Human Movement Science 20 (2001) 213±241
www.elsevier.com/locate/humov
The dynamics of rhythmical aiming in 2D task space:
Relation between geometry and kinematics under
examination
Denis Mottet
a
a,b,*
, Reinoud J. Bootsma
a
UMR Movement and Perception, CNRS, University of the Mediterranean, Case Postale 910, 163,
Avenue de Luming, 13288 Marseille Cedex 09, France
b
Faculty of Sport Sciences, University of Poitiers, Poitiers, France
Abstract
We explored a two-dimensional task space variant of the classical rhythmical Fitts' task in
which participants were asked to sequentially cross four targets arranged around the extreme
points of the major axes of an ellipse. Fitts' law was found to adequately describe the changes
in movement time with the variations in task diculty (ID), but the 1/3 power-law relating
curvature and tangential velocity of the trajectory did not resist the increase in ID. Kinematic
analyses showed that the behavioral adaptation to the ID resulted in an increase in the
contribution of non-linear terms to the kinematics along the two axes of task space. Moreover,
a limit cycle model (combining Rayleigh damping and Dung sti€ness, as in one-dimensional
Fitts' task) captured such a behavior. In such a context, Fitts' law and the 1/3 power law
appear as surface relations that emerge from parametric changes in a dynamical structure that
captures the nature of Fitts' task. Ó 2001 Elsevier Science B.V. All rights reserved.
PsycINFO classi®cation: 2300; 2330
Keywords: Aiming; Kinematics; Coordination
*
Corresponding author. Tel.: +33-04-91-172250; fax: +33-04-91-172252.
E-mail address: [email protected] (D. Mottet).
0167-9457/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 9 4 5 7 ( 0 1 ) 0 0 0 3 8 - 0
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D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
1. Introduction
A characteristic property of two-dimensional drawing movements is that
tangential velocity is related to the curvature of the path: E€ector velocity
drops in the curved parts of the trajectory, much like a driver slowing
down when negotiating a curve. Extensive exploration of this relation (e.g.,
Lacquaniti, Terzuolo, & Viviani, 1983; Viviani & Flash, 1995; Viviani &
Terzuolo, 1982) has led to the identi®cation of a power law relating
tangential velocity (V) to the radius of curvature (R) of the trajectory,
according to V …t† ˆ K R…t†b , where K is a (path-length-dependent) constant
referred to as the velocity gain factor. As the experimentally observed value
of the exponent b is usually close to 1/3, this relation has come to be
known as the 1/3 power law. This lawful relation between kinematic and
geometric properties of two-dimensional movements has been shown to
hold for a variety of drawing tasks, regardless of the orientation of the
plane of motion, rate of movement and amplitude of movement (For a
review, see Viviani & Flash, 1995). Moreover, the law has been found to
hold under isometric conditions (Massey, Lurito, Pellizzer, & Georgopoulos, 1992) and to have correlates at the level of motor cortex activation
(Schwartz, 1994). These ®ndings, in conjunction with the observation that
average velocity scales with trajectory length, have been taken to imply
that the 1/3 power-law re¯ects movement planning with reference to an
internal representation of the entire intended trajectory (Viviani & Flash,
1995).
Without undermining the importance of the empirical results that relate
radius of curvature to tangential velocity in drawing movements, it has been
suggested that, rather than re¯ecting central planning, the power law might
re¯ect the operation of coupled sinusoids (Wann, Nimmo Smith, & Wing,
1988). Indeed, the motion along the elliptic shape, obtained by coupling two
orthogonal harmonic oscillations, mathematically results in the 1/3 power
law (Lacquaniti et al., 1983). From a pure geometric point of view, combining two sinusoids of equal frequency results in an ellipse, whose shape is
characterized by the length ratio of its long and short axes. Such geometrical
properties have been used to model cursive handwriting as resulting from
frequency, amplitude and phase modulation of orthogonally coupled harmonic oscillators (Hollerbach, 1981).
However, empirically observed deviations of the exponent value from 1/3
indicated that Hollerbach's (1981) model was insucient. In ellipse drawing,
the value of the exponent b has been found to be age dependent, increasing
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
215
with development toward the 1/3 value that was attained only for adults
(Viviani & Schneider, 1991). Because coupled sinusoids lead to a value of b
that is always equal to 1/3, the observed deviations from the 1/3 power law
with age cannot be understood with this harmonic model. At ®rst glance, this
result might appear as a strong limitation of the validity of an oscillatory
model of movement generation. Yet, this limitation only concerns harmonic
motion, which is but a special, albeit important, case governed by a linear
equation of motion that follows Hooke's law. Because non-linear sti€ness
functions, for example, can result in exponent values that di€er from 1/3 (e.g.,
Viviani & Schneider, 1991, Appendix), a possible account of the power law
might be found in the operation of a set of coupled oscillators that can
demonstrate deviations from harmonicity.
Analysis of the motion produced in a one-dimensional rhythmical aiming
task demonstrated that the level of diculty of the task, operationalized
through Fitts (1954) index of diculty (ID), systematically in¯uenced the
harmonicity of movement (Guiard, 1993). Recently, Mottet and Bootsma
(1999) proposed a limit-cycle model that captures the main features of the
kinematic patterns observed in such one-dimensional rhythmical aiming
tasks. Accounting for an average 95% of the variance observed, this model
demonstrates a rising in¯uence of the non-linear (Rayleigh) damping and
(Dung) sti€ness components with increasing accuracy constraints.
Although this type of modeling opens new routes towards an understanding
of the reasons underlying the emergence of another well-known relation
between geometry and kinematics, namely Fitts' Law, for the present purposes we simply note that graded deviations from harmonic motion can be
obtained through manipulation of the relevant task constraints (i.e., inter
target distance and target size).
Extension of the classical one-dimensional aiming paradigm to a two-dimensional task space (Fig. 1) has been demonstrated to preserve the dependence of movement time (MT) on the combined diculty of the two
constituent aiming tasks (Mottet, Bootsma, Guiard, & Laurent, 1994). As the
two-dimensional aiming task gives rise to two-dimensional motion patterns,
this paradigm allows examination of the 1/3 power law under di€erent
conditions of task diculty.
The goal of the present study was thus twofold. First, because we wished
to address the origins of the power law in the light of a task-dynamic approach (Saltzman & Kelso, 1987), we sought to characterize the dynamics of
the component oscillators (and their coupling) operating along the two dimensions of task space. To this end, each component oscillator was analyzed
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D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
Y axis
X axis
Fig. 1. Illustration of the two-dimensional Fitts' task. In the case represented, the long axis of motion is
horizontal (i.e., along the X-axis of recording). Although the path is prescribed at the targets only,
participants spontaneously chose an elliptic trajectory. Note that the pen did not leave a trace on the
surface.
using the same methodology (based on the W method developed by Beek &
Beek, 1988) as the one we applied to one-dimensional aiming movements
(Mottet & Bootsma, 1999). This component analysis was complemented by
an analysis of the phase relations between constituent components, so as to
characterize their coupling. Second, we examined the power-law relation
between tangential velocity and radius of curvature, seeking to understand in
what way the identi®ed deviations from harmonic motion (with respect to
both the sti€ness and the damping terms) in¯uenced the coecients and,
more generally, the validity of the power law in such a spatially constrained
task.
2. Method
2.1. Participants
Participants were seven right-handed male volunteers (age 24±41 years),
with normal or corrected to normal vision (1 with glasses and 1 with contact
lenses). Three of them were members of the laboratory sta€ and the other
four were graduate students.
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
217
2.2. Task
The task was to sequentially pass counter-clockwise through each of the
4 targets marked on a model sheet, crossing as many targets as possible in
15 seconds. Each target consisted of a line segment, whose length represented
the prescribed target width, centered on a 1-mm dot and oriented perpendicularly to the tangent of the ellipse at that point (Fig. 1).
Task diculty ± computed after Fitts (1954) as ID ˆ log2 (2D/W) ± was
manipulated independently along the two axes of motion by changing target
size (W) while maintaining a constant inter-target distance (D). Thus, the
targets were centered on the same positions for all experimental conditions,
with D ˆ 180 mm for the long axis and D ˆ 108 mm for the short axis. Task
diculty along each axis (ID ˆ 3; 4; 5) was manipulated by varying the
corresponding target size (for the 180-mm axis: W ˆ 45, 22.5, 11.25 mm; for
the 108-mm axis: W ˆ 27, 13.5, 6.75 mm).
2.3. Procedure
Each participant performed a total of 18 trials (one trial for each experimental condition) in a single experimental session that lasted for about 40
minutes. The 18 experimental conditions resulted from a factorial design with
three factors: ID along the X-axis (IDx 3, 4, or 5), ID along the Y-axis (IDy
3, 4, or 5) and orientation of the ellipse (long axis along the X- or Y-axis of
the tablet).
At the start of each trial, a model sheet (A4) was ®xed on the digitizing
tablet that rested on a tabletop. A trial started with an initial pointing at the
target centers, so as to calibrate the system. Participants were asked to
comfortably orient the tablet (within a range of about 30±45°) and all of them
chose an orientation where the forearm was roughly parallel to the vertical
axis of the model sheet. Then, participants began the warm up part of the
trial. When they felt they were complying with the constraints (i.e., going as
fast as possible with less than 5% errors), they informed the experimenter
who started the data acquisition about 1 second later. Participants were instructed to continue their movement until the computer sounded a bell. At
the end of each trial, information about performance was provided (total
number of target hits, error rate for each target). The authorized overall error
rate was between 0% and 5%. If the number of errors on any single trial was
more than 5%, or if the participant produced more than two consecutive
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D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
series with 0% error, the trial was rerun with the injunction to adapt the
speed.
2.4. Recording system and data processing
The recording system consisted of an Oce Graphics G6453 digitizing tablet
connected to an Apple Macintosh PowerBook computer. This tablet reads
the position of a non-marking stylus pen input device within a distance of
2.5 mm from its surface and provides two-dimensional position at a rate of
163 Hz with a spatial accuracy better than 0.5 mm.
Position time series were ®ltered at a cut-o€ frequency of 8 Hz with a dualpass second-order Butterworth ®lter using a padding point extrapolation at
the end of the series. From the ®ltered data, the ®rst and second time
derivatives were computed using the ®rst central di€erence method.
To examine the changes in kinematics that occurred when task diculty
was manipulated, we used the W method introduced by Beek and Beek
(1988) and used by Mottet and Bootsma (1999) to derive their dynamical
model of behavior in a one-dimensional Fitts' task. A basic assumption of
this method is that the observed motion is the result of attractor plus noise
dynamics, where the attractor is a limit cycle for rhythmical movements and
noise summarizes any higher dimensional (e.g., Mitra, Riley, & Turvey, 1997)
or random ¯uctuation.
Using a graphic approach, the ®rst step is to determine the types of nonlinearity to be included in the limit cycle model. All the plots that served for
the (qualitative) graphic analysis were those of the average cycle (Mottet &
Bootsma, 1999), which is the best estimate of the shape of the attractor as
random ¯uctuations are canceled by averaging. 1 Moreover, as the point of
interest is the shape of the kinematic patterns and the changes in this shape
with the changes in task diculty, all the plots were normalized in time and
space to ensure that the di€erent experimental conditions were comparable.
In fact, this normalization cancels the e€ects of di€erent movement amplitudes and movement times: Time is rewritten in units of average cycle duration and distance is rewritten in units of half inter-target distance. Hence,
after normalization, the time taken to perform a full cycle is always 1, and the
targets are located at 1. It is important to note that this re-scaling simply
1
Due to the ®xed duration of a trial, the number of cycles available for averaging decreased with
increasing task diculty. Over all participants and conditions, the number of cycles was between 6 and 31.
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
219
aims at making the shape of the trajectory and its degree of harmonicity more
visible. By canceling the e€ects of absolute movement amplitude and absolute
movement time, the scales of position, velocity and acceleration become
comparable. For a pure harmonic process, the phase portrait (velocity vs.
position) is a circle of radius 1, and the Hooke portrait (acceleration vs.
position) a line segment of slope )1.
The second step is to assess the values of the parameters of the model terms
using multiple linear regression. Regression of the linear and non-linear
components onto acceleration yields a measure of how well the model ®ts the
data, together with estimates of the parameters values and their change over
experimental conditions.
Following Fitts (1954), MT was measured as the average time taken to go
from one target to its opposite, which is half the mean period of the motion
(or the mean half-cycle duration). MT was analyzed using a three-way
ANOVA with repeated measures on the factors IDx (3, 4, 5), IDy (3, 4, 5)
and Orientation (Horizontal, Vertical).
As the average normalized cycles that are under examination do not
contain signi®cant noise dynamics ± removed through ®ltering and averaging
± a simple measure of the contribution of the linear terms to the observed
dynamics is provided by the r2 of the regression of acceleration onto position,
the two basic terms in a harmonic process. This r2 estimates the amount of
variance attributed to simple harmonic oscillation in the motion and, as the
total observed variance is 100%, the percentage of variance attributed to the
non-linear components is quanti®ed as NL ˆ 1 r2 .
Since the measurement of NL can be performed independently on the Xaxis and on the Y-axis, we introduced an axis factor in the statistical analysis.
This axis factor (X vs. Y, Fig. 1) is partially redundant with the orientation
factor (horizontal vs. vertical), and the combination of the axis (A) and
orientation (O) factors can be rearranged in a motion length (L) factor. For
example, when NL is measured on the X-axis and the ellipse is oriented vertically, the motion is on the short axis of the ellipse. As illustrated in Table 1,
Table 1
The combination of the axis (of movement) and orientation (of the long axis of movement) factors can be
represented in a di€erent way as a length (of movement) factora
Axis
X
X
Y
Y
Orientation
Length
Horizontal
Long
Vertical
Short
Horizontal
Short
Vertical
Long
a
See Fig. 1 for the de®nition of X-axis and Y-axis of measurement.
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D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
the only consequence of this rearrangement is to transpose the A O e€ect on
the L factor, and to transpose the e€ect of O on the A L interaction.
Consequently, the ANOVA was performed with a repeated measures design
with four factors: IDx (3, 4, or 5), IDy (3, 4, or 5), Orientation (H or V) and
Length (L: Long vs. Short). Combining IDx and IDy in an average ID (IDa:
3.0, 3.5, 4.0, 4.5, 5.0) yielded a 3-way ANOVA with repeated measures
(5 IDa 2 O 2 L).
Signi®cance of the experimental e€ects was assessed using Greenhouse±
Geisser corrected probabilities, with a set to 0.05. To assess the amount of
variance attributed to each signi®cant e€ect, the e€ect size (ES) was also
determined (Abdi, 1987).
3. Results
3.1. Qualitative graphic approach
The ®rst step in the W method is to gain a qualitative insight in the nonlinear components to be included in the model of the data. This can be done
by portraying the data in various ways, seeking for typical patterns (Fig. 2)
that signal the presence of typical conservative and dissipative non-linear
terms (Beek & Beek, 1988).
Portraying the average normalized cycles in the Hooke plane (acceleration vs. position) yields a powerful representation of the e€ects of the
conservative terms that constitute the sti€ness function. Fig. 4 illustrates the
changes in the Hooke portraits when IDx and IDy are varied. In this
®gure, one can see (e.g., in the Vertical and Long quadrant) that increasing
IDx or IDy resulted in Hooke portraits that tended towards an inverted N
shape. In other words, with increasing task diculty, local sti€ness tended
to increase at the extremes and to decrease (or reverse) in the middle, which
indicates that the nature of this sti€ness is that of a hardening spring (Fig. 2,
right).
Assessing the damping from the portraits of the data is far more dicult
because the dissipative terms might contain opposite non-linearities (e.g.,
Rayleigh and Van der Pol components that act as sine and cosine skewing on
the portraits in opposite directions, Fig. 2: left and middle). However, the
sum of these terms expresses itself as an asymmetry between the acceleration
and deceleration parts of the same half-cycle, and provides useful information on the nature of the dominant dissipation function (Mottet & Bootsma,
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
Rayleigh
Van der Pol
.
x
Duffing
.
x
.
x
x
x
x
..
x
..
x
..
x
x
.
x
x
x
.
x
t
221
.
x
t
t
Fig. 2. Portraits of classical non-linear oscillators in the phase plane (velocity vs. position, upper row), the
Hooke plane (acceleration vs. position, middle row) and as velocity pro®les (lower row). All units were
normalized to emphasize the shape. The equations used to obtain the plots were x ‡ x x_ ‡ x_ 3 ˆ 0 for the
Rayleigh oscillator, x ‡ x x_ ‡ x2 x_ ˆ 0 for the Van der Pol oscillator and x ‡ x ‡ 4x3 ˆ 0 for hardening
spring Dung oscillator.
1999). When present, detailed analysis of this asymmetry revealed that zero
acceleration (i.e., peak velocity) occurred in the ®rst half of the motion (e.g.,
Vertical and Long quadrant for IDx ˆ IDy ˆ 4). Hence, a self-sustaining
Rayleigh oscillator seems to be the main component of the dissipative terms
to be included in the model (Fig. 2, left).
To summarize, the graphical analysis indicated that a minimal limit
cycle model of the observed dynamics should include a Dung hardening
spring sti€ness function and a self-sustaining Rayleigh damping. These
non-linear terms are those of the Rayleigh + Dung (RD) model, which
222
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
was shown to capture motion dynamics in a one-dimensional Fitts' task
(Mottet & Bootsma, 1999). The equation of motion of the RD model
reads
x ‡ c10 x
c30 x3
c01 x_ ‡ c03 x_ 3 ˆ 0;
…1†
where the dot indicates di€erentiation with respect to time, x is the spatial
deviation from the origin in normalized time and space and cij is the coecient of the corresponding xi x_ j term.
Eq. (1) comprises four terms with an in¯uence on acceleration that can
be divided in two groups. The ®rst two terms (c10 x and c30 x3 ) are sti€ness
terms that in¯uence primarily the frequency of the motion. The corresponding coecients indicate the relative in¯uence of linear (c10 ) and nonlinear Dung sti€ness (c30 ). Due to the sign convention in the RD model, a
negative value for c30 indicates a hardening spring Dung (as in Fig. 2
right, where c10 ˆ 1 and c30 ˆ 4). The last two terms in Eq. (1) (c01 x_ and
c03 x_ 3 ) are damping terms that determine the stability of the motion. Here,
c01 is the linear coecient and c03 is the non-linear Rayleigh coecient. An
important constraint on the damping coecients is that both c01 and c03
should be positive to obtain limit cycle dynamics (as in Fig. 2 left, where
c01 ˆ c03 ˆ 1).
3.2. Quantifying the dynamics on both axes
Having gained a qualitative idea of the changes in kinematics that occur
with the changes in diculty on the two axes of task space, we now aim at
quantifying these changes. In a ®rst step, we assess the e€ect of task
diculty on movement time and on the global contribution of non-linear
terms to the observed movement. In a second step, we show how a
dynamical model can capture the kinematics and its changes with task
diculty.
3.2.1. Movement time
Any oscillatory process can be considered as the sum of a linear (harmonic) and a non-linear contribution, and a ®rst-order approximation for
any rhythmical motion is a harmonic process of equivalent amplitude and
movement time (Beek & Beek, 1988). In the previous section, the time and
space normalization canceled the e€ects of the changes in MT to point out
the changes of non-linear nature. Yet, of itself, MT contains important
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
223
information regarding the tuning of the (linear equivalent) sti€ness to the
experimental condition. 2
Movement time increased with IDx (F(2, 12) ˆ 48.164, p ˆ 0:0001,
ES ˆ 30.37%) and IDy (F(2, 12) ˆ 293.749, p ˆ 0:0001, ES ˆ 32.37%). A signi®cant IDx IDy interaction (F(4, 24) ˆ 10.615, p ˆ 0:0028, ES ˆ 3.63%)
indicated that increasing one ID resulted in a lower e€ect of the other ID, but
the intensity of this e€ect was an order of magnitude smaller than that of IDx
or IDy (Fig. 6A). The O IDx interaction (F(2, 12) ˆ 9.868, p ˆ 0:0032,
ES ˆ 1.72%) and the O IDy interaction that tended toward signi®cance
(F(2, 12) ˆ 4.843, p ˆ 0:062, ES ˆ 2.06%) indicated that the e€ect of the ID
on one axis was more pronounced when the long oscillation was along the
other axis.
Averaging IDx and IDy allows the derivation of a global measure of
task diculty (IDa ˆ 3.0, 3.5, 4.0, 4.5, 5.0). As expected, MT increased with
IDa (F(4, 24) ˆ 121.611, p ˆ 0.0001, ES ˆ 79.80%), with a signi®cant linear
regression (MT ˆ 0:270 IDa 0:459, r2 …18† ˆ 0:88, F(1, 16) ˆ 117.72, p ˆ
0:0001). More detailed post hoc mean comparisons revealed that, for a same
average ID, MT was lower when the ID on the two axes was the same (e.g.,
IDx ˆ IDy ˆ 4 was faster than IDx ˆ 3 and IDy ˆ 5). This result indicates
that the precise measure of diculty in a two-dimensional task space is
probably more complicated than the simple average of the ID on the two
dimensions (Mottet et al., 1994). Nevertheless, IDa accounted for close to
80% of the variance, which shows that MT in a two-dimensional Fitts' task
globally follows Fitts' Law and that a good initial estimate of the overall task
diculty is provided by the average of IDx and IDy.
3.2.2. Global contribution of non-linear terms
Having shown that the global timing properties of the system followed
Fitts' law, we now address the issue of re®ning our understanding of these
global changes by quantifying the contribution of non-linear terms (NL) to
the observed dynamics. Because the non-linear terms express themselves
through deviations from pure harmonic motion, their contribution is assessed by analyzing the average normalized cycles. It should be noted that
changes in MT that result from changes in (global) linear sti€ness do not
2
The results concerning MT were previously discussed in a di€erent contribution (Mottet et al., 1994).
However, as the original data set was completely re-analyzed with a slightly di€erent methodology, and for
the sake of coherence with the other analyses reported in the present article, the new results of the MT
analysis are reported.
224
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
show up in the average normalized cycles presented in Figs. 3, 4, 5, because of
the normalization procedure. Hence, the deviations from pure harmonic
motion visible in these ®gures re¯ect changes in local sti€ness, brought about
by non-linearities.
The contribution of non-linear terms to the motion (NL) increased with
IDx (F(2, 12) ˆ 17.274, p ˆ 0:0018, ES ˆ 13.00%) and IDy (F(2, 12) ˆ 34.735,
p ˆ 0:0004, ES ˆ 15.26%). A signi®cant IDx IDy interaction (F(4, 24) ˆ
4.179, p ˆ 0:0238, ES ˆ 1.10%) showed that the e€ect of the ID on one axis
was stronger when the ID on the other was lower, although the e€ect size of
the interaction was again an order of magnitude smaller than the e€ect size of
the main e€ects (Fig. 6B). Thus, these results replicate the ®ndings with
respect to MT, which indicates that the reasons underlying the changes in
MT rely on subtle but systematic changes in the kinematics, and that the
latter are of non-linear nature.
.
x
Horizontal
Vertical
IDy = 3
1
x
-1
1
IDy = 5
Short
IDy = 4
IDy = 3
IDy = 5
Long
IDy = 4
-1
IDx = 3
IDx = 4
IDx = 5
IDx = 3
IDx = 4
IDx = 5
Fig. 3. Average normalized phase portraits of the long and short oscillations along the X- and Y-axes. The
nine portraits in each quadrant illustrate the combination of the 3 levels of task diculty on each axis. IDx
and IDy denote the index of diculty along the horizontal and vertical axes, respectively (Fig. 1). The
motion corresponding to the same trials are on the diagonal (i.e., when the long oscillation is along the Xaxis, the short one is along the Y-axis). Increasing task diculty resulted in phase portraits that tend to
¯atten or crush at the midpoint. Peak velocity is usually attained in the ®rst part of the motion, which
denotes the in¯uence of Rayleigh damping.
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
Vertical
1
IDy = 4
-1
Horizontal
x
IDy = 5
Short
IDy = 4
IDy = 3
IDy = 5
Long
-1
..
x
IDy = 3
1
225
IDx = 3
IDx = 4
IDx = 5
IDx = 3
IDx = 4
IDx = 5
Fig. 4. Average normalized Hooke portraits of the long and short oscillations along the X- and Y-axis.
The organization of this ®gure is the same as for Fig. 3. Increasing task diculty resulted in Hooke
portraits that tended to an inverted N shape. Comparing the quadrants along the diagonals shows that
increasing IDx in¯uenced the motion on the Y-axis and increasing IDy in¯uenced the motion on the
X-axis, but this e€ect is stronger on the long axis.
NL was higher along the long axis (F(1, 6) ˆ 8.781, p ˆ 0:0252, ES ˆ 2.73%)
and the signi®cant L O interaction (F(1, 6) ˆ 13.176, p ˆ 0:011, ES ˆ 4.61%)
indicated that horizontal motion was less harmonic than vertical motion.
Increasing IDx resulted in a stronger increase in NL for the vertical orientation (O IDx: F(2, 12) ˆ 8.905, p ˆ 0:0044, ES ˆ 2.62%) and increasing
IDy resulted in a stronger increase in NL for the horizontal orientation
(O IDy: F(2, 12) ˆ 9.391, p ˆ 0:0129, ES ˆ 5.93%). A similar di€erence was
found for NL on the long and short axes of the ellipse: The ID e€ect was more
pronounced on the long axis of the ellipse (L IDx: F(2, 12) ˆ 10.021,
p ˆ 0:0135, ES ˆ 1.57% and L IDy: F(2, 12) ˆ 23.132, p ˆ 0:0001, ES ˆ
1.84%). Finally, the L O IDx interaction (F(2, 12) ˆ 8.71, p ˆ 0:0179,
ES ˆ 2.88%) showed that the increase in NL with IDx was stronger for the
long axis, but only for the vertical orientation and the L O IDx interaction
(F(2, 12) ˆ 4.552, p ˆ 0:0355, ES ˆ 0.58%) indicated that the increase in NL
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
Vertical
t
MT
IDy = 5
Short
IDy = 4
IDy = 3
IDy = 5
Long
-1
Horizontal
.
x
IDy = 3
1
IDy = 4
226
IDx = 3
IDx = 4
IDx = 5
IDx = 3
IDx = 4
IDx = 5
Fig. 5. Average normalized velocity pro®les of the long and short oscillations along the X- and Y-axes.
The organization of this ®gure is the same as for Fig. 3. With the increase of task diculty, velocity pro®les
tended to ¯atten or become two-peaked, with peak velocity reached in the very ®rst part of the motion.
Fig. 6. Movement time (MT: panel A) and global contribution of non-linear terms (NL: panel B) as a
function of the index of diculty along the horizontal and vertical axes (respectively, IDx and IDy, see
Fig. 1). MT was measured as average half-cycle duration and NL as the r2 not explained by a linear model.
Note that MT and NL increase in the parallel with IDx and IDy, but that increasing one ID results in a
lower e€ect of the other ID.
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
227
with IDy was stronger on the long axis, but only for the horizontal orientation
(Fig. 7). Overall these results indicate that the e€ect of the ID on one axis was
stronger for the motion along the other axis and that this e€ect was stronger
for the long axis (e.g., the IDx e€ect is stronger on the Y-axis, Fig. 7). Hence,
adaptation to the accuracy constraints on one axis primarily results from
changing the kinematics on the opposite axis. In other words, when allowed to
use two dimensions of motion, participants took advantage of this larger task
space.
Collapsing IDx and IDy in an average index of diculty (IDa) gave rise to
simpler results, with a (linear) increase in NL with IDa (F(4, 24) ˆ 27.74,
p ˆ 0:0003, ES ˆ 42.34%) that was stronger along the long axis (L ID:
F(4, 24) ˆ 14.684, p ˆ 0:0003, ES ˆ 4.67%).
3.2.3. Nature of the observed changes
Given the increase in the contribution of the non-linear terms to the motion on both axes of task space with task diculty, the next step is to assess
the capability of the RD model to capture the observed changes. To quantify
the e€ects that were observed in the Hooke portraits (Fig. 4), the parameters
Long
0.25
Short
0.20
NL
0.15
0.10
0.05
0.00
X-axis
Y-axis
3 4 5
3 4 5
IDx
X-axis
Y-axis
3 4 5
3 4 5
IDy
Fig. 7. Global contribution of non-linear terms (NL) as a function of axis length (L), index of diculty
along the horizontal (IDx) and vertical axes (IDy). The ®gure shows that the e€ect of one ID is stronger on
the opposite axis and on the long axis.
228
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
of the RD model were assessed using multiple linear regression of the four
model components onto acceleration. While the RD model is minimal in the
number of its parameters, it captured the general shape of the dynamics
explaining 95.52% of the variance on average. An ANOVA showed that the
obtained r2 decreased somewhat when task diculty increased (IDx:
F(2, 12) ˆ 9.707, p ˆ 0:0115, ES ˆ 11.27% and IDy: F(2, 12) ˆ 37.171, p ˆ
0:0007, ES ˆ 17.67%) because the dynamics were more complex when task
diculty increased (e.g., for the Hooke portrait in the long and horizontal
quadrant, and for IDx ˆ IDy ˆ 5, Fig. 4). However, goodness of ®t remained
high, with only 2 of the 36 r2 values below 0.90, indicating that the RD model
adequately captures the observed changes in the organization of movement
(Table 2).
This satisfactory ®t allows addressing the question of the changes in the
parameters with the experimental factors. Due to the time and space normalization (i.e., with amplitude ˆ 2 and MT ˆ 1/2), the parameters that we
obtain do not re¯ect the absolute changes in time and space (but switching
back to absolute metrics is easy, see Mottet & Bootsma, 1999). However,
they quantify the changes in the shape of the motion, which is the point of
interest here.
Inspection of Table 2 reveals that the two dissipative and the two conservative coecients changed together. This is a consequence of the design of
the RD model, in which the relative values of the c10 and c30 coecient determine the N shape in the Hooke portrait and the absolute values of the c01
and c03 coecients de®ne the asymmetry of the acceleration±deceleration
phases.
Four-way repeated measures ANOVAs showed signi®cant main e€ects of
the axis length factor for all the terms in the RD model. The value of c10 was
higher for the long axis (F(1, 6) ˆ 104.162, p ˆ 0:0001, ES ˆ 26.03%) as was
the value of c30 (F(1, 6) ˆ 103.143, p ˆ 0:0001, ES ˆ 25.00%), which indicates
that the relative contribution of non-linear sti€ness terms was higher for the
larger amplitudes of movement. Conversely, the value of c01 was lower on the
long axis (F(1, 6) ˆ 7.607, p ˆ 0:0329, ES ˆ 2.04%) as was the value of c03
(F(1, 6) ˆ 7.276, p ˆ 0:0357, ES ˆ 1.71%) indicating that the dissipative terms
contributed less to the kinematics for the long axis. Taken together, these
results indicate that the dynamics along the long and short axes were parameterized in di€erent ways. However, the e€ect size is 10 times higher for
the sti€ness terms, which indicates that the di€erence between the long and
short axes mainly relies on sti€ness changes, with a more linear dynamical
organization along the shorter axis.
Table 2
Average coecient values and goodness of ®t …r2 † of the RD model for the 18 experimental conditions and the two axes of motion (X,Y: axis of
motion, see Fig. 1; IDx, IDy: index of diculty along the corresponding axis; H, V: horizontal or vertical orientation of the long axis of the ellipse)
IDx: 3
Y
5
3
4
5
3
4
5
3
4
5
H
c10
c30
c01
c03
r2
1.014
)0.006
)0.091
0.122
0.993
0.713
)0.418
)0.249
0.369
0.985
)0.118
)1.602
)0.201
0.346
0.940
0.870
)0.193
)0.130
0.183
0.987
0.568
)0.605
)0.078
0.121
0.971
)0.032
)1.482
)0.135
0.228
0.930
0.846
)0.235
)0.183
0.257
0.954
0.556
)0.623
)0.086
0.130
0.956
0.068
)1.308
)0.104
0.163
0.883
V
c10
c30
c01
c03
r2
1.030
0.009
0.092
)0.120
0.967
0.959
)0.110
0.318
)0.444
0.953
1.247
0.223
0.619
)0.777
0.900
0.870
)0.222
0.240
)0.325
0.971
0.835
)0.288
0.422
)0.579
0.948
1.120
0.067
0.538
)0.672
0.899
0.373
)0.903
0.266
)0.405
0.944
0.670
)0.494
0.395
)0.583
0.948
0.686
)0.509
0.551
)0.741
0.877
H
c10
c30
c01
c03
r2
0.847
)0.220
0.049
)0.067
0.986
0.698
)0.439
0.132
)0.188
0.986
0.627
)0.565
0.323
)0.455
0.958
0.588
)0.644
0.288
)0.424
0.969
0.581
)0.639
0.314
)0.472
0.963
0.352
)0.967
0.320
)0.501
0.950
)0.165
)1.706
0.021
)0.037
0.945
)0.055
)1.533
0.209
)0.359
0.933
)0.076
)1.559
0.211
)0.338
0.919
V
c10
c30
c01
c03
r2
1.154
0.175
)0.152
0.192
0.985
1.180
0.190
)0.292
0.364
0.975
0.749
)0.374
)0.153
0.220
0.958
1.195
0.186
)0.107
0.131
0.952
1.089
0.062
)0.137
0.184
0.974
0.851
)0.252
)0.053
0.106
0.966
1.484
0.559
)0.041
0.053
0.945
1.030
)0.026
0.065
)0.072
0.942
0.913
)0.189
)0.081
0.137
0.938
229
IDy
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
X
4
230
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
The signi®cant L O interactions for c01 (F(1, 6) ˆ 8.44, p ˆ 0:0271, ES ˆ
32.65%) and for c03 (F(1, 6) ˆ 8.572, p ˆ 0:0264, ES ˆ 33.94%) indicated that
the e€ect of axis length depended on the orientation. These interactions
suggest that damping di€ered along the X and Y axes, with positive c01 values
(and negative c03 values) for the Y-axis and negative c01 values (and positive
c03 values) for the X-axis. This latter result points out one limitation of the W
method: Because the regression process does not include constraints on the
sign of the coecients, it can lead to an unstable model with a negative c01
coecient (Beek, Schmidt, Morris, Sim, & Turvey, 1995). However, as
Rayleigh and Van der Pol oscillators act in an opposite fashion, an unstable
Rayleigh is similar to a stable Van der Pol limit cycle. Consequently, the
values of damping coecients indicate that Rayleigh damping was dominant
along the vertical oscillation (Y-axis) while Van der Pol damping dominated
along the horizontal oscillation (X-axis).
The e€ects of IDa on the coecients in the model are illustrated in Fig. 8.
Increasing IDa resulted in a decrease of the values of the c10 and c30 coecients (F(4, 24) ˆ 40.940, p ˆ 0:0001, ES ˆ 23.11% and F(4, 24) ˆ 44.033,
p ˆ 0:0001, ES ˆ 24.98%), but this e€ect was stronger on the long axis
(F(4, 24) ˆ 23.065, p ˆ 0:0002, ES ˆ 6.77% and F(4, 24) ˆ 23.792, p ˆ 0:0002,
ES ˆ 6.86%). For the dissipative terms, increasing IDa was accommodated
by a slight but systematic increase in c01 (F(4, 24) ˆ 6.799, p ˆ 0:0102,
Long Axis
1.50
Short Axis
Coefficient value
1.00
0.50
0.00
-0.50
c10
c01
-1.00
-1.50
3.0
3.5
4.0
IDa
4.5
5.0
3.0
3.5
4.0 4.5
IDa
c30
c03
5.0
Fig. 8. Coecient values in the RD model as a function of average ID (IDa) in the 2D task space, for the
long (left) and for the short oscillation (right). This ®gure illustrates that the two sti€ness terms (c10 and
c30 ) tend to co-vary, as do the two damping terms (c01 and c03 ). The e€ect of IDa is approximately linear,
and stronger for the sti€ness terms.
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
231
ES ˆ 4.39%) and c03 (F(4, 24) ˆ 6.732, p ˆ 0:01, ES ˆ 3.99%), with no signi®cant di€erences between the long and short axes. This result shows that
the increase in NL with IDa was primarily a consequence of local sti€ness
changes. As mentioned earlier, the fact that the linear damping term
remained slightly below zero until IDa ˆ 3.5 indicates that the corresponding
model is unstable, which denotes a stronger Van der Pol than Rayleigh
damping. It is worth noting that the critical ID is close to 4, similar to the
®ndings reported with respect to 1D space (Mottet & Bootsma, 1999).
When trying to decompose the IDx and IDy e€ects, a similar pattern is
found. For the conservative terms, increasing IDx or IDy resulted in a
monotonic decrease in the magnitudes of c10 (IDx: F(2, 12) ˆ 22.139,
p ˆ 0:0007, ES ˆ 6.34% and IDy: F(2, 12) ˆ 81.295, p ˆ 0:0001, ES ˆ 6.11%)
and c30 (IDx: F(2, 12) ˆ 23.577, p ˆ 0:0008, ES ˆ 6.82% and IDy: F(2, 12) ˆ
101.594, p ˆ 0.0001, ES ˆ 6.69%). For the dissipative terms, no signi®cant
e€ects of IDx were found, but an increase in IDy resulted in an increase in c01
and c03 (F(2, 12) ˆ 26.828, p ˆ 0:0005, ES ˆ 2.17% and F(2, 12) ˆ 16.44,
p ˆ 0:0033, ES ˆ 1.53%).
For the conservative terms, that account for most of the NL increase with
ID, signi®cant higher-order interactions were found. The L O IDx interactions were signi®cant (for c10 , F(2, 12) ˆ 23.902, p ˆ 0:0001, ES ˆ 8.27%;
for c30 , F(2, 12) ˆ 24.067, p ˆ 0:0001, ES ˆ 8.43%), as were the L O IDy
interactions (F(2, 12) ˆ 24.562, p ˆ 0:0007, ES ˆ 9.12% and F(2, 12) ˆ 24.99,
p ˆ 0:0006, ES ˆ 8.65%). Recalling that the L O interaction is similar to a
direction factor (i.e., horizontal vs. vertical oscillation), these interactions
indicated that the e€ect of increasing ID on one axis primarily in¯uenced the
kinematics on the opposite axis. This e€ect replicates the ID e€ects on MT,
but at the parameter level. Hence, it clearly indicates that the observed
changes in MT rely on local changes in sti€ness as well as on the global
changes canceled through the time normalization. Finally, the L O IDx IDy interaction was signi®cant for c10 and c30 , but associated with a
negligible e€ect size (F(4, 24) ˆ 4.18, p ˆ 0:0275, ES ˆ 0.99% and F(4, 24) ˆ
4.082, p ˆ 0:0302, ES ˆ 0.95%).
3.3. Coordination of the two oscillations
An important issue in a task involving two degrees of freedom is that of
their coordination, often taken to be central for the understanding of behavior (Kelso, 1995). In the present context, however, coordination of the
motions along the X- and Y-axes of task space is heavily constrained by the
232
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
targets drawn on the model sheet, which impose a relative phase of 90° for a
smooth trajectory passing through the four target centers. Indeed, because
the X- and Y-axes are orthogonal, the phase lag between the X- and Ycomponents of the motion de®nes the slanting of the elliptic shape drawn.
However, the high degree of non-linearity that was observed in the motion
along the X- and Y-axes might result in variations in phase lag for trajectory
sections between consecutive targets, where the path is not spatially constrained. To evaluate this possibility, we measured the (continuous) relative
phase (RP) of the X- and Y-components of the observed movement and its
intra-trial variability (SDRP). RP was de®ned as the mean phase di€erence
(over trial duration) between the X- and Y-motion, and SDRP as the corresponding (intra-trial) standard deviation. 3
Over all experimental conditions and participants, on average relative
phase was 87:49° and the corresponding standard deviation 0:125°. These
results show that relative phase was fairly close to the expected 90° prescribed
by the four targets, but the most striking result is perhaps the high degree of
intra-trial stability which indicates a strong phase locking of the motions
along the two dimensions of task space.
The fact that relative phase was not exactly 90° indicates a slight slanting
of the overall trajectory relative to the frame of reference of the targets (see
Table 3). A three-way ANOVA with repeated measures (2 O 3 IDx 3 IDy) on the average intra-trial relative phase (RP) revealed that the shape
drawn was less slanted with the increase of IDy (F(2, 12) ˆ 25.882, p ˆ 0.0014,
ES ˆ 19.70%). The decrease of slant with the increase of IDy was also less
important when IDx increased (F(4, 24) ˆ 4.347, p ˆ 0:0345, ES ˆ 2.54%). As
participants were allowed to adapt orientation of the graphic tablet in the
most comfortable way, and because they spontaneously chose a slanting of
about 30°, this result indicates that they tended to underestimate the necessary slanting of the main axis of the workspace relative to their body.
Whatever the experimental conditions, motion along the two axes demonstrated an almost perfect phase locking, as revealed by the consistently
high degree of intra-trial stability of relative phase (see Table 3). A three-way
ANOVA with repeated measures (2 O 3 IDx 3 IDy) showed that SDRP
increased somewhat with IDy (F(2, 12) ˆ 5.526, p ˆ 0:0397, ES ˆ 6.14%). The
signi®cant O IDx and O IDy interactions (F(2, 12) ˆ 4.375, p ˆ 0:0386,
3
To obtain comparable scales for position and velocity, angle in phase space was computed after time
normalization in units of movement time.
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
233
Table 3
Relative phase (RP) and standard deviation of the relative phase (SDRP) in the 18 experimental conditions
(IDx, IDy: index of diculty along the corresponding axis of motion, see Fig. 1; H; V : horizontal or
vertical orientation of the long axis of the ellipse)
IDx: 3
4
5
IDy
3
4
5
3
4
5
3
4
5
H
RP
SDRP
86.23
0.135
87.55
0.113
89.44
0.146
85.71
0.106
87.61
0.118
89.15
0.142
86.17
0.122
87.26
0.115
88.81
0.150
V
RP
SDRP
83.88
0.124
87.72
0.116
91.33
0.113
84.74
0.116
87.32
0.117
89.32
0.118
86.97
0.137
88.58
0.135
88.46
0.137
ES ˆ 2.46% and F(2, 12) ˆ 9.665, p ˆ 0:0061, ES ˆ 7.37%) and post hoc
comparisons indicated that relative phase variability signi®cantly increased
with IDx in the vertical condition only and with IDy in the horizontal
condition only. Thus, relative phase variability increased slightly when ID
along the long axis of the ellipse was raised, but the most noteworthy point
remains that SDRP was always less than 0:15°.
3.4. Trajectory vs. kinematics relationships
In addition to examining the dynamics of motion along the two axes of the
task space de®ned by a two-dimensional Fitts' task, a second goal of the
present study was to evaluate the validity of the power law linking the instantaneous tangential velocity (V) to the radius of curvature (R) of the
trajectory for such a spatially constrained task. A straightforward test of the
1/3 power law is provided by linear regression of the natural logarithms of V
onto R. The validity of a power-law relation is re¯ected in the percent of
variance explained by the regression and estimates of K and b are obtained
according to 4
log…V † ˆ K ‡ b log…R†:
…2†
In a ®rst step, we performed a qualitative analysis of the power law by
plotting log…V † as a function of log…R†, separately for each trial. In line with
4
For a wider applicability of this relation (e.g., when the overall path includes di€erent units of action
or in¯ection points), R is to be replaced by R ˆ R=…1 ‡ aR†, with a being usually about 0.05 (Viviani &
Flash, 1995). As typical trajectories in the present experiment were elliptical, the simplest expression was
sucient.
234
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
the results obtained in graphic tasks (Viviani & Flash, 1995), when the spatial
constraints were low, the 1/3 power law adequately captured the relationships
between trajectory and kinematics. Yet, increasing the accuracy requirements
resulted in noisier log…V † vs. log…R† plots, where systematic structure could
hardly be found (Fig. 9). Hence, in the two-dimensional Fitts' task, tangential
velocity seems to be related to the curvature of the trajectory for the lowest
diculties only.
A quantitative analysis of the power law was also performed on the basis
of Eq. (2), which allows assessing the values of r2 , K and b for each trial. On
average, the ®t of the recorded data to the power law was relatively poor, the
linear regression of log…V † onto log…R† rendering an average r2 of 0.58. A
three-way ANOVA with repeated measures (2O 3 IDx 3 IDy) showed
that the goodness of ®t decreased with increases in IDx and IDy
(F(2, 12) ˆ 24.499, p ˆ 0.0004, ES ˆ 10.80% and F(2, 12) ˆ 28.358, p ˆ 0.0001,
ES ˆ 15.03%), but did not change signi®cantly with orientation (F(1, 6) ˆ
2.862, p ˆ 0.1417). The signi®cant O IDx and O IDy interactions
IDx = 4
IDx = 3
IDx = 5
V (cm/s)
IDy = 3
100
10
V (cm/s)
IDy = 4
100
10
V (cm/s)
IDy = 5
100
10
1
2
10
R (cm)
10
4
1
2
10
R (cm)
10
4
1
2
10
R (cm)
10
4
Fig. 9. Representation of tangential velocity (V) as a function of radius of curvature (R) in logarithmic
scale for 9 typical trials (Participant 5, vertical orientation). Increasing the index of diculty along the
horizontal axis (IDx) or along the vertical axis (IDy) resulted in a noisier plot, indicating that tangential
velocity is no longer related to the curvature by a power-law.
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
235
(F(2, 12) ˆ 20.697, p ˆ 0.0003, ES ˆ 12.67% and F(2, 12) ˆ 10.959, p ˆ 0.0136,
ES ˆ 11.20%) and post hoc comparisons indicated that the e€ect of IDx
existed in the vertical orientation only and that of IDy in the horizontal
orientation only. Moreover, the e€ect of one ID was stronger when the value
of the other was lower (F(4, 24) ˆ 7.037, p ˆ 0.004, ES ˆ 3.89%). The power
law accounted for less than 45% of the variance for the most dicult experimental condition (Fig. 10A). A similar analysis conducted on the value of
the exponent b yielded a comparable pattern of results. The value of b decreased with IDx and IDy (F(2, 12) ˆ 23.477, p ˆ 0.0003, ES ˆ 8.27% and
F(2, 12) ˆ 123.921, p ˆ 0.0001, ES ˆ 20.67%), but the e€ect of IDx existed in
the vertical orientation only (F(2, 12) ˆ 25.308, p ˆ 0.0001, ES ˆ 13.47%) and
that of IDy in the horizontal orientation only (F(2, 12) ˆ 28.262, p ˆ 0.0008,
ES ˆ 14.80%). As for r2 , the e€ect of one ID was stronger when the other was
lower (F(4, 24) ˆ 7.014, p ˆ 0.0088, ES ˆ 2.40%) and the value of b was reasonably close to 1/3 in the easiest case only (i.e., 0.30 for IDx ˆ IDy ˆ 3, see
Fig. 10B).
These results converge to indicate that the power-law relating the
geometrical and kinematic aspects of planar limb movements is valid only
when minor spatial constraints are imposed on the trajectory (e.g.,
Thomassen & Teulings, 1985; Wann et al., 1988). This is the case in most
drawing movements, but not in a two-dimensional Fitts' task where the path
β
Fig. 10. Goodness of ®t of the power law to the recorded data (r2 : Panel A) and corresponding exponent
value (b: Panel B) as function of the index of diculty along the horizontal (IDx) and vertical axes (IDy).
With the increase of ID, the ®t of the power law decreased while the exponent value departed from the
typical 1/3 value.
236
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
is constrained at the 4 targets (Fig. 1). In the present experiment, the validity
of the power law was inversely related to the level of non-linearity in the
motion (compare Fig. 10B and Fig. 6B). Because a level of non-linearity
indicates that the motion is less smooth (i.e., higher harmonics play a
signi®cant role in the motion), these results indicate that the validity of the
power law is mainly related to the smoothness of the actual trajectory
(Todorov & Jordan, 1998).
4. Discussion
In this contribution, we addressed the problem of trajectory formation
in two-dimensional aiming following a task-dynamic approach (Saltzman
& Kelso, 1987). While the idea of orthogonally coupled functional oscillators as a basis for drawing movements is not new (e.g., Hollerbach,
1981), the development of novel conceptual tools for the understanding of
coordination (Haken, Kelso, & Bunz, 1985) has led to a renewed interest in
the characteristics of the component dynamics and their coupling (e.g.,
Semjen, Summers, & Cattaert, 1995). Here, we have shown that the
behavioral adaptation to task diculty (i.e., adaptation to the con¯icting
constraints of speed and accuracy) resulted in an increase in the global
contribution of non-linear terms to the kinematics along the two axes of
task space. As in one-dimensional task space (Guiard, 1993), this nonlinearity increased in a very systematic fashion with task diculty. An
important result was that a 4-parameter limit-cycle model was able to
capture the overall topology of movement organization and its changes
with task diculty. As this limit-cycle model is the same as the one we
proposed to model the dynamics in one-dimensional task space (Mottet &
Bootsma, 1999), we take these results as evidence that the RD model
captures the very logic of Fitts' task.
With an average of 95% of the observed variance accounted for by the RD
model in both the one- and two-dimensional tasks, our results indicate that
the nature of the underlying dynamics is identical in one- and two-dimensional task space. In fact, when the number of dimensions of task space is
increased from one to two, the general changes in the kinematics with ID
follow the same pattern: An increase in ID results in an increase in the
contribution of non-linear dynamics to the motion. The systematic changes
observed in the dynamics point out two important points. First, the covariations of NL and MT indicate that the changes in MT are a consequence
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
237
of parametric changes in the dynamic of the task, while the nature of the
dynamics does not change. In this light, MT is a macroscopic property that
emerges as consequence of parametric changes in task dynamics. Second, the
evolution of the coecients of the RD model with task diculty, in onedimensional (Mottet & Bootsma, 1999) and two-dimensional (present study)
task spaces, reveals a characteristic pattern. The observed changes in this
characteristic pattern with task diculty are twofold. On the one hand, increasing task diculty leads to an increasing contribution of the dissipative
terms, with velocity-dependent Rayleigh damping becoming dominant when
task diculty is increased over 3±4 bits. Notwithstanding the methodological
diculties associated with the identi®cation of dissipative terms in the absence of external perturbations, this result highlights the role of velocity information in the emergence of spatial accuracy. On the other hand,
increasing task diculty leads to an increasing contribution of the conservative terms, with sti€ness becoming more and more non-linear. A point to
note is that the sign of the non-linear sti€ness coecient c30 is reversed in oneand two-dimensional task spaces. With increasing diculty in one-dimensional task space, the sti€ness becomes that of a softening spring, while it
develops into that of a hardening spring in the two-dimensional task space.
Mottet and Bootsma (1999) suggested that the decrease in local sti€ness
serves to lower the system's relaxation time in the neighborhood of the targets (to ensure that the local spatial variability stays in the allowed range)
while keeping the overall sti€ness as high as possible (i.e., to minimize
movement time). In the one-dimensional case, this function is thus subserved
by a softening spring. In the two-dimensional case, the hardening spring
character leads to a decrease in local sti€ness in the neighborhood of the
oscillation center, corresponding to the moment that the e€ector crosses the
target on the other axis.
Indeed, the task in Fig. 1 is di€erent from the one-dimensional task where
the target interval is parallel to the axis of motion. In the two-dimensional
task used in the present experiment ± with the task being to ``walk through''
the target(s) ± at the point at which the stylus crosses a target, the movement
is actually in the orthogonal direction. When considering the present task in
this light it is not surprising that the e€ect of the ID on one axis primarily
in¯uenced the dynamics along the other axis. This result is important because
it indicates that, as the use of the available degrees of freedom is driven by
coordination principles (Buchanan, Kelso, & de Guzman, 1997; de Guzman,
Kelso, & Buchanan, 1997), the action system can take advantage of the
multiple dimensions of the task space to achieve its goal. According to this
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D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
logic, the principles underlying trajectory formation are to be found at the
behavioral level (i.e., de®ned in terms of task-related variables), because it is
at the level of the task that the relevant dimensions of movement systems are
determined (Mottet, Guiard, Ferrand, & Bootsma, in press; Saltzman &
Kelso, 1987).
An interesting aspect of the present two-dimensional aiming task is that
the presence of the four targets critically constrains the coordination of the
constituent components, driving the system towards a 90° phase lag
between the two oscillatory motions. The ®nding that this coupling resisted
the important non-linearities in the component oscillations is a cue indicating a true phase locking of the motion along the two axes of task space.
Indeed, the only kinematic aspect that appeared to be invariant over
orientation and task diculty was the relative phase of the motion along
the X - and Y -axes.
Taken together, these results lead us to reformulate the statement that
``trajectory determines movement dynamics'' (Viviani & Terzuolo, 1982,
p. 431) as trajectory constraints determine movement dynamics, at the level of
the (task de®ned) component oscillators, as well as at the level of their
coupling. The important di€erence between these two formulations lies in
their implications for the locus of organizational principles. For Viviani and
Terzuolo (1982), the transformation from an intended trajectory to a plan of
movement is achieved by invoking organizational principles, such as the
power-law relating tangential velocity and radius of curvature. For us,
organizational principles play their role in the assembly of task-speci®c
dynamical systems, that may or may not give rise to lawful relations between
geometry and movement kinematics. We take the ®nding that the one-third
power law of human drawing movements did not resist the increase in task
diculty in the two-dimensional Fitts' task as support for our position. In
the present experiment, the power relation between the curvature and the
tangential velocity is strongly related to the harmonic-like dynamics that
characterizes unconstrained movements. With the increase of task diculty,
not only the exponent value decreased (which can be due to a hardening
spring sti€ness, Viviani & Schneider, 1991), but also the regression of log…V †
onto log…R† strongly departed from linear. In a way, this result replicates
what was found in the developmental data, where the ®t of the power law
increased linearly from 88% to 100% as the value of b increased from 0.25 to
0.34 (Fig. 7 in Viviani & Schneider, 1991).
According to Todorov and Jordan (1998), the convergence noted by
Viviani and Flash (1995) between the power law and a minimum jerk
D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
239
movement organization criterion is a consequence of the similarity between a
minimum jerk and a maximally smooth trajectory. Because a maximally
smooth oscillation (i.e., a sinusoid) is obtained with linear (second-order)
dynamics, it is not surprising that, in the present study, the power law was
found to hold for the conditions that gave rise to quasi-linear dynamics, that
is, the conditions with minimal spatial accuracy constraints. Increasing the
spatial accuracy constraints, however, resulted in a rising in¯uence of nonlinearities in the dynamics, leading to movements that no longer satisfy the
power law.
From the perspective of two coupled RD oscillations, it must be pointed
out that the e€ects of the non-linear damping and non-linear sti€ness terms
are di€erent. Simply introducing a Dung sti€ness term results in a motion
following the power law, but with an exponent value that di€ers from 1/3 (a
value lower than 1/3 indicates a softening spring and a value higher than 1/3 a
hardening spring, as already demonstrated in Viviani & Schneider, 1991).
Such a non-linear sti€ness strongly in¯uences the MT but, because it does not
a€ect the symmetry of the velocity pro®le ± acceleration and deceleration
remain symmetric ± it is not responsible for the deviations from a power law
in our data. In the logic introduced with the RD model, the deviations from a
power law are related to the rising in¯uence of the Rayleigh damping function, which skews the velocity pro®les towards a shorter acceleration than
deceleration phase.
The 1/3 power law in human drawing movements and Fitts' law in
human aiming movements are among the few quantitative and lawful
regularities in experimental psychology. An important theoretical issue
concerns the reasons underlying Fitts' law and the 1/3 power law. From
the perspective of a task-dynamic approach, Fitts' law is a consequence of
the re-description in the space of the parameters of the speci®c constraints
that are de®ned at the interface of the actor-environment, the interface
being shaped by the constraints of the task. In other words, task constraints (or intention) act as limits on the morphogenetic capabilities of
the perception-movement coupling that is the dynamical system. The
power law seems to be more a by-product of an action system spontaneously moving in a harmonic fashion (when no spatial constraints occur)
than a movement generating principle that is employed by the central
nervous system to constrain trajectory formation. Hence, one strength of
the task-dynamic approach is to provide a consistent and uni®ed account
of Fitts' law, the 1/3 power law and other kinematic aspects of the
behavior.
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D. Mottet, R.J. Bootsma / Human Movement Science 20 (2001) 213±241
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