ArticlePDF Available
Velocity Estimation Algorithms for Audio-Haptic
Simulations Involving Stick-Slip
Stephen Sinclair, Marcelo M. Wanderley, Member, IEEE, and Vincent Hayward, Fellow, IEEE
Abstract—With real-time models of friction that take velocity as input, accuracy depends in great part on adequately estimating
velocity from position measurements. This process can be sensitive to noise, especially at high sampling rates. In audio-haptic acoustic
simulations, often characterized by friction-induced, relaxation-type stick-slip oscillations, this gives a gritty, dry haptic feel and a raspy,
unnatural sound. Numerous techniques have been proposed, but each depend on tuning parameters so that they may offer a good
trade-off between delay and noise rejection. In an effort to compare fairly, each of thirteen methods considered in the present study was
automatically optimized and evaluated; finally a subset of these were compared subjectively. Results suggest that no one method is
ideal for all gain levels, though the best general performance was found by using a sliding-mode differentiator as input to a Kalman
integrator. An additional conclusion is that estimators do not approach the quality available in physical velocity transduction, and
therefore such sensors should be considered in haptic device design.
Index Terms—Haptics, friction, velocity estimation
Ç
1INTRODUCTION
COMPLEX oscillations arising from non-linear mechanics
are commonly found in the real world, leading to high-
frequency behaviour. For instance, a dry finger tip sliding
on an otherwise smooth surface generates wideband noise
[1]. In the case of musical instruments, non-linear behaviour
is fundamental to their operation, and must be taken into
account during simulation; such simulations may be used
to train skills, create music and sounds, study their physics,
or study human sensorimotor behaviour [2]. Wideband,
audible properties of such phenomena imply high simula-
tion rates, viz. 20 to 40 kHz, exacerbating any noise issues
due to sampling and differentiation.
In simulation, audio and haptic feedback signals may be
generated synchronously for interactive applications; in such
a case, noise issues are not only annoying, but may affect the
haptic properties of the simulated material (e.g., modifying
perception of hardness [3]), or even result in low-quality
audio synthesis which defeats the purpose of the simulation.
For simulating the bowed string, at least two parameters,
velocity and pressure, must be accounted for to control the
friction-induced vibrations between a bow and a string [4].
In implementing such a simulation, we encountered the
challenge that friction-driven dynamics depend on velocity,
but force feedback devices almost invariably employ dis-
placement sensors, necessitating differentiation. This led to
an unacceptable level of noise, destroying realism of both
the sound and feel of the simulation.
One solution, employed in past efforts, is to include an
inertial element in the mechanical model, which acts as a fil-
ter to attenuate the noise portion of the signal [4]. However,
filtering in general adds delay and decreases the capabilities
of the simulation, for instance, by limiting the stiffnesses
that can be represented. Noise removal thus comes with a
trade-off, precluding certain types of simulations. Like in
most engineering systems, it is greatly preferable to elimi-
nate noise at its source rather than attempt to filter it out.
In this work, a variety of velocity estimation methods are
compared with velocity sensing in terms of this trade-off. A
stochastic multi-objective optimisation was used to tune
parameters in order to eliminate manual tuning and human
partiality. We emphasize that although the performance/
quality trade-off is well-known in the haptics community,
evaluation is often in the context of simple simulations such
as virtual walls, where damped interaction occurs only briefly
during transients, and noisiness at the threshold may be
mostly ignored, or even considered to contribute serendipity
to a wall’s feeling of being “crisp” and hard, or vice versa con-
tribute to softness [3]. The unilateral virtual spring is a useful
test because it forms the basis of a variety of force feedback
rendering methods. However, in the present study we con-
sider that friction-driven dynamics represents a class of inter-
action with particular challenges for haptic rendering: large
bandwidth response, velocity dependence, and a fundamen-
tal connection to sound synthesis; this latter point forces con-
sideration for high frequency simulation. The bowed string is
an example, but these requirements also apply to scratching
textures, or stick-slip action on sticky surfaces.
2LIMITATIONS TO VELOCITY ESTIMATION
FROM A POSITION SIGNAL
Indirect velocity acquisition implies choices in numerical
estimation methods and sensing apparatus. Individual
S. Sinclair and V. Hayward are with Institut des Syst
emes Intelligents et
de Robotique, UPMC University Paris 06, Paris 75005, France.
E-mail: stephen.sinclair@isir.upmc.fr, vincent.hayward@upmc.fr.
M.M. Wanderley is with the Input Devices and Music Interaction Labora-
tory (IDMIL) at McGill University, Montr
eal, QC H3A 0G4, Canada,and
the Centre for Interdisciplinary Research in Music Media Technology
(CIRMMT) , Montr
eal, QC H3A 1E3, Canada.
E-mail: marcelo.wanderley@mcgil.ca.
Manuscript received 16 Sept. 2013; revised 23 July 2014; accepted 27 July
2014. Date of publication 7 Aug. 2014; date of current version 15 Dec. 2014.
Recommended for acceptance by J.-H. Ryu.
For information on obtaining reprints of this article, please send e-mail to:
reprints@ieee.org, and reference the Digital Object Identifier below.
Digital Object Identifier no. 10.1109/TOH.2014.2346505
IEEE TRANSACTIONS ON HAPTICS, VOL. 7, NO. 4, OCTOBER-DECEMBER 2014 533
sensor signals can be processed in a variety of ways, and
multiple sensors and estimates can also be combined.
Estimation from a lower derivative necessarily adds some
delay; position is delayed by one time step relative to the force
command signal, and so velocity, since it must take into
account previous position samples, is delayed by at minimum
two time steps [6]. More generally, for discrete position-
controlled systems it is necessary to consider the Courant-
Friedrichs-Lewy condition, which says that for explicit finite
difference schemes, velocity may only be known within a
quantum defined by time and space resolution:
D>v
CT; vC<D
T;(1)
where Dis the spatial resolution, Tis the temporal resolu-
tion, and vCis the critical velocity of one quantum D=T .
Thus, sampling faster may improve velocity resolution, but
only if position resolution is sufficient [7], leading to
demanding hardware specifications.
The effect of signal delay principally impacts impedance
range [8]. Reduced delay has been shown to improve the
impedance range of a damped virtual wall in several cases,
including two-sliding observer [9] and adaptive windowing
[10] estimation methods. Sensing higher derivatives may
allow for less delay, since more information can be inferred
about the same instant in time.
3VELOCITY-COUPLED BOWED STRING
In sound synthesis, a well-recognized method for bowed
string physical modeling called the digital waveguide was
proposed by Smith and is based on D’Alembert’s traveling
wave solution to the wave equation [5]. In this formulation,
traveling waves in a 1-DOF mechanical system are repre-
sented by a delay line loop, with linear losses aggregated
into a filter, and propagation time modeled using pure
delay. By placing a non-linear element correctly, a variety of
instruments, namely the bowed string and single-reed bore
(e.g., clarinet), can be simulated very efficiently in real time.
In previous work, the possibility of bypassing the veloc-
ity estimation problem by modifying Smith’s formulation to
leverage position-dependent friction methods has been
explored [11], [12]. Although such tricks were shown to
work, limitations were apparent [12], particularly for
extreme playing gestures—force/velocity combinations
outside the normal range of playing. In the current work, a
physically-correct coupling to the digital waveguide, based
on velocity, was therefore used to increase realism.
As Smith’s formulation is intended as an audio synthesis
model, it samples the string velocity for listening purposes,
but does not include a string-on-bow force output needed
for haptic feedback. Our proposed coupling includes a
Device block representing a one-sample delay, having a
velocity output and a force input, shown in Fig. 1. Force on
the device (the “bow friction force,” Fb) is calculated from
the string velocity vsand the bow-string impedance Rb. This
implies that the velocity of the device must be available,
which is the subject of the current work.
This coupling is based on the observation [13],
Fb¼RbðvDÞvD;(2)
where vDis the velocity difference between bow and string.
It can be shown [14] that this solution corresponds with the
implicit junction coupling method for digital waveguides
[15]. Note that Rbis itself a function of velocity, therefore
the impedance is time-varying. It must be selected to corre-
spond with r, the bow junction transmission coefficient,
which is also a velocity-dependent relation.
4EQUIPMENT
4.1 Haptic Device
We used the Ergon_X Transducteur Gestuelle R
etroactif
(TGR), designed at ACROE/ICA, INPG, Grenoble [16]. It is
a modular device composed of several linear sensor/actua-
tor pairs which can be coupled in a variety of ways. For the
current work, we used the device in a side-ways, 1-DOF
configuration, as seen in Fig. 2.
Fig. 1. Bowed string digital waveguide from [5], enhanced with a string-on-bow force output to provide haptic feedback. The Device block represents
the impedance haptic device, which outputs estimated velocity and takes a force command as input.
Fig. 2. The TGR device in a horizontal 1-DOF configuration, coupled to
the Trans-Tek Series 100 LVT tachometer. The PCB Piezotronics accel-
erometer, not visible here, was adhered to the backside of the handle
using wax recommended for the purpose.
534 IEEE TRANSACTIONS ON HAPTICS, VOL. 7, NO. 4, OCTOBER-DECEMBER 2014
A dedicated signal processor board provides a variable-
frequency internal interrupt clock, and features 16-bit ana-
log input and output. The motors can support impulsive lin-
ear forces up to 200 N, or 60 N sustained.
For our application, the digital waveguide simulation can
run at 35 kHz, although lower rates were used during test-
ing in order to execute multiple estimators simultaneously.
4.2 Sensors
The TGR device features built-in LVDT displacement sen-
sors for each motor. In addition, a tachometer and an accel-
erometer were attached to the end effector. The tachometer
was the Series 100 linear velocity transducer (LVT), model
0112-0000, from Trans-Tek. It has a flat frequency response
up to 500 Hz. The moving magnet has a mass of 15 g. We
measured a noise amplitude of 0.115 mm/s after analog-to-
digital conversion when at rest.
The accelerometer was the model 352C22 from PCB
Piezotronics. It weighs 0.5 grams, has a measurement
range of 4;900 m/s2(500 gravities), and a frequency
range up to 10 kHz. We measured a constant RMS noise
level of 0.22 m/s2(0.021 gravities) from this sensor when
at rest.
For the LVDT sensors, we measured an RMS amplitude
of 15 mm in the sampled analog signal when at rest. The
total displacement range is 2.2 cm.
All sensors were verified to give a Gaussian normal dis-
tribution when at rest. A photo of the device coupled to the
velocity sensor can be seen in Fig. 2.
5DIFFERENTIATORS
This section describes several methods for determining
velocity based on a sampled position signal. Although the
following collection cannot be said to be complete, several
estimator and observer approaches are included that have
been previously proposed in the haptics literature.
5.1 Backward Difference
Defining xkas the signal xðtÞat time t¼kt, where tis the
sample period, the differential dx
dt as can be approximated,
^vk¼xkxkz
zt;(3)
where ^
vkis the velocity estimate for sample k, and z1is
some number of time steps. Precision can thus be had at the
expense of time.
Since this defines velocity as the change in position over
time, it is effectively a poor average over the sample period
[17]. One solution is to use averaging or some other low-
pass filter on the ^vkestimate [8]. In this work we employ a
second-order Butterworth filter as a comparison case.
5.2 Least Squares Fit
The averaging approach of backward difference is based on
an assumption of a constant slope of a linear fit over the
window. It follows that a higher-order fit may provide an
improved approximation by taking the derivative at the
most recent sample time.
A linear least squares estimator can be expressed as a set
of finite impulse response (FIR) filter coefficients [17]. The
differential of an N-order polynomial is,
d^
xk
dt ¼c1þ2c2tkþþNcNtN1
k;(4)
^
x¼Ac: (5)
Therefore, Ais matrix of size NM, representing a lin-
ear combination of the last Msamples, and the sum of
squares of the error can be minimized by,
c¼ðATAÞ1ATx¼Ayx: (6)
A vector _
q¼½012M3M2 NMN1can be used to
take the derivative with respect to time,
d^
x
dt ¼_
qTAyx¼_
hTx¼_
vk;(7)
where _
hTrepresents a linear combination of previous sam-
ples, i.e. the desired FIR coefficients [17].
5.3 Adaptive Windowing
The first-order adaptive windowing filter (FOAW) itera-
tively compares a noise estimate against a given expected
margin to dynamically select the smallest acceptable win-
dow size for a linear fit at each sample [10]. In pseudocode,
for iin 1..N,
wi¼½xkixk
Li¼line_fit(wi)
if max(jwi-Lij)<e:
continue
else
return slope(Li1)
for some a priori estimate of expected signal noise e. The
best-fit method, where the line_fit routine is a least
squares linear fit, was used [10].
5.4 Levant’s Differentiator
The two-sliding method as a differentiator was proposed by
Levant [18] as a “robust” and “exact” differentiator, hence we
refer to it as Levant’s differentiator, following Chawda et al.
[9]. r-sliding mode, or higher-order sliding modes (HOSM),
are a model-free method to maintain a constraint up to its rth
derivative [19]. Therefore two-sliding mode control uses a
two-stage process to impose finite-time convergence for con-
straint s¼_
s¼0,where,foranobserverwon signal x,
s¼wxðtÞ;(8)
_
s¼u_
xðtÞ;(9)
where uis an observer on the differential. If _
w¼u, then uðtÞ
can be taken as an estimate of _
xðtÞ.
Levant [18] gives the control laws for the differentiator as,
_
w¼u; (10)
u¼u1jwxðtÞj1=2sgn ðwxðtÞÞ;(11)
SINCLAIR ET AL.: VELOCITY ESTIMATION ALGORITHMS FOR AUDIO-HAPTIC SIMULATIONS INVOLVING STICK-SLIP 535
_
u1¼asgn ðwxðtÞÞ;(12)
where a;>0, and u1is an additional observer state.
Here, ais some proportion of C, and some proportion
of ffiffiffiffi
C
p, where C>0is the Lipschitz constant. This guaran-
tees local differentiability of _
xðtÞif its derivative stays under
a limit, i.e. if the following inequality holds:
j_
xðtkÞ_
xðtk1Þj  Cjtktk1j:(13)
Recommended choices [18] are,
a¼1:1C; (14)
¼ffiffiffiffi
C
p;(15)
which we use in our implementation [9].
It is noted that a significant advantage for this technique
is that performance increases with sampling rate, as
opposed to backward-difference techniques which worsen
[9]. However, switching noise can be detrimental and it is
recommended to follow with a low-pass filter [18], poten-
tially harming this advantage. This method was therefore
tested with and without this post-filter, again employing a
second-order Butterworth.
5.5 Sensor Fusion
Two methods to combine position and accelerometer sig-
nals were evaluated. The complementary filter [20] is a pair of
filters of complementary frequency bands applied to differ-
ent sensor signals and summed to arrive at a complete spec-
trum. Low- and high-frequency filters are respectively
applied to the differentiated position signal, to remove high
frequency noise, and the integrated accelerometer signal, to
remove low-frequency bias.
Another approach is to make use of statistical maximiza-
tion to estimate the most likely true signal value at a given
time. The Kalman filter is such a technique, which at each
step updates a prediction based on a system model, and
then corrects the prediction using an optimal combination
of weighted error judgements based on the expected covari-
ance of each input with the measurement error and model
uncertainty [21].
5.6 Kalman Filter
The haptic device is modeled as a mass driven by an
unknown external force. Using notation from Bishop and
Welch [21], a linear system model of a driven mass is,
^
x
k¼A^
xk1þBuk1;(16)
P
k¼APk1ATþQ; (17)
where Aand Bare the linear model coefficients, ^
x
kand ^
xk
are the predicted and estimated states, ukis the command
signal, P
kand Pkare the predicted and estimated error
covariance, and Qis the provided process covariance.
The state prediction ^
x
kis updated according to a process
model, and this prediction is corrected by measurements
according to reliability expressed by covariance R. The mea-
surement reliability along with the predicted error covariance
determines an optimal gain Kon the residual [21]. We do not
model the human input, and therefore set B¼0.
We have ^
xkand ^
x
kas three-vectors of the form ½x_
x
xT
and Ais a 33matrix. Since we have a discrete system, A
must express an update for ^
x
kover time t, where time
t¼kt, and similarly Qmust describe the discrete propaga-
tion of noise through the model. From Bar-Shalom et al. [20,
p. 274], for a third-order system,
xkþ1¼FxkþGvk;
where vkis the process noise, then,
F¼
1t1
2t2
01 t
00 1
2
43
5G¼
1
2t2
t
1
2
43
5:(18)
For (16) and (17) we set A¼F, and,
Q¼Gs2GT¼
1
4t4 1
2t3 1
2t2
1
2t3t2t
1
2t2t1
2
43
5s2;(19)
where sis the power spectral density of the process noise.
The Kalman filter has been applied to velocity estimation
for applications in speed control: Kim et al. [22] applied this
method to the two estimates available from timer- and event-
driven encoder count measurement; B
elanger et al. [23] used
a time-varying Kalman filter to update state estimates at inter-
vals driven by both the encoder and clock events.
5.7 Hybrid Solutions
Although the Kalman estimator should perform optimally,
our accelerometer measurement does not adhere to its bias-
free assumptions. Correction of measurement bias at the
expense of noise should therefore be of interest.
Two so-called hybrid methods were constructed, using
noisy position-based acceleration estimates. Double-appli-
cation of a differentiator was added to high-passed accelera-
tion measurements before feeding the mixed data to the
Kalman update. The FOAW and Levant estimators were
both tried.
We therefore define four hybrid conditions as:
KALLEV:Kx;LðLðxÞÞðÞ
KALLEVACC:Kx;LðLðxÞÞ þ Hð
xÞðÞ
KALFOAW:Kx;FðFðxÞÞðÞ
KALFOAWACC:Kx;FðFðxÞÞ þ HðxÞðÞ;
where xis the position measurement,
xis the acceleration
measurement, Kis a Kalman filter with position and acceler-
ation as input, Lis the Levant differentiator, Fis the FOAW
differentiator, and His a 20 Hz high-pass filter to remove
accelerometer DC error. No matching low-pass filter was
applied to the position signal; rather, the Kalman update is
intended to deal with the zero-centered position noise.
This method leverages on the one hand the improved
noise and delay performance of the non-linear differentia-
tors, and on the other hand, the optimal estimation prop-
erties of the Kalman filter. To verify that accelerometer
data is used as intended, hybrid methods were evaluated
with and without adding accelerometer measurements, as
defined above.
536 IEEE TRANSACTIONS ON HAPTICS, VOL. 7, NO. 4, OCTOBER-DECEMBER 2014
6OPTIMISATION AND NUMERICAL EVALUATION
Several example signals were recorded while interacting
with the bowed string model at increasing friction feedback
gain. The LVT tachometer was used to drive the velocity
input, and therefore this signal was used as the comparison
case for evaluating the estimators.
Using an appropriate error metric, estimators were
numerically tuned and compared according to noise/delay
criteria, described below.
6.1 Signal Recordings
The TGR was mounted in the 1-DOF horizontal orientation
as described in Section 4.1. Fifteen recordings of bowing
gestures were made at increasing levels of friction force
gain. Position, velocity, and acceleration were recorded for
7 seconds per recording at 5 kHz.
Since the system was 1-DOF, friction gain was controlled
with reference to Fmax, the maximum available friction
force, rather than discussing a “friction coefficient,” which
would depend on normal force. In the case of the bowed
string model, this refers to,
Fmax ¼max Fb¼max mRbðvDÞvD:(20)
Thus Fmax implicitly determines a gain m:The 15 recordings
spanned values of Fmax from 0 to 12.1 N in an approximately
logarithmic distribution: 0, 0.3, 0.5, 0.8, 1.1, 1.4, 1.6, 1.9, 2.2,
3.3, 4.4, 5.5, 7.7, 9.9, 12.1 N.
6.2 An Error Metric for Signal Comparison
Many works (e.g. [9], [17], [24], [25]) make use of root mean
square error (RMSE) to evaluate signal differences. This is
calculated for a digital signal as,
RMSEðx1;x
2Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
nX
n1
k¼0ðx1½ktx2½ktÞ2
v
u
u
t;(21)
where tis the sample period, and operator y½tis time-quan-
tized access into signal yat time step bt=tc.
In practice we found that using RMSE to compare the
performance of velocity estimators often produced an unfair
advantage to noisy-but-fast estimators, leading to unsatis-
factory choices. Since delay leads to large error during tran-
sients, the bowed string signal, which consists of a series of
transients, is particularly sensitive to this issue.
Considering noise and delay as two distinct sources of
error that are confounded by RMSE, an approach is to dis-
tinguish them by estimating delay and removing it before
calculating the error due to noise. Then, the two objectives
can be weighted appropriately during optimisation.
Delay was determined by peak cross-correlation between
the estimate and the measured velocity. This delay was
then removed by time shifting, and the RMSE between the
base signal and the delay-corrected signal was measured,
providing the noise measurement.
This procedure can be summarized as,
De
rðxÞ¼argmax
kðvr$~
vrÞ½k;(22)
Ee
rðxÞ¼RMSEðvr;shiftDe
rðxÞ(23)
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
nX
n1
k¼0vr½kt~vrtkþDe
rðxÞ2
v
u
u
t;(24)
vr¼
y0
r;(25)
~
vr¼ex;
yr;
y00
r;(26)
where Dis the relative global delay in samples between two
signals detected via a maximum cross-correlation analysis
corrected by a time-shift operator SHIFT. The reference vris
the measured LVT signal for recording r, and ~
vris the esti-
mated velocity signal determined by some estimator ðe; xÞ
applied to position and acceleration measurements r, where
eis the estimator algorithm and xis the argument vector to
the estimator. Sensor measurements for position (LVDT),
velocity (LVT), and acceleration (accelerometer) for record-
ing rare denoted
yr;
y0
r;
y00
r;respectively.
This method was applied to artificially delayed and fil-
tered examples based on the recordings and verified that
sufficient levels of accuracy in Eand Dwere achieved [14].
Although a filter imposes frequency-dependent phase
delay, we consider Dto be an acceptable scalar measure of
“global” signal delay, in the sense that the time-location of
impulsive events are preserved after correcting for it. In the
work that follows, Eand Dfor each recording are seen as a
set of objectives to be minimized.
6.3 Estimator Tuning and Comparison
The above error metric was used to determine an optimally
tuned parameter set for each estimator based on our data
recordings, minimizing both the delay and the delay-
corrected error. A stochastic global optimisation approach
was used in combination with an objective sum strategy for
combining these two measures [26]. A block diagram
describing the parameter optimization process can be found
in Fig. 3.
The objective sum is a special case of the weighted sum
method for global objective Fg,
Fig. 3. Error evaluation and parameter optimization process.
SINCLAIR ET AL.: VELOCITY ESTIMATION ALGORITHMS FOR AUDIO-HAPTIC SIMULATIONS INVOLVING STICK-SLIP 537
FgðxÞ¼X
j1
i¼0
wiðFiðxÞþoiÞ
si
;(27)
where all weights wi¼1:Here, j¼2nfor nrecordings since
we have two objectives per recording, and oiand siare nor-
malization offsets and scaling factors per objective. Other
choices of weights would allow expression of preference for
some objectives over others, but we wish to balance our cri-
teria evenly. Objectives Fiare Erand Drfor recording r,
FiðxÞ¼ E0ðxÞD0ðxÞ  En1ðxÞDn1ðxÞ½:(28)
An adaptive normalization procedure, defined below,
ensures that these objectives are comparable. An optimiser
for each estimator ethen finds the parameter set xethat min-
imizes the global objective Fe
gfor that estimator.
6.3.1 Normalization
In order for wi¼1to be effectively true, the objectives Fi
must be of comparable magnitude. Not only are error and
delay of different units, but between recordings we can
expect the estimates of these values to vary; signals recorded
with higher friction gain have stronger transients and there-
fore are more sensitive to delay. The scaling and offset were
therefore adjusted adaptively according to the range defined
by the best and worst results from a previous iteration.
6.3.2 Procedure
On iteration k, we compute Ee
rand De
r, the error and delay
for estimator efor recording r, and compute normalized
objectives Fe
i,
Fe
2rðk; KÞ¼ Ee
rxeðkÞmineEe
rxeðKÞ
maxeEe
rxeðKÞmineEe
rxeðKÞ(29)
¼Ee
rxeðkÞþO2rðKÞ
S2rðKÞ;(30)
Fe
2rþ1ðk; KÞ¼ De
rxeðkÞmineDe
rxeðKÞ
maxeDe
rxeðKÞmineDe
rxeðKÞ(31)
¼De
rðxeðkÞÞ þ O2rþ1ðKÞ
S2rþ1ðKÞ;(32)
where ðSi;O
iÞare scalings and offsets for error and delay for
each recording across all estimators. This normalizes error
and delay functions for nrecordings, 0r<n1, into the
range ½0;1at iteration k, based on the bounds of a different
iteration K: The global objective Fe
g¼P2n1
i¼0Fe
iwas mini-
mized for each estimator, and used to produce a ranking.
Though the scale changes dynamically, it is of course
necessary to keep it the same when comparing across itera-
tions. Comparisons therefore have the form Fe
gðk; bkÞ<
Fe
gðk1;b
kÞ;where bkis the iteration of the best-so-far Fe
g
at iteration k. Stated algorithmically, the steps are as
follows:
1) At iteration k:¼0, for each estimator,
2) Select parameter set xeð0Þ:
3) Compute Ee
rand De
rfor all estimators on all
recordings.
4) Let b0¼0;and memorize all Siðb0Þ;O
iðb0Þ:
5) For every subsequent iteration k:¼kþ1,
6) Select parameter set xeðkÞ:
7) Compute Ee
rand De
rfor all estimators on all
recordings.
8) If Fe
gðk; bkÞ<F
e
gðk1;b
kÞ;let bkþ1¼kand memo-
rize all SiðkÞ;O
iðkÞ:Otherwise, let bkþ1¼bk.
Comparisons are therefore based on the same scaling,
but the scale is adjusted as estimator performance improves.
At each iteration the parameters of a selected estimator are
perturbed, and retained only if the global objective shows
an improvement. We halt the procedure if no improvements
are found in 1,000 new parameter sets for any estimator.
6.3.3 Parameter Selection
Since objectives Fe
iin the space of xeare in general non-
convex and noisy for most e, we avoided gradient-based
search strategies. Instead, pure adaptive search [27] was
employed, a model-free stochastic sampling approach. Ran-
dom sampling of the region around the best-so-far discov-
ered point with increasing density allows to “zoom in” on
the best parameters. Though slow, this method ensures a
thorough, unbiased sampling of the objective function with-
out involving tuning of optimiser hyperparameters.
7NUMERICAL RESULTS
Results of the optimisation give a parameter set and a nor-
malized scalar for each estimator, from which we can calcu-
late a global ranking as well as evaluate individual
performance for each recording. Labels in Table 2 will be
used throughout the remainder of this section for discussion
of optimisation results.
7.1 Parameter Selection
Final parameter selection based on the global criterion is
given in Table 1. Observations on parameter choices follow:
TABLE 1
Final Parameters for Each Estimator Based on Global
Optimisation across All Recordings
Estimator Parameters
COMP1LPF ¼317 Hz HPF ¼317 Hz
COMP2LPF ¼256 Hz HPF ¼2;254 Hz
FOAW Size ¼17 Noise ¼1:001 104m/s
KALPOS Q¼0:191
KALMAN Q¼108 Ra¼6:28 104
KALLEV Q¼6:38 104Ra¼195
KALLEVACC Q¼2;954 Ra¼64:6
KALFOAW Q¼77:7Ra¼8:06
KALFOAWACC Q¼88:5Ra¼3:85
LEASTSQ N¼2M¼33
LEVANT C¼19:7
LEVANTLP C¼3:6 LPF ¼293 Hz
LOWPASS LPF ¼256 Hz
Position covariance Rp¼21010 was set a priori, based on measurement.
Accelerometer input was pre-filtered with HPF ¼20 Hz.
538 IEEE TRANSACTIONS ON HAPTICS, VOL. 7, NO. 4, OCTOBER-DECEMBER 2014
FOAW The LVDT position noise was measured to have a
maximum error of 6:4105m. The optimal error
margin was selected as 1:001 104, about twice
larger than expected, but within the correct order of
magnitude.
LEVANT The Lipschitz bound on acceleration, C, poses a
problem for optimisation across recordings, since
the maximum expected acceleration changes with
friction gain, which reaches about 200 m/s2for the
larger gain settings. A worst-case approach would
be used in manual tuning, and a value of C>100
or more should be expected, but the optimiser
instead allows some temporary drift as C20 is
exceeded, which we regard as erroneous behav-
iour. As demonstrated in Fig. 4, in LEVANTLP the fil-
tering effectively “covers up” these divergences,
as well as problems with large switching noise,
allowing smaller Cto be selected.
KALMAN While it is possible to set the measurement covari-
ance parameters Rbased on measured noise
amplitude, the process covariance Qis more diffi-
cult to tune and it is common to determine it using
an automatic optimisation procedure [21]. Since
modifying both Rpand Racan lead to multiple
equivalent solutions, position covariance was held
constant at Rp¼s2
p¼21010, where sp¼6:4
105m/s was the measured noise amplitude.
Raselected by the optimiser was not of the same
order of magnitude as measured s2
a¼0:048
(sa¼0:22 m/s2), perhaps reflecting problems due
to low-frequency error, since Kalman assumes a
zero-centered noise source. This seems confirmed
by noticing that Rais lower in the hybrid cases
(KALLEV,KALFOAW), and there is additionally a small
decrease in Rawhen accelerometer data is
included (KALLEVACC,KALFOAWACC). In general,
selected Qshows the opposite trend, compensating
for distrusted Raand vice-versa.
7.2 Performance Grouping
Error and delay estimates varied across recordings, but
good consistency was found within recordings across esti-
mators. Results were therefore normalized by expressing
them in terms of percentage of the LOWPASS results.
TABLE 2
Complete List of Estimators Tested, with Description and a List of Parameters to Be Tuned
Estimator Description Parameters
COMP1 Sum of a second-order Butterworth low- and high-pass
complementary filter pair, configured with identical cut-off frequencies.
LPF ¼HPF
COMP2 Sum of a second-order Butterworth low- and high-pass complementary
filter pair, with independently-optimised cut-off frequencies.
LPF, HPF
FOAW First-order adaptive windowing filter, best-fit method. Max. window size, noise margin
KALPOS Second-order Kalman filter with a double-integrator process model,
pos. measurement, pos. covariance set to Rp¼21010.
Process covariance Q.
KALMAN Third-order Kalman filter with a double-integrator process model,
pos. and accel. measurement, pos. covariance set constant at Rp¼21010 .
Process covariance Q, accel.
covariance Ra.
KALLEV Like KALMAN, with accel. measurement replaced with double-differentiated
pos. using LEVANT.
Process covariance Q, accel.
covariance Ra.
KALLEVACC Like KALMAN, with accel. measurement added to double-differentiated pos.
using LEVANT,C¼100.
Process covariance Q, accel.
covariance Ra.
KALFOAW Like KALMAN, with accel. measurement replaced with double-differentiated
pos. using FOAW of size 12, noise margins 104m and 0.5 m/s.
Process covariance Q, accel.
covariance Ra.
KALFOAWACC Like KALMAN, with accel. measurement added todouble-differentiated pos.
using FOAW of size 12, noise margins 104m and 0.5 m/s.
Process covariance Q, accel.
covariance Ra.
LEASTSQ Least squares polynomial fit expressed as a set of FIR filter coefficients. Poly. order N, window size M.
LEVANT Levant’s differentiator, a two-sliding observer driven to follow pos. Max. accel. C
LEVANTLP The LEVANT estimator followed by a second-order low-pass Buttworth filter. Max. accel. C, LPF cut-off freq.
LOWPASS Second-order Butterworth LPF. LPF cut-off freq.
Fig. 4. Examples of Levant’s differentatiator behaviour. For both graphs,
Fmax ¼7:67 N, LEVANT C¼19:7,LEVANTLP C¼3:6;LPF ¼293 Hz.
(a) Switching noise exceeds slip amplitude. (b) Lipschitz bound is
exceeded for short intervals, e.g. from t¼3:480 to 3:487 s. In both cases,
the low-pass post-filtering effectively removes switching noise and cov-
ers up temporary divergence, allowing for a smaller choice of C. The
trade-off is an increase in delay and a reduction of the sharpness of the
peaks.
SINCLAIR ET AL.: VELOCITY ESTIMATION ALGORITHMS FOR AUDIO-HAPTIC SIMULATIONS INVOLVING STICK-SLIP 539
An exception was the two highest-gain recordings,
Fmax ¼9.9 and 12.1 N, where estimator performance was
inconsistent with other gain settings, e.g. rankings dif-
fered. For this reason they were excluded from the analy-
sis as outliers, but optimisation was performed a second
time including only the higher-gain half of the data set
from 2.2 up to 12.1 N.
Results for these two groups are seen in Figs. 5a and 5b
respectively. A Wilcoxon rank-sums test [28] was used to
determine significance between pairs, with significance
determined by probability threshold of two distributions
being the same of p<0:05. It can be seen that the multi-
objective optimisation has correctly balanced the two crite-
ria: the top performers on the global criterion also per-
formed best for both error and delay.
7.3 Validation
A separate data set was recorded with the same friction val-
ues and very similar gestures as our test set, and kept aside
for validation. Using the parameter sets from Table 1, we
evaluated error and delay for the second data set.
Taking the difference between results at matching friction
levels, the average percentage of absolute error difference
was 10.3 percent, with a standard deviation of 7.76 percent.
The average delay difference was 2.60 samples with a
standard deviation of 2.02 samples. This represents a large
percentage difference (75.6 percent), but it is comparable to
delay differences across recordings in the original dataset,
reflected in the large variance seen in the delay results of
Fig. 5. In other words, the delay estimator can vary a fair
amount between recordings, even for the same friction gain.
Therefore we ascribe delay differences primarily to imperfect
delay estimation rather than to differences between data sets.
An additional observation is that differences in mean
delay between data sets were not significant across estima-
tors, but there was a correlation between the mean delay
absolute difference and standard deviation, (Pearson’s
r¼0:72,p<0:05:) Small mean delays received smaller
deviation in delay, meaning that time-accurate estimators
remained accurate for the validation data set. We conclude
therefore that the performance of estimators on data outside
the training set is acceptable, and over-fitting did not occur.
7.4 Discussion
Fig. 5a shows that for all but the highest gain settings, most
approaches performed as well as or better than LOWPASS on both
error and delay criteria, in some cases with a median improve-
ment of about 30 percent on error, and 50 percent on delay.
Fig. 5. Global objective performance, left, with error and delay evaluation, middle and right, for each recording using optimised parameter sets for
each estimator, non-parametric box plot form. Results are given in left-to-right order of best (lowest) to worst (highest). Significance was determined
using the Wilcoxon rank-sums test with threshold of p<0:05:Note that LEVANT error results are present above the graph limits, with a median of 560
percent. a) 13 recordings with Fmax ¼0to 7:7N were included. b) High-gain case: seven recordings with Fmax ¼2:2to 12:1N were included.
540 IEEE TRANSACTIONS ON HAPTICS, VOL. 7, NO. 4, OCTOBER-DECEMBER 2014
The third-order Kalman estimators were the best per-
formers on both criteria, although differences between them
were generally not significant. The performance of the
hybrid approaches—the Kalman filter combined with non-
linear estimators—indicate some possible trends toward
improvement in median delay, but variance is too high to
draw conclusions.
The second-order least squares fit (LEASTSQ) turns out to
be a very effective estimator in terms of delay. Although its
error performance is not as good as other methods, its delay
performance achieved a very competitive global ranking.
In comparison, the FOAW estimator had very good
error performance, but suffered badly from delay. This
was surprising because the FOAW algorithm is designed
specifically to improve on delay and accuracy during
transients—if we consider that our data consists of con-
tinual small transients due to stick-slip, we expected the
window to be small on average. For FOAW to be useful
here, short windows should trigger at roughly the period
of the waveform, but this was not the case. Instead, the
FOAW window was at maximum for over 98 percent of
time steps: the adaptive filter was thus not reacting to
stick-slip transients. We conclude that our data is patho-
logical for this algorithm, indicating a distinction
between the current case and more typical virtual wall
scenarios.
Overall, no significant differences could be found for the
best six estimators on both error and delay criteria simulta-
neously. Therefore we can conclude that the KALMAN estima-
tor provides the best global results, since it is not
significantly different than LEASTSQ in delay, and beats it by a
large margin in error performance. No significant improve-
ments could be found by including measured acceleration,
nor by means of using the FOAW or Levant methods to
improve measured acceleration via the hybrid estimators.
The high-gain results, Fig. 5b, show one statistically sig-
nificant benefit of using the hybrid Kalman methods—the
KALFOAW estimator does no better than KALMAN, but KALFOA-
WACC is significantly better on error. However, the KALLEV
and KALLEVACC are the best performers and are not different,
therefore advantages of including accelerometer informa-
tion remain inconclusive. Notably, the Levant-Kalman esti-
mators are the only ones to beat LOWPASS on both error and
delay criteria in the high-gain case.
8SUBJECTIVE EVALUATION
In the previous section, some estimation and measurement
methods were shown to reduce signal noise while introduc-
ing less delay than a low-pass filter. According to teleopera-
tor theory, lower delay should lead to an improvement in
the impedance range of the display, while to the benefit of
audio-haptic interaction, it should minimize amplitude of
perceived noise in the velocity signal that stimulates the
acoustic model.
To verify these predictions, the subjective noise perfor-
mance was rated by human operators; secondly, participants
were asked to determine the friction impedance range.
1
The perceptual rating task was necessary because we
found, informally, that measurements of stationary noise
did not well-reflect the relative perceived noisiness of the
estimators. This may be due to differing spectral signatures
of estimator noise [14]. We opted therefore to judge per-
ceived quality directly using human participants.
8.1 Methodology
The study consisted of two parts, a rating task and an adjust-
ment task. These were performed consecutively, with the rat-
ing task performed first, and the adjustment task second.
There were nine participants recruited from the univer-
sity lab between the ages of 25 and 35, where seven were
male and two were female. All self-reported to have normal
hearing and normal cutaneous feeling. Of these, four partic-
ipants played an instrument, though only one played a
stringed instrument, the violin. None played an instrument
professionally.
In the rating task, participants rated the “noisiness” of
eight estimators on a normalized scale, freely switching
between them to compare. Participants were not instructed
how to interpret the word “noisiness,” however they were
allowed to browse the conditions and discover the noisier
ones themselves, making the concept evident. They were
encouraged to consider both the sound and feel in their
judgement. They were asked to determine the “best” and
then the “worst” estimator, establishing the highest and
lowest knob positions respectively, and then to rate the
remaining estimators within this subjective range. All par-
ticipants finished this task in less than 5 minutes.
For the adjustment task, participants were given control
over a single variable, the friction gain, during each trial.
The objective was to determine the maximum gain at which
the model behaviour “breaks down” and no longer per-
forms as a string simulation, which we identify with the
point of marginal stability. This is explained further below.
8.2 Apparatus
The same apparatus as used for our data recordings, the
Ergon_X TGR device in a horizontal configuration, was
employed. A photo of a participant performing the experi-
ment is presented in Fig. 6. For both tasks, participants
Fig. 6. A participant interacting with the device in experimental condi-
tions. The left hand is manipulating a knob while the right hand interacts
with the haptic device. The device is seen in its horizontal configuration,
coupled to the velocity transducer.
1. This study was performed with the approval of McGill Uni-
versity’s Research Ethics Board, REB #105-0908.
SINCLAIR ET AL.: VELOCITY ESTIMATION ALGORITHMS FOR AUDIO-HAPTIC SIMULATIONS INVOLVING STICK-SLIP 541
listened to the string model using headphones.
2
Participants
were seated and grasped the handle with their right hand,
while their left hand controlled a set of eight MIDI knobs.
3
In the first part of the experiment, moving a knob
resulted in the associated condition being selected, while
also adjusting the rating for that condition.
For the second part, participants were given control over
a single MIDI knob controlling Fmax in order to perform the
adjustment task. Each trial started with Fmax ¼0, and was
increased logarithmically by turning to the right until dis-
covering where unexpected behaviour began, fine-tuning it
by compensating hysteresis if necessary. Six samples per
condition per participant were taken.
8.3 Stimulus
For the rating task, participants interacted with the model
running at 5 kHz by moving the handle along its single free
axis, and listened to the sound of the string as well as any
background noise produced by the selected estimator. Fric-
tion forces were exerted with a constant Fmax ¼1N.
For the adjustment task, we characterise the impedance
range for the estimator as the maximum displayable friction
gain, Fmax such that the display remains stable. Participants
were told to find the margin of stability, apparent due to
device oscillation and clear distortion in the sound.
Participants were encouraged to use the breakdown of
the string behaviour as a cue to differentiate stable and
unstable conditions. Specifically, they were told to “find the
point immediately before the model no longer feels or
sounds like a bowed string.” A few training trials were
used to demonstrate.
8.4 Conditions
The estimators used in this experiment were the subset
from Table 2 that depend only on position: these were
LOWPASS,LEASTSQ,FOAW,LEVANT,LEVANTLP,KALPOS,KALLEV,
KALFOAW,aswellasLVT, a direct usage of the tachometer
signal. Accelerometer-based methods were not included.
Estimators were executed concurrently on the Toro hard-
ware to allow quick switching between conditions. Under
this computational load, the system executed at a maximum
of 5 kHz, with the exception of KALFOAW, which was only
able to run at 4 kHz.
4
For this reason, KALFOAW was left out
of the rating task, but included in the impedance task.
9RESULTS
The friction gain adjustment results are given in Fig. 7, and
the rating task is analysed in Fig. 8. A scatter plot compari-
son is available in Fig. 9.
9.1 Maximum Impedance Judgement
We notice firstly, in Fig. 7, that the tachometer, condition
LVT, features nearly twice the impedance range compared to
any estimator. Secondly, although it was very noisy, LEVANT,
the non-filtered Levant differentiator, performed second
best, agreeing with the hypothesis that delay, independent
of noise, is an important factor for the range of stability.
Confirming this, the FOAW estimator performed worst, and
was also a weak performer on the delay criterion in Fig. 5.
However, the next-best impedance ranges after LEVANT
belong to KALPOS and LOWPASS, which did not perform excep-
tionally well on the delay criterion in Fig. 5. This is unex-
pected, since it does not correspond with numerical delay
estimates, and it therefore suggests that there are other fac-
tors affecting behaviour at high gain.
We note that variance in Fig. 7 is mostly due to inter-sub-
ject differences, while intra-subject variance was on average
61 percent lower. Participants may have used different crite-
ria to judge marginal stability, but distribution differences
were nonetheless statistically significant, therefore normali-
zation was not necessary. Furthermore, individual rankings
were strongly correlated to the global ranking, (Spearman’s
r>0:9for all participants.) We additionally note that vari-
ance may represent a real, ambiguous range in which a
higher likelihood of instability is gradually approached.
9.2 Subjective Noisiness Ratings
The subjective ratings, Fig. 8, are more easily interpreted.
The best and worst performers for noisiness, correspond-
ing to LVT and LEVANT, were selected unanimously. The
Fig. 7. Maximum friction force Fmax for bowed string interaction. Significance is determined by Wilcoxon’s rank-sums test, with p<0:05 indicating
confidence that distributions are not the same. We see that LEVANT performs best after the tachometer, agreeing with the hypothesis that delay is a
significant factor for the impedance range, however the next best estimators are KALPOS and LOWPASS, which did not perform exceptionally well for
delay. Therefore we must assume there are additional factors influencing performance at high gain.
2. Bose QuietComfort 15 noise-cancelling headphones.
3. Akai LPD-8, providing a 7-bit potentiometer.
4. Prior to seeing the computational load implied by concurrent exe-
cution, this study was to run at a rate of 20 kHz, to allow a more natu-
ral-sounding bowed string; at 5 kHz, the sound is degraded, but
recognizable as a string.
542 IEEE TRANSACTIONS ON HAPTICS, VOL. 7, NO. 4, OCTOBER-DECEMBER 2014
preference for KALLEV is indicated clearly, which confirms
its performance on the noise criterion as predicted by
Fig. 5.
10 DISCUSSION
These results show exceptionally good performance for
direct measurement using the tachometer, and also show
that some estimators are indeed preferable. KALLEV seems to
allow better noise rejection than LOWPASS while featuring
similar impedance range. Finally, it is clear that a 250 Hz
low-pass filter actually ranks fairly well in practice.
We note that rankings derived from noisiness rating
medians did not correlate strongly with numerical delay-
corrected error medians, nor did similar median rankings
from max. impedance ratings correlate strongly with rank-
ings based on numerically-judged delay, (Spearman’s
r¼0:61 and r¼0:43, respectively.) However, this may be
due simply to lack of significance between the best-rated
conditions in Fig. 5.
11 CONCLUSION
In this article we addressed the question of noise/delay
trade-off in velocity estimation, as applied to audio-haptic
interaction with an acoustic bowed string model.
From our selection of differentiators, the numerical
results indicate that for friction-coupled audio-haptic inter-
action, velocity is best estimated using a third-order Kalman
approach; a non-linear two-sliding observer-based differen-
tiator driving a statistically optimal linear observer to
remove switching noise gave the best performance on our
objective sum. We may assume that a more detailed process
model taking device characteristics into account might
bring further improvements.
The inclusion of measured accelerometer data only
contributed significantly in the high-gain case, and only
provided mild improvements. This indicates either that
accelerometer data could not provide significantly better
information, or more likely that in the high-frequency
region where accelerometer data can contribute, the small
transient stick-slip peaks are too small relative to noise to be
correctly measured by our criteria. These high-frequency
transients are important to audio perception of the bowed
string acoustics, therefore it is possible that better measures
are needed to properly optimise for acoustic models.
For instance, an objective function based on models of
human perception could be relevant for future work.
It is interesting to note that methods such as FOAW that
have previously shown good results for virtual wall models
do not perform particularly well in this scenario, similarly
due to the small transients. This implies that some estimators
may be more appropriate to specific classes of interaction.
Our choice of the objective sum method is not the only
possible optimisation strategy. Since estimators were
allowed to optimise on either axis, there is no consistency
between estimators on any one measure, leading to difficul-
ties in subjective comparison. For example, one cannot lis-
ten to each optimised estimator and judge its noise
qualities, since another parameter selection for the same
estimator may reduce noise at the expense of delay. An
alternative could be to use a unilateral constraint on one
variable while optimising the other.
We performed a two-axis evaluation with participants to
determine how numerical results corresponded with subjec-
tive quality. Unexpectedly, numerical delay estimates did
not well-predict the ranking of estimators for impedance
performance, nor did error estimates well-predict noisiness
ratings. This further supports the need for better predictive
models of the practical effects of noise and delay on human
perception and machine performance in the context of
wideband signals.
The study indicated that no estimator could approach the
velocity transducer in terms of perceived noise and attain-
able impedance, therefore such sensing hardware should be
considered for integration into force feedback devices.
Fig. 8. Subjective ratings of noisiness for several estimators. Significance is determined by Wilcoxon’s rank-sums test, with p<0:05 indicating confi-
dence that distributions are not the same. The small deviations of LVT and LEVANT are due to instructions that the noisiest and least noisy condition be
maximized and minimized within the range—the choices of best and worst were unanimous. The KALFOAW estimator was not included in these ratings.
Fig. 9. Scatter plot of median subjective results from Figs. 7 and 8. Esti-
mators closer to the top-left corner are better. Results are reflective of
the estimators with the tuned parameter set. Each point is one sample
(the “best” according to our criteria) of a contour in this space as estima-
tor parameters are adjusted.
SINCLAIR ET AL.: VELOCITY ESTIMATION ALGORITHMS FOR AUDIO-HAPTIC SIMULATIONS INVOLVING STICK-SLIP 543
ACKNOWLEDGMENTS
The work was funded by the Natural Sciences and Engi-
neering Research Council of Canada. The authors would
like to thank Gary Scavone (CAML/CIRMMT, McGill Uni-
versity) for his participation in development of the bowed
string model coupling, as well as Jean-Loup Florens
(ACROE) for extensive discussion of bowed string haptic
simulation, his inspiring previous work on the subject, and
his expertise with the TGR.
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[24] E. Kilic, O. Baser, M. Dolen, and E. I. Konukseven, “An enhanced
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mation using incremental position encoders,” in Proc. Int. Conf.
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[25] S. P. Chan, “Velocity estimation for robot manipulators using neu-
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Stephen Sinclair received the PhD degree in
music technology from McGill University,
Montreal, QC, Canada, in 2012. He is currently a
postdoctoral fellow at the Universit
e Pierre et
Marie Curie (Paris VI), Paris, France, in the Insti-
tut des Syst
emes Intelligents et de Robotique.
His current research interests include haptic ren-
dering systems, sensory integration, human-
machine interaction, signal processing and con-
trol, and robotics.
Marcelo M. Wanderley (M’XX) received the PhD
degree from the Universit
e Pierre et Marie Curie
(Paris VI), Paris, France, in acoustics, signal
processing, and computer science applied to
music. He is currently a William Dawson scholar
and an associate professor of music technology
at the Schulich School of Music, McGill Univer-
sity, Montreal, QC, Canada, where he directs the
Input Devices and Music Interaction Laboratory
and the Centre for Interdisciplinary Research in
Music Media and Technology. He was the chair
of the 2003 International Conference on New Interfaces for Musical
Expression and coauthored the textbook New Digital Musical Instru-
ments: Control and Interaction Beyond the Keyboard (A-R Editions). His
current research interests include gestural control of sound, and input
device design and evaluation. He is a member of the IEEE.
Vincent Hayward (F’08) received the Dr-Ing.
degree from the University of Paris XI, Paris,
France, in 1981. He was a postdoctoral fellow
and then as a visiting assistant professor at
Purdue University, in 1982, and joined CNRS,
Paris, France, as Charg
e de Recherches in
1983. In 1987, he joined the Department of
Electrical and Computer Engineering at McGill
University, Montreal, QC, Canada, as an
assistant, associate and then full professor in
2006. He was the director of the McGill Center for
Intelligent Machines from 2001 to 2004 and held the “Chaire
internationale d’haptique” at the Universit
e Pierre et Marie Curie
(UPMC), Paris, France, from 2008 to 2010. He is currently a professor
at UPMC. His current research interests include haptic device design,
haptic perception, and robotics. He is a fellow of the IEEE.
"For more information on this or any other computing topic,
please visit our Digital Library at www.computer.org/publications/dlib.
544 IEEE TRANSACTIONS ON HAPTICS, VOL. 7, NO. 4, OCTOBER-DECEMBER 2014
... In [21], Koul et al. proposed a dual-rate sampling scheme that decouples the position and velocity control loops, and employs a slower sampling rate for velocity control loop to reduce the quantization effects in FDM based velocity estimation to improve the Z-width performance. Sinclair et al. [22] presented a comparative experimental evaluation of a variety of velocity estimation algorithms including several hybrid combinations of more than one algorithm. ...
... In related work, Sinclair et al. [22] experimentally compared a variety of velocity estimation algorithms. They optimized each algorithm's relevant parameters using a pure adaptive search method to minimize a multi-objective criterion that took into account both the delay and the error in delay-corrected velocity estimations. ...
Article
Full-text available
This paper comparatively evaluates the effect of real-time velocity estimation methods on passivity and fidelity of virtual walls implemented using haptic interfaces. Impedance width, or Z-width is a fundamental measure of performance in haptic devices. Limited accuracy of velocity estimates from position encoder data is an impediment in improving the Z- width in haptic interfaces. We study the efficacy of Levant's differentiator as a velocity estimator, to allow passive implementation of higher stiffness virtual walls as compared to some of the commonly used velocity estimators in the field of haptics. We first experimentally demonstrate feasibility of Levant's differentiator as a velocity estimator for haptics applications by comparing Z-width performance achieved with Levant's differentiator and commonly used Finite Difference Method (FDM) cascaded with a lowpass filter. A novel Z-width plotting technique combining passivity and fidelity of haptic rendering is proposed, and used to compare the haptic device performance obtained with Levant's differentiator, FDM+lowpass filter, First Order Adaptive Windowing and Kalman filter based velocity estimation methods. Simulations and experiments conducted on a custom single degree of freedom haptic device demonstrate that the stiffest virtual walls are rendered with velocity estimated using Levant's differentiator, and highest wall rendering fidelity is achieved by First Order Adaptive Windowing based velocity estimation scheme.
... Zhou et al. (2008), however, argue that at higher sampling rates the adaptive windowing technique may be biased in selecting certain window lengths preferentially over others and consequently yield poor velocity estimates. A similar observation is made by Sinclair et al. (2014) during stick-slip rendering in coupled audio-haptic simulations. Due to the stickslip phenomenon, the adaptive windowing was expected to have mostly assumed small window lengths, however, for nearly 98% of time steps the algorithm used the maximum window length and thus induced time delay in the loop. ...
Article
Full-text available
This work presents an exhaustive analysis of the impact of time delay on the stability of dual-rate haptics controllers using an exact discrete-time method. The mathematical formulation of such controllers leads to higher order state-space models, in particular, for higher values of time delay and sampling rates. A balanced truncation based model order reduction framework is therefore utilized for obtaining reduced order models, while preserving the stability properties of the original higher order models. The likely order of the reduced models is selected on the basis of the Hankel singular values which represent the contribution of the system states towards the overall system energy. Using this framework, it is empirically found that third order reduced models yield exactly the same stability ranges as given by the allied full order models. This empirical finding is backed up by a rigorous analysis of the controller while considering a wide range of values for the time delay spanning both the realistic and worst-case application scenarios. This result is hitherto unknown in the haptics literature and is for the first time reported in this paper. For comparison purposes, the stability ranges of the dual-rate controller are also obtained using an equivalent continuous-time method, and numerical simulations. This work generalizes the results of previous works available in the literature for uniform-rate sampling scheme both for delayed and non-delayed haptics controllers. The results demonstrate that for a time-delayed dual-rate haptics controller, an increase in the sampling rate leads to an enhancement in the stable range of virtual wall parameters as long as the value of time delay relative to the sampling rate (non-dimensional time delay) remains small. For higher values of the non-dimensional time delay, higher sampling rates do not necessarily lead to performance enhancement.
... Though a body of work was developed over the years focusing on measurements, models and applications, musical force-feedback has never become as widespread; the game-changing force-feedback musical application is yet to come. Despite this situation, in recent years a number of works have addressed several aspects of this topic, proposing software platforms and simulation models with the potential to provide popular and/or advanced force-feedback tools for musical applications [1,15,16,23,24]. ...
... Though a body of work was developed over the years focusing on measurements, models and applications, musical force-feedback has never become as widespread; the game-changing force-feedback musical application is yet to come. Despite this situation, in recent years a number of works have addressed several aspects of this topic, proposing software platforms and simulation models with the potential to provide popular and/or advanced force-feedback tools for musical applications [1,15,16,23,24]. ...
Article
Full-text available
Haptics, audio and human–computer interaction are three scientific disciplines that share interests, issues and methodologies. Despite these common points, interaction between these communities are sparse, because each of them have their own publication venues, meeting places, etc. A venue to foster interaction between these three communities was created in 2006, the Haptic and Audio Interaction Design workshop (HAID), aiming to provide a meeting place for researchers in these areas. HAID was carried out yearly from 2006 to 2013, then discontinued. Having worked in the intersection of these areas for several years, we felt the need to revive this event and decided to organize a HAID edition in 2019 in Lille, France. HAID 2019 was attended by more than 100 university, industry and artistic researchers and practitioners, showing the continued interest for such a unique venue. This special issue gathers extended versions of a selection of papers presented at the 2019 workshop. These papers focus on several directions of research on haptics, audio and HCI, including perceptual studies and the design, evaluation and use of vibrotactile and force-feedback devices in audio, musical and game applications.
... 8.3. This device was recently used by Sinclair et al. [46] to investigate velocity estimation methods in the haptic rendering of a bowed string. ...
Chapter
Full-text available
Haptics, and specifically vibrotactile-augmented interfaces, have been the object of much research in the music technology domain: In the last few decades, many musical haptic interfaces have been designed and used to teach, perform, and compose music. The investigation of the design of meaningful ways to convey musical information via the sense of touch is a paramount step toward achieving truly transparent haptic-augmented interfaces for music performance and practice, and in this chapter we present our recent work in this context. We start by defining a model for haptic-augmented interfaces for music, and a taxonomy of vibrotactile feedback and stimulation, which we use to categorize a brief literature review on the topic. We then present the design and evaluation of a haptic language of cues in the form of tactile icons delivered via vibrotactile-equipped wearable garments. This language constitutes the base of a “wearable score” used in music performance and practice. We provide design guidelines for our tactile icons and user-based evaluations to assess their effectiveness in delivering musical information and report on the system’s implementation in a live musical performance.
Thesis
Full-text available
Acoustic models driven by real-time velocity signals can suffer unduly from quality issues due to sampling and differentiation, especially at high sampling rates. In audio-haptic friction interaction, as found in a bowed string simulation for example, this noise appears as a gritty or dry feel, and is audible in the sound. In this thesis, two approaches to this problem are proposed: firstly, reduction of the sensitivity of the model to velocity noise by the application of a position-dependent friction model; secondly, the improvement of velocity estimation by means of filtering and enhanced sensing. Several estimators are compared, by means of parameter optimisation, to direct veloc- ity measurement in order to find a good trade-off between filter-imposed delay and noise rejection. Optimised estimators are then compared by subjects in an online scenario to test their respective effect on the impedance range and noise qualities of a bowed string friction display.
Article
Full-text available
In this paper, an enhanced velocity and acceleration estimation method is presented based on the discrete-time adaptive windowing technique that exclusively employs incremental optical position sensor readings. In a previous work proposed by, a first-order adaptive windowing method (FOAW) was shown to be optimal in the sense that it minimizes the velocity error variance while maximizing the accuracy of the estimates. Since the method lacks the ability of accurate acceleration estimation, a new model is developed to estimate the acceleration from signals generated by quantized position sensors. Hence, a numerical solution of an enhanced adaptive windowing (EAW) method is derived and simulated for velocity and acceleration estimation. Effectiveness of the proposed method is validated by experiments done on a one degree of freedom haptic device.
Article
This paper suggests a Kalman-filter approach to the estimation of angular velocity and acceleration from (quantized) shaft-encoder measurements. Finite-difference estimates deteriorate as sampling rates are increased. For small sampling periods, we show that the filtering problem is the dual of the cheap control problem, and we jus tify the use of all-integrator models. We investigate Kalman filtering with constant sampling rate, and also with measurements triggered by encoder pulses. Simulation and experimental results are given.
Article
The main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises. The proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal. Such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for Lipschitz’s constant of the derivative.
Article
A haptic musical instrument is an electronic musical instru- ment that provides the musician not only with audio feed- back but also with force feedback. By programming feed- back controllers to emulate the laws of physics, many hap- tic musical instruments have been previously designed that mimic real acoustic musical instruments. The controller programs have been implemented using finite difference and (approximate) hybrid digital waveguide models. We present a novel method for constructing haptic musical instruments in which a haptic device is directly interfaced with a con- ventional digital waveguide model by way of a junction el- ement, improving the quality of the musician's interaction with the virtual instrument. We introduce both the explicit digital waveguide control junction and the implicit digital waveguide control junction.
Book
From the Publisher: "Estimation with Applications to Tracking and Navigation treats the estimation of various quantities from inherently inaccurate remote observations. It explains state estimator design using a balanced combination of linear systems, probability, and statistics." "The authors provide a review of the necessary background mathematical techniques and offer an overview of the basic concepts in estimation. They then provide detailed treatments of all the major issues in estimation with a focus on applying these techniques to real systems." "Suitable for graduate engineering students and engineers working in remote sensors and tracking, Estimation with Applications to Tracking and Navigation provides expert coverage of this important area."--BOOK JACKET.