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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],

Fb¼RbðvDÞvD;(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 ¼c1þ2c2tkþþNcNtN1

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,

wi¼½xkixk

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

_

u1¼asgn 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 Hð€xÞðÞ;

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 mRbðvDÞvD:(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

k¼0ðx1½ktx2½ktÞ2

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)

¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1

nX

n1

k¼0vr½kt~vrtkþDe

rðxÞ2

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.

REFERENCES

[1] M. Wiertlewski, C. Hudin, and V. Hayward, “On the 1/f noise

and non-integer harmonic decay of the interaction of a ﬁnger slid-

ing on ﬂat and sinusoidal surfaces,” in Proc. IEEE World Haptics

Conf., 2011, pp. 25–30.

[2] J.-L. Florens, A. Razaﬁndrakoto, A. Luciani, and C. Cadoz,

“Optimized real-time simulation of objects for musical synthesis

and animated image synthesis,” in Proc. Int. Comput. Music Conf.,

1986, pp. 65–70.

[3] Y. Visell, B. L. Giordano, G. Millet, and J. R. Cooperstock,

“Vibration inﬂuences haptic perception of surface compliance

during walking,” PLoS one, vol. 6, no. 3, p. e17697, 2011.

[4] J.-L. Florens, “Expressive bowing on a virtual string instrument,”

in Proc. 5th Int. Gesture Workshop Gesture-Based Communication in

Human-Computer Interaction, Apr. 2003, vol. 2915, pp. 487–496.

[5] J. O. Smith, “Efﬁcient simulation of the reed-bore and bow-string

mechanisms,” in Proc. Int. Comput. Music Conf., 1986, pp. 275–280.

[6] V. Hayward and K. MacLean, “Do it yourself haptics: Part I,”

Robot. Autom. Mag., vol. 14, no. 4, pp. 88–104, 2007.

[7] G. Campion and V. Hayward, “Fundamental limits in the render-

ing of virtual haptic textures,” in Proc 1st Joint Eurohaptics Conf.

Symp. Haptic Interfaces Virtual Environ. Teleoperator Syst., 2005,

pp. 263–270.

[8] J. E. Colgate and J. M. Brown, “Factors affecting the z-width of a

haptic display,” in Proc. IEEE Conf. Robot. Autom., 1994, pp. 3205–

3210.

[9] V. Chawda, O. Celik, and M. K. O’Malley, “Application of

Levant’s differentiator for velocity estimation and increased Z-

width in haptic interfaces,” presented at the World Haptics,

Istanbul, Turkey, Jun. 2011,

[10] F. Janabi-Shariﬁ, V. Hayward, and C.-S. J. Chen, “Discrete-time

adaptive windowing for velocity estimation,” IEEE Trans. Cont.

Sys. Technol., vol. 8, no. 6, pp. 1003–1009, Nov. 2000,

[11] S. Sinclair, G. Scavone, and M. M. Wanderley, “Audio-haptic

interaction with the digital waveguide bowed string,” in Proc. Int.

Comput. Music Conf., Montreal, QC, Canada, 2009, pp. 275–278.

[12] S. Sinclair, M. M. Wanderley, V. Hayward, and G. Scavone, “Noise-

free haptic interaction with a bowed-string acoustic model,” pre-

sented at the World Haptics Conf., Istanbul, Turkey, Jul. 2011,

[13] J. O. Smith. (2010). Physical audio signal processing: For virtual musi-

cal instruments and audio effects. W3K Publishing [Online]. Avail-

able: http://ccrma.stanford.edu//jos/pasp/

[14] S. Sinclair, “Velocity-driven audio-haptic interaction with real-

time digital acoustic models,” Ph.D. dissertation, Schulich School

of Music, McGill Univ., Montreal, QC, Canada, 2012.

[15] E. Berdahl, G. Niemeyer, and J. Smith, III, “Using haptic devices

to interface directly with digital waveguide-based musical

instruments,” in Proc. 9th Int. Conf. New Interfaces Musical Expres-

sion, 2009, pp. 183–186.

[16] J. Florens, A. Luciani, C. Cadoz, and N. Castagn

e, “ERGOS: Multi-

degrees of freedom and versatile force-feedback panoply,” in

Proc. Eurohaptics, 2004, pp. 356–360.

[17] R. H. Brown, S. C. Schneider, and M. G. Mulligan, “Analysis of

algorithms for velocity estimation from discrete position versus

time data,” IEEE Trans. Ind. Electron., vol. 39, no. 1, pp. 11–19, Feb.

1992,

[18] A. Levant, “Robust exact differentiation via sliding mode

technique,” Automatica, vol. 34, no. 3, pp. 379–384, 1998.

[19] A. Levant, “Principles of 2-sliding mode design,” Automatica,

vol. 43, no. 4, pp. 576–586, 2007.

[20] Y. Bar-Shalom, T. Kirubarajan, and X.-R. Li, Estimation with Appli-

cations to Tracking and Navigation. New York, NY, USA: Wiley,

2002.

[21] G. Bishop and G. Welch, “An introduction to the Kalman ﬁlter,” in

Proc. SIGGRAPH 2001 Course Notes, 2001, pp. 1–82.

[22] H. Kim, J. Choi, and S. Sul, “Accurate position control for AC

servo motor using novel speed estimator,” in Proc. Int. Conf. Ind.

Electron., Control Instrum., 1995, vol. 1, pp. 627–632.

[23] P. B

elanger, P. Dobrovolny, A. Helmy, and X. Zhang, “Estimation

of angular velocity and acceleration from shaft-encoder meas-

urements,” Int. J. Robot. Res., vol. 17, no. 11, pp. 1225–1233, 1998.

[24] E. Kilic, O. Baser, M. Dolen, and E. I. Konukseven, “An enhanced

adaptive windowing technique for velocity and acceleration esti-

mation using incremental position encoders,” in Proc. Int. Conf.

Sig. Electron. Sys., Gliwice, Poland, Sep. 2010, pp. 61–64.

[25] S. P. Chan, “Velocity estimation for robot manipulators using neu-

ral network,” J. Intell. Robot. Syst., vol. 23, no. 2, pp. 147–163, 1998.

[26] R. Marler and J. Arora, “Survey of multi-objective optimization

methods for engineering,” Struct. Multidisciplinary Optim., vol. 26,

no. 6, pp. 369–395, 2004.

[27] N. Patel, R. Smith, and Z. Zabinsky, “Pure adaptive search in

monte carlo optimization,” Math. Program., vol. 43, no. 1, pp. 317–

328, 1989.

[28] R. L. Ott and M. Longnecker, An Introduction to Statistical Methods

and Data Analysis,6th ed. Belmont, CA, USA: Brooks/Cole Cen-

gage Learning, 2010.

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.

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