Search trajectory with twisting motion for dual peg-in-hole assembly

Abstract

This paper proposes a search trajectory for a dual peg-in-hole assembly in the presence of uncertainty. To assemble a dual peg into a dual hole, the peg should be guided to the hole by coinciding with both position and orientation. Regarding this requirement, the trajectory of the dual peg is designed by the following new concepts: insertion guarantee region and hole insertion condition. First, the insertion guarantee region is a three-dimensional region based on the clearance between the dual peg and hole, which serves as the goal for the trajectory. Second, the trajectory is designed to satisfy the hole insertion condition, which is a geometric constraint. In this respect, the trajectory has distinct advantages of robust search performance and feasibility of implementation regardless of the shape of the components. The performance of the trajectory was experimentally verified using a 7-degree-of-freedom torque-controlled manipulator, and 100% success rate was reported in the guidance of the peg into the hole.

I

This appendix considers a case of fitting a rectangular peg into a corresponding hole as an extension of the geometric analysis given in the paper above. Firstly, the dimension of the rectangle peg-in-hole assembly will be explained. Then, the insertion guarantee region will be computed as it is in Sect. 2. Figure 15 describes the coordinate system of the single rectangular peg-in-hole. In this figure, O$_H$ and O$_P$ are the centers of the hole and the peg coordinate systems, respectively. $w_H$ and $w_P$ are the width of the peg and the hole, $h_H$ and $h_P$ are the height of the peg and the hole. Figure 16 represents the top view of the single rectangular peg inside the single rectangular hole. The outer rectangle marks the hole and the inner rectangle marks the peg. $\beta$ is the angle of $\angle AO_PB$, which is constant. $\varDelta x$ and $\varDelta y$ are the translations of $O_P$ from the $O_H$.

$$\begin{aligned} \begin{aligned} ||\varDelta x||, \ ||\varDelta y||&\le d \end{aligned} \end{aligned}$$

(20)

where d is the position margin and

$$\begin{aligned} d = \frac{w_H-w_P}{2} = \frac{h_H-h_P}{2} \end{aligned}$$

(21)

Fig. 15

Coordinates and geometric information of a single rectangular peg and a single rectangular hole

Fig. 16

Top view of the rectangular peg inside a rectangular hole

Fig. 17

Illustrations to derive the insertion guarantee region. a The right-top corner contact; b the left-top corner contact; c the left-bottom corner contact; d the right-bottom corner contact

Fig. 18

An example of the insertion guarantee region when $w_H = 60\, \mathrm{mm}$, $h_H = 80\,\mathrm{mm}$, and $d = 0.1\, \mathrm{mm}$ where $w_H$ and $h_H$ are along x-axis and y-axis, respectively

By geometric interpretation, Fig. 17 shows how to compute the angle margin for the insertion guarantee region. $C_i$ means the contact points between the peg and the hole in each case. $\alpha _i$ is the angle of $\angle BO_PC_i$. $\theta _{u_i}$ means the candidates of the yaw angle margin, and each $\theta _{u_i}$ can be computed as

$$\begin{aligned} \theta _{u_i} = {\left\{ \begin{array}{ll} \alpha _i - \beta \qquad \text {when}\ \alpha _i > \beta \\ \beta - \alpha _i \qquad \text {when}\, \alpha _i \le \beta \end{array}\right. } \end{aligned}$$

(22)

For example, $\theta _{u_1}$ is

$$\begin{aligned} \theta _{u_1} = \text {asin}\bigg (\frac{h_H - 2\varDelta y}{2L}\bigg ) - \beta \end{aligned}$$

(23)

where

$$\begin{aligned} L = \sqrt{\bigg (\frac{w_P}{2}\bigg )^2+\bigg (\frac{h_P}{2}\bigg )^2} \end{aligned}$$

(24)

Since this geometric interpretation is same as the analysis given in Sect. 2, the boundary of the allowed angle margin is

$$\begin{aligned} \begin{aligned} \theta _u&= \min (\theta _{u_i}) \\ \theta _l&= \max (\theta _{l_i}) \end{aligned} \end{aligned}$$

(25)

where $i = 1, 2, 3, 4$. Consequently, the insertion guarantee region S is defined as

$$\begin{aligned} S = \{(\varDelta x, \varDelta y, \theta _s)| \ ||\varDelta x||, ||\varDelta y|| \le r_C, \ \theta _l \le \theta _s \le \theta _u\}\nonumber \\ \end{aligned}$$

(26)

where $\theta _s$ is a yaw rotation that allows the assembly task to be possible. Figure 18 shows an example of the insertion guarantee region when $w_H = 60$ mm, $h_H = 80$ mm, and $d = 0.1$ mm.