Design of a Novel Polishing Tool Mechanism with 3-axis Compliance

Abstract

In this paper, a novel polishing tool mechanism with 3-axis compliance is presented, which consists of 2-axis rotational and 1-axis linear compliances in series. The 2-axis rotational compliance mechanism is made up of four cantilever beams for adjusting rotational stiffness and one flexure universal joint at the center for constraining the z-axis deflection. The 2-axis rotational compliance can mechanically adjust the polishing tool to machined surfaces. The polishing press force can be simply controlled by using a linear spring along the z-axis. The 2-axis rotational and 1-axis linear compliance design is decoupled. The stiffness analysis of the 2-axis compliance mechanism was performed based on link compliance matrix and rigid body transformation. A 3-axis polishing tool was designed by configuring the 2-axis compliance mechanism and one linear spring.

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. The direction along the z-axis is the direction in which the polishing tool applies force to the workpiece, and the directions about the x- and 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 f_z, and the reaction moments are defined as n_x, n_y. The 2-axis rotational compliance mechanism is shown in Fig. 2. Additionally, the frame {B_i} is B_i - x_iy_iz_i for i = 1.2.3.4. Fig. 3 [11] is modeled as Euler-Bernoulli beams with the frame ki as the origin. The compliance matrix of a cantilever beam can be obtained by Eq. (1).

\(\begin{align}F _ { k i } = \left[ \begin{array} { c c c c c c } { \frac { L ^ { 3 } } { 3 E I _ { y } } } & { 0 } & { 0 } & { 0 } & { \frac { L ^ { 2 } } { 2 E I _ { y } } } & { 0 } \\ { 0 } & { \frac { L ^ { 3 } } { 3 E I _ { x } } } & { 0 } & { - \frac { L ^ { 2 } } { 2 E I _ { x } } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { \frac { L } { A E } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - \frac { L ^ { 2 } } { 2 E I _ { x } } } & { 0 } & { \frac { L } { E I _ { x } } } & { 0 } & { 0 } \\ { \frac { L ^ { 2 } } { 2 E I _ { y } } } & { 0 } & { 0 } & { 0 } & { \frac { L } { E I _ { y } } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { \frac { L } { G I _ { p } } } \end{array} \right]\end{align}\)        (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

Fig. 3 Elastic model of link ki

where A and L denote link area and length; E and G are the modulus of elasticity and shear modulus; and I_x, I_y and I_p are the area moments of inertia about the x- and y-axes, and the polar area moment of inertia, respectively.

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 L are combined with others in another reference frame G by introducing 6 × 6 rigid body transformations.

\(\begin{align}{ }^{G} E_{L}=\left[\begin{array}{cc}{ }^{G} R_{L}{ }^{G} \hat{\boldsymbol{p}}_{L}{ }^{G} R_{L} \\ 0 \; { }^{G} R_{L}\end{array}\right]\end{align}\)       (2)

where ^GR_L is the rotation matrix in frame G of frame L, \(\begin{align}{ }^{G} \hat{\boldsymbol{p}}_{L}\end{align}\) is the vector in frame G from origin G to origin L expressed as a 3 × 3 skew-symmetric matrix.

Referring to Fig. 4(a), the compliance matrix ^BiF_1i for the i^th cantilever beam in frame {B_i} can be expressed in frame {B} by the following transformation.

\(\begin{align}{ } ^ { B } F _ { 1 i } = { } ^ { B } E _ { B i } { } ^ { B i } F _ { 1 i } { } ^ { B } E _ { B i } ^ { T } \quad \text { for } i = 1,2,3,4\end{align}\)       (3)

Fig. 4 Frame definitions of a 2-axis rotational compliance mechanism

Since the four cantilever beams are connected in parallel to the moving platform, the stiffness matrix by four cantilever beams, ^BK_B = ^BF_B^-1 is obtained by

\(\begin{align}{ }^{B} F_{B}^{-1}=\sum_{i=1}^{4}{ }^{B} F_{1 i}^{-1}\end{align}\)       (4)

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 ^B5jF_2j for the j^th small beam in frame {B_5j} can be expressed in frame {B} by the following transformation.

\(\begin{align}{ } ^ { B } F _ { 2 j } = { } ^ { B } E _ { B 5 j } { } ^ { B 5 j } F _ { 2 j } { } ^ { B } E _ { B 5 j } ^ { T } \quad \text { for } j = 1,2\end{align}\)       (5)

Since two small beams are connected in parallel, the compliance matrix of one rotational axis of a flexure universal joint is obtained by

\(\begin{align}{ }^{B} F_{2}^{-1}=\sum_{j=1}^{2}{ }^{B} F_{2 j}^{-1}\end{align}\)       (6)

Since the two perpendicular rotational axes of a universal joint are connected in series, the resulting compliance matrix of a flexure universal joint, ^BF_U can be obtained by the sum of two compliances.

\(\begin{align}{ } ^ { B } F _ { U } = { } ^ { B } F _ { 2 } + { } ^ { B } R _ { z } { } ^ { B } F _ { 2 } { } ^ { B } R _ { z } ^ { T }\end{align}\)       (7)

where ^BR_z is the rotational matrix about the z-axis by 90°.

Therefore, the stiffness matrix of the 2-axis rotational compliance mechanism can be obtained as follows.

\(\begin{align}{ } ^ { B } K = { } ^ { B } F _ { B } ^ { - 1 } + { } ^ { B } F _ { U } ^ { - 1 }\end{align}\)       (8)

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 ^BK for a 2-axis rotational compliance mechanism is expressed by

\(\begin{align}{ } ^ { B } K = \left[ \begin{array} { c c c c c c } { k _ { 11 } } & { 0 } & { 0 } & { 0 } & { k _ { 15 } } & { 0 } \\ { 0 } & { k _ { 22 } } & { 0 } & { k _ { 24 } } & { 0 } & { 0 } \\ { 0 } & { 0 } & { k _ { 33 } } & { 0 } & { 0 } & { 0 } \\ { 0 } & { k _ { 42 } } & { 0 } & { k _ { 44 } } & { k _ { 45 } } & { 0 } \\ { k _ { 51 } } & { 0 } & { 0 } & { k _ { 54 } } & { k _ { 55 } } & { 0 } \\ { 0 } & { 0 } & { 0 } & { 0 } & { 0 } & { k _ { 66 } } \end{array} \right]\end{align}\)       (9)

In the case of p_ix = p_iy of the position vector, ^Bp_i = [p_ix, p_iy, p_iz]^T in Fig. 4(a), the stiffness matrix becomes k₄₅ = k₅₄ = 0. 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, p_5jz of ^Bp_5j in Fig. 4(b), the off-diagonal elements k₁₅ = k₅₁ and k₂₄ = k₄₂ 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 {B} is at the center of the flexure universal joint and p_ix = p_iy. The design variables for a diagonal stiffness matrix are p_ix = p_iy = 50.5mm and p_5jz = 7.5mm.

Fig. 5 Off-diagonal elements in the stiffness matrix, ^BK according to p_5jz

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 (k_z). Since the 2-axis rotational compliance mechanism is connected in series with the linear spring, the total stiffness can be obtained by

\(\begin{align}{ } ^ { B } K _ { T } = \operatorname { diag } [ k _ { 11 } , k _ { 22 } , k _ { 33 } ^ { \prime } , k _ { 44 } , k _ { 55 } , k _ { 66 } ]\end{align}\)       (10)

Fig. 6 3D modeling of the 3-axis polishing tool

where k^'₃₃^-1 = k₃₃^-1 + k_z^-1. The total compliance matrix (^BF_T = ^BK_T^-1) in frame {B} can be obtained by

\(\begin{align}{ } ^ { B } F _ { T } = \operatorname { diag } [ f _ { 11 } , f _ { 22 } , f _ { 33 } , f _ { 44 } , f _ { 55 } , f _ { 66 } ]\end{align}\)       (11)

The total compliance expressed in frame {B} is shown in Table 1. The total compliance matrix of the 3-axis compliance mechanism in frame {C} at the polishing tool tip can be expressed by

\(\begin{align}{ } ^ { C } F _ { T } = { } ^ { C } E _ { B } { } ^ { B } F _ { T } { } ^ { C } E _ { B } ^ { T }\end{align}\)       (12)

Table 1. Diagonal elements of ^BF_T

The total compliance matrix in frame {C} 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, f₁₁ and f₂₂ are increased due to the distance from {B} to {C} and the rotational compliances of f₄₄ and f₅₅. The proposed polishing tool has finite 2-axis rotational compliance values of f₄₄ and f₅₅ and 1-axis linear compliance value of f₃₃. The other diagonal compliance elements have smaller values as expected.

Table 2. Diagonal elements of ^CF_T

Table 3 shows the error between the analytical calculation and finite element method (FEM) analysis for f₄₄ and f₅₅, and the error is less than 5%.

Table 3. Compliance analysis accuracy

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.

Fig. 7. Prototype of the 3-axis polishing tool

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.