in parallel, the compliance matrix of one rotational axis of a flexure universal joint is obtained by
Fig. 4 Frame definitions of a 2-axis rotational compliance mechanism
\({ } ^ { B } F _ { 2 } ^ { - 1 } = \sum _ { j = 1 } ^ { 2 } { } ^ { B } F _ { 2 j } ^ { - 1 }\)
Since the two perpendicular rotational axes of a universal joint are connected in series, the resulting compliance matrix of a flexure universal joint, can be obtained by the sum of two compliances.
\({ } ^ { B } F _ { U } = { } ^ { B } F _ { 2 } + { } ^ { B } R _ { z } { } ^ { B } F _ { 2 } { } ^ { B } R _ { z } ^ { T }\)
where is the rotational matrix about the z-axis by 90o.
Therefore, the stiffness matrix of the 2-axis rotational compliance mechanism can be obtained as follows.
\({ } ^ { B } K = { } ^ { B } F _ { B } ^ { - 1 } + { } ^ { B } F _ { U } ^ { - 1 }\)
3. Design of a 3-axis Polishing Tool
In this chapter, a polishing tool with 3-axis compliance was designed based on the stiffness analysis in the previous chapter. The 3-axis compliance can be designed by the 2-axis rotational compliance mechanism and the 1-axis linear compliance with compression springs.
The stiffness matrix for a 2-axis rotational compliance mechanism is expressed by
\({ } ^ { B } K = \left[ \begin{array} { c c c c c c } { k _ { 11 } } & { 0 } & { 0 } & { 0 } & { k _ { 15 } } & { 0 } \\ { 0 } & { k _ { 22 } } & { 0 } & { k _ { 24 } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { k _ { 33 } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { k _ { 42 } } & { 0 } & { k _ { 44 } } & { k _ { 45 } } & { 0 } \\ { k _ { 51 } } & { 0 } & { 0 } & { k _ { 54 } } & { k _ { 55 } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { k _ { 66 } } \end{array} \right]\)
In the case of of the position vector, in Fig. 4(a), the stiffness matrix becomes . For numerical calculation, the four cantilever beams (material: SUS304) are selected with 52 mm length, 15 mm width, and 1.5 mm thickness. The overall cantilever beam dimensions are designed considering 100×100mm polishing tool size. Depending on the z coordinate, of in Fig. 4(b), the off-diagonal elements and can converge to 0. The z coordinate indicates the center of the universal joint. Therefore, the stiffness matrix become a diagonal matrix when the frame is at the center of the flexure universal joint and . The design variables for a
Fig. 5 Off-diagonal elements in the stiffness matrix, according to
Fig. 6 3D modeling of the 3-axis polishing tool
diagonal stiffness matrix are mm and mm.
In Fig. 6, 3D modeling for the 3-axis polishing tool is presented. In addition, it is possible to attach a displacement sensor along the compression springs and to calculate polishing press force by spring constant (). Since the 2-axis rotational compliance mechanism is connected in series with the linear spring, the total stiffness can be obtained by
\({ } ^ { B } K _ { T } = \operatorname { diag } [ k _ { 11 } , k _ { 22 } , k _ { 33 } ^ { \prime } , k _ { 44 } , k _ { 55 } , k _ { 66 } ]\)
where ′ . The total compliance matrix ( ) in frame can be obtained by
\({ } ^ { B } F _ { T } = \operatorname { diag } [ f _ { 11 } , f _ { 22 } , f _ { 33 } , f _ { 44 } , f _ { 55 } , f _ { 66 } ]\)
Table 1. Diagonal elements of
The total compliance expressed in frame is shown in Table 1. The total compliance matrix of the 3-axis compliance mechanism in frame at the polishing tool tip can be expressed by
\({ } ^ { C } F _ { T } = { } ^ { C } E _ { B } { } ^ { B } F _ { T } { } ^ { C } E _ { B } ^ { T }\)
The total compliance matrix in frame has off-diagonal elements. Off-diagonal elements are much smaller than diagonal ones. In Table 2, only diagonal elements of the total compliance matrix at the tool tip are presented. Comparing Table 2 with Table 1, and are increased due to the distance from to and the rotational compliances of and . The proposed polishing tool has finite 2-axis rotational
Table 2. Diagonal elements of
Table 3. Compliance analysis accuracy
Fig. 7. Prototype of the 3-axis polishing tool
compliance values of and and 1-axis linear compliance value of . The other diagonal compliance elements have smaller values as expected.
Table 3 shows the error between the analytical calculation and finite element method (FEM) analysis for and , and the error is less than 5%.
Based on the design method, a prototype of the 3-axis polishing tool is developed and attached at the UR10e robot as shown in Fig. 7.
4. Conclusions
In this paper, a novel 3-axis compliance mechanism is proposed for a robotic polishing tool. It is noted that the 2-axis rotational and 1-axis linear compliance design is decoupled. The analytical stiffness analysis for the 2-axis rotational compliance mechanism
is performed and verified through FEM analysis. The prototype of the polishing tool with 3-axis compliance is developed. In future research, precision polishing control based on the force/moment measurements will be conducted.
Acknowledgements
This research was supported by "Regional innovation Strategy (RIS)" through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(MOE) (2021RIS-003)
References
- Nobuaki E. et al, "Application of Model Predictive Control to Polishing Robot for Pushing Operation," ICCAS, pp. 518-522, (2022).
- Yuxin D. et al, "A Review of Robot Grinding and Polishing Force Control Mode," IEEE ICMA, pp. 1416-1418, (2022).
- Zhiyun H. et al, "Design of a Novel Force Controlled Polishing Device Based on a Coupled Active-Passive Compliant Mechanism," IEEE ICMA, pp. 986-991, (2023).
- Chuntao Z. et al, "Design of an Industrial Robot Polishing Control System," IEEE ICMRA, pp. 14-18, (2022).
- Hafidz M. D. and Hsien-I L. "Improvement of Force Performance in Robotic Polishing Tasks with Position Compensation," ICSSE, pp. 412-417, (2023).
- Srinivasan L. et al, "An adaptive framework for robotic polishing based on impedance control," IJAMT, pp. 401-417, (2021).
- Kim H. S. "Design of a 6-axis Compliance Device with F/T sensing for Position/Force Control," KSIC, vol. 21, no. 2, pp. 63-70, (2018).
- Kim H. S. "Kinestatic Control using Six-axis Parallel-type Compliant Device," KSMTE, vol. 23, no. 5, pp. 421-427, (2014). https://doi.org/10.7735/ksmte.2014.23.5.421
- Park S. B. and Kim H. S. "Position/force Control using 6-axis Compliance Device for Chemical Coupler Assembly," KSIC , vol. 25, no. 5, pp. 909-915, (2022).
- Kim G. S., Jeong S. H., Kim H. S. "Study on the Stability of Force Control using a 6-axis Compliance Device with F/T Sensing," KSIC, vol. 26, no. 1, pp. 211-215, (2023).
- Kim H. S. and Lipkin H. "Stiffness of Parallel Manipulators With Serially Connected Legs," Transactions or the ASME, vol. 6, pp. 1-9, (2014). https://doi.org/10.1115/1.4026333