Process System
Resume:
*** **** ************ ** ******** AND AUTOMATION, VOL. 19, NO. 2, APRIL 2003
State Space Modeling of Dimensional Variation
Propagation in Multistage Machining Process
Using Differential Motion Vectors
Shiyu Zhou, Qiang Huang, and Jianjun Shi
VD&T Vectorial dimensioning and tolerancing.
Abstract In this paper, a state space model is developed to de-
scribe the dimensional variation propagation of multistage ma-, Position and orientation deviation of LCS
chining processes. A complicated machining system usually con- from nominal positions w.r.t. RCS.
tains multiple stages. When the workpiece passes through mul-,, Three elements of .
tiple stages, machining errors at each stage will be accumulated
Stage index, .
and transformed onto the workpiece. Differential motion vector, a
Total number of newly generated feature at
concept from the robotics field, is used in this model as the state
vector to represent the geometric deviation of the workpiece. The stage .
deviation accumulation and transformation are quantitatively de-,, Column vectors of a rotational matrix.
scribed by the state transition in the state space model. A system-
Skew symmetric matrix obtained from
atic procedure that builds the model is presented and an experi-
vector .
mental validation is also conducted. The validation result is satis-
Extension of a vector . .
factory. This model has great potential to be applied to fault di-
agnosis and process design evaluation for complicated machining Feature index for newly generated feature
processes. at stage . .
Index Terms Differential motion vector, multistage machining Feature index for the primary datum
process, state space model, variation propagation., secondary datum, and tertiary
datum of the th stage.
Feature index for the generated feature up
NOMENCLATURE
to stage (not include stage ). is an
FCS, FCS Nominal and actual fixture coordinate integer from 1 to total number of generated
system. features up to stage .
HTM Homogeneous transformation matrix., Actual and nominal vector of the origin of, Actual and nominal HTM between RCS LCS expressed in RCS.
and LCS .,, Three elements of .
Deviation HTM defined by and .
Differential motion vector representing the, by identity matrix and by zero
deviation of LCS in RCS. It is a stack of
matrix, respectively.
and and is expressed in LCS .
LCS th local coordinate system.
State vector., Total number of key features and ma-
Stack of differential motion vectors of
chining stages, respectively.
newly generated features at stage .
RCS Reference coordinate system., Differential transformation matrix corre-, Actual and nominal rotational matrix be-
sponding to and .
tween RCS and LCS ., Actual and nominal Euler rotational angles, Transformation matrices for datum-in-
between RCS and LCS .
duced error.,, Three elements of .
Transformation matrix for fixture error.
th element of a vector in the bracket.
Cross product of two vectors.
Manuscript received July 23, 2001; revised April 15, 2002. This paper was
recommended for publication by Associate Editor M. Wang and Editor N.
I. INTRODUCTION
Viswanadham upon evaluation of the reviewers comments. This work was
supported by the National Science Foundation Engineering Research Center
P RODUCT variation reduction is an important engineering
for Reconfigurable Machining Systems under Grant EEC95-92125 at the
University of Michigan, and the valuable input from the Center s industrial objective in both design and manufacturing. For a mul-
partners.
tistage machining process, the product variation at certain
S. Zhou is with the Department of Industrial Engineering, University of Wis-
stages consists of two components: the variation brought by the
consin, Madison, WI53706, USA (e-mail: *****@****.****.***).
Q. Huang and J. Shi are with the Department of Industrial and Operations volumetric error of the current machine stage and the variation
Engineering, University of Michigan, Ann Arbor, MI 48109 USA (e-mail:
brought by the datum feature error produced from previous
******@*****.***; *******@*****.***).
stages. The second component exists because we have to use
Digital Object Identifier 10.1109/TRA.2003.808852
1042-296X/03$17.00 2003 IEEE
ZHOU et al.: STATE SPACE MODELING OF DIMENSIONAL VARIATION PROPAGATION IN MULTISTAGE MACHINING PROCESS 297
machining process where a part will be machined through dif-
part features produced by previous stages as the machining
ferent setups when it passes through this process. It is not neces-
datum in current operation. The variation from previous stages
sary that a multistage machining process contains multiple ma-
will be accumulated onto current operation.
chining stations. If there are different setups on only one ma-
At each single stage, there are many types of volumetric
chining station, this machining process is still considered as a
error sources, such as the geometric and kinematic errors,
multistage machining process.
thermal errors, cutting force induced errors, and fixturing
When a workpiece passes through certain stage of a multi-
errors. A huge body of literature can be found on the error
stage machining process, the machining error and fixturing error
modeling and compensation on a single machining stage. A
of this stage will be accumulated on the workpiece. These errors
review of these papers can be found in Ramesh et al. [1], [2].
will again affect the machining accuracy of the following stages
However, due to the complicated interactions between different
if the datum used by following stage is produced at current stage.
variation errors at different stages, very few attempts have been
Since the workpiece carries all the machining error information,
made on the variation propagation analysis for a multistage
a representation of accuracy of a workpiece is required to study
machining process. Mantripragada and Whitney [3] adopted
the complicated interaction of errors among different stages.
the concept of output controllability from control theory to
evaluate and improve the automotive body structure design.
A. Workpiece Geometric Deviation Representation
Lawless et al. [4] and Agrawal et al. [5] investigated variation
transmission in both assembly and machining process by To regulate the deviations of part features, researchers have
using an AR(1) model. Jin and Shi [6] proposed a state space developed standards for geometric dimensioning and toler-
model to depict the variation propagation in a multistage body ancing (ISO 1101 (1983) or ANSI Y14.5(1982)). However,
assembly process. Their approach cannot be applied directly these conventional geometric tolerances are originated from
to machining processes. Huang et al. [7] proposed a variation the hard gauging practice. They are not suitable for the working
propagation model for multistage machining process. However, principle of Coordinate Measurement Machine (CMM) that
that model is an implicit nonlinear model. Djurdjanovic and Ni is now a standard measurement equipment for machining
[8] extended Huang s formulation to obtain a linear model. In process. In addition, the representation of part feature in the
their derivation, a norm vector and a position vector are used conventional geometric tolerances does not conform to the part
to represent the geometric error of the workpiece and Taylor representations used in CAD/CAM systems.
series expansion is used to linearize the model. Although the Recently, some researchers [10] proposed a vectorial dimen-
final model is in linear form, the explicit expression of each sioning tolerancing (VD&T) strategy. The principle of VD&T
system matrix is not given. The physical insights we can obtain is based on the concept of substitute elements or substitute. A
are limited. substitute feature is an imaginary geometrical ideal feature (e.g.,
In this paper, an explicit analytical variation propagation plane, circle, line) whose location, orientation, and size (if ap-
model is developed for a multistage machining process. This plicable) are calculated from the measurement data points of
model is in the state space form. Differential motion vector, a the workpiece surface. Substitute features are represented by
concept widely used in robotics [9], is used as the state vector the location vector, orientation vector, and size(s). The location
to describe the workpiece geometric deviation. The theory of vector indicates the location of a specified point of the substitute
homogeneous transformation is heavily used in the derivation feature. The substitute orientation vector is a unit vector that is
and explicit expressions for all the system matrices are given. normal to the substitute plane or parallel to the substitute axis
The process and production information are quantitatively (cylinder, cone, etc). The size is available for some features. For
integrated together in the system matrices of this model. This example, the diameter is the size of a circular hole. The VD&T
model can be used for process evaluation in design and root workpiece feature representation follows the working principle
cause identification in manufacturing for multistage machining of CMM and CAD/CAM systems. The measurement data from
processes. CMM can be analyzed and compared with the design model di-
The terminologies, representations of part features, and as- rectly. The difference between the true feature and the design re-
sumptions of this model are introduced in Section II. The model quirement can be fedback to the manufacturing process directly.
itself is derived in Section III. Section IV presents the experi- It is a better tolerancing method for manufacturing process con-
mental validation results for this model. Finally, the conclusions trol [11].
and some discussion of the applications of this model are given In this paper, we adopt a vectorial feature representation pro-
in Section V. posed by Yau [12], [13]. The difference between his represen-
tation and the ordinary vectorial representation is in the orien-
tation representation. Instead of using a unit direction vector,
he used a vector that consists of three Euler rotating angles to
II. WORKPIECE GEOMETRIC DEVIATION REPRESENTATION AND
represent the orientation of the substitute feature. The represen-
MODEL ASSUMPTIONS
tation of using unit direction vector makes it difficult to desig-
A machining process is used to remove materials from the nate tolerance on the orientation for a general 3-D geometric
workpiece to obtain higher dimensional accuracy, better surface element. Moreover, the direction vector representation violates
finishing, or a more complicated surface form which cannot be certain functional requirements that VD&T intends to capture.
obtained by other processes. A complicated machining process Another advantage of angular representation of orientation is
is usually a multistage machining process, which refers to a that there are many mathematical tools available in the fields
298 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 19, NO. 2, APRIL 2003
The workpiece deviation before stage is represented by state
vector . is a stack of differential motion vectors of all
key features of the workpiece. If there are key features on a
workpiece, the dimension of vector will be by 1. Before
the feature is produced, the corresponding differential motion
vector is set to be zero. The initial value of state vector is
zero before the first stage. After the workpiece passes the first
stage, the differential motion vectors corresponding to the fea-
tures produced at the first stage are set to be nonzero values as
the dimensional deviations generated at the first stage. By this
way, state vector is generated. Similarly, the state vector
Fig. 1. Illustration of feature representation.
after the th stage can be generated. The workpiece de-
viation at stage comes from three sources: the datum-induced
of robotics and kinematics for this representation. Therefore, in deviation caused in previous stages, the machining inaccuracy
this paper, a location vector and a vector that consists of three at the current stage, and unmodeled noise. The deviation propa-
rotating Euler angles are used to represent a workpiece feature. gation can be written in the following linear discrete state space
Since the size of a feature is usually formed at one machining format [14]:
stage, it is not considered in the following derivation.
The part representation is illustrated in Fig. 1. O X Y Z
is the reference coordinate system (RCS). Plane 1 and Hole 2
(1)
are represented by the local coordinate systems (LCSs) that are
attached on them, respectively. The position and orientation of where represents the deviations of previously ma-
Plane 1 can be represented by . is the location vector chined features and the deviation of newly machined features
that points to the origin of LCS . is a vector contains roll, that is only contributed by the datum error, repre-
pitch, and yaw Euler rotating angles between coordinate system sents the workpiece deviation caused by the relative deviation
LCS and RCS. is in Fig. 1. Similarly, Hole 2 can between the workpiece and the cutting tool (this deviation is
be represented by . is in Fig. 1. caused by the fixture error and the imperfection of the tool path),
To describe the accuracy of a machined workpiece, we need is the measurement, is the unmodeled system noise,
to study the relationships between different features. Since and is the measurement noise. Two assumptions are im-
LCS is used to represent each feature, the relationships among plied in this formulation: 1) machining error on single stage is
different features can be described by the relationships of cor- modeled as a tool path deviation from its nominal path and 2)
responding coordinate systems. Homogeneous transformation only position and orientation error are considered. Profile error
matrix (HTM) as a mathematical tool is used to study the trans- is not included.
formation between different coordinate systems. Appendix I
The rationale of the first assumption is as follows. Many types
listed the basic notation and results of HTM used in this paper.
of machining error sources can affect the accuracy of the work-
If the feature index is, we denote the corresponding LCS as
piece on a single stage. According to Ramesh [1], they can
LCS . The deviation of the feature is described by the deviation
be categorized as quasistatic errors and dynamic errors. Quasi-
of the corresponding actual LCS from the corresponding nom-
static errors are the static or slow varying errors between the
inal LCS. As shown in Appendix I, the deviation of a LCS can
tool and the workpiece. They include the geometric and kine-
be represented by a differential motion vector . If the matic errors, thermal errors, cutting force induced errors, tool
actual location vector for the th element is and the corre- wear induced error, fixturing error, etc. Quasi-static errors ac-
sponding nominal location vector is, then . count for about 70% of overall machining errors. Dynamic er-
However, the orientation vector does not have this property, i.e., rors are caused by sources such as spindle error, machine struc-
in general, . The reason is that the multiplica- ture vibration, controller error, etc. They are more dependent on
tion of rotating matrix does not commute in general cases. particular operating conditions of the machine. The state vector
In summary, a location vector and an orientation angular of this model is the manifestation of all machining error sources
vector are used in this paper to represent a feature in the RCS. on the workpiece. It does not directly map to certain machining
The relative position and orientation of a feature is described error. Using the homogeneous transformation representation of
as a homogenous transformation matrix. The deviation of a the feature deviation, all the position errors and orientation er-
feature from its nominal value is represented by a differential rors can be included in this model. The detailed modeling of
motion vector. The rationale of this representation is that it certain machining errors for the whole working space of a ma-
conforms with the working principles of CMM and CAD/CAM chine stage is quite involved. For example, the geometric error
models. Moreover, many mathematical tools are available for that forms one of the largest sources of machining inaccuracy
analyzing this representation. contains 21 interrelated error components for a three-axis ma-
chine. To simplify the problem, the machining error input to the
B. Model Assumptions is represented as a deviation of the tool path from
model
Based on the workpiece geometric deviation representation, its nominal path. It is possible to map certain machining error
a multistage machining process can be described by Fig. 2. components to the tool path deviation.
ZHOU et al.: STATE SPACE MODELING OF DIMENSIONAL VARIATION PROPAGATION IN MULTISTAGE MACHINING PROCESS 299
Fig. 2. Diagram of a multistage machining process.
Fig. 3. Transition of differential motion vector: case 1.
Fig. 4. Transition of differential motion vector: case 2.
The rationale for the second assumption is that this model fo-
B. Model Derivation
cuses on describing the machining variation propagation among
different stages. Since most the profile errors are generated on a For the sake of convenience, the definitions of the four coor-
single stage and keep unchanged throughout the whole process, dinate systems involved in the derivation are listed.
it is not necessary to build a model describing its transforma-
1) LCS represents the features of the workpiece.
tion and propagation. Therefore, profile errors are not included
2) RCS is the reference for the features of the workpiece.
in this model.
The primary datum feature, which is explained in fol-
lowing section, is often selected as RCS. RCS is also
III. DERIVATION OF THE STATE SPACE MODEL FOR MULTISTAGE
called Part Coordinate System in some literature.
MACHINING PROCESSES
3) Fixture Coordinate System (FCS) is determined by the
To setup a state space model for the variation propagation actual fixture setup.
in a multistage machining process, we need to find a general 4) Nominal Fixture Coordinate System FCS is deter-
expression for the system matrices in (1). First, useful general mined by the ideal fixture setup. FCS is also called
properties of differential motion vector are introduced. Machine Coordinate System in some literature.
The state vector is defined as a stack of differential mo-
A. Properties of Differential Motion Vectors tion vectors corresponding to each feature w.r.t. RCS. It is clear
One common scenario of transition of differential motion that the reference feature does not have any deviation by defini-
vectors is given in Fig. 3. In this case, we know the deviation tion. There are three major components in :
of feature 1 w.r.t. the reference and the deviation of feature 2 1) Machining error, which is defined as the deviation of the
w.r.t. feature 1, we want to calculate the deviation of feature 2 cutting tool from its nominal path w.r.t. FCS. Since the
w.r.t. the reference. This problem is solved by Corollary 1. The LCS of the newly generated feature is determined by the
proof is given in Appendix II. cutting tool path, the machining error can be represented
Corollary 1: Consider a RCS and two features 1 and 2. Given by the deviation of LCS with respect to FCS., and the deviation of feature 1 w.r.t. RCS, and the 2) Fixturing error, which is caused by the imperfection of
deviation of feature 2 w.r.t. feature 1,, then the locators. It is represented by the deviation of FCS with
respect to FCS.
3) Datum error, which is the deviation of FCS with respect
(2)
to RCS.
Another important transition of differential motion vectors is The relationships among these coordinate systems and errors are
shown in Fig. 4. In this transition, we know the deviation of shown in Fig. 5.
The machining error is represented by a differential motion
feature 1 w.r.t. reference and the deviation of feature 2 w.r.t.
vector that describes the deviation of LCS w.r.t. FCS. It can be
reference, we want to calculate the deviation of feature 2 w.r.t.
feature 1. This problem is solved in Corollary 2. The proof is further decomposed into thermal, geometric error, and/or other
machining errors sources. To limit the scope of this article, this
also given in Appendix II.
Corollary 2: Consider a RCS and two features 1 and 2. Given part is not included here.
1) Analysis of Datum-Induced Error: The most commonly
and the deviation of feature 1 w.r.t. RCS, and the
used fixture scheme in practice is the 3-2-1 fixturing scheme. A
deviation of feature 2 w.r.t. RCS,, then
general 3-2-1 layout is shown in Fig. 6(a).
Surface ABCD defines the primary datum plane, which con-
(3) strains two rotational and one translational motion. ADHE is the
secondary datum plane, which constrains one rotational and one
These two corollaries are very useful when the RCS switches translational motion. CDHG is the tertiary datum plane, which
to another coordinate system. constrains the last translational motion. They are represented
300 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 19, NO. 2, APRIL 2003
F F is the X axis of FCS, Z axis is perpendicular to the pri-
mary datum plane. The LCS of the primary datum plane (sur-
face ABCD) is taken as the RCS. The secondary datum plane is
denoted as feature 2 and its deviation is represented as . The
tertiary datum plane is denoted as feature 3 and its deviation is
represented as . Given and, the deviation of FCS w.r.t.
RCS,, can be obtained by
(4)
where and are determined by the nominal positions of
feature 2, 3, and the fixture locating pins. and can be
obtained by the following general procedure.
Let and be the datum points touching the secondary
Fig. 5. Composition of overall feature deviation.
datum and be the datum point touching the tertiary datum.
The nominal coordinates of these three points in FCS are de-
noted as,, and . Denoting, we have
(5)
To guarantee that these points touch with the corresponding
part surface, the coordinates of these points in LCS are zeros
(the direction is defined as the normal direction of the sur-
face). Consider the first equation. Noting
and, the left-hand side of the first equation
of (5) changes to
(6)
(a)
The third element of is zero to guarantee touching, therefore
(7)
Under nominal conditions,,, and should touch with
datum plane. Hence, . Equation (7) changes
to
(8)
Denoting
we have
(b)
Fig. 6. 3-2-1 fixturing setup by a plane and three locators. (a) General 3-2-1
setup. (b) Variation of (a).
by LCS of O X Y Z, O X Y Z, and O X Y Z, respec-
(9)
tively. F, F, and F are the perpendicular projection points of
the locators P, P, and P on the primary datum. The fixture co- and
ordinate system (FCS) is shown in Fig. 6 as O X Y Z . F F
(10)
is the Y axis of FCS, the line passing F and perpendicular to
ZHOU et al.: STATE SPACE MODELING OF DIMENSIONAL VARIATION PROPAGATION IN MULTISTAGE MACHINING PROCESS 301
Similarly, we can get other two equations for and . Fi-, and, the solution
nally, we have of (11) can be obtained as (4), where
and
(11)
The procedure presented in this section can be used to study
the datum-induced error for a general 3-2-1 fixture setup. Fol-
lowing the same deviation, similar results can be obtained for
Although there are six parameters in, only three of them
other fixture setups, such as that used in turning operations.
are unknown nonzero values. By solving the above equation
2) Analysis of Fixture Errors: In a general 3-2-1 fixture
system, we can obtain the differential motion vector and
scheme as shown in Fig. 6(a), the workpiece position is
put it in the form of (4) by rearranging the terms. To guarantee
located by six locators P P P L L L . Assume that
the inverse of the matrix exists, the line passing the two datum
the nominal coordinates of these six points in the nominal
points on the second datum plane cannot be perpendicular to the
fixture coordinate system FCS are,,
primary datum plane. If so, the first and the second rows of the,,, and,
coefficient matrix of in the left-hand side of (11) will be the
respectively. (Note that we assume that the coordinates of
same.
P and P are the same to simplify the problem). If there are
The expression of and for a general 3-2-1 setup is very
small deviations on these six locators and
complicated. However, if datum fixtures 1, 2, and 3 are orthog-, where, the actual fixture
onal to each other, which is a very common case in practice,
coordinate system FCS will deviate from its nominal FCS.
and can be significantly simplified. Given the nominal posi-
Cai et al. [15] gave an analytical infinitesimal error analysis
tions for features 2 and 3 and locating pins in Fig. 6(a) as
for a rigid body locating scheme with general six points.
The fixture error can be derived based on their results. In
Fig. 6, the surface norm vector for L, L, and L is (0, 0,
1), for P and P is ( 1, 0, 0), and for P is (0, 1, 0).
With the locators position vectors, [15, eq. (3.1.6)] can
be applied to obtain the deviation of FCS w.r.t. FCS as,, and,, where
the solution of (11) can be obtained as (4), where
we have the equation shown at the bottom of the next page, and
.
We need to know as well. Note that is
an identity matrix,
. On the other hand,
. It is clear that
. From (A8), we have
(12)
For the fixture setup shown in Fig. 6(b), the nominal coordi-
nates of P, P, and P in FCS are (0, 0, 0), (0,, 0), and (0,
0, 0), respectively. Substituting these values into the expression
of, we can obtain another matrix for the particular setup
shown in Fig. 6(b) as
A very common variation of the general 3-2-1 fixture setup is
shown in Fig. 6(b). The six degree of freedoms of the workpiece
are constrained by the plane ABCD (two rotational and one
translational motion), a circular short hole P and P (two trans-
lational motion), and a slot P (one rotational motion). Given
the nominal positions for features 2 and 3 and locating pins in
Fig. 6(b) as
The fixture error analysis procedure for a general 3-2-1 fix-
ture scheme is presented in this section. For other fixture sys-
tems, the analysis can follow a very similar procedure.
302 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 19, NO. 2, APRIL 2003
Fig. 7. Steps of the derivation of variation propagation model.
3) Procedures for Variation Propagation Modeling: Fig. 7
. . . . . . .
shows the steps of the derivation of the variation propagation . . . . . . .
. . . . . . .
model. The first step is to re-locate the workpiece at current
stage . The second step calculates the datum-induced error,
. . . . . . .
. . . . . . .
which is caused by dimensional errors generated in previous . . . . . . .
stages. The third and fouth steps calculate the contribution of
. . . . . . .
fixture error and machining error that are only related with cur- . . . . . . .
. . . . . . .
rent stage. The fifth stage combines all errors together to ob-
tain the deviation of the newly generated features. Finally, at
the sixth step, the newly generated features are combined with
.
other features to form . The detailed derivations are as .
.
follows.
S1. Using Corollary 2, transform from a stack of
.
.
to a stack of, . is the .
feature index of the primary datum of stage . Let
be the stack of, then .
.
.
(13)
where
.
. where is an integer from 1 to the total number of features gen-
.
erated up to stage . In (13),,
. If primary datum is not changed between stage
.
.
. and stage, equals .
S2. Following the procedure of datum-induced error analysis
. in Section III, find datum error as
.
.
(14)
ZHOU et al.: STATE SPACE MODELING OF DIMENSIONAL VARIATION PROPAGATION IN MULTISTAGE MACHINING PROCESS 303
S6. Adding the newly generated features with the others
where . The dimension
yields
of is 6 by . as follows:
S3. Denoting the fixture imperfection as, we can obtain
(21)
the fixture error following the procedure of fixture error analysis
in Section III,, as where A is a selector matrix in the form
(15)
. . . . .
. . . . .
. . . . .
where .
S4. Because the machine tool is calibrated based on FCS,
. . . . .
the tool path imperfection is often represented as a deviation . . . . .
. . . . .
w.r.t. FCS. Denoting the machining error of newly generated
feature, is from 1 to, as and stack them up . . . . .
. . . . .
. . . . .
as . Using Corollary 1, we can find . Denote
as a stack up of,, yields . . . . .
. . . . .
. . . . .
(16)
where and Its dimension is by and it places the deviations
of the newly generated features on the corresponding location
when they are added to .
.
Considering (13) (21), we can get the state transition equa-
tion for the state space model
In (16),,,
and .
S5. Based on from step 2 and from step 4, we
(22)
can use Corollary 1 again to obtain the deviations of the newly
generated features with respect to RCS,, .
The detailed derivation can be found in Appendix III. In
In more detail, denoting as a stack of yields (1),,
and .
(17)
The derivation of the observation equation in (1) is straight-
Note that in (17) an approximation forward. The measurement stage can be viewed as a special ma-
chining stage. Assume that feature is used as the mea-
surement reference feature and denote as the stack of, following the same derivation of Step 1, we can obtain
(23)
(18) where has the same format as in (13) except that
is used in the place of . Hence, the observation
equation can be written as
is used because and are both small values. The
(24)
details are as follows. First note that
. Since where is a selector matrix that is similar to the format of
in (21). selects the measured features among all,,
the available features. Therefore, in (1), the observation equa-
or see (19), shown at the bottom of the page. Hence
tion is
(25)
(20)
Substituting (20) into the right-hand side of (18) and neglecting The above model describes the deviation propagation among
the second-order small values yields the approximation result of a multistage machining process. An experimental validation of
(18). this model is presented in Section IV.
(19)
304 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 19, NO. 2, APRIL 2003
Fig. 9. Operation sequence.
(a)
Fig. 10. Fixture locating schemes in three stages.
reference coordinate is the intersection point between the center
line of hole B and the plane determined by rough datum X,
X, and X, the y axis is the line connecting the centers of hole
B and hole C, the z axis coincides with the centerline of hole
B and points from the cover face to the joint face, the x axis
is determined by the right-hand principle. From Table III, it is
straightforward to obtain the nominal relationship between any
two features.
Following the procedure shown in Section III, a state space
model can be obtained for this three-stage machining process.
(b)
The state space model can be obtained as
Fig. 8. Final product. (a) Joint face of the engine head. (b) Cover face of the
engine head.
(26)
IV. EXPERIMENTAL VALIDATION OF THE STATE SPACE (27)
VARIATION PROPAGATION MODEL
The details of the system matrices are listed in Appendix IV.
The deviation propagation model is validated on a multistage
machining process developed at the NSF Engineering Research C. Comparison Between the Real Measurement and the Model
Center for Reconfigurable Machining Systems. The process is Prediction
introduced as follows. To validate the model, a workpiece is machined in this
testbed. In the machining, a fixture error is intentionally added
A. Experimental Machining Process
to the process at each stage. The inputs to the model that
The product is a V-6 automotive engine head. Its key features corresponds to the fixture error are
are shown in Fig. 8. H101 H108 are eight bolt holes on the joint
face. H101 and H104 are also called the locating holes B and C.
X, X, X, Y, Y, Z are the cast rough datum. H451 H458
are eight bolt holes on the cover face. S1 S12 are twelve screw
holes on the cover face. The nonzero values in correspond to the magnitudes of
There are three operations in this process. Each operation and the fixture errors. All the inputs corresponding to the machining
corresponding datum setup are shown in Figs. 9 and 10 and error are set as zeros. The workpiece is measured on a
Table I. CMM after machining. The measurement results from CMM are
The major tolerances on these key features are shown in listed in Table IV. For a plane feature, the CMM measurement
Fig. 11 and Table II. In Fig. 11, surface D is defined by the gives the coordinates of a point on the surface and the norm
rough data X, X, and X . direction vector
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