Error Analysis for the In-Situ
Fabrication of Mechanisms
Sanjay Rajagopalan
e-mail: [email protected]
Mark Cutkosky
e-mail: [email protected]
Center for Design Research,
Stanford University,
Palo Alto, CA 94305-2232
1
Fabrication techniques like Solid Freeform Fabrication (SFF), or Layered Manufacturing, enable the manufacture of completely pre-assembled mechanisms (i.e. those that
require no explicit component assembly after fabrication). We refer to this manner of
building assemblies as in-situ fabrication. An interesting issue that arises in this domain
is the estimation of errors in the performance of such mechanisms as a consequence of
manufacturing variability. Assumptions of parametric independence and stack-up made in
conventional error analysis for mechanisms do not hold for this method of fabrication. In
this paper we formulate a general technique for investigating the kinematic performance
of mechanisms fabricated in-situ. The technique presented admits deterministic and stochastic error estimation of planar and spatial linkages with ideal joints. The method is
illustrated with a planar example. Errors due to joint clearances, form errors, or other
effects like link flexibility and driver-error, are not considered in the analysis—but are
part of ongoing research. 关DOI: 10.1115/1.1631577兴
Introduction
The last decade of the millennium has seen the widespread
adoption of new ‘‘freeform’’ fabrication techniques. Called by
various names 共Rapid Prototyping, Layered Manufacturing, Solid
Freeform Fabrication etc.兲, this technology builds a part directly
from its digital 共CAD兲 representation by ‘‘slicing’’ the part model,
and building it incrementally by selectively adding and removing
material 关1兴 关2兴 关3兴.
Of particular interest for the purposes of this paper is the capability of these processes to fabricate assemblies 共e.g., mechanisms
with mating components兲 in-situ. In conventional fabrication,
each component of a device is individually fabricated and then
assembled together. For in-situ fabrication, the entire device is
built encapsulated in a sacrificial support material. This support
material is removed 共by etching, melting, or dissolving it away兲 to
yield the final part with operational mating and fitting features
共see Fig. 1兲.
This paper examines the manner in which manufacturing errors,
specifically errors in the spatial location 共position and orientation兲
of joints, affect the performance of mechanical devices fabricated
in-situ. It is well established in theoretical kinematics that the
primary determinant of mechanism behavior 共for rigid body
mechanisms兲 is the spatial location of its joints 关4兴. Consequently,
the focus of mechanism error analysis techniques on parametric
variability 共e.g. link length兲 is an artifact of the manufacturing
techniques used to fabricate these mechanisms.
In this paper, an assumption is made that the joint location
variability is process-specific, and is taken as the primary exogenous factor to the analysis. Given this assumption, the techniques
presented in this paper are not unique to any specific fabrication
process, nor are they only limited to the analysis of planar
mechanisms.
rors in the 1800s 关5兴. The problems typically occur due to inaccuracies in mechanism dimensions, poor joints, out-of-plane flexibility in links and assembly issues. These problems are
exacerbated in the construction of spatial mechanism prototypes.
The advent of Solid Freeform Fabrication 共SFF兲 could revolutionize the manner in which mechanisms are designed and fabricated 关6兴 关7兴 关8兴. In-situ technology allows for precision components, sensors, actuators and electronics to be directly integrated
into the mechanism frame during fabrication. Alternately, highprecision joints may also be directly built by freeform processes at
a specified location. Figure 2 shows examples of some mechanisms recently fabricated at Stanford University. Others have been
built at Rutgers University 关9兴 and Laval 关10兴. Similar devices are
found in the realm of microelectromechanical systems 共MEMS兲
which also use an incremental layered manufacturing technique,
with much smaller feature sizes 共see Fig. 3兲.
Whether at microscopic or macroscopic scales, in-situ manufacturing practices have a process flow 共Fig. 4兲 that is fundamentally
different from either traditional ‘‘craftsman’’ manufacturing or
conventional mass production. As described in the following sections, the difference in process flow leads to differences in the way
that dimensional errors are generated and accumulate, requiring a
different approach to tolerance analysis. We begin with a brief
review of classical tolerance analysis for mechanisms and use it as
a point of departure for the modified approach that is the main
contribution of this paper.
1.1 Scope of the Paper. This paper is concerned with the
study of general 共i.e. planar or spatial, open-chain or multi-loop兲
mechanisms, fabricated using freeform techniques.
As students in kinematics classes soon learn, the fabrication of
precise mechanism prototypes can be a complex, time-consuming
and sometimes, frustrating task. It is believed that Charles Babbage’s mechanical computing engine, a good example of a complex spatial mechanism, failed mainly because of the inability of
its fabricators to avoid accumulated component dimensional erContributed by the Mechanisms and Robotics Committee for publication in the
JOURNAL OF MECHANICAL DESIGN. Manuscript received March 2002; revised
April 2003. Associate Editor: J. S. Rastegar.
Journal of Mechanical Design
Fig. 1 Conventional versus in - situ fabrication
Copyright © 2003 by ASME
DECEMBER 2003, Vol. 125 Õ 809
Fig. 4 Comparing conventional and in - situ manufacturing
methods—process flow chart. Actions that impart accuracy to
the mechanism are specifically identified.
Fig. 2 In - situ mechanism prototypes fabricated via Shape
Deposition Manufacturing †3‡: „a… a polymer insect-leg prototype with embedded pneumatic actuator, pressure sensor and
leaf-spring joint „b… a hexapedal robot with integrated sensors,
actuators and electronics „c… an ‘‘inchworm’’ mechanism, with
integrated clutch components, „d… a slider-crank mechanism
made from stainless steel. Images courtesy the Stanford Center for Design Research and Rapid Prototyping Laboratories.
1.2 Introducing Mechanism Error Analysis. The modern
scientific treatment of mechanism error estimation dates to the
early 1960’s 关11兴 关12兴. In the several decades since, many alternative approaches to error analysis for mechanisms have been
proposed—each with various simplifying assumptions and different levels of complexity 关13兴 关14兴 关15兴 关16兴 关17兴. All approaches,
however, attempt to solve the same basic problem—to predict the
nature and amount of performance deterioration in mechanisms
as a result of non-ideal synthesis, fabrication, materials or
componentry.
In this paper the focus is on kinematic performance. In other
words, we assume that we are always able to describe the desired
task in terms of an output equation of the form:
y⫽ f 共 ⌽,⌰兲
(1)
where y denotes the (m⫻1) vector of output end-effector locations, coupler-point positions or output link angles, ⌰ is a (k
⫻1) vector of known driving inputs, and ⌽ is a (n⫻1) vector of
independent mechanism variables—including deterministic or
randomly distributed geometric parameters and/or dimensions.
The function f (•) is called the kinematic function of the mechanism and is, in general, assumed to be a continuous and differentiable 共i.e. smooth兲 non-linear mapping from the mechanism parameter space to an output space 共e.g. a Cartesian workspace兲. In
the absence of higher-pairs 共i.e. joints that have line and point
contact, as opposed to surface contact, between their member
links兲 and multiple-contact kinematics, the smoothness assumption generally holds true.
1.3 Conventional Mechanism Error Analysis. Conventional error analysis deals with degradation in the performance of
a mechanism as a result of parametric or dimensional variations,
and play in joints. The parameters typically considered are link
lengths for planar linkages, or some form of the DenavitHartenberg 关18兴 parameters for spatial linkages. Error in the performance of known mechanisms can be estimated analytically if
certain assumptions are made, rendering the underlying mathematical treatment more tractable. For example:
• Mechanism dimensions and parameters have a known, given
variability characteristic—either deterministic, or stochastic.
• Dimensional/parametric variations and clearance values are
significantly smaller than their nominal values.
• Individual component variations are independent, uncorrelated and identically distributed.
• The output is, at most, a weak non-linear function of the
mechanism parameters at the operating configuration of interest.
As a result of these assumptions, it becomes possible to approximate the actual error by lower-order estimates. Other assumptions 共e.g. negligible variability of the clearance value itself,
Normal or Uniform distribution of component parameters etc.兲,
which either eliminate unnecessary model complexity or enable
analytical tractability, are also commonly made.
Fig. 3 Micromechanisms and devices built using in - situ fabrication techniques. Images courtesy Sandia National Laboratories, SUMMiT„tm… Technologies, www.mems.sandia.gov.
Used with permission.
810 Õ Vol. 125, DECEMBER 2003
1.3.1 Sensitivity Analysis. Sensitivity analysis is based on
the Taylor-series expansion of the output function. As stated in Eq.
共1兲, the end-effector position, coupler path or output angle of a
mechanism can be expressed as:
y⫽ f 共 ⌽,⌰兲
(2)
Transactions of the ASME
where ⌰⬅ 关 ␪ 1 , ␪ 2 , ¯ , ␪ k 兴 T are the k known driving inputs, and
⌽⬅ 关 ␾ 1 , ␾ 2 , ¯ , ␾ n 兴 T are the n mechanism parameters 共or dimensions兲 subject to random, or worst-case deterministic, variability. Since ⌰ is assumed static for a given mechanism configuration 共i.e. the driving inputs are held perfectly to their nominal
values兲, it is dropped from the equation for notational simplicity.
The previous equation is re-written as:
y⫽ f 共 ⌽兲
(3)
Expanding this function in Taylor-series around the nominal
values
of
the
mechanism
parameters
(⌽ nom
nom
nom
nom T
⬅ 关 ␾ 1 , ␾ 2 , ¯ , ␾ n 兴 ), we get:
n
y⫽ f 共 ⌽nom兲 ⫹
兺
i⫽1
⫺ ␾ inom兲 2 ⫹
⳵f
⳵␾i
册
共 ␾ i ⫺ ␾ inom兲 ⫹
nom
⳵2 f
兺 ⳵␾ ⳵␾
i⬎ j
i
j
册
1
2!
n
兺
i⫽1
⳵2 f
⳵ ␾ i2
册
共␾i
nom
共 ␾ i ⫺ ␾ inom兲共 ␾ j ⫺ ␾ nom
j 兲 ⫹¯
nom
(4)
or, using a more concise notation:
y⫽ f 共 ⌽nom兲 ⫹
⳵f
⳵⌽
册
共 ⌽⫺⌽nom兲 ⫹
nom
1 ⳵2 f
2! ⳵ ⌽2
册
共 ⌽⫺⌽nom兲 2 ⫹¯
nom
(5)
For small, independent variations about the nominal configuration, a linear approximation can be made—thereby rewriting the
above equation as:
⳵f
⳵⌽
y⬇ f 共 ⌽nom兲 ⫹
or
⌬ y⬇
⳵f
⳵⌽
册
册
共 ⌽⫺⌽nom兲
(6)
nom
⌬⌽
(7)
nom
The quantity ⳵ f / ⳵ ⌽] nom is known as the sensitivity Jacobian of
the mechanism, evaluated at the nominal configuration. This Jacobian relates the component variability (⌬ ⌽) in the mechanism
parameter space to the output variation (⌬ y) in Cartesian space.
This is classical sensitivity analysis, where all variational effects
are bundled into a simple parametric space, and all higher order
effects are neglected.
Equation 共7兲 is used as the basis for error analysis and tolerance
allocation. For error analysis, the component variability (⌬ ⌽) and
sensitivity Jacobian ( ⳵ f / ⳵ ⌽) are known for a given mechanism
configuration. The output error (⌬ y) is then a simple calculation.
The component variability can either be expressed as worst-case
values, or as stochastic variations in link parameters. Each of
these approaches is discussed in the next sections.
For tolerance allocation problems, the maximum permissible
output error (⌬ max
y ) and sensitivity Jacobian are known. Equation
共7兲 forms the basis for the constraint equations, and the objective
is to maximize the overall variability 共i.e. ⌬ ⌽), given the constraints. Greater allowable variability typically means lower
manufacturing and inspection costs, and thus, is preferred. One
simple formalization of the tolerance allocation problem is as
follows:
n
minimize Z⫽
兺
i⫽1
1
⌬⌽
(8)
Here, an assumption is made that each component variability
parameter is weighted equally in the cost function, which may not
always be true. Some manufacturing parameters may be easier to
control accurately than others 共e.g. hole size can typically be held
to tighter tolerances than center-distance between holes兲. Additionally, zero tolerance 共or close-to-zero tolerance兲 for some parameters, which is permissible for the above formalization, is infeasible for real manufacturing processes. Non-homogeneous
manufacturing capability within the mechanism workspace is also
not considered in this system.
The optimization problem can be solved using standard methods of parametric programming—Lagrange multipliers, or Powell’s conjugate direction method 共i.e. unconstrained optimization
of a penalty function兲 关19兴. An example of these optimization
techniques applied to mechanism tolerance allocation can be
found in 关20兴.
1.3.2 Deterministic, Worst-Case Error Estimation. In worstcase error estimation, each parameter ␾ i is assumed to take 共exclusively兲 one of two deterministic values ␾ imin and ␾ imax . Furthermore, it is assumed that ␾ imin⭐␾inom⭐ ␾ imax ;i⫽1,2, . . . ,n, where
␾ inom is the nominal value of the ith parameter.
The objective of this kind of error estimation is to determine the
worst case envelope of the mechanism performance error. Except
for applications where performance within specified limits is absolutely critical, the worst-case analysis results in conservative
estimates of error 共and thereby, over-design of components兲. Since
the worst performance can occur for any combination of minimum and maximum component parameter values, the technique
proceeds by exhaustive calculation of total error for each combination of individual error values. For n parameters, this leads to a
search space of 2 n combinations for each mechanism configuration. If the objective is to find the worst-case performance within
the entire workspace of the mechanism, then this calculation has
to be repeated at each incremental driver position.
An alternative approach is to use dynamic programming 关21兴
关22兴 to estimate the maximum error without computing the total
error for every possible combination. The assumption made while
using this technique is that the global optimization problem can be
re-stated as a multi-stage optimization problem, with the nth stage
solution related to the (n⫺1)th stage solution through a functional equation. While this technique results in significant reduction of the computational burden involved, it is not guaranteed to
find the global optimum when the underlying monotonicity assumptions do not hold.
1.3.3 Stochastic Error Estimation. Statistical error estimation proceeds by assigning a probability distribution function
共PDF兲 to each variable parameter ␾ i . The component dimension
under consideration is assumed to be a random variable, distributed according to the characteristics of its underlying PDF, denoted as p ⌽i ( ␾ i ). The cumulative distribution function 共CDF兲 of
the output functions can then be estimated using standard techniques for stochastic analysis. If certain assumptions can be made
共e.g. linearity, independence, identical distribution etc.兲, the estimation of the distribution and moments of the output function is
highly simplified.
The error equation 共Eq. 共7兲兲 can be replaced by an equivalent
equation for the stochastic estimation of each output CDF, as
follows:
P Y j共 y j 兲 ⫽
冕
␾1
...
⫺⬁
冕
␾n
⫺⬁
y j .p ⌽1 . . . ⌽n 共 ␾ 1 , . . . , ␾ n 兲 d ␾ 1 . . . d ␾ n
where j⫽1,2, . . . ,m
subject to:
g 共 ⌽兲 ⬅⌬ max
y ⫺
⳵f
⳵⌽
册
Journal of Mechanical Design
and for independent and uncorrelated ␾ i
⌬ ⌽⭐0 and
n
nom
g i 共 ⌽兲 ⬅ ␾ i ⭓0; i⫽1,2, . . . ,n.
(10)
(9)
p ⌽1 . . . ⌽n 共 ␾ 1 , . . . , ␾ n 兲 ⫽
兿p
i⫽1
⌽i 共 ␾ i 兲
(11)
DECEMBER 2003, Vol. 125 Õ 811
In general, the complete analytical evaluation of the integrals in
Eqs. 共10兲 and 共11兲 are not simple, or even tractable. However, it
may not always be necessary to evaluate the error CDF. Given
certain assumptions, it is possible to determine the mean and variance of the output distribution directly from the mean and variance ( ␮ ␾ i , ␴ ␾2 i ) of the individual components. To do this, the output 共Eq. 共3兲兲 is expanded in a Taylor series about the mean values
( ␮ ␾ i ) of the component dimensions as follows:
n
1
⫹
2!
⫹
兺
i⬎ j
n
i⫽1
i⫽1
册
⳵2 f
兺 ⳵␾
⳵f
兺 ⳵␾
y⫽ f 共 ␮ ␾ i ;i⫽1,2, . . . ,n 兲 ⫹
共 ␾ i⫺ ␮ ␾i兲
2
i ␮
⳵2 f
⳵ ␾ i⳵ ␾ j
册
i
册
共 ␾ i⫺ ␮ ␾i兲
␮
2
共 ␾ i ⫺ ␮ ␾ i 兲共 ␾ j ⫺ ␮ ␾ j 兲 ⫹¯
(12)
␮
Assuming that the output is approximately linear for small
variations of the random variables about their mean values, the
higher-order terms in the above equation can be dropped, and the
equation re-written as:
n
y⬇a⫹
⳵f
兺 ⳵␾
i⫽1
i
册
共 ␾ i⫺ ␮ ␾i兲
(13)
␮
where a⬅ f ( ␮ ␾ i ;i⫽1,2, . . . ,n), and the partials are evaluated at
the mean value of the parameters. Equation 共13兲 can be written in
terms of the proxy 共difference兲 variables ⌬ y and ⌬ ␾ i 共see Eq. 共7兲兲
as:
n
⌬ y⬇
⳵f
兺 ⳵␾
i⫽1
i
册
⌬ ␾i
␴ 2y ⫽
兺冉
i⫽1
⳵f
⳵␾i
冊
(15)
2
␮
␴ ␾2 i
where ␮ y and
denote the mean and variance, respectively, of
the output function.
The full derivation of Eq. 共15兲 is given in Appendix A, as the
treatment is important for the extension of this model to the case
of in-situ fabrication. A key assumption in this treatment—that of
parametric independence—fails in the case of in-situ fabrication,
and Eq. 共15兲 needs modification.
The specific probability of the output falling within a given
range y1 ⭐y⭐y2 can either be estimated using the standard tables
共for normal distributions兲, or the Chebychev inequality 共for a symmetric range兲. Since the linearized equation approximates the output error as a weighted sum of the component variation, a normal
output distribution can be assumed either when the individual
component variations are each normally distributed, or when the
Central Limit Theorem can be applied with Liapunov’s condition
关23兴 关20兴. Thus, the validity of the linear approximation is a fundamental defining assumption in this type of analysis, since no
simple general technique 共other than numerical simulation兲 is
␴ 2y
812 Õ Vol. 125, DECEMBER 2003
1
␮ y⫽a⫹
2
兺 冉 ⳵␾ 冊
n
␴ 2y ⫽
where ⌬ y and ⌬ ␾ i are zero-mean random variables with all
higher-order moments identical with y and ⌽i respectively. In
other words, by studying the variance properties of Eq. 共14兲, we
are in effect studying the variance properties of the original equation 共i.e. Eq. 共13兲兲.
If the parameters ⌬ ␾ i are assumed to vary independently, then it
can be shown 共see Central Limit Theorem 关23兴兲 that the output y
follows an approximately Normal distribution 共for n⬎5), with the
mean and variance of the distribution given as follows:
n
available for the estimation of the probability distribution of a
complex non-linear function of random variables.
In the event that the assumption of weak non-linearity of the
output function does not hold, then a second order estimate of the
mean a variance may yield better results. This is given as 共derivation follows from results in Appendix A兲:
(14)
␮
␮ y⫽a
Fig. 5 Modified Denavit-Hartenberg representation for spatial
linkages „Lin and Chen, 1994…. Note that the specific mechanism shown here is irrelevant—used for illustrative purposes
only.
⳵f
i⫽1
⫹
兺冉
i⫽ j
n
兺
i⫽1
冉 冊
⳵2 f
⳵ ␾ i2
2
i ␮
1
2
␴ ␾2 i ⫹
⳵2 f
⳵ ␾ i⳵ ␾ j
冊
n
兺
i⫽1
␮
␴ ␾2 i
冉 冊
⳵2 f
⳵ ␾ i2
2
␮
␴ ␾2 i
2
␮
␴ ␾i␴ ␾ j
(16)
1.3.4 Kinematic Representations. The preceding sections
present a generic treatment of error estimation where no assumption is made regarding specific parameter assignments to the
mechanism geometry. Mechanisms can be described using dimensions of geometric elements 共e.g. link length for planar linkages兲
or using mechanism parameters 共e.g. link length, link angle, offset
and twist for spatial linkages兲. Typically, the assumptions made in
the sensitivity calculations detailed above will fail for certain
mechanism instances, depending upon the specific representation
used 关24兴.
A widely accepted parametric representation for spatial mechanisms is the Denavit-Hartenberg 共or D-H兲 representation 关18兴, and
the extensions thereof 关25兴. In this representation 共see Fig. 5兲 a
spatial mechanism is described in terms of four parameters for
each link i in the linkage. These parameters are termed the linkangle ( ␪ i ), link-length (a i ), link-offset (d i ), and twist-angle ( ␣ i ).
In a mechanism with revolute, prismatic and cylindrical joints, the
link-lengths and twist-angles typically remain static during operation, and the link-angles and link-offsets vary 共depending upon the
type of joint兲.
The Denavit-Hartenberg representation presents difficulties for
error analysis when mechanisms have parallel or nearly parallel
joint axes. Small variations in the D-H parameters result in large
errors in the output function. Various modifications have been
proposed 关25兴 关24兴 that rectify this problem. For this paper we
adopt the representation of 关26兴 that adds an extra parameter (l i ),
resulting in a better representation of the link shape 共see Fig. 5兲.
The extra parameter does not add anything to the kinematic description of the mechanism but is advantageous for error analysis.
Transactions of the ASME
Fig. 6 The effects of link length variation in an assembled 4-bar mechanism
2
Worst-Case Error Analysis for In-Situ Fabrication
The conventional error models presented in Sec. 1.3 cannot be
directly applied to in-situ fabrication since this fabrication technique differs from conventional sequential shape-and-assemble
fabrication techniques in some fundamental ways. Primarily, the
differences are:
• In-situ fabrication is blind to conventional component boundaries. Consequently, the input to the analysis is not the dimensional variability in links, but the absolute position and orientation
variability in joints. As the assembly is built, joints are created
directly or embedded within a surrounding matrix of part and
support material. Links are formed around the joints. Parametric
variability is therefore a function of joint placement accuracy.
• Tolerance stack-up due to dimensional/parametric errors in
components is not an issue for in-situ fabrication. Instead, joints,
and other features such as coupler points or end-effectors, are
placed in the workspace with a known absolute accuracy.
• Gaps and clearances in joints are manifest directly in the
geometry of the support structure. In conventional fabrication, the
gap geometry is a consequence of the interaction amongst
complementary mating/fitting feature geometries.
• Conventional error analysis does not explicitly allow for the
consideration of variable accuracy within the manufacturing
workspace. But when entire mechanisms are fabricated In-situ, the
build configuration 共or pose兲 can be chosen to make best use of
the manufacturing error characteristics.
These differences are accounted for in the general abstract
model for in-situ fabrication and the associated error analysis
techniques presented below.
2.1 An Abstract Model for In-Situ Fabrication. The main
difference between conventional error analysis, and error analysis
for in-situ fabrication lies in the form of the inputs into the model.
Conventional error analysis treats parametric variability 共i.e. variability in link-lengths etc.兲 as a given constant input. In-situ error
analysis estimates parametric variability for each build configuration from the location variability of the joints that make up the
linkage. The parametric variability is determined by the sensitivity
of each parameter to the joint positions and orientations at a given
build pose. An important observation is that the mechanism parameters that result from such fabrication are not independent, but
pair-wise correlated. This is because multiple 共adjacent兲 parameters depend upon the same independent inputs 共i.e., the positions
and orientations of their shared joints兲. Although several parameters can all be adjacent to each other if they share a common
joint, their correlation is still taken pair-wise since covariance is
defined on random variable pairs. The degree of correlation depends upon the configuration in which the mechanism is fabricated 共also called the build pose兲. The output variability, in turn, is
determined by the sensitivity of the output function to the mechanism parameters at each operating configuration. Figures 6 and 7
illustrate the fundamental differences between the two scenarios,
for the simple case of a four-bar mechanism.
2.2 Frames and Notation. We assign a global workspace
datum frame (OXY Z) and local datum frames (o i x i y i z i ) associated with each feature of interest, 共see Fig. 8兲. Without loss of
generality, it can be assumed that the z-axis of the global frame is
aligned with the process growth direction 共e.g. vertical, or spindleaxis兲. If the feature of interest is a joint, then it is assumed that the
local joint z-axis (z i for the ith joint兲 is aligned with the jointfreedom axis 共i.e. nominal pin/shaft axis for revolute joints, direction of translational motion for prismatic joints etc.兲. The direction
of the x-axis of the ith frame (x i ) is taken as that of the common
normal between the ith and 共adjacent兲 i⫺1th nominal joint axes.
Typically, the position and orientation of each feature frame is
specified in the global frame, and the feature geometry is specified
in the local frame. The nominal location of the origin in the ith
local frame is represented as the position vector pi in the global
frame 共or alternately, as the homogeneous coordinates
关 x i ,y i ,z i ,1兴 ), and the nominal orientation of the ith frame is represented by the direction vector zi 共with direction numbers
关 l i ,m i ,n i 兴 ). Alternately, the z-axis of the joint frame can be
uniquely represented in a global frame in terms of its Plücker 关27兴
coordinates (Qi ,Q⬘ i ), where:
Qi ⬅ 关 q 1i ,q 2i ,q 3i 兴
(17)
are the direction numbers, and:
⬘ ,q 2i
⬘ ,q 3i
⬘兴
Q⬘ i ⬅pi ⫻Qi ⬅ 关 q 1i
(18)
2
is the moment vector of the line. Furthermore, we can let q 1i
2
2
⫹q 2i ⫹q 3i ⫽1 without any loss of generality, making these coordinates the same as the direction cosines of the line.
Fig. 7 The effects of joint location variation for an in - situ fabricated 4-bar
mechanism
Journal of Mechanical Design
DECEMBER 2003, Vol. 125 Õ 813
SLS, Stereolithography etc. The variability region is simply a
worst-case or stochastic characterization of the variation in frame
position and orientation, given its nominal location and other
process-specific parameters. While this methodology extends to
the general spatial scenario, it is illustrated here with a simple
planar example.
2.3.1 Planar Example. In the planar case, the orientations of
the joint axes 共i.e. zi ) are discarded, as all joint axes are assumed
parallel. Given a nominal joint location pnom⬅(x nom,y nom), the
precision function returns a region R as follows:
R⫽ ␶ 共 x nom,y nom, ␲ 兲
(21)
In deterministic worst-case analysis, this function returns the extremal positions of the region in which the actual joint lies, as
follows:
R⫽ 关 worst⫺case,x min,x max,y min,y max兴
Fig. 8 Frames and notation for the abstract model of in - situ
fabrication
Similarly, in stochastic analysis, the function returns a probability
distribution that describes the position of the point as a random
variable, as follows:
R⫽ 关 normal, ␮ x , ␴ 2x , ␮ y , ␴ 2y 兴
Thus, using this representation, the nominal configuration
(C nom) of a mechanism can be represented in terms of the local
frame positions and orientations as:
C nom⬅ 兵 共 pinom ,zinom兲 其 ; i⫽1,2, . . . n
(19)
or alternately, in terms of the joint-axis Plücker coordinates as:
C nom⬅ 兵 共 Qi ,Q⬘ i 兲 其 ; i⫽1,2, . . . n
(20)
Fabrication proceeds by constructing or embedding non-ideal
joints at the given nominal locations. By quantifying the extent of
these errors, it is possible to predict overall performance errors in
mechanisms fabricated in-situ. The complete procedure is described in later sections 共Secs. 2.4 and 3兲.
2.3 Heterogeneous Workspace Modeling. For modeling
variable fabrication accuracy within the process workspace, we
assume that we have a precision function 共␶兲 that returns the variability region R of a joint in the build space, given the nominal
position and orientation, and other process parameters 共␲兲. Note
that the precision function ␶ is process-specific and needs to be
empirically determined for each process, such as SDM, FDM,
(22)
(23)
In the most general case, R is a closed region of arbitrary geometry within which the actual joint position (x,y) lies with a known
probability distribution. By applying the precision function ␶ to all
the joint and coupler points (x inom ,y inom) in a planar mechanism,
we get joint variability regions R i as:
R i ⫽ ␶ 共 x inom ,y inom , ␲ 兲
(24)
In other words, the regions R i determine the characteristics of the
interval or random values that represent the variable nature of the
joint locations. The mechanism parameters ␾ i are functions 共e.g.
distance function of the form ␾ i ⫽ 兵 兺 (x i ⫺x j ) 2 其 1/2) of the positions and orientations, and the parametric variability is a function
of the joint variability regions (R i ), all at the given build configuration (C b ):
⌬ ␾ i ⫽⌬ ␾ i 共 R 1 ,R 2 , . . . R n 兲 ;
i⫽1,2, . . . n
(25)
Error analysis involves estimating the variability in the link
parameters ␾ i using the above equation, and then applying sensitivity analysis techniques to determine the error in the output
function 共at various operating configurations兲 for a mechanism
Fig. 9 Actual and schematic diagrams of the planar 4-bar crank-rocker mechanism used as an
example in this paper. The parameter values are: L 1 Ä15 cm, L 2 Ä5 cm, L 3 Ä25 cm, L 4
Ä20 cm, L 5 Ä7.5 cm, L 6 Ä20 cm „after Mallick and Dhande, 1987…. Stochastic simulations on
the example are performed with a positional variance ␴ x2 k Ä0.01 cm2 . Worst case simulations
are performed with a positional variability of 0.3 cm, equivalent to the 3␴ stochastic error.
814 Õ Vol. 125, DECEMBER 2003
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Fig. 10 Multiple positions of the example 4-bar mechanism, corresponding
to 30 deg increments of the input angle, ␪
that is fabricated in-situ. In the following sections, this process is
described, and illustrated using the specific planar 4-bar mechanism shown in Fig. 9. The mechanism parameter values were
chosen to allow checking of results with earlier published work
关28兴. However, in that case, the authors consider a clearance error
of 0.05 cm in the joints, along with a 0.5 percent error in link
length. Since neither link variability or clearance are variables
in our analysis, the comparison is qualitative. To aid with discussion of the results, the mechanism is also shown in various
configurations 共i.e. specific values of the driving angle ␪兲 in Fig.
10. In Sec. 4, the analysis is extended to cover general spatial
mechanisms.
2.4 Error Estimation. Worst case error estimation for insitu fabrication proceeds in two stages. First, at a candidate build
pose C b , all the worst case parameter values ( ␾ iWC ) are evaluated
by choosing, in sequence, all possible combinations of the worstcase fabrication input values 兵 (piWC ,ziWC ) 其 .
The precision function 共Eq. 共24兲兲 returns these extremal values
of the position of each joint, given the mechanism nominal build
pose. For k fabrication input variables, this process generates 2 k
candidate mechanisms at each pose C b . Figure 11 shows the example of a mechanism with 3 mm square precision regions and a
candidate build configuration.
In the second stage, the error in the output function 共y兲 is evaluated for each one of the candidate mechanisms produced in the
first stage. This calculation is repeated for all operating angles, for
every build pose. Overall, if c operating and build positions are
considered for a mechanism with m independent degrees of freedom, and k independent fabrication variables, the determination
of worst-case error boundaries for the output has computational
complexity O(2 k mc 2 ). Dynamic programming approaches 关22兴
can significantly improve upon the computational complexity, but
need to be re-stated appropriately for each specific problem.
Figure 12 illustrates the results of the worst-case error estimation for the example 4-bar mechanism for a few candidate build
poses. The coupler-point location is shown as a cloud of points in
the vicinity of the nominal coupler-point, with each point corresponding to one combination of worst-case joint locations. Figure
13 plots the worst-case variability of the coupler-point location
共i.e. half the perimeter of the bounding box for each cloud in Fig.
12兲 as a function of the build configuration. Of the four build
configurations evaluated, the one corresponding to ␪ ⫽180 deg is
evidently best for minimizing the worst-case errors in coupler
position.
3
Fig. 11 An example build configuration and worst-case variations in joint and coupler point locations
Journal of Mechanical Design
Stochastic Error Analysis for In-Situ Fabrication
The worst-case method presented in the previous section is both
overly conservative, and computationally expensive for most applications. By contrast, a stochastic approach results in superior
DECEMBER 2003, Vol. 125 Õ 815
Fig. 12 Worst case coupler-point positional error, plotted on the coupler path
error estimates in constant-time 共as opposed to exponential or linear time for worst-case methods兲. However, the conventional approach to stochastic error estimation needs modification in order
to be applicable to in-situ fabrication.
In this analysis, we assume that the joint coordinates 共positions
and orientations兲 are independent random variables with known
distributions. Given the nominal location of a joint i, the precision
function 共Eq. 共24兲兲 returns the appropriate distribution for its actual location. Mechanism parameters 共like link-lengths, joint
angles, joint offsets and skew angles兲 are functions of the independent, random joint coordinates. This, in turn, makes the parameters themselves random variables which are pairwise correlated
共being jointly dependent on the same independent variables兲. The
output, then, is a complex function of correlated random variables.
The probability distribution 共i.e. PDF兲 of a known function of
random variables can, in principle, be derived exactly from the
given, analytically specified, distributions of the original random
variables. However, in practice, the exact derivation is intractable
in the absence of certain simplifying assumptions, due to the complexity of the algebra involved. For a weakly non-linear function
of independent and uncorrelated random variables, the mean and
variance of the function can be approximated directly from the
mean and variance of the underlying random variables, as illustrated in Eqs. 共15兲 and 共16兲. When the simplifying assumptions
共i.e. independent and uncorrelated兲 do not hold, the function properties need to be determined analytically by integrating the jointPDF 共see Eq. 共28兲兲, by modifying the approximation techniques to
include the effects of correlation, or by using Monte Carlo simulation techniques. In general, the analytical technique is not tractable for all but the simplest of cases. In the following section, an
improved approximation technique for the estimation of the moments of a weakly-nonlinear function of correlated random vari-
Fig. 13 Total worst-case coupler-point positional errors, plotted against operating angle for four
different build configurations, corresponding to different values of the input angle. The error values
can be compared with 3␴ stochastic errors „see Fig. 17….
816 Õ Vol. 125, DECEMBER 2003
Transactions of the ASME
Fig. 14 First order estimates of the link-length variance compared to the results of a Monte Carlo
simulation
ables is developed and applied to the problem of stochastic error
estimation for mechanisms that are fabricated in-situ. The results
are compared to those obtained by Monte Carlo simulation.
3.1 Estimating the Parametric Variance. Equation 共15兲
can be applied directly to the mechanism parameters ( ␾ i ), given
the stochastic properties 共i.e. mean and variance兲 of the joint variables (x k ). The parameters are simple functions 共i.e. sums, products and differences兲 of the joint variables, which are assumed
independent and uncorrelated. Moreover, the variance in any joint
variable can be assumed to be much smaller than its mean 共for
macro-scale devices兲, since the precision of fabrication equipment
is typically several orders-of-magnitude smaller than the part
dimensions. This implies that the variability in the mechanism
parameters can be approximated as a linear function 共weighted
by the sensitivity coefficients兲 of the variability in the input, as
follows:
␴ ␾2 i ⬇
兺冉 冊
⳵␾i
⳵xk
k
2
␮
␴ x2k ; i⫽1,2, . . . n
(26)
where ␴ ␾2 i is the variance of the ith mechanism parameter, and x k
represents the kth joint variable, and ␴ x2k represents the variance
of the kth joint variable. If the joint variables follow Normal distributions 共typical for most physical random processes involving
many noise factors兲, then the parameters too will follow a Normal
distribution.
The parameters ␾ i , however, are correlated random variables.
The correlation coefficients ( ␳ i j ) of each parameter pair ( ␾ i , ␾ j )
can be approximated using the sensitivity coefficients as follows:
兺 冉 ⳵x 冊
⳵␾i
␳i j⬇
k
k ␮
冉 冊
⳵␾ j
⳵xk
␴ ␾i␴ ␾ j
␮
␴ x2k
3.2 Estimating the Output Variance. In the previous section, we have established a method for efficiently estimating the
variance and correlation coefficients of the parameters of a
mechanism that has been fabricated in-situ. Our real interest in
this treatment, however, is in the behavior of the output function
共y兲 during operation. As indicated earlier, the output is a function
of the mechanism parameters which, being dependent functions of
the given independent random variables 共i.e. the joint variables兲,
are themselves correlated random variables. Thus, the simplifying
assumptions which could be made for the estimation of parametric variability are not applicable for the estimation of output variability. No simple analytical technique exists for the determination of the distribution of a general function of correlated random
variables. In theory, the cumulative distribution function of the
output can be evaluated as follows:
PY共 y 兲⫽
冕 冕
␾1
⫺⬁
¯
␾n
⫺⬁
f 共 ␾ 1 , . . . ␾ n 兲 .p ⌽1 . . . ⌽n 共 ␾ 1 , . . . ␾ n 兲
⫻d ␾ 1 . . . d ␾ n
(28)
(27)
Figure 14 compares the first order estimate of link-length variability against that obtained by Monte Carlo simulation, for the
Journal of Mechanical Design
links in the example 4-bar in Fig. 9. 共Since the mechanism is built
in-situ, the link length variation is a consequence of variations in
joint location.兲
Figure 15 compares the pairwise correlation coefficients obtained for the approximation in Eq. 共27兲 against those obtained by
Monte Carlo simulation, for the same four links of the example
4-bar. In both cases, the approximation yields results that are very
close to the simulation—illustrating the validity of the assumption
of independence. Note also that these results hold for an example
with tolerances that are looser than is common in macroscopic
devices. As a percentage of the link lengths, the tolerances are
more characteristic of MEMS devices.
However, the joint distribution function p ⌽1 . . . ⌽n ( ␾ 1 , . . . ␾ n ) is
not easy to determine when the random variables ␾ i are correlated. Furthermore, the upper limits of the multiple integral need
DECEMBER 2003, Vol. 125 Õ 817
Fig. 15 Pairwise correlation coefficients of the link lengths—first-order results compared to the Monte
Carlo simulation
to be expressed in terms of the output variables, which is not
analytically feasible except for the simplest of cases.
The assumption that makes this problem tractable, once again,
is that of weak-nonlinearity in the output function. In other words,
if we can assume that the second and higher-order terms in the
Taylor Series expansion of the output function can be discarded,
then it is possible to derive an expression that directly produces an
approximate estimate for the output variance, given the variance
( ␴ ␾2 i ) and correlation coefficients ( ␳ i j ) of the mechanism parameters. Furthermore, if the total number of parameters is large 共i.e.
n⬎5), then, according to the Central Limit Theorem, the output
function will follow an approximately Normal distribution, regardless of the individual parameter distributions 关23兴. Thus, by
making the linear approximation, we completely side-step the
evaluation of the extremely problematic multiple integral in Eq.
共28兲. The derivation of the approximation equation is given in
Appendix A, and the final result is summarized below:
兺 冉 ⳵␾ 冊
n
␴ 2y ⬇
i⫽1
⳵f
2
i ␮
␴ ␾2 i ⫹2
⳵f
兺 兺 ⳵␾
i
j
i
册 册
␮
⳵f
␳ ␴ ␴
⳵ ␾ j ␮ i j ␾i ␾ j
(29)
where i⫽1,2, . . . n and j⫽i. In the special case where only adjacent parameters share a joint variable, ␳ i j ⫽0 for non-adjacent
parameters, and the above equation needs to be evaluated only for
the cases where j⫽i⫺1. Note that all the sensitivity coefficients
in the above equation are evaluated at the nominal operating configuration 共␮兲 of the mechanism. Comparison of Eq. 共29兲 and Eq.
共15兲 reveals that they differ only in the second term on the RHS.
This term, then, is the adjustment term that accounts for the correlation effect that results from the co-dependence of the mechanism parameters on the same joint coordinates.
Summarizing, the first order approximations are the only tractable, general purpose estimates of the output function variability.
Equation 共29兲 indicates that the output error depends upon the
output function sensitivity coefficients 共evaluated at the nominal
operating configuration兲, the parametric variances, and the pairwise correlation coefficients of the parameters. The parametric
818 Õ Vol. 125, DECEMBER 2003
variances and the correlation coefficients are functions of the
mechanism build pose, during in-situ fabrication. Equation 共29兲
succinctly relates the fabrication workspace to the operational
workspace, thereby presenting us with a method for evaluating the
optimal build pose, given an operational tolerance specification.
This issue is explored in more detail in 关29兴.
Figure 16 compares the first order estimated coupler-point error
for the example 4-bar fabricated in-situ against the Monte Carlo
simulations of the same quantity. Also included are the estimates
using the conventional approach, which does not include the consideration of correlation effects. Comparisons can also be made
between these results, and those of the worst case error estimate
presented earlier 共see Fig. 13兲. The worst-case and stochastic estimates for a specific build angle are compared in Fig. 17. It is
clear from the comparison that the worst-case method is significantly more conservative in its estimation of output error.
Figure 18 plots the simulated coupler-point variance against the
number of random trials. This helps with the estimation of the
minimum number of trials needed in order for the random estimates to converge to a steady value 共between 4000 and 10,000 in
this case兲.
4
Extension to Spatial Parameters
While the detailed treatment of spatial error analysis, with supporting numerical results, is beyond the scope of this paper, the
theoretical extension of the error analysis techniques presented
above to spatial systems is straightforward once the essential concepts have been established. Spatial systems are traditionally described in terms of the Denavit-Hartenberg parameters 共see Section 1.3.4兲, or modifications thereof. Spatial error analysis is the
process of relating variability in the spatial parameters to errors in
the output function.
For in-situ fabrication, parametric variability is not directly
available, but is a function of the position and orientation variability in joint placement. Earlier sections in this paper have dealt
with the issue of estimating the output variance, given the stochastic characteristics of the joint variables. The approach has been
Transactions of the ASME
Fig. 16 First order estimates of coupler-point variance for the an in - situ fabricated
crank-rocker mechanism „Fig. 9… using conventional stochastic analysis, analysis
modified to for in - situ fabrication, and direct Monte Carlo simulation
illustrated using a planar example, and the technique is extended
here to cover general spatial mechanisms. The basic issue that
remains to be addressed for the spatial case is that of explicitly
expressing the spatial parameters illustrated in Fig. 5 in terms of
the joint-frame positions illustrated in 8. This is a fairly simple
problem in the analytical geometry of three dimensions 关30兴.
Given the origin coordinates (pi ,p j ,pk ) and the direction numbers (zi ,z j ,zk ) of the axes of three adjacent spatially located
joints, the modified Denavit-Hartenberg parameters of the jth
joint can be expressed in terms of the joint Plücker coordinates
共see Section 2.1兲 of the three joint axes (Qi ,Q⬘ i ), (Q j ,Q⬘ j ) and
(Qk ,Q⬘ k ), and those of the two common normals (Qi j ,Q⬘ i j ) and
(Q jk ,Q⬘ jk ). This notation is illustrated in Fig. 19. The direction
coordinates of the common normal are given as:
Qi j ⬅ 关 q 1i j ,q 2i j ,q 3i j 兴 where
q 1i j ⫽q 2i q 3 j ⫺q 3i q 2 j
and the moments of the common normal between axes i and j are
given as follows 共this can be extended to j and k by symmetry兲:
⬘ j ,q 2i
⬘ j ,q 3i
⬘ j 兴 where
Q⬘ i j ⬅ 关 q 1i
⬘ j⫽
q 1i
Qi j • 关共 q 2i j q 3 j ⫺q 3i j q 2 j 兲 Q⬘ i ⫺ 共 q 2i j q 3i ⫺q 3i j q 2i 兲 Q⬘ j 兴
储 Qi j 储 2
⬘ j⫽
q 2i
Qi j • 关共 q 3i j q 1 j ⫺q 1i j q 3 j 兲 Q⬘ i ⫺ 共 q 3i j q 1i ⫺q 1i j q 3i 兲 Q⬘ j 兴
储 Qi j 储 2
⬘ j⫽
q 3i
Qi j • 关共 q 1i j q 2 j ⫺q 2i j q 1 j 兲 Q⬘ i ⫺ 共 q 1i j q 2i ⫺q 2i j q 1i 兲 Q⬘ j 兴
储 Qi j 储 2
(31)
q 2i j ⫽q 3i q 1 j ⫺q 1i q 3 j
q 3i j ⫽q 2i q 3 j ⫺q 3i q 2 j
Journal of Mechanical Design
(30)
The modified Denavit-Hartenberg parameters for link j can
now be written as:
DECEMBER 2003, Vol. 125 Õ 819
Fig. 17 Comparison of 3␴ stochastic and worst-case „deterministic… error estimates
for crank-rocker mechanism
mation matrices 共three translations and two rotations兲, that transform one local coordinate frame to the adjacent frame 共the jth
frame to the kth frame in this case兲, as follows:
␣ j ⫽arcsin共 储 Q jk 储 兲
a j⫽
Q j •Q⬘ k ⫹Qk •Q⬘ j
sin共 ␣ i 兲
Akj ⫽T共 0,0,d j 兲 ⫻R共 z j , ␪ j 兲 ⫻T共 a j ,0,0 兲 ⫻R共 x j , ␣ j 兲 ⫻T共 0,0,l j 兲
(33)
共 Q jk ⫻Q j 兲
d j ⫽ 共 pk ⫺p j 兲 •
储 Q jk 储 2
l j ⫽ 共 p j ⫺pk 兲 •
Next, the first-order Taylor Series approximation of the transformation matrix is written as follows:
共 Q jk ⫻Q j 兲
储 Q jk 储 2
␪ j ⫽arcsin共 储 Q jk 储 ⫻ 储 Qi j 储 兲
(32)
Since the mechanism parameters are now known in terms of the
joint positions and orientations, it is possible to estimate the error
in output function given the variability in joint location using
techniques similar to those outlined for the planar case earlier. The
process proceeds by writing the product of homogeneous transfor-
⌬ Ak ⫽
⳵ Akj
⳵ Akj
⳵ Akj
⳵ Akj
⳵ Akj
⌬ d j⫹
⌬ ␪ j⫹
⌬ a j⫹
⌬ ␣ j⫹
⌬
⳵d j
⳵␪ j
⳵a j
⳵␣ j
⳵l j lj
(34)
The parameter variabilities 共i.e. ⌬ d j ,⌬ ␪ j ,⌬ a j ,⌬ ␣ j and ⌬ l j ) are
now either interval 共for worst-case analysis兲 or random 共for sto-
Fig. 18 Convergence rates of Monte Carlo simulations for different operational
angles
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Transactions of the ASME
• There is the freedom to choose a build configuration that will
minimize the output variability when the mechanism is in its
operating configuration.
• Important functional gaps and clearances can be controlled
directly, by controlling the dimensions of sacrificial support
material between mating parts, rather than being a consequence of the mating of independently fabricated parts.
We surmise that it will be particularly important to take advantage of these characteristics in fabricating MEMS and meso-scale
mechanisms, for which the process variability is typically a larger
percentage of the feature size than for macroscopic devices.
These topics are the subject of ongoing investigation. Some
results on the treatment of clearances and on build pose optimization are provided in 关29兴.
Acknowledgments
Fig. 19 Notation for the derivation of modified DenavitHartenberg parameters from joint Plücker coordinates.
chastic analysis兲 parameters, the variances and correlation coefficients of which can be obtained using the relationships derived in
Eq. 共32兲.
5
Conclusions and Future Work
A framework has been presented for reasoning about errors in
the performance of mechanisms that are slated to be built using
the increasingly popular ‘‘freeform’’ fabrication techniques. This
is achieved by formulating an abstract model for the in-situ fabrication of mechanisms, and solving the problem of analytical
estimation of the variance of the kinematic function, in the presence of correlated random parameters. The fundamental assumptions in this treatment of error analysis are:
• The desired performance of the mechanism is specified in
terms of a kinematic output function, which is a continuous and
differentiable mapping from a parameter space to the operational
workspace 共usually a Cartesian space兲. This assumption limits the
application of the methods presented to linkages with lower pairs
and ‘‘well-behaved’’ higher pairs only.
• The output is a weakly non-linear function of the inputs. This
enables a first-order Taylor Series approximation of the error at
the points of interest.
• In-situ fabrication is abstracted as a process of independent
insertions of joints 共which could have internal clearances兲 into a
fabrication workspace, with a known accuracy. The inaccuracy is
specified as worst-case limits on position and orientation 共for deterministic error analysis兲 or variances with known distributions
共for stochastic error analysis兲.
Note that no assumptions of planarity or of homogeneity in workspace characteristics are made anywhere in the methodology.
Analysis of parametric errors in spatial mechanisms has also been
covered in the theoretical formulation.
This paper demonstrates that differences in the manufacturing
process flow for in-situ fabrication leads to fundamental differences in how process input variability is manifested in the kinematic output of a mechanism.
For stochastic analysis, the essential result is that we must account for correlations among adjacent links. In this paper we have
presented a modified stochastic analysis that accounts for the correlations and shown that it compares favorably with numerical
Monte Carlo simulations.
Although the need to consider correlations in the variabilities of
link parameters somewhat complicates the analysis, in-situ fabrication also affords some important advantages over conventional
fabrication for reducing output variability, notably:
• Tolerances do not accumulate along serial chains.
Journal of Mechanical Design
We gratefully acknowledge the support of the National Science
Foundation 共MIP 9617994兲 and the Alliance for Innovative Manufacturing 共AIM兲 at Stanford. We are also grateful for the assistance provided by members of the Stanford Center for Design
Research and Rapid Prototyping Laboratories. Sanjay Rajagopalan also thanks Jisha Menon for her generous support during the
formulation of this work—some of which constitutes the basis for
his Ph.D. thesis at Stanford University.
Appendix: Estimation of Mean and Variance
Here, we are concerned with the approximate estimation of the
mean and variance of an output y, described in terms of its output
function f (•) and a set of n random parameters ⌽
⬅ 关 ␾ 1 , ␾ 2 , . . . , ␾ n 兴 as follows:
y⫽ f 共 ⌽兲
(35)
where f (•) is, in general, a continuous and differentiable nonlinear mapping, and the parameters ⌽ are random variables with
no assumptions made about their distributions, correlations or independence. It is assumed, however, that the function f (•) is only
weakly non-linear 共i.e. high-order terms in it’s Taylor Series expansion can be neglected兲 and that the mean and variance of the
parameters ␾ i are known, and denoted as ( ␮ ␾ i , ␴ ␾2 i ).
We begin by expanding the output function in its Taylor Series,
about the mean values of the parameters, as follows:
n
y⫽ f 共 ␮ ␾ i ;i⫽1,2, . . . ,n 兲 ⫹
⫹
1
2!
n
⳵2 f
兺 ⳵␾
i⫽1
册
⳵f
兺 ⳵␾
i⫽1
共 ␾ i⫺ ␮ ␾i兲
2
i ␮
2
⫹
i
册
共 ␾ i⫺ ␮ ␾i兲
␮
⳵2 f
兺 ⳵␾ ⳵␾
i⬎ j
i
j
册
共 ␾ i ⫺ ␮ ␾ i 兲共 ␾ j
␮
⫺ ␮ ␾ j 兲 ⫹¯
(36)
With a little bit of rearrangement, the above equation can be rewritten in terms of proxy variables ⌬ ␾ i as:
n
y⫽ f 共 ␮ i ;i⫽1,2, . . . ,n 兲 ⫹
⫹
兺兺
i
j
⳵2 f
⳵ ␾ i⳵ ␾ j
册
⳵f
兺 ⳵␾
i⫽1
i
册
⌬ ␾i
␮
⌬ ␾ i ⌬ ␾ j ⫹O 3
(37)
␮
where ⌬ ␾ i ⫽ ␾ i ⫺ ␮ ␾ i are zero-mean random variables, with all
higher order moments identical with ␾ i . The term O 3 stands for
all terms in the Taylor Series expansion that are of third degree or
more, and are usually negligible.
We now go about the task of estimating the mean and variance
of the output 共the LHS term兲, using the above equation. In this
DECEMBER 2003, Vol. 125 Õ 821
regard, we make use of the following results, which are based on
elementary applications of theorems in the area of Mathematical
Statistics 关23兴:
E 兵 f 共 ⌽兲 其 ⫽ f 共 ␮ ␾ i 兲
References
E 兵 ⌬ ␾ i 其 ⫽0
Var兵 y其 ⫽E 兵 共 y⫺ ␮ y兲 2 其
Cov兵 ⌬ ␾ i ,⌬ ␾ j 其 ⫽E 兵 ⌬ ␾ i ⌬ ␾ j 其 ⫺E 兵 ⌬ ␾ i 其 E 兵 ⌬ ␾ j 其
(38)
where E 兵 • 其 stands for the expected value, Var兵 • 其 stands for the
variance and Cov兵 • 其 stands for the covariance. For notational
simplicity, we denote the expected value, or mean, by the symbol
␮ 共with the appropriate subscript兲, and the variance by the symbol
␴ 2 . In addition, we use the covariance coefficient ( ␳ i j ), which is
defined as follows:
␳i j⬅
Cov兵 ⌬ ␾ i ,⌬ ␾ j 其
(39)
␴ ␾i␴ ␾ j
Note that ⫺1⭐ ␳ i j ⭐1, and that ␳ i j ⫽1 when i⫽ j and ␳ i j ⫽0 for
independent or uncorrelated ␾ i and ␾ j . From the above equations, it is also apparent that:
Cov兵 ⌬ ␾ i ,⌬ ␾ j 其 ⫽E 兵 ⌬ ␾ i ⌬ ␾ j 其 ,
E 兵 ⌬ ␾i⌬ ␾ j其 ⫽ ␳ i j ␴ ␾i␴ ␾ j
(40)
and
Returning to the output expansion in Eq. 共37兲, and using the results detailed above, we are able to write the expression for the
expected value of the output function as follows:
E 兵 y其 ⬅ ␮ y⬇ f 共 ␮ ␾ i 兲 ⫹0⫹
兺兺
i
j
⳵2 f
⳵ ␾ i⳵ ␾ j
册
册
␮
or, using Eq. 共40兲:
E 兵 y其 ⬅ ␮ y⬇ f 共 ␮ ␾ i 兲 ⫹
⳵2 f
兺 兺 ⳵␾ ⳵␾
i
j
i
j
E 兵 ⌬ ␾ i ⌬ ␾ j 其 (41)
␮
␳ i j ␴ ␾i␴ ␾ j
(42)
Equation 共42兲 is a general expression for the approximation of the
mean of a function f (•) of random variables, which are—in
general—correlated.
In a manner similar to the earlier analysis, we can use Eq. 共37兲
to write an expression for the output variance as follows:
再冉兺 册
册 冊冎
n
Var兵 y其 ⬅ ␴ 2y ⫽E 兵 共 y⫺ ␮ y兲 2 其 ⫽E
⫹
兺兺
i
⫽
⳵2 f
⳵ ␾ i⳵ ␾ j
j
⳵f
兺 兺 ⳵␾
i
j
i
兺冉 冊
n
i⫽1
⳵f
⳵␾i
2
␮
␴ ␾2 i ⫹2
␮
⳵f
E 兵 ⌬ ␾ i ⌬ ␾ j 其 ⫹O 3
⳵␾ j ␮
⳵f
j
⌬ ␾i
␮
2
兺 兺 ⳵␾
i
⳵f
⳵␾i
⌬ ␾i⌬ ␾ j
Combining Eq. 共43兲 with Eq. 共40兲,
␴ 2y ⬇
i⫽1
册 册
␮
i
(43)
册 册
⳵f
␳ ␴ ␴ , i⫽ j
⳵ ␾ j ␮ i j ␾i ␾ j
␮
(44)
Equation 共44兲 is a general expression for the approximation of
the variance of a function of correlated random variables. The first
term in the RHS expression is the variance assuming independent and uncorrelated parameters. The second term applies an
822 Õ Vol. 125, DECEMBER 2003
adjustment to the variance estimate from the first term, accounting
for any correlative effects.
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