A Variational Method of Robust Fixture Configuration Design for 3-D Workpieces

Wayne Cai*

S.Jack Hu

J. X. Yuan

Department of Mechanical Engineering and Applied Mechanics,

The University of Michigan, Ann Arbor, Ml 48109

A Variational Method of Robust Fixture Configuration Design for 3-D Workpieces Fixtures are used to locate and hold workpieces during manufacturing. Because workpiece surface errors and fixture set-up errors (called source errors) always exist, thefixtured workpiece will consequently have position and/or orientation errors (called resultant errors). In this paper, we develop a variational method for robust fixture configuration design to minimize workpiece resultant errors due to source errors. We utilize both first-order and second-order workpiece geometry information to deal with two types of source errors, i.e., infinitesimal errors and small errors. Using the proposed variational approach, other fundamental fixture design issues, such as deterministic locating and total fixturing, can be regarded as integral parts of the robust design. Closed-form analytical solutions are derived and numerical examples are shown. By employing the nonlinear programming technique, simulation software called RFixDesign is developed. This paper presents a new procedure for robust fixture configuration design that contributes especially to fixture designs where deformation is not influential.

1 Introduction Fixtures are used to locate and hold workpieces during manu

facturing operations, such as machining, assembling, and measuring. Traditional fixture design usually requires human designers to study the geometrical features and working conditions, then use experience-based rules such as the 3-2-1 principle. For complex workpieces and non-trivial working conditions, prototyping is usually required to achieve a valid design.

Research in fixture design started inevitably from "closure analysis," which can be dated back to 1885 when Reuleaux studied the form-closure mechanism for 2-D workpieces. Lakshminarayana (1978) proved mathematically that a minimum of seven points is required for form closure of 3-D work-pieces. Nguyen (1988) studied the force-closure mechanism for robotics hand grasps, while Asada and Kitagawa (1989) studied the form closure of robotics hands for both convex and concave workpieces. Generally speaking, the form closure emphasizes the kinematic analysis and the force closure studies the static equilibrium of the workpiece.

The emerging computer-integrated manufacturing system requires automated fixture design. Work in this area includes artificial intelligence-based software by Ferreira et al. (1985), Grippo et al. (1987), Nee et al. (1987), Pham and de Sam Lazaro (1990), and rule-based fixture design methodologies by Mani and Wilson (1988), Menassa and DeVries (1989), Linder and Cipra (1993), etc. Although their work presented automated fixture design procedures, fixture design issues such as locating, restraining, etc., are dealt with using design rules rather than analyses or syntheses.

For over a decade, a popular fixture design tool has been "screw theory," which was first proposed by Ball (1900) and elaborated by several other researchers, including Waldron (1972) and Ohwovoriole and Roth (1981). As a workpiece in 3-D can be represented by a translational motion in one direction and the rotation along the translational axis, it is analogous to a screw. Based on screw theory, Salisbury and Roth (1983) studied seven different types of finger contact and recommended

* Currently with Lamb Technicon of Western Atlas Inc, Warren, Michigan. Contributed by the Manufacturing Engineering Division for publication in the

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received

Aug. 1995; revised May 1996. Associate Technical Editor: S. G. Kapoor.

the finger configuration to completely immobilize a gripped object. Ohwovoriole (1987) analyzed grasping with friction in terms of complete or partial constraint of a rigid body using extended screw theory. Chou et al. (1989) formulated mathematical theories for automatic configuration of machining fixtures for prismatic parts. Bausch and Youcef-Toumi (1990) developed a motion-stop method for fixture configuration synthesis that examines the ability of each fixture contact to prevent or stop the reciprocal screw motions of workpieces. Weill et al. (1991) considered the influence of fixture positioning errors on the geometric accuracy of mechanical parts using a small screw model. DeMeter (1994) studied the surface contact and friction issues in restraint analysis for grippers or fixtures. Sa-yeed and DeMeter (1994) developed fixture design and analysis software that considers kinematic restraint, total restraint, and tool path clearance requirements. As one can see, screw theory can be applied to fixture design to deal with problems of deterministic locating and total restraint (Chou et a l , Bausch and Youcef-Toumi, Sayeed and DeMeter), the quality of locating (Weill et al.), and contact type and friction issues (DeMeter).

Another distinctive fixture design approach is the homogenous transformation technique, as first used by Asada and By (1985) to study kinematic problems, including deterministic locating, total restraining, accessibility and detachability. King and Hutter (1993) used a similar approach to develop an optimal design procedure for prismatic parts considering the work-piece stiffness, resistance to slip, and stability.

Research work considering the workpiece and/or fixture element rigidity was first performed by Shawki and Abdel-Aal (1965), who studied the effects of fixture rigidity and wear on dimensional accuracy based on experimental results. Finite element analysis on workpiece can be credited to Lee and Haynes (1987), fixture position optimization for machining operations was done by Menassa and DeVries (1991), and fixture optimization for deformable sheet metal parts was studied by Cai, Hu and Yuan (1996). These researchers addressed deformation issues in fixture design when manufacturing loads are influential.

All of the above-mentioned research work, much of that has been reviewed by Cohen and Mittal (1989), Trappey and Liu (1990), and Hazen and Wright (1990), addressed fixture design issues such as deterministic locating, total restraint, deforma-

Downloaded From: http://manufacturingscience.asmedigitalcollection.asme.org/ on 09/30/2014 Terms of Use: http://asme.org/terms

tion, interference, clamping sequence, etc. However, one of the important issues, the quality of a design, has not been studied intensively. This issue must be considered in fixture design because it can tell the best one from among virtually infinite sets of feasible design scenarios. Among a few who did research on this topic are Shawki and Abdel-Aal (1965) and Weill et al. (1991). Shawki and Abdel-Aal (1965) used experiments to determine the influence of clamping force and fixture wear on workpiece positional accuracy. Using screw theory, Weill et al. (1991) analyzed workpiece positional errors where the "function comparison method" (an optimization method for discrete variables) was employed to choose the best locating set-up from randomly selected ones, based on the locally linearized workpiece geometry.

In this paper, we develop a variational method to conduct robust fixture design to minimize the workpiece positional errors due to workpiece surface and fixture set-up errors. The proposed robust design has some apparent advantages. First, it is an analytical approach with complete mathematical derivations, compact closed-form solutions, and easy physical interpretations. Second, it considers both first- and second- order information of workpiece geometry, dealing adequately with both prismatic and non-prismatic parts. Third, it employs a variational method, where fundamental fixture design issues, e.g., deterministic locating (kinematics restraint), total fixturing (static equilibrium), and robust design (sensitivity analysis) can be dealt with in a unified way. Three basic assumptions are used.

Firstly, we assume that the workpiece is a rigid body enclosed by a set of piecewise differential surfaces. Secondly, a locator is a point contact on the workpiece surface. The assumption for point contact is valid only for mathematical abstractions. In real applications, line and surface contact locators are used predominantly. These types of locators, however, can always be represented by a collection of point contact locators for analysis purposes, which is not discussed in this paper. Thirdly, the clamp is considered as an applied force with controllable magnitude. In Section 2, it will be shown that the deterministic locating condition can be achieved if the rank of the Jacobian of the constraint equations equals the degrees of freedom of the workpiece, and the total fixturing condition can be achieved if the workpiece maintains the deterministic locating condition under loading. In Section 3, the robustness of the design is analyzed to achieve an optimal workpiece fixturing. Computational algorithms are depicted in Section 4, and examples are demonstrated in Section 5 to validate the proposed approach for robust fixture design.

2 Fundamental Issues for Fixture Design

This section addresses fundamental issues for fixture design. It will be shown that deterministic locating is a prerequisite for robust design, and that total fixturing ensures that the robustly located workpiece can be totally restrained during process operations.

N o m e n c l a t u r e

A = direction cosine orientation matrix (or transformation matrix) from body-fixed system O'X'Y'Z'(O'X'Y') to global system OXYZ(OXY)

d = degrees of freedom of a workpiece

e0,d, e2, e3 = Euler parameters representing rotational degrees of freedom

FA = applied force vector, expressed as [FA, FA, FA]T in 3-D and [FA

X, FA ]T in 2-D

FA,FA,FA = the X, Y, Z components ofF*

F c = the resultant constraint/

contact force vector Ff = the constraint/contact

force vector for the ith locator

p>. = the function describing workpiece boundary on which the ith locator resides, represented in body-fixed coordinate system

J = Jacobian of constraint equations, an acronym f o r * *

J, = first-order derivative of ith constraint equation $, w.r.t. q0

N„f = number of reference points used in objective function evaluation

n, (n/ ) = m e n o rmal vector of workpiece boundary on which the ith locator resides, represented in global/body-fixed coordinate system

N(0, a2) = normal distribution with 0 mean and a standard w, deviation

nix, niy, niz = the X, Y, Z components &t,, <5yr-, 6z, of normal vector n,

OXY(Z) = global Cartesian coordinate system in (2D(3D)-dimensional) space

0'X'Y'(Z') = body-fixed Cartesian co-ordinate system (2D(3D)-dimensional) space

q0 = a vector represents all the degrees of freedom of a rigid workpiece, denoted in compact form as [ r r n j ] 7 , or in full as [xo,y>o,Zo, eu e2, e 3 ] r i n 3-D and [x0,y0, <M r in 2-D

r0 = the origin of the body-fixed coordinate system representing the position of the workpiece, expressed as [xo, yo, ZoV in 3-D and [x0, y0]

T in 2-D r, ( r / ) = m e position vector of the

ith locator in global/ body-fixed coordinate system, expressed as [x,, y i . z . ] r / [ * ; , y / \ * / ] r i n 3-D and [x,, yiYlUl, y , T i n 2 - D

R = collective locator position vector, expressed as [rf, r r , . . . , r r ] r i n 3-D and [ r [ , rl r r r i n 2 - D .

TA(TA) = applied torque vector, expressed as [TA,TA,TA]' in 3-D and TA in 2-D

= weights used at the ith locating point

= three components of translational errors

Ki = workpiece boundary curvature at the ith locating point

\ = Lagrange multiplier vector, expressed as [ku X2> . . . , X 6 ] r in3-Dand [\u

X.2> \ 3 ] r i n 2 - D $ = collective constraint equa

tion vector formed from $ , , i = 1, . . . , m

$, = the ith constraint equation derived from the ith locator

$ R = first-order derivative of constraint equation vector w.r.t. R

n 0 = the orientation of the workpiece, expressed by the rotation angle <f>0 in 2-D and Euler parameters [<?i, e2, ^ ^ i n 3-D

594 / Vol. 119, NOVEMBER 1997 Transactions of the ASME

Locator 1

Fig. 1 Locating scheme for a generic 3-D workpiece

2.1 Mathematical Representation of Locator Constraints. A generic locating scheme with m locators is shown in Fig. 1 for a 3D workpiece. The OXYZ represents the global coordinate system, and the O'X'Y'Z' represents a body-fixed coordinate system. In this paper, parameters with primes are in the body-fixed system and those without primes are in the global system.

Suppose the ith locator is in contact with the differentiable workpiece surface F',(x', y', z') = 0, ;' = 1, 2, . . . , m at ith contact point, then the tangent plane at that point is

n / 7 - r ' - n / r - r / = 0 (2.1.1)

where n/ = [dFj/dx', dF\ldy', dF\ldz'Y\^\,y\,z\)is t he normal vector at the locating point r,! = [x\, y\, z'Y- Equation (2.1.1) can be transformed to

n / r A r ( r - r0) - n, ' r-r , ' = 0 (2.1.2)

where A is the direction cosine orientation matrix (transformation matrix) from the body-fixed system to the global system. Because the ith locator r; = [*,, y,, ztV must satisfy Eq. (2.1.2), we have the constraint equation for the ith locator:

$,. = n , , 7 A T ( r , - r 0 ) - n , ' r - r , ' = 0, i = l , . . . , m (2.1.3)

For a locating scheme with m locators, the collective constraint equation is

# (q 0 ) = [ * „ * 2 , . . . , # m ] r (2.1.4)

where q0 = [ r j , n j ] = [x0, y0, zQ, eu e2, e 3 ] r denotes the parameters of the six degrees of freedom of the workpiece. Note here we intentionally choose three parameters, (ex ,e2,e3), from Euler Parameters (e0, eu e2, e3), to represent three rotational DOFs of the workpiece. This is so because any set of three independent parameters, such as Euler angles, will introduce transformation singularity at q0 = 0 (Appendix A), and Euler parameters are the only set of non-singular parameters.

2.2 Deterministic Locating

Definition If a workpiece under a certain locating scheme can not make an infinitesimal motion while maintaining contact with all the locators, then the workpiece is deterministically located or the

Journal of Manufacturing Science and Engineering

( • $ . * » )

o x ( • $ . * » )

ZU fiA Fig. 2 Deterministic locating for a rectangular workpiece

deterministic locating of the workpiece is achieved.

If we denote the Jacobian of the constraint equations as

J - $ q „ - [ J , , . . . , J , , . . . , J , „ ] r (2.2.1)

_ r ^ a*, d%_ o^ d^_ a*,] |_ dx0 dy0 dzo dex de2 9e3 J

then we have the following (see Appendix B for proof):

A workpiece can be deterministically located if the rank of the Jacobian of the constraint equations equals six (3-D) or three (2-D).

Without loss of generality, we can always assume that the body-fixed coordinate system is identical to the global coordinate system. Then Eq. (2.2.2) can be expressed explicitly as (Appendix A),

for 2-D: J, = [~nlx, —niy, niyx, — nixy(] (2.2.3)

for 3-D: J, = [ -«„ , -niy, -niz,

2(nlzy, - riiyZi), 2(nixZi - nizx,), 2(niyx, - «,>y,)] (2.2.4)

An example of a 2-D rectangular workpiece is shown in Fig. 2 to illustrate the deterministic locating. Three locators have

Fig. 3 A generic locating scheme for a sphere

NOVEMBER 1997, Vol. 1 1 9 / 5 9 5

been arranged, with two locators (L, & L2) on one horizontal edge and the third locator (L3) on a vertical edge. We have,

J = 0 — 1 -Xi

0 -1 -x2 -1 0 y?,

(2.2.5)

| J | = *i - xj (2.2.6)

Equation (2.2.6) shows that if x\ = JC2, the Jacobian is zero. This implies that locators L\ and Li should not be coincident as required by the deterministic locating condition.

Some workpieces can never be deterministically located, such as a sphere (Fig. 3). This is so because the Jacobian for the sphere is

J = -*, - y , - Z l 0 0 0

•xm -ym -zm 0 0 0 (2.2.7)

Therefore, a sphere has at least three rotational DOFs under any locating scheme.

2.3 Total Fixturing

Definition If a deterministically located workpiece, under the influence of clamping and machining forces, maintains contact with all the locators, then the workpiece is totally fixtured or the total fixturing of the workpiece is achieved.

In Section 2.2, the condition for deterministic locating was identified assuming that the workpiece is known to be in contact with all the locators. This is a kinematic condition because machining load and clamping forces are not involved. Because a valid fixture design should ensure consistent contact of the workpiece with the locators during processing, clamps are required to restrain the workpiece. In this paper, clamps are regarded as applied forces (clamping forces). Therefore, total fixturing for a 3-D workpiece is to select the clamping position and intensity to totally restrain the workpiece under the influence of manufacturing process load.

Contact between the workpiece and locators is achieved when the reaction forces at locating points are non-negative. The reaction forces can be calculated from static analysis by using a degenerated form of the constrained Euler-Newton equations, as stated in Appendix C, which, at the ;'th locating point, is

Ff = dx0 " dy0 dz0

K (2.3.1)

where the Lagrange multiplier vector \ is introduced as -p/t-

(2.3.2)

Here, FA and TA are the resultant applied force and torque at

Fig. 4 Total fixturing for a rectangular workpiece

596 / Vol. 119, NOVEMBER 1997

the origin, respectively, which may include gravity, machining forces/torques, and clamping forces, and F c is the resultant reaction forces at contact points. Note here that the workpiece has to be deterministically located, as discussed in Section 2.2, so that the transpose inverse of the Jacobian in Eq. (2.3.2) holds, i.e., J is nonsingular.

A 2-D rectangle workpiece is used to illustrate the total fixturing concept, as shown in Fig. 4. Suppose the resultant applied force is F" = [FA, FA)T and the resultant applied torque is TA, then the Jacobian J can be obtained directly from Eqn. (2.2.5). Following Eqn. (2.3.2), we have

\ = [X.1, X-2> ^3]

-y3FA + x2F

Ay - TA y,FA - x{F

A + TA „

X\ X2 X\ X*i

and then from Eq. (2.3.1), we have

Ff = n n \3 , F 2

C = , Ff = \3

A-i h 0

(2.3.3)

(2.3.4)

Therefore, in order to totally fixture the workpiece, the condition

\ a 0, ( = 1, 2, 3 (2.3.5)

must be satisfied. A quick conclusion from condition (2.3.5) is that the resultant force FA must be less than zero to ensure contact of L3 with the workpiece, which is intuitively correct. To better comprehend condition (2.3.5), two special cases, as shown in Fig. 5, are studied, which agree with the existing literature, such as Reuleaux and Chou et al.

Case 1: The resultant force has only a horizontal component FA as in Fig. 5(a) .

From condition (2.3.5), we have y3 = 0, i.e., the workpiece is stable only if the resultant force is of the same height as the locator L3.

Case 2: The resultant force has only a vertical component FA as in Fig. 5(b).

From condition (2.3.5), we have Xj > 0 and x2 < 0, implying that the workpiece is only stable when the resultant force is applied between locators L[ and L2, i.e., within the shaded area.

3 Robust Fixture Configuration Design Definition Robust design in this context refers to the minimi

zation of workpiece positional and/or orientation errors (resultant errors) due to workpiece surface and/or fixture set-up errors (source errors) in a deterministic locating environment.

Section 2.2 described conditions of deterministic locating and Section 2.3 described conditions for total fixturing. An important issue missing so far is the quality of the locating scheme. As there may exist an infinite number of feasible locating schemes for a workpiece according to the deterministic locating and total fixturing conditions, a "best" locating scheme is required. We propose in this section a robust design methodology that can best compensate for source errors that exist in real industrial applications. In other words, the robust design generates a locating scheme with minimal workpiece positional variations due to source errors. We first introduce the robust design based on an infinitesimal error approach (linear analysis), and then present the small error approach (second order analysis) for generic 3-D workpieces.

3.1 Infinitesimal Error Analysis (IEA). For a generic workpiece, we can perform the robust design based on first-order information of workpiece surfaces, i.e., the workpiece surface is linearized at locating points. This is called an infinitesimal error approach because the analysis is valid only at the infinitesimal neighborhood of the locating points.

Transactions of the ASME

a) Horizontal clamping b) Vertical clamping

Fig. 5 Total fixturing under clamping

For a 3-D deterministically located workpiece, Eq. (2.1.3) can be rewritten as

# , (q 0 , r ,) = n / r A r ( r , - r„) - n,'7'-r,' = 0,

i = 1, 6 (3.1.1)

Define a collective locator position vector R = [ r [ , r\, . .., rl]T, then Eq. (3.1.1) can be expressed in vector form as

$ ( q 0 , R ) = 0 (3.1.2)

Form the variation equation

*,o<5qo + *R<5R = 0 «• J6q0 + #R6R = 0 (3.1.3)

n[ 0 0 0 0 0 0 nl 0 0 0 0

0 0 0 0 0 n6

(5R = [<5r[, 6vl &rlV

(3.1.4)

(3.1.5)

denote the source errors. Then we form the sensitivity equation for robust design

<5q0 - J ' * R 5 R

where the Jacobian J is nonsingular when the workpiece is deterministically located.

In this way, the workpiece resultant errors fiq0 are linked to source errors <5R. Although Eq. (3.1.6) can be expressed explicitly by using symbolic mathematical tools such as MAPLE, pages of equations would recommend numerical procedures.

We use a 2-D workpiece here (Fig. 6) to show procedures

7T£ ST Fig. 6 Robust design of a 2-D workpiece

Let the tangent at the third locating point be represented by

ax + by + c = 0

(3.1.6) T h e n

J = — 1 —JCi

— 1 —Xi —b ay3 — bx?,

| J | = a(x, - x2)

(3.1.7)

(3.1.8)

(3.1.9)

If a * 0, i.e., the tangent of the third surface is not parallel of robust design through the analytical approach, while numerical to the first two edges, and x, * x2, i.e., the first two locators examples are reserved for Section 5. Assuming that two locators, are not coincident, then the deterministic locating condition is Lj and la, are on the same horizontal edge, the robust locating satisfied. By following Eqs. (3.1.4) through (3.1.6), the follow-design is to identify the position of the third locator L^. ing relationship can be derived

y&t - y,5y2 + ^ + b / x^c, ^ _ x^c, ^ + ^

(X \X\ X2 X\ X2

(3.1.10)

Journal of Manufacturing Science and Engineering NOVEMBER 1997, Vol. 119 /597

From Eqs. (3.1.9) and (3.1.10), we have the following observations:

(1) If bla = 0, i.e., the third locator L3 is positioned on the workpiece with tangent parallel to the Y axis, then the error 6x0 is minimized. This can be generalized to show that the 3-2-1 principle is the best locating scheme for 3-D workpieces by selecting three mutually perpendicular planes as locating datums.

(2) The term (x, - x2), i.e., the distance between Lx and Z^, plays an important role. The two locators should be arranged as far apart as possible to reduce resultant errors.

(3) The error Sx0 is minimized when y3 = 0, which implies that L3 is better positioned at the same Y level as the origin of the coordinate system, or the reference point (see more explanations on reference point selections in Section 3.3).

3.2 Small Error Analysis (SEA). In Section 3.1, we addressed the robust design problem by considering infinitesimal error components which linearize the workpiece boundary. In practice, source errors by no means are infinitesimal, although they may be of extremely small magnitudes. Therefore, considering second-order boundary information to perform the small error analysis is very important, especially for nonprismatic workpieces, i.e., those which do not have plane surfaces. This notion can be easily accepted by the fact that some workpiece regions may have such significant geometry changes that positioning a locator there introduces more error and should not be recommended. Noticing that the second-order description for a workpiece boundary is curvature K in 2-D or the curvature index K derived from Gaussian curvature in 3-D (Appendix D), we propose using weights in the objective function for optimization to balance the source errors. The weights are defined as

w, = K,R + 1 (3.2.1)

with i = 1, 2, 3 for 2-D and i = 1, . . . , 6 for 3-D, and R denotes the mean dimension of the workpiece. In Eqn. (3.2.1), weight Wt is unitless, equaling one if a locator is placed on a planar surface, and greater than one otherwise.

To perform the small error analysis in robust design, we need to modify Eqn. (3.1.5) to

SR = [w,<5rf, w28rl, ..., w66rT6]

T (3.2.2)

and then follow the robust design procedures as described in Section 3.1.

3.3 Practical Issues for Robust Design. So far, we have discussed the generic procedures for robust fixture design, considering either infinitesimal errors (Section 3.1) or small errors (Section 3.2). For a particular design case, we usually need to consider several practical issues, as described below.

(1) Deterministic or statistical approach. If we want to study resultant errors based on deterministic or known source errors, then the design is called a deterministic design. On the other hand, if we only know variations or tolerances of error sources, we usually consider source errors as statistical variables and therefore statistical design is achieved. For example, in Eq. 3.1.10, under the assumption that error sources follow independent normal distribution N(0, a2), variances of resultant errors will be

(2) Selection of resultant error components. We can choose to study one or more resultant positional errors, depending on applications. For example, if we want to study primarily the rotational behavior of the workpiece as shown in Fig. 6, then the component 6<f>0 in Eq. (3.1.10) or var (6<fi0) in (3.3.1) is most critical. If we choose to evaluate the overall locating performance, then all the translational and rotational error sources should be included. Note that the translational and rotational resultant errors are of different units, i.e., the unit of length and the unit of radian, respectively.

(3) Selection of reference points. In most cases, we are interested in resultant errors for a set of points or a whole workpiece instead of one single point as discussed so far. Therefore, a robust design procedure should usually consider resultant errors for certain representative points simultaneously. This is particularly important when we notice that the resultant translational and rotational error components are not measured by the same units, and thus cannot be added up directly. Thus, in this paper, when we formulate the objective function, we select translational errors for several extreme points on the workpiece to represent the overall workpiece errors.

4 Computational Algorithms

Section 3 detailed analytical procedures for robust design. It has been found that unless the workpiece geometry is very simple, analytical solutions are difficult to obtain and hard to interpret. Therefore, we develop a simulation module called RFixDesign to carry out the robust fixture design for generic 3-D workpieces.

4.1 Flowchart for RFixDesign. The flowchart for RFixDesign software is shown in Fig. 7. The workpiece geometry is represented by a set of enclosed surfaces. Initial locator positions are selected by users and represented in Cartesian coordinates. The clamp positions/forces are used for the total fixturing analysis. The program first goes to the deterministic locating test, which is the prerequisite for the subsequent analyses. If the test is not satisfied, the program must restart from a new design scenario. Otherwise, the nonlinear programming procedure will be continued by forming the objective function for the optimization. If the objective function is minimized (<5q0 is minimum), then the optimal locating condition is achieved. At this moment, the total fixturing condition is checked to make sure the clamp positions/forces are adequate. If not, users have to modify the clamping conditions (positions and/or forces) until the total fixturing condition is satisfied and the optimum fixturing is achieved.

var (<5q°) var (Sx0) var (Sy0) var (6</>0)

i yi i X\ Xi

X i + xi (x -x2)

(*i - x2y

+ i + l -

Workpiece Geometry

Initial Locator Positions

Clamp Positions /Forces

where X is a vector representing the design variables, or locator coordinates, F ( X ) is the objective function for optimization, v stands for variance of the resultant error, d is the dimension of the workpiece and NreS is the number of reference points used in the robust design (Section 3.3). Equation (4.1) is the pooled standard deviation of the resultant errors with length in units. G i ( X ) represents equality constraints for design variables, i.e., locators must always satisfy workpiece surface equations. G 2 (X) represents inequality constraints of the design variables, i.e., workpiece surface regions where locators can move. In this research, F ( X ) and G ^ X ) are usually nonlinear functions of design variables X, and G 2 (X) are linear functions of X. For optimization, we use VMCON (Crane et al., 1980), a sequential quadratic programming-based software package.

5 Examples

As discussed in Section 3.3, different applications can choose different objective functions, e.g., the error components that are most critical to particular applications. In the following examples, we chose the objective function as the summation of all three resultant translational error components from several reference points, and used the statistical approach (variance) to evaluate the resultant errors. Both infinitesimal or small error analysis approaches are considered. The workpieces surfaces are represented by closed-form mathematical formulae.

Fig. 7 Numerical procedures for robust fixture configuration design

4.2 Nonlinear Programming. In this research, the optimization procedure is carried out through a nonlinear programming technique, with the following format

minimize F(X) 1

W ,e f / = r f

^ v , i . , w <1J)

subject to Gi (X) = 0 (4.2)

G 2 ( X ) > 0 (4.3)

Example 1. This example shows the robust design of a 2-D ellipse in Fig. 8. The ellipse is represented by x2/4002 + )>2/2002 = 1. Four reference points are included in the objective function formulation, i.e., A, B, C and D. Results from this example show significant reductions of resultant errors through robust design. Using the SEA approach, locators tend to not move near poles because of larger weights there.

Example 2. This example shows the robust design of a 3-D block as shown in Fig. 9. The block, is of 500 X 300 X 200 mm3, and its six surfaces are represented by six equations, i.e., x = ±250 (right and left); y = ±150 (front and back); z

A Initial Locator

j ^ Optimal Locator (SEA)

J^ Optimal Locator (IEA)

• Optimization Direction

Fig. 8 Robust design for an ellipse workpiece

Journal of Manufacturing Science and Engineering NOVEMBER 1997, Vol. 1 1 9 / 5 9 9

A Initial Locator A Optimal Locator • Optimization Direction

Fig. 9 Robust design for a block workpiece

= ±100 (top and bottom). The locating scheme is 3-2-1. Eight reference points, i.e., eight vertices, are included in the objective function formulation. As the block is made of plane surfaces, results from both IEA and SEA are identical. This example shows that the commonly used 3-2-1 principle gives only a qualitative design criterion, while the robust design procedure suggests the much better 3-2-1 locating scheme.

Example 3. This example shows the robust design of an ellipsoid, which is represented by x2/4002 + y2/2002 + +z2/ 1002 = 1. In Fig. 10, the ellipsoid is cut in halves, i.e., the top half and bottom half, for better visualization. Eight poles are selected as reference points.

Figure 10 shows that when initial locators are clustered relatively together, die IEA robust design tends to move them apart, while the SEA design tends to give trade-off locator positions.

Table 1 gives a summary of results. In the table, Parameters d and JVref define the dimension of the workpiece and the number of reference points used in the objective function (see Section 3.3), respectively. Initial Locator Positions axe, defined in a Cartesian coordinate system. Because robust designs are performed twice for each case using infinitesimal error approach (IEA) and small error approach (SEA), respectively, there are two sets of Optimal Locator Positions for each case. Consequently, two sets of initial and optimal Resultant Errors exist corresponding to each of the different approaches. The resultant error represents the pooled standard error of the workpiece in three translational directions when unit standard error exists at each locating point. Resultant errors from IEA and SEA, however, are generally not comparable because different weights are used in the objective function formulation.

A Initial Locator

A Optimal Locator (SEA)

Optimal Locator (IEA)

Optimization Direction

Fig. 10 Robust design for an ellipsoid workpiece

Table 1 Robust design examples using RFixDesign

Parameter! Initial Locator Positions (mm)

Optimal Locator Positions (nun) Resultant Error

( m m ) Case

# d N«f

Initial Locator Positions (mm) Infinitesimal Error

Analysis (IEA) Small Error

Analysis (SEA) Initial (IEA) Initial (SEA)

OptimalQEAJ Optimal(SEA)

1 2 4 (-100.0,-193.7.0.0) (100.0,-193.7,0.0) (-100.0,193.7,0.0)

(-359.7,-87.4,0.0) (272.5,-146.4,0.0) (-377.2,66.5,0.0)

(-306.4.-128.5,0.0) (206.4,-171.3,0.0) (-329.0,113.8,0.0)

4.5 1.2 1 2 4

(-100.0,-193.7.0.0) (100.0,-193.7,0.0) (-100.0,193.7,0.0)

(-359.7,-87.4,0.0) (272.5,-146.4,0.0) (-377.2,66.5,0.0)

(-306.4.-128.5,0.0) (206.4,-171.3,0.0) (-329.0,113.8,0.0) 6.4 2.5

(-150.0,0.0,-100.0) (-100.0.-50.0.-100.0) (150.0,0.0,-100.0) (-50.0,150.0,0.0) (50.0,150.0,0.0) (-250.0.-50.0.50.0)

(-250.0,150.0,-100.0) (87.0,-150.0,-100.0) (250.0.150.0,-100.0) (-250.0,100.0.50.0) (250.0,100.0,-50.0) (-250.0.0.0,0.0)

same as left

7.8 1.6

(-150.0,0.0,-100.0) (-100.0.-50.0.-100.0) (150.0,0.0,-100.0) (-50.0,150.0,0.0) (50.0,150.0,0.0) (-250.0.-50.0.50.0)

(-250.0,150.0,-100.0) (87.0,-150.0,-100.0) (250.0.150.0,-100.0) (-250.0,100.0.50.0) (250.0,100.0,-50.0) (-250.0.0.0,0.0)

same as left

same as above same as above

(-200.0,-100.0.-70.7) (200.0,-130.0,-57.2) (34.0.100.0,-86.2) (-290.0,0.0,68.9) (-100.0.-145.5,63.9) (-75.0.-190.0,25.0)

(-372.0.-73.6,0.0) (294.2.-130.5,-18.3) (307.2.120.1,-22.3) (-367.6.78.8,1.5) (-59.9.148.3,65.4) (-111.9.-141.2,65.1)

(-304.6.-100.1.-41.2) (286.9,-123.8,-32.0) (96.2,145.8,-64.1) (-297.1,122.0,27.7) (-82.1,103.8,34.9) (-90.2,-156.1,58.3)

15.8 1.1

(-200.0,-100.0.-70.7) (200.0,-130.0,-57.2) (34.0.100.0,-86.2) (-290.0,0.0,68.9) (-100.0.-145.5,63.9) (-75.0.-190.0,25.0)

(-372.0.-73.6,0.0) (294.2.-130.5,-18.3) (307.2.120.1,-22.3) (-367.6.78.8,1.5) (-59.9.148.3,65.4) (-111.9.-141.2,65.1)

(-304.6.-100.1.-41.2) (286.9,-123.8,-32.0) (96.2,145.8,-64.1) (-297.1,122.0,27.7) (-82.1,103.8,34.9) (-90.2,-156.1,58.3)

28.4 2.6

6 Conclusions

This paper addresses the robust design issue in fixture configuration design using a variational approach. Through robust design, the influence of workpiece surface errors and fixture set-up errors can be minimized by appropriately arranging the fixture configuration, leading to a best design from among an infinite number of feasible fixture schemes. It has been shown mathematically that deterministic locating is only the prerequisite for robust design. By knowing the design input, i.e., initial fixture positions and workpiece geometry, the robust design procedure can be carried out either analytically (for 2-D simple workpieces) or numerically (for complex workpieces) through the RFixDesign software, without any graphical aids. Results show that the optimal fixture scheme leads to significant enhancement of workpiece positional accuracy. The robust fixture design methodology proposed in this paper is especially suitable in such areas as measuring, robust-grasping, light-duty machining where workpiece/fixture deformation is not influential.

Acknowledgment

Funding of this research is sponsored in part by the NSF-I/ UCRC for Dimensioning Measurement and Control in Manufacturing at The University of Michigan, and the NSF/ARPA Machine Tools-Agile Manufacturing Research Institute.

References Asada, H., and By, A. B., 1985, "Kinematic Analysis of Workpiece Fixturing

for Flexible Assembly with Automatically Reconfigurable Fixtures,'' IEEE Journal of Robotics and Automation, Vol. RA-1, No. 2, pp. 86-94.

Asada, H., and Kitagawa, M., 1989, "Kinematic Analysis and Planning for Form Closure Grasps by Robotic Hands," Robotics and Computer-Integrated Manufacturing, Vol. 5, No. 4, pp. 293-225.

Ball, R. S., 1990, A Treatise on the Theory of Screws, Cambridge University Press, Cambridge.

Bausch, J., and Youcef-Toumi, K., 1990, "Kinematic Methods for Automated Fixture Reconfiguration Planning," IEEE Conference of Robotics and Automation, pp. 1396-1401.

Cai, W., Hu, S. J., and Yuan, J. X„ 1996, "Deformable Sheet Metal Fixturing: Principles, Algorithms and Simulations,'' ASME JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING, August.

Chou, Y-C, Chandra, V„ and Barash, M. M., 1989, "A Mathematical Approach to Automatic Configuration of Machining Fixtures: Analysis and Synthesis," ASME JOURNAL OF ENGINEERING FOR INDUSTRY, Vol. I l l , pp. 299-306.

Cohen, P. H„ and Mittal, R. O., 1989, "A Methodology for Fixturing and Machining of Prismatic Components," Advances in Manufacturing Systems Integration and Process, Proceedings of 15th Conference on Production Research and Technology, Jan. 9-13.

Crane, R. L„ Hailstorm, K. E„ and Minkoff, M., 1980, VMCON User's Manual, Argonne National Laboratory.

DeMeter, E. C, 1994, "Restraint Analysis of Fixtures which Rely on Surface Contact," ASME JOURNAL OF ENGINEERING FOR INDUSTRY, Vol. 116, No. 2, pp. 207-215.

Ferreira, P.M., Kochar, B., Liu, C. R., and Chandra, V., 1985, "AIFIX: An Expert System Approach to Fixture Design," ASME Winter Annual Meeting on Computer Aided/Intelligent Process Planning, Miami Beach, FL.

Gray, A., 1993, Modern Differential Geometry of Curves and Surfaces, CRC Press, Boca Raton.

Grippo, P. M., Gandhi, M. V., and Thompson, B. S„ 1987, "The Computer-Aided Design of Modular Fixturing Systems," International Journal of Advanced Manufacturing Technology, Vol. 2, No. 2, pp. 75-88.

Haug, E. J., 1992, Intermediate Dynamics, Prentice Hall, New Jersey. Hazen, F. B„ and Wright, P. K., 1990, "Workholding Automation: Innovations

in Analysis, Design and Planning," Manufacturing Review, Vol. 3, No. 4. King, L. S. B., and Hutter, I., 1993, "Theoretical Approach for Generating

Optimal Fixturing Locations for Prismatic Workparts in Automated Assembly," Journal of Manufacturing Systems, Vol. 12, No. 5, pp. 409-416.

Lakshminarayana, K., 1978, "Mechanics of Form Closure," ASME Technical Paper, 78-DET-32.

Lee, J. D., and Haynes, L. S., 1987, "Finite Element Analysis of Flexible Fixturing Systems," ASME JOURNAL OF ENGINEERING FOR INDUSTRY, Vol. 109, pp. 134-139.

Linder, K. J., and Cipra, R. J., 1993, "A Strategy for Determining the Stability of Parts Fixtures Under Three-Point Frictional Constraint," ASME Journal of Mechanical Design, Vol. 115, December, pp. 863-870.

Mani, M., and Wilson, W., 1988, "Automated Design of Workholding Fixtures Using Kinematic Constraint Synthesis,'' Proceedings of the Sixteenth North American Manufacturing Research Conference, pp. 437-444.

Menassa, R., and DeVries, W., 1989, "Locating Point Synthesis in Fixture Design," Annals of the CIRP, Vol. 38, No. 1, pp. 165-169.

Menassa, R., and DeVries, W., 1991, "Optimization Methods Applied to Selecting Support Positions in Fixture Design," ASME JOURNAL OF ENGINEERING FOR INDUSTRY, Vol. 113, pp. 412-418.

Nee, A. Y. C , Bhattacharyya, N„ and Poo, A. N„ 1987, "Applying Al in Jigs and Fixtures Design," Journal of Robotics and Computer-Integrated Manufacturing, Vol. 3, No. 2, pp. 195-200.

Nguyen, V., 1988, "Constructing Force-Closure Grasps," International Journal of Robotics Research, Vol. 7, No. 3, pp. 3-16.

Ohwovoriole, M.S., and Roth B., 1981, "An Extension of Screw Theory," ASME Journal of Mechanical Design, Vol. 103, pp. 725-734.

Ohwovoriole, E. N., 1987, "Kinematics and Friction in Grasping by Robotic Hands," ASME Journal of Mechanisms, Transmissions and Automation in Design, Vol. 109, pp. 389-404.

Pham, D. T., and de Sam Lazaro, A., 1990, "AUTOFIX—An Expert CAD System for Jigs and Fixtures," International Journal of Machine Tool Design and Research, Vol. 30, No. 3, pp. 403-411.

Reuleaux, F., 1885, Kinematics of Machinery, Macmillan, London. Salisbury, J. K., and Roth, B., 1983, "Kinematic and Force Analysis of Articu

lated Mechanical Hands," ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 105, pp. 35-41.

Sayeed, Q. A., and DeMeter, B.C., 1994, "Machining Fixture Design and Analysis Software," International Journal of Production Research, Vol. 32, No. 7, pp. 1655-1674.

Shawki, G. S. A„ and Abdel-Aal, M. M., 1965, "Effect of Fixture Rigidity and Wear on Dimensional Accuracy," International Journal of Machine Tool Design and Research, Vol. 5, pp. 183-202.

Trappey, J. C , and Liu, C. R., 1990, "A Literature Survey of Fixture Design Automation," International Journal of Advanced Manufacturing Technology, Vol. 5, pp. 240-255.

Waldron, K. J., 1972, "A Method of Studying Joint Geometry," Mechanism and Machine Theory, Vol. 7.

Weill, R., Darel, I., and Laloum, M., 1991, "The Influence of Fixture Positioning Errors on the Geometric Accuracy of Mechanical Parts," Procedings of CIRP Conference on PE & MS, pp. 215-225.

Wittenburg, J., 1977, Dynamics of Systems of Rigid Bodies, Stuttgart, Teubner.

A P P E N D I X A

This appendix derives Eq. {22A). Prior to the derivation, the direction cosine matrix A must be defined. Because there are three rotational degree of freedom for a 3-D rigid body, a set of three

Journal of Manufacturing Science and Engineering NOVEMBER 1997, Vol. 1 1 9 / 6 0 1

independent orientation parameters, such as Euler angles, can be chosen to define the transformation matrix. Euler angles, however, suffer from having a singular orientation that must be avoided in practice if computational difficulties are to be circumvented. This is so because no single set of three independent orientation parameters has ever been found, and arguments have been made that none exist that are both independent and have no singular configurations (Wittenburg, 1977; Haug, 1992). To avoid this discrepancy, a set of four dependent orientation parameters called Euler parameters, e0, eu e2, e3, has been used to define the A which has no singular points. The transformation matrix defined by Euler parameters is expressed as,

el + e\ - \ e,e2 - e0e3 exe3 + e0e2

e,e2 + e0e3 el + e\-\ e2e3 - e0ei

e,e3 - e0e2 e2e-i + e0e, el + e\ - j_

where the Euler parameter normalization constraint must be applied

e20 + e\ + e\ + e\ = 1 (A.2)

If we want to use the (eu e2, e3) as three independent Euler parameters, we have

= ±Vl - {e\ + e\ + el) (A.3)

Note that in Eq. (A.3), either sign may be selected to achieve the same orientation.

In this paper, we are most interested in the condition when A is an identity matrix. Considering only the positive sign in Eq. (A.3), we have a unique set of Euler parameter

(e0,eue2,e3) = ( 1 , 0 , 0 , 0 ) (A.4)

when A becomes an identity matrix. Substituting (A.l) into Eq. (2.1.3) and subsequently Eq. (2.2.2), we have

J, 9ro

dXi - n ' r A

n / ' — r ; , n; dex

de2 r • n ' '

de- f, (A.5)

Evaluating Eq. (A.5) at (x0, yo, Zo» ei, e2, e3) = 0, and noticing that the global and body-fixed systems are coincident, Eqn. (2.2.4) can be obtained accordingly.

A P P E N D I X B Theorem For a general nonlinear equation <&(q) =

[* i (q ) , * 2 ( q ) , • • •, * m ( q ) J T where q = [qu . . . , qm]T, let q° satisfy $(q°) = 0. Then the solution q° is unique near the neighborhood of q° if the Jacobian

l*,(q°)l »! (q° ) '

d1j * 0 (B.l)

Proof: For the ith constraint equation $ , (q ) = 0, we have

$ , . ( q ) - * i ( q ° ) + i ^ ^ A % = 0

" 1 ^ = 0 (B.2)

This is a set of m linear equations of m variables (AqJt 7 = 1, . . . , m), and non-trivial solutions for Ag;s exist if the condition

(B.l) holds. Note that A^s are assumed to be of infinitesimal magnitude.

A P P E N D I X C This appendix explains how Eqs. (3.1.1) and (3.1.2) have

been derived. Denote * q o = [<&ro $n 0 ] . where q0 = [ r j , n j ] r = [x0, v0, zo, eu e2, e3]

T, then the equilibrium conditions for a single rigid body can be derived from constrained Newton-Euler equations of motion as in the global coordinate system,

-F* = - # £ \

where the vector X is the Lagrange multiplier; F" and TA are resultant applied forces and torques, respectively, which may include gravity, machining force and clamping force; and F c , T c are resultant constraint/contact forces and torques, respectively.

Equation (C.2) can further be expressed as:

FC = - * I , \ = .= 1 OXo

-I 7%. dza X,

= [F f ,F yc , F f ] J (C.3)

with each Fcx , Fy , Ff consisting of six individual reaction

force components at six locators. Equation (3.1.2) can be derived directly from Eq. (C.l) by solving the 6 x 6 linear equations for unknown \ of six components, and Eq. (3.1.1) can be obtained by expressing the resultant contact force in Eq. (C.3) into six components.

For a detailed analysis on the variational approach for rigid body kinematics and dynamics, see Haug (1992).

A P P E N D I X D This appendix gives the formulae for curvatures and Gaussian

curvatures as described in differential geometry theory such as in Gary (1993).

(1) For a parameterized regular curve a(t) = (x(t), y(t)) in 2-D, the curvature is given by

x'(t)y"(t) - x"(t)y'(t) (x'2(t) + y ' 2 ( 0 ) 3 ' 2

(2) For a parametrized regular surface S(u, v) in 3-D, the Gaussian curvature is given by

K = eg-f2

EG - F (D.2)

E — S„ • S„

F = S„ • Sv and <

G = S„ • S„

(S„USUS„)

VEG - F2

(S„„S„S„)

4EG - F2

(SmSuS„)

4EG - F2

Note in Eq. 3.2.1, we actually use

as the curvature index for 3-D workpieces.

A Variational Method of Robust Fixture Configuration Design for 3-D Workpieces

Documents

Transcript of A Variational Method of Robust Fixture Configuration Design for 3-D Workpieces

2'x4' (nominal) lighting fixture - Town of North Castle

SpeedStar Configuration 2.1 Installation

Visibility Analysis for Assembly Fixture Calibration Using Screen Space Transformation

WIRING HARNESS CONFIGURATION DIAGRAMS

Initial Configuration - hservers.org

Module 13: WLAN Configuration

Variational transition-state theory

Complex Lie Symmetries for Variational Problems

Variational approach to the Bogomolny separation

Variational theory of hot nucleon matter

Fixture del Mundial 2014 en Excel

Basic Router Configuration

Boundary Regularity in Variational Problems

Variational Quantum Evolution Equation Solver - arXiv

Replica Cluster Variational Method

Configuration guide - Support

SAP Controlling Configuration

Prototype/Sheet Metal - Checking Fixture/Gage - It works!

A flexible variational registration framework

SAP PROFITABILITY ANALYSIS CONFIGURATION