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1. 1060 zyxwvutsrqponm
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 19, NO. zyxwvu
5, SEPTEMBER/OCTOBER 1989
An Evaluation of Motor Models
of Handwriting
R~JEAN
PLAMONDON, SENIOR MEMBER, IEEE, AND FRANS J. MAARSE zyxw
Abstract -A general method is presented for describing and analyzing
biomechanical handwriting models. Using Laplace’s transform theory, a
model can be represented in what we call the neural firing-rate domain.
Consistent terminology is also proposed to facilitatemodel evaluation and
comparison. An overview of previouslypublishedmodels suggeststhat they
could be described using this method, with second- and third-orderlinear
model representation. Fourteensimplifiedtheoretical models are simulated
in an experiment designed to study the parameter domain in which
handwriting is controlled by the nervous system to gain insight into which zyxwvutsr
type of model provides the best reconstruction of natural handwriting.
Results show that velocity-controlled models produce the best outputs. No
significant difference exists between second- and third-order systems. In
handwriting, fine motor behavior is, in the first instance, velocity-con-
trolled. These findings agree with other recent automatic signature verifi-
cation resultsand are of interestfor a numberof applications,frompattern
recognitionto handwritingeducation.
I. INTRODUCTION
HE PROCESSES involved in handwriting are rather
Tcomplex and have been analyzed over the past decades
from several perspectives by various groups of researchers:
physicists, engineers, computer scientists, experimental
psychologists, neurologists, cognitive scientists, and
graphoanalysts. Because it is one of the basic human-
specific slulls, handwriting has been analyzed to study
motor behavior, control and learning, to detect neurologi-
cal disease, to develop psychodiagnostic tests for remedial
teaching and even to design anthropomorphic robots
[1]-[4]. Being one of the natural channels used by humans
to communicate, handwriting has also been analyzed to
develop new man-computer interfaces, character recogniz-
ers, signature verifiers, text authenticators, and handwrit-
ing generators [5]-[8].
These studies and applications may, however, be so
specialized that researchers in one area are unaware of
developments in related areas. In fact, most of these pro-
Manuscript received November 3, 1987; revised February 12, 1989.
This work was supported in part by CRSNG of Canada under Grant
OGP-0000915, in part by FCAR, Quebec, Canada, under Grand AS-
2240F, and in part by a grant from The Netherlands Organization for the
Advancement of Pure Research.
R. Plamondon is with the Laboratoire Scribens,DCpartement de Genie
Electrique, Ecole Polytechniquede MontrCal, C.P. 6079, Succursale“ A ,
MontrCal, PQ, Canada H3C 3A7.
F. J. Maarse is with the Nijmegen Institute for Cognition Research and
Information Technology, P. 0. Box 9104, 6500 HE Nijmegen, The
Netherlands.
IEEE Log Number 8927729.
jects rely directly or indirectly upon a basic knowledge of
the handwriting processes themselves. From the time work
began in ths field, handwriting modelization has been the
foundation upon which some of the basic research projects
and development applications have been built. Nowadays,
several models are used in a large spectrum of projects,
from fundamental studies aimed at understanding the
biomechanical or neuropsychological systems to immediate
applications at the feature extraction level in the design of
pattern recognizers. Some models are more oriented to-
ward handwriting analysis, others toward handwriting gen-
eration.
Since these models were not all developed in the same
context nor with a common goal, it is not always easy to
analyze their differing perspectives. Comparing one model
with another often means coping with problems of basic
vocabulary. Neither is it always evident how to evaluate
and discuss the impact of the simplifying assumptions
associated with each model.
The purpose of this paper is first to provide a general
context within which most of the biomechanical models
proposed to date can be analyzed and reviewed. This study
is limited to physical models, that is, mathematical models
that can be used directly to analyze or generate a piece of
handwriting. Conceptual models, more oriented toward the
understanding of high-level motor programming, are not
included here. Nevertheless the expectation is that the
terminology used here will be consistent with the terminol-
ogy used in this important category of research work on
modelization.
Secondly,its purpose is to gain further insight into what
variable is controlled by nervous system during handwrit-
ing. Stein [9]has suggested the following control variables:
force, velocity, length (spatial target), stiffness, viscosity,
more than one of the previously mentioned, or none of the
previously mentioned. By using simplified models, three
control variables are tested in this study by means of
computer simulation [lo]: acceleration, velocity, and spa-
tial target (length).
Finally, we are hopeful that ths overview will shed
significant light on the basic problems related to develop-
ing the computer applications of handwriting. For example
cursive script recognition [ll]and signature verification [5]
systems are still limited owing to inadequate answers to
fundamental pattern recognition questions.
0018-9472/89/0900-lO60$01.00 01989 IEEE

2. PLAMONDON AND MAARSE: AN EVALUATION OF MOTOR MODELS OF HANDWRITING 1061 zyx
Top-down
approaches
Motor program:
grapheme level
allograph level
parameter level
Activationmechanisms
Motor program
extraction zyxwvutsrqp
J
!
I
Nerves
Information
transmission
Muscles
Movement
activation
Pen/paper
Trajectory
Memorization zyxwvutsr
Izyxw
I
Bottom-up
approaches
NeNe-muscleinterface
> 1 zyxwvut
5
'order system
Hand-paperinterface
(2"! order system)
Fig. 1. Study of handwriting zyxwvut
as motor behavior
Using a proper representation space for signals and
having a good model to describe their generation can lead
to more efficient segmentation protocol, feature selection,
signal coding, and compression. Similarly,better definition
and representation of strokes can be a great help in design-
ing software for (remedial) teaching of handwriting and
also in developing a measure of fluency and ballisticity in
handwriting, for use in psychophysiological tests and ex-
periments.
Starting with a commonly accepted view of the biologi-
cal processes involved in handwriting, a standardized clas-
sification of the previously published models is proposed
according to the Laplace transform theory of system de-
scription. In this way, the models may be classified in
terms of the order of the system used to generate the
movement. Consistent terminology is also proposed to
facilitate model evaluation and comparison. The practical
interest of this standard approach and terminology is
demonstrated in the last part of this paper, where several
models are simulated [lo] to gain further insight in which
variable is controlled by the nervous system.
11. HANDWRITING
PROCESSES zyxwvutsrq
A . General Overview
Several conceptual, physical, and empirical models have
been developed for studying and understanding handwrit-
ing. Although the models may differ greatly in their de-
scription of the phenomena, depending on the context and
purposes for which they have been developed, most can be
viewed as emanating from a commonly accepted view of
the processes involved.
The center column of Fig. 1 is a block diagram of the
organizing functions involved in handwriting. This schema
also reflects somebasic hypotheses generallyassumed when
one studies these biomechanical processes. Like any highly
skilled motor process, fast handwriting is considered a
ballistic phenomenon, that is, a motion controlled without
instantaneous position feedback, the product of a learned
motor program [12]-[14]. At the beginning of a writing
segment, the whole tt.ajectoryof that movement is defined.
No extra control is applied during execution. According to
this model, somecentral nervous systemmechanismswithn
the brain fire, with a predetermined intensity and duration,
the nerve network which activates the proper muscles in a
predetermined order. The motion of the pen on the paper,
resulting from muscle contraction/relaxation, leaves a par-
tial trace of the pen-tip trajectory.
Although there is no clear-cut boundary between fast
and slow handwriting movements (a slower writing process
is probably a matter of position and visual feedback [15]),
ths representation has resulted in a stepwise analysis,
involving several stages, from movement planning to mus-
cle activation. Severalmodels have been proposed to study
these mechanisms and, depending on the emphasis placed
on the behavior of the brain or of the hand, two comple-
mentary approaches have been followed to study handwrit-
ing: top-down and bottom-up. This complementarity is
illustrated in the left- and right-hand parts of Fig. 1.
The top-down approach has been developed mainly by
psychologists and researchers interested in the study and
application of the motor program itself: the fundamental
unit of movement coding, code sequency and retrieval,
movement control and learning, and task complexity [lo],

3. 1062 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. zyxwvuts
19, NO. zyxwvut
5, SEPTEMBER/OCTOBER 1989
[16]. In this context, research is concentrated on high-level
representation and utilization of information. The biophys-
ical phenomena are grouped within a black box dedicated
to movement generation, which in turn activates the proper
nerve and muscle pairs to generate the movement corre-
sponding to the grapheme/allograph/parameter represen-
tation of the programmed information.
Indeed, according to such a conceptual approach, hand-
writing would be produced in at least three steps reflected
by the additive effects on reaction time experiments [17],
[18]: first the abstract motor program is retrieved from
long-term motor memory, then parameters like actual size,
accuracy, and speed are fed into ths program to make it
more concrete, and finally, this information (program and
parameters) is translated for the recruitment and activa-
tion of the proper muscles.
The bottom-up approach has been used by those, mainly
physicists and engmeers, interested in the analysis and
synthesis of the biomechanicalprocesses. Their goal was to
produce handwriting forms and not to simulate the psy-
chomotor process. Several models of the muscle-activated
hand motion have been proposed [13], [14], [19]-[27]. In
their simplest form, these models represent the study of a
point mass under the action of muscular forces. Some
refinements have incorporated, at least theoretically, the
viscous force [13],[14], [19],[22], the external friction force
[14], the elastic force [20], [21], [24], and also, to some
extent, nerve-muscle interface phenomena [22], [26].
A study of these models reveals that research efforts are
concentrated on the mathematical description of the
nerve-muscle-pen-paper system: choice of differential
equations, parameter extraction, movement analysis, and
synthesis. This system is assumed to be controlled by a
program that can be described as a sequence of input
stimuli.
B. Basic Equations
Although both top-down and bottom-up models have
been found helpful in many studies on handwriting, only
the bottom-up approaches are considered here. These rely
on mathematical description of the different subsystems
depicted in the right-hand part of Fig. 1.
The hand-pen-paper system is represented by a point
mass M whose motion along one linear direction zyxwvuts
r can be
described by
M d 2 r ( t )
dt zyxwvutsrq
+ A T
d r ( t ) zyxwvutsrqp
+Kr(t)+f , N ( t ) =F,.(t) (1)
equivalent mass of the hand-pen system,
intrinsic viscosity coefficient of the hand,
stiffness coefficient of the hand,
extrinsic friction coefficient between the pen-tip
and the writing surface;
component of the writing pressure normal to the
writing surface,
q ( t ) muscular force applied to the equivalent point
mass.
At least two of these equations are needed to produce
two-dimensional handwriting movements.
In many studies, simple second-order equations are used
for describing and simulating handwriting. A nerve-muscle
interface of zero order is thus assumed in this case. In
more fundamental studies, the nerve-muscle interface has
been thoroughly studied and several models have been
proposed. In its simplest representation, this interface can
be described by a first-order system 1281, [29]:
where
F,
a, b
g,( t ) activation level,
u ( t )
This system acts as a pure force generator if the constant
maximal isometric force in the operating-length
region of the muscle,
constants specific to the muscle under study,
contraction velocity of the muscle.
b is substantially greater than the contraction velocity:
F , ( t ) = F,g,(t), ifb>> u ( t ) . (3)
Since it is generally assumed that the activation level
g,(t) can be described at least as a first-order response to
the neural firing rate [29], the followingis obtained for the
nerve-muscle interface: zyxw
(4)
where u r ( t ) is the neural firing rate and C is the fitting
constant.
The brain-nerve interaction has been studied in the
context of motor program representation. This top-down
approach has not resulted in any mathematical description
of the mechanisms involved and it is not clear thus far if
such a representation can be developed or would be signif-
icant.
Moreover this basic knowledge, although expressed
mathematically, should not mask the various hypotheses
from which it has been developed. It is assumed, for
example, that the coefficients of these differential equa-
tions are constants, at least over one handwriting stroke,
and that the complex set of muscles involved in handwrit-
ing production along a principal direction behaves like a
single nerve-muscle interface. In addition all these subsys-
tems are assumed to be linear and stationary.
111. CLASSIFICATION
AND STANDARDIZATION
As will be seen in Section V, several other simplifying
assumptions have been proposed by other authors, to
make this theoretical description more practical for spe-
cific applications. Several models have been documented
to date, and, to understand their relevance better, a stan-

4. PLAMONDON AND MAARSE: AN EVALUATION OF MOTOR MODELS OF HANDWRITING zyxwvutsrqpo
1063 zy
Motor program
1 zyxwvutsrqpo
i domain
I
I
L______________________
J
(b)
Fig. 2. (a) General form of handwriting models. (b) Third-order approximation.
dard representation might be helpful. Fig. 2(a) summarizes
the basic principles upon which the standardization pro-
posed in this paper is based.
1) A generation channel can be described by an nth-
order linear system using Laplace transform representa-
tion, the overall system being described by a cascade of zyxwvu
m
simpler subsystems to be studied separately or as a whole,
according to the equivalenceof the cascaded linear system.
2) Whatever the domain of analysis for which it has
been used, the model is transposed to a mathematical
representation where the system output is in the space
versus time domain and the input motor program zyxwvuts
U ( S )is
represented by a sequence of abstract input step functions zyxwvuts
(V,(S ) ) whose amplitude reflects the neural firing rate:
where A , is the amplitude of the step function, K, = +1 or
-1,to take into account positive and negative stimuli, and
U,(S)= K,A,e-TS/S is the unit step function occurring
at zyxwvutsrqpo
T,.
The choice of the input function and the output domain
of analysis has been guided by established practice and
convenience. Indeed, early works on handwriting modeling
use the step function as the system input, and the ultimate
test for any model is whether the proper pen-tip trajectory
can be generated in the space-time domain. In this con-
text, the order of a model is defined as the order of the
differential equation used to describe it, when it is fired
with a standard abstract input U ( S )to produce the proper
displacement r ( S ) .The system order must be viewed as a
relative definition and not as an absolute one, since there
is no way of knowing what the real input function is that
represents a motor program within the brain. Substituting,
(4) and (3) into (1)and using Laplace transform represen-
tation (with no initial conditions), the minimal order of the
equation theoretically representing the behavior of the
handwriting generation system is obtained:
This (6) makes it possible to represent any motor program
in what we will henceforth call the abstract neural firing-
rate domain [U(s)].
It also shows that at least a third-order
linear system would be necessary to code the handwriting
movement in that abstract domain, if the other simplifying
assumptions made in this study are justified. Fig. 2(b)
shows, for one generation channel, a block diagram of this
minimum theoretical model described by the following
equation:
1
. (7)
-- -
r ( 4
u(s) [s+4 a s *+BS +y +S / r ( s ) ]
For simplicity, in this equation the gain of the system is
fixed at unity since there is no way of measuring it. A
first-order nerve-muscle system is also assumed. In a more
simplified approach, this interface can be supposed to be
of zero order, so that the equation would therefore be
reduced to a second-order system.
IV. TERMINOLOGY
The standard representation proposed in the previous
section for biomechanical handwriting models also leads to
some practical definitions [30].

5. 1064 zyxwvutsrqpon
Axis zyxwvutsrqponmlkjihg
I
Axis zyxwvutsrqponmlkj
2 -
AXIS 3
(pressure)
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 19. NO. 5. SEPTEMBER/OCTOBER 1989
MOTOR PROGRAM REPRESENTATION
-
=
=
I
I
* Word b
Fig. 3. Terminology resulting from motor program representation as function of time. zyxwv
1) Stimulus: A stimulus is any change of state in an
input function U( zyxwvutsrqp
S).This change may be characterized by
its time of occurrence and its signed variation in ampli-
tude. The overall gain of the system is included in ths
amplitude.
2) Motor Program: A motor program is a set of stimulus
sequences that can generate handwriting when fed into a
proper biomechanical model of the process under study.
As shown in Fig. 3, a program is at least composed of
three sequences, two for producing the plane movement of
the pen-tip and one for the generating of pressure or
pen-up/pen-down signals.
3) Stroke: A stroke is a sequence of stimuli which
results in the production of a fundamental unit of hand-
writing movement.
4) Component: A component is a sequence of strokes
produced during a continuous pen-down signal. It may be,
for instance, a letter or an allograph.
5) Word: A word is a sequence of components resulting
in a handwritten discontinuous curve normally having a
few specific semantic values. At this level, handwriting is
analyzed in a linguistic context.
In Fig. 3, a schematic representation of a motor program
is given. The timing is represented, in addition to the
amplitude of stimuli and for the three movement genera-
tion systems (axes 1-3). The first two axes refer to stimuli
that might be needed to generate plane movement (coded
in an zyxwvutsrqpo
X-Y Cartesian reference system, or in any other
which might be more powerful [26]). Stimuli for move-
ments in the thrd dimension, those resulting in pen-
up/pen-down movements and pen pressure, are contained
in axis 3. The definitions of “component” and “word”
derive from this third dimension. These three motor pro-
grams, when fed into three movement generation subsys-
tems, result in a typical handwriting output.
V. REVIEW
OF HANDWRITING
MODELS
With ths representation and terminology, all the bot-
tom-up models proposed to date may be classified accord-
ing to the order of the system used to describe the biome-
chanical generators. It must be remembered however that
these models were not necessarily developed for the same
purpose and that the proposed classification in a biome-
chanical context is introduced only for purposes of com-
parison. In spite of the differences, most of them may be
interpreted from a physiological point of view using this
standard method. Some models dealing with a representa-
tion and segmentation of handwritten images have also
been proposed, mainly for off-line character recognition
[31].Since these studies were not concerned with temporal
simulation, they are not included in this review.
A. Second-Order Systems
Table I summarizes the properties of the second-order
models that have been published to date, using the stan-
dard terminology defined in Fig. 2. The first model in ths
table was proposed by Denier van der Gon et al. [32]and
is the product of experimental studies in handwriting simu-
lation based on four principles: 1) fast handwriting is a
ballistic phenomenon, 2) movements are caused by two
independent and perpendicular groups of muscles, 3) these
muscles apply forces to hand and pencil which are consid-
ered as a mass with some internal friction, and 4) once
applied, the force increases to a certain fixed value and
then remains more or less constant, the duration of its

6. PLAMONDON AND MAARSE: AN EVALUATION OF MOTOR MODELS OF HANDWRITING zyxwvutsrq
1065
TABLE I
LISTOF SECOND-ORDER MODELS zyxwvuts
REMARKSzyxwv
AUTHORS STANDARDIZED MODEL zyxwvutsrq
1 no external friction
I negligible elasticity
I
I
Denier Van der Gon I I 1 I
Thuring and Strackee I ?*I ~ I
-
>
zyxwvut
X(S) or y(s) 1 two identical axes
(19621 1 ux(S) 1zyxwvutsrqponmlkjihg
as2 + BS ] 1 subsystem
)or uY(s) I orthogonal model
I I
I
Eden I
(1962) I
I
Eden I
(19641 I
I
I
(1981) I
I
I
I
I
i
Mermelstein and I
Hollerbach
1 external and internal friction
I neglected
I predominance of elasticity
I phenomena
I orthogonal model
I horizontal movement divided into
I oscillatory and trend components
I
I
I
I
I
I
I
I
Dooijes
( 1983
I
I
I external friction
I internal friction
I also neClected in some
I reconstruction experiments
I
I
I
I
I
I
I
I
I
I
I
I non-orthogonal model (X', Y'1
I horizontal movement
-
I 1 I I
-%I ~ h
uXl(S) 1zyxwvutsrqponmlkjihg
as2 + BS I I divided into oscillatory
I
> X'(S) 1 and trend components
- I
-&I __ + I
UxZ(S) I s I I
I
I 1 I 4-
I I
'
Plarnondon
( 1987-88)
I
I
I
I
I
I
I
I
I
I
1 external friction
1 neglected
I invariable with rotation
n
1 1 I I
I s I
1 - r(S) 1 velocity generators
1 independent of any fixed
I reference axis
I
I
I

7. 1066 zyxwvutsrqponm
IEtE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 19, NO, zyxwv
5 , SEPTEMBER/OCTOBER 1989
application being related the magnitude of the movement.
In this model, strokes of different lengths were produced
by different force timings, the amplitude of the forces
being constant for a specific writing size. Using our stan-
dard approach, ths model becomes a zero-order system
for the nerve-muscle interface and is thus a second-order
system where external friction and elasticity effects consid-
ered negligible.
Three authors have proposed undamped versions of the
second-order system. Their general view is based on a
harmonic oscillator description of the muscle action [20],
[21], [24] and uses velocity equations to describe momen-
tum impulses or strokes. Eden [21] proposed the first of
these models. Sinusoids were used as the input function to
an integrator. With the appropriate difference in oscilla-
tion frequency of the orthogonal zyxwvutsrq
X and Y axes, plus a
proper phase shft, handwriting could be generated, pro-
vided also that a linear trend was added to the horizontal
displacement. Mermelstein and Eden [20] modified ths
representation to incorporate a different velocity ampli-
tude for the rise and fall of input excitation. They found
that in this case handwriting could be better simulated
with the help of a larger set of fitting parameters: X and Y
velocity amplitude, frequency and phase shift between X
and Y velocities, and timing of velocity changes.
Hollerbach [24], using a similar approach, carried out an
extensive study of the practical interest of the oscillatory
model in controlling the shape, height, and slant of hand-
writing. He analyzed the basis of such a representation in
the context of a muscle-spring model, as well as studying
the effects of a nonorthogonal writing axis, a variable
spring constant, internal friction, etc. In the context of our
general representation, all these models can be schema-
tized by a second-order undamped system,with an integra-
tor added in the zyxwvutsrq
x branch to superimpose a linear trend
movement.
Two additions were proposed for the Denier van der
Gon model by Dooijes [23], [29]. The first is the assump-
tion that the principal directions of motion, i.e., those
directions emerging if either of the antagonistic muscle
pairs are independently excited, form an oblique reference
system instead of an orthogonal one. The orientation of
these two principal axes (X’-Y’) may be determined with
the help of two procedures based either on the Lissajous
transformation or on Hilbert transform pairs. The second
is the assumption that the movement along the horizontal
direction is the results of two independent mechanisms: a
uniform trend upon which a second-order generator is
superimposed. The time-based component was extracted
by linear regression analysis. In his analysis Dooijes also
neglected internal friction terms and was dealing with a
purely ballistic model.
Plamondon [26], [27]recently proposed model based on
intrinsic representation of handwritten curves. Using dif-
ferential geometry, the handwriting process is biomechani-
cally represented, without reference to any fixed-axis sys-
tem. In its simplest form, the generation of handwriting is
thus reduced to the production of two types of displace-
ment: a linear displacement ( a ) of the pen-tip along the
curvilinear abscissa and an angular displacement that re-
sults in the proper change of direction of the pen-tip. The
whole system behaves like a speed generator and the brain
has to control the magnitude and the orientation of the
pen-tip velocity. The trajectory is integrated via the action
of pen-paper contact. zyxwv
B. Third-Order System
Using the same approach, Table I1 depicts the third-
order models. Apart from a single-axis model developed
by Plamondon and Lamarche [22], these models use two
sets of similar equations for both X and Y displacement.
Therefore, one axis is described here, identified by r(S).
The first two of these are direct modifications of the
model proposed by Denier van der Gon [13], and were
designed to improve its performance in simulating hand-
writing. MacDonald [19],using a similar set of equations,
has suggested the use of trapezoidal acceleration pulses to
feed a second-order system, the slopes of the pulse corre-
sponding to the rate of change of muscular force per unit
of time. Yasuhara [14] has demonstrated that exponential
transitions for the stimulus pulse are even more powerful.
This model to a certain extent incorporates external fric-
tion components.
Looking at the general representation proposed in this
paper, these improvements can be seen as adding a first-
order systemin front of the second-order model previously
proposed by Denier van der Gon [13]. Indeed starting
from abstract rectangular step function in the nerve
firing-rate domain, a first-order system produces an expo-
nential muscle force function. A trapezoTda1 one is ob-
tained if the time constant of ths stage is assumed to be
null. These changes, that is, incorporation of the nerve-
muscle interface in the biomechanical interpretation of the
processes, clearly constitute an ingenious way of working
with higher order systems.
In a very different context, Morass0 et al. [25] have
developed a model that incorporates some dynamic infor-
mation to allow a complete generation of a handwritten
trajectory from basic segments. Basic strokes (described by
curved segments of given length, tilt angle, and angular
change) are used to reconstruct human handwriting perfor-
mance with the constraint that each stroke was generated
with a symmetricbell-shaped velocity profile, centered at a
given instant of time. Assuming a proper mechanism for
transforming spatial sequences into angular/muscular se-
quences, a motor program coded with these segments can
generate handwriting, provided there is a proper overlap
between two successive segments.
Ths space-oriented model may be considered to be
halfway between the muscle-oriented model and the con-
ceptual top-down model. Their representation in the neu-
ral firing-rate domain is not straightforward. A third-order
transfer function, which can be used to generate a bell-
shaped velocity profile from a step input, is shown in
Table 11. The output of this first stage, when integrated,

8. PLAMONDON AND MAARSE: AN EVALUATION OF MOTOR MODELS OF HANDWNTING zyxwvutsr
1067
TABLE I1
LISTOF THIRD-ORDER MODELS
I
AUTHORS zyxwvutsrqponm
I STANDARDIZED zyxwvutsrqp
MIDEL
I
RDIARKS
I
MacDonald
(1966)
1 no external friction
I trapezofdal acceleration patterns
I -
'
I
1-4 - H -
I 1 1 1 1 I I negligible elasticity zyxw
er(S)
I
~~ ~
I I
1 external friction
I --
Yasuhara I I 1 1 1 1 I 1 partly integrated in internal
(1975) 1-4 - H - +-> r(S) 1 friction term
I ur(S) I S + zyxwvutsrqp
B I I zyxwvutsrq
as2 + BS I I negligible elasticity
'
U I exponential acceleration patterns
I
I
I
I orthogonal model
I I
-- I segment defined with bell-shaped
I velocity profile and specific
Morass0 and I I U2S2 + 4 - 7 5 I I 1 I
Mussa Ivaldi I -
4 ~ +
> r(S) I length, tilt angle and angular
(1982) 1 Ur(S) I S(S2 + 4 - 7 5 I 1 s I I change
I
--
I I
--
I
I
Plamondon I
Lamarche I ---->-I--- H -
I
1 external friction
I neglected
1 simplified to a second-order
I 1 1 1 1 I
+
> r(S)
( 1986 I ur(S) I as2 + BS I I S 1 I system for test experiments
U - I single axis model
Maarse
(1987
I
I purely ballistic model
--
1 1 1 1 1 1 I
+
> r(S) 1 orthogonal model
->_I - H -
Ur(S) I s 1 I as2 1 I
'
U I triangular acceleration patterns
I
I several other input
1 patterns also studied
I
IzyxwvutsrqponmlkjihgfedcbaZ
I
results in segment generation.Proper geometricparameters
could be controlled by initial conditions, timing, and am-
plitude parameters.
Yet another environment has been used by Plamondon
and Lamarche [22] to develop a model. Applying the
transfer function of the dc motor used in the Vredenbregt
and Koster's handwriting simulator [33], they described
the handwriting process with a speed generator system fed
with a rectangular pulse voltage that directly represents the
neural firing rate. Unidirectional movement was analyzed
in various simulation experiments with a simplified sec-
ond-order version of this model. It was demonstrated that
firing-rate domain. This scheme is used in the following
section to perform some control variable analysis.
Higher order models have also been proposed recently
[34] that can be fitted to the same general scheme using
fifth- and seventh-order systems. Their interpretation, in
terms of the peripheral psychomotor model of Fig. 1,led
to the use of third- or fifth-order systems to describe the
nerve-muscle interface. These models, based on the dy-
namic minimization of the jerk or snap, allow reconstruc-
tion of the kinematic movement from the shape of the
pen-tip trajectory only.
at least a third-order linear system was necessary to simu-
late human handwriting performance.
VI. HANDWRITING SIMULATION EXPERIMENT
Finally, Maarse [lo]in a model comparison experiment,
has used a cascade of integrators to process triangular
acceleration pulses. This approach is equivalent to assum-
ing a purely ballistic third-order system in the neural
The formal mathematical representation described in the
previous sections is very helpful in comparing, theoreti-
cally, the various models proposed to date but may also be
helpful in studying the variable used by the brain to

9. 1068 zyxwvutsrqponm
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 19. NO. zyxwvut
5, SEPTEMBER/OCTOBER zyxw
1989
Simulation zyxwvutsrqponmlkj
I zyxw
2 3 zyxwvutsrq
4 5 6
Fig. 4. Displacement, velocity, and acceleration used for simulations 1-6. Acceleration inputs were set. Velocity and
displacementwere computed from these inputs.
control these systems and in evaluating which model is the
best for producing and reconstructing natural handwriting.
Indeed, computer simulations can possibly answer this
question. To implement all the different models with opti-
mal values for each specific parameter would be a lengthy
task. Maarse [lo] showed that simplified pure ballistic
models with different inputs can produce good reconstruc-
tions of natural handwriting, although (external) friction
and elasticity were neglected (which is the case for many of
the models presented in Tables I and 11). Provided that
handwriting models are linear, system parameters can be
transferred to the input of a pure ballistic linear system,
and rather reliable simulations can be performed by choos-
ing adequate inputs. As previously mentioned, the sum of
the equivalent order of the inputs and the number of
integrators used yields to the order of the whole model.
In ths context, inputs fed to two integrators represent
strokes in the acceleration domain and are called force
impulses. Inputs or strokes defined in the velocity domain
are the so-called momentum impulses. When there is no
integrator at all, the inputs represent strokes in the spatial
domain or represent pieces of handwriting and are called
segments [30].This domain of analysis has been used, for
example, by Morass0 [25] who reconstructs handwriting
from circular segments.
By defining inputs in the three domains mentioned, an
experiment can be performed to study what type of input
yields the best reconstruction. If inputs defined in a certain
domain give significantly better results than others, it may
be an indication that handwriting is controlled by vari-
ables in that domain. Or in other words, insight into the
control variables (force, velocity, and spatial target or
lengths) can be gained. Let us look at a more detailed
description of the experiment performed, the reconstruc-
tion techniques, the inputs, and the results [lo]. zyx
A. Experimenl
Reconstructions were made of handwriting samples us-
ing 14 different simulations. Differences between these
reconstructions and the original samples were computed to
discover whch input(s), most accurately reconstructed nat-
ural handwriting.
The handwriting samples were recorded on a Calcomp
9240 X-Y tablet connected to a PDP-11/45 computer.
The handwriting signal was sampled with a sampling fre-
quency of 105.2 Hz. The pen was equipped with a stan-
dard ball-point refill. Four subjects were required to copy,
in their normal handwriting, a text of eight lines on a sheet
of paper. The horizontal writing direction was coincident

10. PLAMONWN AND MAARSE: AN EVALUATION OF zyxwvutsrqpo
MOTOR MODELS OF HANDWRITING 1069 zyx
Ve l o c zyxwv
it y
1.6 m/s2 zyxwvutsrqponmlkj
- - - - - - -
0.0 ‘r]
l
iz
p:
- - - -
t/
- - -
f-l$l-
ii_A c c e l e r a t i o n
-1.6 m/s2- - - - - - _ -
1 zyxwvutsrqponmlkji
250 S zyxwvutsrqponmlk
1
S i m u l a t i o n 7 E 9 10 I 1 12
Fig. 5. Displacement, velocity, and acceleration used for simulations 7-12. Velocity inputs were set. Acceleration and
displacement were computed from these inputs.
with the horizontal zyxwvutsrqp
X direction of the X - Y tablet. Follow-
ing data acquisition and filtering with a low-pass finite-
impulse response (FIR) filter with a cutoff frequency of 10
Hz and a transition band of from 10 to 25 Hz, the
simulations were then done on a VAX-11/750 computer
connected to an interactive graphics display and a plotter.
B. Reconstruction Techniques
I ) Finding Moments in Time and Length of Strokes: In
the 14 simulations performed, three types of strokes were
derived from the original handwriting specimen. In the
acceleration domain, strokes or force impulses were found
for X, as well as for Y,by minima in the absolute accelera-
tion in the X and Y directions, respectively (simulations
1-6). In the velocity domain, the momentum impulses
were derived from minima in the absolute velocity for X
and Y independently (simulations 7-12). The strokes or
segments in spatial simulation 13 were derived from the
minima in the absolute velocity. In simulation 14, minima
as well as maxima in the absolute velocity were used. For
the three types of strokes,the length and the velocity at the
beginning of a stroke were computed from the original
handwriting. For the momentum impulse, the initial veloc-
ity was, of course, zero.
2) Znputs Used: Based on stimulus timing, stroke length,
and starting velocity, a new synthetic stroke was com-
puted. For the force impulses, the shapes of the six inputs
used are given in Fig. 4 (simulations 1-6). Simulation 1
shows a rectangular force impulse, as used by Denier van
der Gon [30] and Dooijes [28]. Simulation 2 with its
trapezoidal shape was inspired by MacDonald [13]. The
exponential function of simulation 3 was suggested by
Yasuhara [12]. Simulations4-6, with sinus, bell, and trian-
gle shapes, respectively, were introduced by Maarse [101.
The same shapes were used as input for a first-order
system for the simulations in the velocity domain (7-12 in
Fig. 5 ) as those used for simulations 1-6.
In simulation 13, it was assumed that the segments are
circular. The velocity distribution was supposed to be
bell-shaped. By an overlap in time of 50 percent of the
successive segments, a fluent course of the reconstructed
trace was obtained. The curvature of a segment is derived
from the pen position at the beginning, at the middle, and
at the end of the original segment.
Simulation 14 consists of two linear systems working
independently, one for long segments between minima of
the absolute velocity and one for short segments between
its maxima. Here again the velocity profiles were bell-
shaped for both systems. With the two systems working
simultaneously,the final output had a fluent course.
C. Performed Simulations and Computing Errors
In this pilot experiment, reconstructions of one-minute
handwriting samples were made from the data obtained
from four subjects,using the 14 simulation protocol previ-

11. 1070 zyxwvutsrqpon
O r i g i n a l zyxwvutsrqponmlkjihgf
1 zyxwvutsrqponmlkji
3 zyxwvutsrqponmlkjihgf
5 zyxwvutsrqponmlkjihgf
7
9
11
13
Fig. 6.
IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS, VOL. 19, NO. 5,SEPTEMBER/OCTOBER 1989 zy
p-l
2
4
6
0
10
12
14
Original handwriting and 14 reconstructions of dutch word fuif
(party)written by one subject.
ously described. The partial results of the reconstruction
are shown in Fig. 6 (only one word appears here for
purposes of clarity). Note that the quality of the recon-
i output differs ,from one experiment to another.
An .ojective spatial measurement as defined by Maarse
[lo] was used to compute the error between the original
and the reconstructed stroke. This measurement computes,
for every stroke, the area between the original and the
reconstructed stroke. By dividing this surface area by the
square of the length of the stroke, a measurement indepen-
dent of the size of the handwriting was obtained. For all
handwriting specimens, the average value of this measure-
ment was computed. zyxwvutsrqp
D. Results
Table I11 records the average error for the whole experi-
ment for the four subjects. It will be noted that the
simulations in the acceleration domain yield more or less
the same results, with the exception of simulation 5. A
bell-shaped force impulse seems to produce a simulation
which is significantly less realistic than the others. In the
velocity domain, it is evident that simulation 7 yields the
worse result. Indeed rectangular momentum impulses are
unrealistic. Simulations 8-12 do not differ significantly.
For the simulations in the spatial domain, the output with
long and short segments seems to be less realistic.
With the exception of these three poor results (recon-
structions zyxwvutsrqpo
5, 7, and 14), all the other reconstructions yield
consistent results. In general, reconstructions in the veloc-
ity domain are significantly better, or, in other words,
handwriting reconstructions controlled by momentum im-
pulses are the best ones. However as will be discussed
later, the conclusion that handwriting is controlled by
velocityimpulses only is not evident from these results, but
in these kinds of simulations they are superior to force
impulses and segments.No significant difference in perfor-
mance between second- and third-order linear systems was
actually observed with this approach.
VII. CONCLUSION
The main purpose of this analysis was to propose a
general mathematical context for analyzing handwriting
models and to give an overview of handwriting models
previouslypublished. The proposed model definition seems
to be helpful not only for mathematical modeling and
comparing biophysical handwriting systems, but also for
higher level handwriting modeling. It gves clear defini-
tions of parameters like stimulus, stroke, and components,
and offers the potential for standardizing models. For
representation of the system, Laplace transformation the- zy
ory is used. In the proposed description of handwriting
models, the order of a system is defined between the final
product on paper and an arbitrary step function as input.
Theoretically if a first-order nerve-muscle interface is as-
sumed, the minimum order of a system, between the
abstract neural firing-rate domain and the spatial domain,
should be at least three. When ths assumption is not
made, the order is at least two. The models are assumed to
be linear and stationary.
The overview of previously published handwriting mod-
els indicates that all systems are of second and third order
and are linear and stationary within one stroke. Only for
simulation purposes were first- and fourth-order systems
studied. It appears to be possible to describe all the
previously published models with the proposed mathemati-
cal method.
The secondary purpose of this analysis was to gain
further insight into the psychomotor aspects of handwrit-
ing. From the simulations performed, it seems that second-
and third-order models with stimuli defined in the velocity
domain yield the best results. Solely on the basis of these
simulations, it cannot be concluded that the nervous sys-
tem controls the velocity of handwriting movements, but it
is evident that for simulation, regeneration and generation
purposes, momentum impulses will yield the best results. It
is clear that in these simulations, a first-order system
cannot adequately describe a handwriting system. In gen-
eral, in the acceleration domain, thrd- and fourth-order
systems do not yield better results than do second-order
systems. In the velocity domain, the type of input used
does not lead to significant differences between second-
and third-order systems, although a previous study [22]
suggests that third-order overdamped systems would be
superior to second-order overdamped models. These
third-order overdamped models were not studied here.
Handwriting may be seen as a motor task producing a
certain spatial output within relatively stringent time lim-
its. Controlling velocity is, or seems to be, the simplest way
to perform such a task. By generating momentum impulses
handwriting can be divided into strokes with relatively less

12. PLAMONDON AND MAARSE: AN EVALUATION OF MOTOR MODELS OF HANDWRITING 1071
TABLE zyxwvut
I11
DISTANCE
BETWEEN
ORIGINAL
NATURAL
HANDWRITING
OF FOUR
SUBJECTS
AND 14 RECONSTRUCTIONS
SUBJECT NUMBER
SIMULATION SHAPE OF EQUIVALENT zyxwvutsrqp
NUMBER INPUT SIGNAL ORDER OF THE
SYSTEM zyxwvutsr
1 zyxwvuts
' 2 3 4 Mean zyxw
% % % zyxwvutsr
4
0.
---
Acceleration 1 Rectangular 2 4 . 8 4.3 5 . 8 5 . 1 5 . 0
2 Trapezoidal 3 5 . 2 6 . 0 5 . 3 7 . 0 5.9
domain
3 Exponential 3 5.9 5.5 5.9 6 . 7 6 . 0
4 Sinusoidal 4 5 . 6 6.5 5 . 5 7 . 5 6 . 3
5 Bell-shaped 4 11.0 11.4 1 0 . 5 1 2 . 0 11.2
6 Triangular 3 6 . 2 7.1 5.9 8 . 0 6 . 8
Velocity
domain
Spatial
domain
7 Rectangular 1 7 . 1 6 . 8 1 0 . 8 6 . 4 7 . 8
0 Trapezofdal 2 4 . 0 3.7 7 . 5 4 . 1 4 . 8
9 Exponential 2 3.3 3.3 7 . 0 4.7 4 . 6
10 Sinusoidal 3 3 . 4 3 . 1 6.7 3.9 4.0
11 Bell-shaped 3 3 . 4 3.6 5 . 7 4.3 4.3
12 Triangular 2 3 . 2 3 . 2 7 . 0 4 . 1 4.4
13 Circular strokes 3 5.0 4 . 8 5.6 5 . 3 5 . 2
14 Long and short 3 7 . 3 7 . 3 9.5 8 . 1 8 . 1
strokes
activity at the beginning and at the end. These points can
be seen more or less as spatial targets, and it is easy to
develop motor programs based on such spatial targets. In a
training phase these targets can be used as feedback. If
handwriting is assumed to be force-impulse controlled, the
feedback mechanism is more complex. The beginning and
the end of a force stroke coincide more or less with
velocity extrema.
Moreover other indirect data also support velocity as a
control variable. Indeed, in a recent comparison experi-
ment, it has been suggested that the velocity domain seems
to be the best representation space for automatic signature
verification, as compared to the position and acceleration
domains, for systems using a zyxwvutsrqp
X-Y tablet as input. Com-
bining results from three different signal comparison algo-
rithms, this study showed that smaller error rates were
obtained when X-Y signals were used to represent a
two-dimensional signature signal [35]. If fast fine-motor
human behavior is controlled in the first instance in the
velocity domain, as suggested here, this conclusion could
be of great importance in handwriting education and in
other fine-motor training programs. The task to be per-
formed must contain easily recognizable target points.
During training, these points can be used intentionally for
segmentationin strokes or other smallparts of movements.
Well-chosen target points may ultimately be expected to
reduce the motor program developmentprocedure.
For example, interactive systems are currently being
developed to help children learn handwriting. These sys-
tems display characters on a screen and the student is
asked to reproduce them. A character-recognizermodule
gives the proper feedback about their performance [36].
This approach has the advantage of teaching not only how
to write the image of a character but also the proper stroke

13. 1072 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 19,NO. zyxwv
5, SEPTEMBER/OCTOBER 1989
order. Our conclusion suggests that using stroke definition
characters would result in the most realistic patterns in
terms of the dynamic visualization of the whole generation
handwriting: A computational model,’’ zyxwv
Biol. Cybern., vol. 45, pp.
131-142, 1982.
handwriting generation?” in Proc. zyxwv
3rd Int. Symp. Handwriting and
Computer Applications, Montreal, PQ, Canada, 1987,pp. 11-13.
[27] zyxwvuts
-, “A handwriting model based on differential geometry,” in
Computer Recognition and Human Production of Handwriting,
and representation in the domain to generate [26] R, plamondon, “What does differential geometry tell us about
process. zyxwvutsrqp
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[36]
Rejean Plamondon (M’79-SM85) received the
B.Sc. degree in physics and the M.Sc.A. and
Ph.D. degrees in electrical engineering from Uni-
versitt Laval, Quebec, PQ, Canada, in 1973,1975.
and 1978. respectively.
In 1978, he joined the staff of the Ecole Poly-
technique, Universitt de Montreal. Montreal,PQ,
where he is currently an Associate Professor.
From 1985 to 1986, he was involved in several
research projects while a guest of the Computer
Science Department, Concordia University,
Montreal, the Motor Behavior Laboratory, University of Wisconsin-
Madison, the Department of Experimental Psychology, University of
Nijmegen, The Netherlands, and the Laboratoire de Genie Electrique de
Crtteil, Universitk de Paris Val-de-Marne, France. His research interests
deal with the computer applications of handwriting: biomechanical mod-
els, neural and motor aspects, character recognition, signature verifica-
tion, signal analysis and processing, computer-aided design via handwrit-
ing, forensic sciences, software engineering, and artificial intelligence. He
is the founder and director of the Laboratoire Scribens at the Eole
Poltyechnique de Montrtal, a research group dedicated exclusivelyto the
study of these topics. He is the author or coauthor of numerous publica-
tions and technical reports.
Dr. Plamondon is a member of the board of the International Grapho-
nomics Society and President of the IAPR technical committee on text
processing applications.
Frans J. Maarse received the M S degree in
electncal engineenng from the Technical Univer-
sity of Delft, The Netherlands in 1970 and the
Ph D degree in social sciences in 1987
He joined the Psychologcal Department of
the Umversity of Nijmegen in 1971 He first
promoted the use of computers in expenmental
psychology and was for ten years Head of the
Computer Section of the Psychological Depart-
ment He is now involved in psychomotor and
handwriting research at the Nijmegen Institute
~, . .., r r ~.~...
~.
[25] P: Morass0 and F. A. Mussa Ivaldi, “Trajectory formation and for Cognition Research and Information Technology.