A contribution for increasing workpiece location accuracy in a 3-2-1 fixture system

Abstract

The 3-2-1 fixture design principle is the most used method to locate prismatic workpieces. To further increase location accuracy, the influence of friction and contact stiffness between workpiece and fixture parts should be minimized. This can be achieved by decreasing the structural stiffness of the locators in the tangential direction to the contact. Hence, based on flexure hinges, locators with two different stiffness values are investigated. To analyze the influence of locator stiffness, a special experimental setup is developed to study two locators positioned in the secondary plane of the 3-2-1 fixture. Experiments are carried out to compare the locators with flexure hinges to a rigid one. The displacement and rotation of the workpiece are investigated as a function of the clamping force. The experimental results show that the reduction in the tangential stiffness can improve the position and orientation accuracy of workpieces.

Introduction

In order to increase the positioning accuracy of the 3-2-1 fixture system, several studies have been published in recent decades. The positioning and orientation errors of a workpiece have been modeled as a function of locators’ accuracy by Fallah and Arezoo ¹ and Tang et al. ² In addition, Abedini et al., ³ Nasr et al. ⁴ and Vasundara and Padmanaban ⁵ developed algorithms to determine the best position of locators and clamps to minimize workpiece positioning errors in machining. When machining a workpiece with narrow tolerances is desired, maintaining the workpiece in the same attachment without repositioning is advised. That is because factors such as friction, surface roughness and contact stiffness stop the workpiece from moving to the ideal location. In this regard, kinematic couplings play an important role to locate mechanical components with high accuracy. ⁶ The simplest kinematic coupling consists of three radial V-grooves on one part that match three half spheres on the other part. An interesting work is presented by Li et al., ⁷ in which the improvement of positioning accuracy of kinematic coupling is compared to other types of coupling principles. Therefore, even for these more precise systems, the problem of positioning accuracy exists. Schouten et al. ⁸ pointed out that the positioning accuracy is improved when hysteresis decreases. In the kinematic coupling, the hysteresis $Sv$ can be modeled by the relationship

$$S_v=2·\frac{|F_f|}{k_c}$$

(1)

where $F_f$ is the friction force and $k_c$ is the contact stiffness in the same direction.

Analyzing equation (1), it can be observed that the hysteresis can be reduced by decreasing the frictional force, or rather the coefficient of friction, and/or increasing the contact stiffness. Schouten et al. ⁸ argue that preload variation can lead to a relative movement between the spheres and the grooves. Thus, the same friction force remains. When the surfaces of the V-grooves follow the movement of the spheres, the friction tends to zero, so does the hysteresis. This can be achieved when the V-grooves with elastic hinges are designed. Comparing the V-grooves with elastic hinges with the conventional one, the hysteresis decreased at least 10 times. The same hysteresis phenomenon that occurs in kinematic coupling also occurs in any given fixture systems. However, studies about hysteresis phenomenon in fixture systems have not been addressed. Despite this, the same principle proposed by Schouten et al. ⁸ can be applied to 3-2-1 fixture.

The stiffness model that represents the stiffness due to the contact between the workpiece and the fixture is shown in Figure 1(a). The equivalent stiffness $k_{eq}$ , both in normal and tangential directions, can be calculated based on springs in series as presented in Figure 1(b) and is given by

$$k_{eq}=\frac{1}{\frac{1}{k_{c1}}+\frac{1}{k_l}+\frac{1}{k_{c2}}}$$

(2)

Figure 1.

(a) Stiffness model and (b) the equivalent spring model.

Equation (2) is, therefore, a function of contact stiffness between the workpiece and locator $k_{c1}$ , structural stiffness of locator $k_l$ and contact stiffness between the locator and the base of the fixture system, $k_{c2}$ .

From the above equation, it can be seen that if the one of the stiffness values decreases, the equivalent stiffness also decreases. In the fixture system, this can be achieved by decreasing locator structural stiffness in the tangential direction while the contact stiffness is maintained as high as possible. As a result, the hysteresis decreases and the location accuracy increases.

Experimental setup

Experiments are performed to investigate the influence of the locator stiffness on workpiece position and rotation as a function of clamping forces. The fixture layout used in the tests is shown in Figure 2(a), consisting of a 3-2-1 workpiece location scheme. On the base plate of dimensions 300 mm × 300 mm × 40 mm, locators $L_1$ , $L_2$ and $L_3$ are mounted, forming the primary plane. These locators and clamps $C_1$ and $C_2$ are constructed from 15-mm diameter bearing spheres. Washers and screws are used to hold the spheres in position. In actuators $C_1$ and $C_2$ , the spheres are held using a magnet attached to the rod (Figure 2(c)). To minimize the friction, a thin oil lubricant is applied. Locators $L_4$ and $L_5$ are on the secondary plane, perpendicular to the base. On the tertiary plane, perpendicular to the base and the lateral plane, is locator $L_6$ .

Figure 2.

Experimental setup: (a) the location of the fixture elements, (b) flexible and rigid locator and (c) spheres used as locators.

$L_4$ , $L_5$ and $L_6$ are spherical tipped locators, with an 18-mm radius, 20-mm diameter and 25-mm length. In order to make it interchangeable, an M12 thread is machined on the opposite side of the locator head (Figure 2(b)). They were screwed into columns of dimensions 70 mm × 50 mm × 70 mm, which are fastened to the base plate with four bolts. The coordinates of the locators and actuators are listed in Table 1.

Table 1.

Fixture layout data.

	X (mm)	Y (mm)	Z (mm)
$L_1$	−37.5	−31.5	0
$L_2$	−37.5	31.5	0
$L_3$	37.5	0	0
$L_4$	−37.5	62.5	35
$L_5$	37.5	62.5	35
$L_6$	75	0	35
$C_1$	0	−62.5	35
$C_2$	−75	0	35

Two flexible locators are designed based on a circular notch flexure hinge, one of $2-\mathrm{mm}$ and the other of $4-\mathrm{mm}$ neck thickness. The circular rings have $8-\mathrm{mm}$ radius and their centers are $15\mathrm{mm}$ from top of the locators. They are mounted in $L_4$ and $L_5$ . The locator in $L_6$ is kept rigid. The workpiece used in the tests is a 150 mm × 125 mm × 75 mm block of aluminum, and the locator elements are made of structural steel.

Two pneumatic clamps with an in-line pressure gauge are used to apply up to 375 N. Linear variable differential transformers (LVDTs) with resolution of $0.1\mathrm{\mu }\mathrm{m}$ are positioned 15 mm below locators $L_4$ , $L_5$ and $L_6$ . A data acquisition board model NI USB-6009 and LabVIEW software from National Instruments are used to control the pneumatic clamps and acquire the signal from the LVDTs.

At the beginning of each experiment, the workpiece is in contact with all locators. Then, actuator $C_1$ gradually applies the clamping force up to 375 N. After that, $C_2$ begins to apply force ranging from 10 to 320 N. The experiments to determine workpiece position and rotation errors are performed by repeating the loading and unloading cycles under the same conditions. The measured results are assumed to be normally distributed, and the errors are computed within the limits of the distribution with a 95% confidence interval.

Experimental results

Figure 3 shows the experimental results of workpiece displacement and rotation as a function of clamping force $C_2$ for the three types of locators.

Figure 3.

Workpiece displacement and rotation on X-Y plane versus clamping force $C_2$ .

Considering the workpiece displacement in X-direction, the experimental results show the same tendency of ${\delta }_x$ for the three locator types. It can be observed that as $C_2$ increases, ${\delta }_x$ also increases, tending to a maximum value. As expected, for a given clamping force, workpiece displacement gets larger as the tangential stiffness of locators $L_4$ and $L_5$ decreases. This means that the normal force on locator $L_6$ increases and the workpiece displacement tends to the Hertzian approximation for normal contact deformation (dashed line in Figure 3).

In relation to the workpiece position in the Y-direction ${\delta }_y$ and the rotation around Z-axis ${\theta }_z$ , as $C_2$ increases, both decrease linearly. The stiffness reduction in the locators tends to improve the results in both directions. This occurs because the reaction force on $L_6$ increases, restricting workpiece movement even more.

Figure 4 presents workpiece position and rotation errors as a function of locator type. The position errors in the X-direction decrease from about $3.4\mathrm{\mu }\mathrm{m}$ for the rigid locator to $1.6\mathrm{\mu }\mathrm{m}$ for the $4-\mathrm{mm}$ neck-thickness locator, reaching $1.0\mathrm{\mu }\mathrm{m}$ for the $2-\mathrm{mm}$ neck-thickness locator. This means a significant raise in workpiece position accuracy when locator stiffness decreases. In the Y-direction, the smallest error is about $2.0\mathrm{\mu }\mathrm{m}$ , obtained with the $2-\mathrm{mm}$ neck-thickness locator, while with $4-\mathrm{mm}$ neck-thickness locator and with the rigid locator, the errors are $3.7$ and $3.6\mathrm{\mu }\mathrm{m}$ , respectively.

Figure 4.

Workpiece positioning and orientation errors as a function of locator type.

With respect to workpiece rotation, the result is the opposite. The rigid locator produces the smallest rotation error, about $0.2\mathrm{\mu }\mathrm{rad}$ , while for the $4-\mathrm{mm}$ neck-thickness locator the error increases to $0.3\mathrm{\mu }\mathrm{rad}$ and for $2-\mathrm{mm}$ neck-thickness locator, the error reaches $0.6\mathrm{\mu }\mathrm{rad}$ .

Conclusion

Fixture systems are devices used to repeatedly and accurately locate workpieces. However, friction and contact stiffness affect the position and orientation of the workpieces, leading to errors. In this work, locators are designed based on the principle of flexible elements and used in order to increase workpiece location accuracy.

Two flexible locators are positioned in the secondary plane of an especially designed 3-2-1 fixture. Locators with neck thickness of $2$ and $4\mathrm{mm}$ are compared to a rigid one. From the experimental results, the structural stiffness of the locators is observed to have a significant influence on the final position and orientation of the workpiece. The lower the locator structural stiffness, the lower the workpiece displacement in the direction of the flexible locators and workpiece rotation on the X-Y plane. However, in the direction of the fixed locator, workpiece displacement increases. Regarding workpiece position and orientation variations, it is observed that when the locator stiffness decreases, the workpiece position error in both directions decreases as well, but the rotation error increases.

The best result for the positioning accuracy is achieved for the $2-\mathrm{mm}$ neck-thickness locators. Compared to the rigid ones, the positioning error decreases from $3.4$ to $1.0\mathrm{\mu }\mathrm{m}$ in the X-direction, while in the Y-direction, the error decreases from $3.6$ to $2.0\mathrm{\mu }\mathrm{m}$ . However, the rigid locator shows better results for the rotation error, $0.2\mathrm{\mu }\mathrm{rad}$ against $0.6\mathrm{\mu }\mathrm{rad}$ for the $2-\mathrm{mm}$ neck-thickness locators.