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Two calibration procedures for a gyroscope-free inertial measurement system based on a

double-pendulum apparatus

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2008 Meas. Sci. Technol. 19 055204

(http://iopscience.iop.org/0957-0233/19/5/055204)

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IOP PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 19 (2008) 055204 (9pp) doi:10.1088/0957-0233/19/5/055204

Two calibration procedures for agyroscope-free inertial measurementsystem based on a double-pendulumapparatusP Cappa1,2, F Patane1,2 and S Rossi1,2

1 Department of Mechanics and Aeronautics, ‘Sapienza’ University of Rome, Via Eudossiana,18-00184 Rome, Italy2 Paediatric Neuro-rehabilitation Division, Children’s Hospital ‘Bambino Gesu’ IRCCS,Via Torre di Palidoro, 00050 Passoscuro (Fiumicino), Rome, Italy

Received 8 October 2007, in final form 3 March 2008Published 2 April 2008Online at stacks.iop.org/MST/19/055204

AbstractThis paper presents a novel calibration algorithm to be used with a gyro-free inertialmeasurement unit (GF-IMU) based on the use of linear accelerometers (AC). The analyticalapproach can be implemented in two calibration procedures. The first procedure (P-I) isarticulated in the conduction of a static trial, to compute the sensitivity and the direction of thesensing axis of each AC, followed by a dynamic trial, to determine the AC locations. Bycontrast, the latter procedure (P-II) consists in the calculation of the previously indicatedcalibration parameters by means of a dynamic trial only. The feasibility of the two calibrationprocedures has been investigated by testing two GF-IMUs, equipped with ten and six bi-axiallinear ACs, with an ad hoc instrumented double-pendulum apparatus. P-I and P-II werecompared to a calibration procedure used as a reference (P-REF), which incorporates the ACpositions measured with an optoelectronic system. The experimental results we present in thispaper demonstrate that (i) P-I is able to determine the calibration parameters of the AC arraywith a higher accuracy than P-II; (ii) consequently, the errors associated with translational(a0 − g) and rotational (ω) acceleration components for the two GF-IMUs are significantlygreater using P-II than P-I and (iii) the errors in (a0 − g) and ω obtained with P-I arecomparable with the ones obtainable by using P-REF. Thus, the proposed novel algorithmused in P-I, in conjunction with the double-pendulum apparatus, can be globally considered aviable tool in GF-IMU calibration.

Keywords: accelerometer array, IMU, calibration, navigation, biomechanics

Nomenclature

a linear acceleration of the ACa0 linear acceleration of the origin of the

global reference frameAC accelerometerc optimization indexj Ci rotational matrix from the reference

system i to jE optical encoderF cross-product matrix

g gravity vectorGF gyroscope-free systemIMU inertial measuring unitK rotational sensitivity matrixLSB least significant bitM global sensitivity matrixM∗ pseudoinverse M matrixn, N AC axis unity vector and orientation

matrixN number of AC used in the clusterNc number of calibration measurements

0957-0233/08/055204+09$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK

Meas. Sci. Technol. 19 (2008) 055204 P Cappa et al

Nv number of collected samples in avalidation test

O, O AC offset voltage and offset vectorOS optoelectronic systemp probability value that resulted from

ANOVAP-I first calibration procedureP-II second calibration procedureP-REF reference calibration procedurer, R AC position vector and position matrixrOS, ROS AC position vector and matrix calcu-

lated by means of an OSRMS, RMSE root mean square and root mean square

errorS, S axis sensitivity and sensitivity matrixSNR signal-to-noise ratioSD standard deviationV, V AC output voltage and output vectorαP-II angle between unity vectors calculated

with P-I and P-IIε|a0− g|P-I,P-II,P-REF root mean square error in the

evaluation of the modulus of(a0 − g)

ε|ω|P-I,P-II,P-REF root mean square error in the evaluationof the modulus of angular acceleration

�rP-I,P-II differences between each AC positionobtained with P-I and P-II and the resultsof an optoelectronic system

�SP-II error in the AC sensitivity obtained withP-II with respect to P-I

ω angular velocityω angular acceleration

1. Introduction

Over the years various types of miniaturized siliconmicromachined accelerometers (ACs) have been proposed.Today, having become smaller and cheaper, with low energyconsumption and accurate outputs, they are effectively used inseveral research and application fields.

One of the emerging applications of ACs is in thedevelopment of an inertial measuring unit (IMU) to computethe linear and angular motion of a rigid body using gyroscopes.The IMUs have a multitude of uses including (i) navigation[1–3], (ii) biomechanics [4–9] and (iii) robotics andautomation [10–12]; the cited references represent a few ofthe numerous works available in the literature. In general,high precision gyroscopes, prior to the recently availableMEMS sensors, had many drawbacks, including high cost,high power consumption, large volume and large weight, etc.So there was a big incentive to develop IMUs consisting ofACs only, i.e., the so-called gyroscope-free (GF) system, forlow-cost and low-accuracy applications. Research, whichcontinued for over 30 years, was focused on the proposalof the optimal AC configuration capable, in conjunctionwith a specific postprocessing procedure, of overcoming theinherent calculation error that increases with time and toimprove the global metrological performances. The proposalscan be arranged as follows. In theory any configuration

of six distributed linear ACs yields a complete descriptionof rigid body motion in terms of translational and angularaccelerations; two schemes were given by Morris [13] andby Chen et al [14]. To improve the obtainable metrologicalperformances, the number of ACs was increased to 9; in thatcase the ACs had to be strategically attached to the rigidobject at specific locations and orientations in a 2D [15] or3D [16–18] manner to ensure the proper operation of the ACcluster. A redundant scheme based on the minimum numberof 12 accelerometric axes was successively studied to assure aunique solution, to avoid numerical instabilities and to furtherimprove the metrological performances [19, 20]. Finally,Parsa et al [21, 22] studied more redundant schemes usingtri-axial ACs only, while Cappa et al [23, 24] proposed andexperimentally validated redundant clusters of linear ACs.

In general the error sources of GF-IMU are (i) inducedby the numerical integration [25, 26] and (ii) associatedwith the configuration of sensors (i.e. errors in location andorientation) together with the inherent accuracy and cross-sensitivity error of each AC embedded in the array. To reducethe previously indicated errors and because the metrologicalcharacteristics of low cost MEMS sensors may deviate frompiece to piece, the identification of calibration procedures is ofthe utmost importance. The calibration methodologies usedin GF-IMU can be divided into two classes. In the firstmethodology, the AC cluster is statically positioned in theearth’s gravitational field at given orientations and then thesensor outputs are measured [25]. The latter methodology,however, requires the use of a device capable of imposinga known motion [27–30]. The main disadvantages of thepreviously indicated methodologies are (i) the static calibrationpermits the computation of the sensitivity and the directionof ACs but not of the AC positions and (ii) the dynamiccalibration hypothesizes the known AC positions to determinethe remaining AC parameters.

To tackle the above-mentioned limitations, the authorsof the present paper proposed a novel algorithm capable ofcomputing not only the sensitivity and the direction of anAC cluster, but also its sensor positions. Specifically, weproposed two calibration procedures: the first implies thecarrying out of static and dynamic trials, while the latter isbased on the calculation performed only on dynamic data. Toconduct the two procedures, the authors designed and built anovel bench test, based on an instrumented double-pendulumscheme, capable of imposing, in general, known motions ona GF-IMU system. Specifically, in the present paper wetested two helmets redundantly equipped with six and ten bi-axial linear ACs, respectively. The main characteristic of theinstrumented helmets, described in previous papers [23, 24],is that the ACs are manually placed and oriented on the rigidbody.

From a comparison of the two procedures with a third oneassumed as a reference, we tested whether the calibration thatuses only dynamic trials is capable of assuring a satisfactoryinterval of uncertainties associated with the translational(a0 − g) and rotational accelerations of the AC cluster, or,conversely, whether the calibration procedure must include apreliminary static calibration trial.

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Meas. Sci. Technol. 19 (2008) 055204 P Cappa et al

The present paper is organized as follows. In section 2,we present the theoretical approach that permits the estimationof location, direction and sensitivity for each AC. In section 3,we describe the experimental approach to show, by means ofthe test bench, the feasibility of the two calibration proceduresapplied with two GF-IMU systems that are characterized bya different sensor number. In section 4, we discuss the datacollected in the calibration and the validation trials. Finally,in section 5, we make some concluding remarks.

2. Theoretical approach

To obtain the linear and angular accelerations of an Naccelerometric cluster, the AC positions r = [rxryrz]T , thesensitivities S and the unity vectors n = [nxnynz]T , whichdefine the sensing axis of every accelerometer, are needed.The method to compute (a0 − g) and ω from the AC outputsis reported in the appendix. The solving equation is[

w(a0 − g)

]= M∗(V − O)

w = [ω1 ω2 ω3 ω2

1 ω22 ω2

3 ω2ω3 ω3ω1 ω1ω2]T

(1)

where M∗ is the pseudoinverse of M, defined in equation (2),which is dependent on position r and orientation n of everyaccelerometric axis:

M = S[K NT ] (2)

where K and N matrices are dependent on ACs’ position andorientation components, so the M∗ matrix can be consideredthe final result of a calibration procedure.

The present study focuses on the exploitation of themetrological performances of the two calibration proceduresused to calculate matrix M∗. The first procedure consistsin a static trial capable of determining the AC sensitivitiesand the direction of the sensing axes, followed by a dynamictrial needed to calculate the AC positions; that procedure ishenceforth indicated by P-I. The latter procedure consists indetermining R, N and S (i.e. M∗) with only a dynamic trial,and this calibration procedure is referred to as P-II.

2.1. Static and dynamic calibration procedure (P-I)

The first calibration procedure consists of two steps. In thefirst step the angle between the sensitive axis of the AC andthe direction of gravity vector g is utilized to obtain sensitivityS and unity vector n. The procedure consists in stationarilyplacing the accelerometric cluster with different attitudes inthe gravity field. The linear acceleration of a generic AC is

a = a0 + ω × r + ω × (ω × r) (3)

a = a0 + F(ω)r + F(ω)2r = a0 + Wr (4)

where F is a cross product matrix. The output voltage of theAC is equal to

V = S(nT (a0 − g + Wr)) + O (5)

O being the electrical voltage offset. Since the measurementsare taken from ACs at rest, neglecting the Coriolis component,

the only force affecting the sensor is gravity. So W and a0 areboth equal to zero and equation (5) can be rewritten as

V = SnT (−g) + O (6)

V = [SnT O

] [−g1

]. (7)

To determine sensitivity S, unity vector n and electrical voltageoffset O, at least four equations are needed; then, for a genericnumber of calibration measurements Nc, corresponding todifferent orientations of the AC cluster relative to the gravityfield, equation (7) can be rewritten as follows:

[V1 · · · VNc

] = [SnT O]

[−g1 · · · −gNc

1 · · · 1

](8)

where g1, . . . ,gNcare the gravity vector g0 evaluated in the

relative, i.e., moving reference system:

g1 = 1C0 · g0

· · ·gNc

= Nc C0 · g0.

(9)

The orientation and the sensitivity of each AC are calculatedby means of

[SnT O ] = [V1 · · · VNc

] [−g1 · · · −gNc

1 · · · 1

]∗

Nc � 4. (10)

Equation (10) consists in the RMS minimization of the linearsystem expressed in equation (8) and, applied at each AC,provides sensitivity matrix S and orientation matrix N.

In the latter step, which involves the estimation of the ACpositions, the dynamic trial is instead needed and it consists inimposing a known motion on the GF-IMU. Equation (5) canbe rewritten as

V − SnT (a0 − g) = [SnT W 1]

[rO

]. (11)

To determine the AC positions, as in the static step, at leastfour calibration trials are needed:

[rO

]=

⎡⎢⎣

SnT W1 1... 1

SnT WNc1

⎤⎥⎦

∗

·

⎛⎜⎝

⎡⎢⎣

V1...

VNc

⎤⎥⎦ −[(a0 − g)1 · · · (a0 − g)Nc

]T Sn

⎞⎟⎠ Nc � 4

(12)

where S and nT are calculated by the previously describedstatic trial. Position matrix R is obtained by applyingequation (12) at each AC of the cluster. Once S, N andR are obtained, matrix M∗ can be calculated by usingequation (2), and the GF-IMU is fully calibrated.

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Meas. Sci. Technol. 19 (2008) 055204 P Cappa et al

2.2. Dynamic calibration procedure (P-II)

The second calibration method does not require a prior statictrial but is based on the postprocessing of only dynamic datato estimate the entire set of calibration parameters for theGF-IMU. Equation (5), as reported in the appendix, can beextended to the entire AC cluster as

V = M ·[

w(a0 − g)

]+ O. (13)

Extending the previous equation to the time series of Nc

measurements gathered in the dynamic trial, we obtain

[V1 · · · VNc

] = [M O

]⎡⎣ w1

(a0 − g)1

1· · ·

wNc

(a0 − g)Nc

1

⎤⎦ .

(14)

Orientation matrix N and sensitivity matrix S are calculatedby extracting and normalizing the columns SNT from matrixM:[

M O] = [

SK SNT O] = [

V1 · · · VNc

]·⎡⎣ w1 wNc

(a0 − g)1 · · · (a0 − g)Nc

1 1

⎤⎦

∗

. (15)

Then, position matrix R is obtained using the same solvingequation used for the method P-I, i.e. equation (12), but nowtaking into account N and S calculated with equation (15),instead of equations (10) and (2).

3. Experimental analysis

3.1. Experimental test setup

An ad hoc apparatus was developed in order to impose motionlaws for calibration and validation tests of a GF-IMU system.The calibration test bench herewith proposed consists of aninstrumented double pendulum capable of generating knowntranslational and rotational accelerations to the AC cluster inrepetitive trials; the apparatus is shown in figure 1. The rotationalong the fixed axis is given by an electrical motor coupled witha harmonic steel platen; the second rotation along a mobile axisis imposed by an extension spring.

The maximum range of rotation is about ±40◦ and ±60◦

for the fixed and the mobile axis, respectively. Both rotationsare monitored with two high-resolution optical encoders(5000 ppr 2× interpolation).

The aims of the double pendulum are to provide ameasurement of the GF-IMU orientation in the static trialsand the translational and angular accelerations in the dynamictrials. The estimation of the inaccuracies relative to theabove-mentioned variables was based on the Monte Carlomethod [31]. The identified error sources considered in thesimulation were encoder resolution (0.018◦) and positioningand orientation of GF-IMU in the reference frame of thedouble pendulum (0.5 mm and 0.5◦ for each rotational axis,respectively). Resulting accuracies were 0.5◦, 0.03 m s−2

Figure 1. View of the double-pendulum system used to test theGF-IMU.

and 8 deg s−2. The overall experimental uncertainties can beconsidered acceptable in relation to the foreseen use of theinstrumented helmet.

The GF-IMU here tested was the same as that described ina previous paper [24]; it consists of ten bi-axial accelerometers(ADXL 311) mounted on a lightweight foam bicycle helmet.The sensor’s specifications are sensitivity of 167 mV g−1, fullscale equal to ±2 g and signal-to-noise ratio (SNR) of 0.01 g.This sensor widely covers the range of requirements inmovement analysis for wearable GF-IMUs. The setup of theacquisition system was similar to that used in the previouslymentioned paper, but also included the gathering of the twoencoder outputs equipping the calibration apparatus. Theoutput of the ACs was connected via a shielded cable to anA/D PC-board (LSB = 20 µV) and a 20 channel multiplexerwith an integrated 200 Hz Bessel filter. The output of theencoders was acquired with a 32 bit 40 MHz counter board.The acquisition rate was set to 1 kHz.

A low pass filter was applied to the raw AC and encoderoutputs to reduce the SNR, and the morphology was a second-order recursive (0-lagged) Butterworth filter. The properselection of the cut-off frequency, which depends on the inputfrequency, was based on the consideration reported in thefollowing section.

In order to study the accuracy of the two calibrationmethods in terms of the AC positions we used a six-cameraoptoelectronic system (VICON 512) characterized by aninaccuracy of 1/3000 of the maximum dimension of theworkspace; in the present case the inaccuracy can be estimatedequal to ∼1 mm. A pointer equipped with two markers wasused by the experimenter to measure the 3D position of eachAC origin, i.e. matrix ROS.

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Meas. Sci. Technol. 19 (2008) 055204 P Cappa et al

Figure 2. Scheme for the computation of a global calibration matrixM by means of P-I, P-II and P−REF.

3.2. Experimental procedure

As reported in the previous paragraph, to verify both thecalibration procedures, static and dynamic trials were carriedout.

The static trial consisted of placing the instrumentedhelmet on the double pendulum with different attitudes in thegravity field, so the only force acting on the ACs was gravity.A redundant set of 15 different orientations was used for thestatic calibration to solve equation (10).

The dynamic trial consisted of fixing the helmet on thedouble pendulum and imposing known rotational time lawson the AC cluster along the two perpendicular axes. Thefrequency of the imposed rotations was about 1 Hz for bothaxes, while the range of the amplitude rotation was coincidentwith the maximum range of rotations previously indicated.The maximum value of angular velocity was equal to∼300 deg s−1, while the angular and linear accelerations wereof ∼3000 deg s−2 and ∼10 m s−2. The kinematic variables ofthe imposed input were obtained by a numerical differentiationof the encoder outputs, low pass filtered at 10 Hz; the cut-offfrequency was selected on the basis of the frequency of theimposed rotation. The time length of each dynamic trial was∼20 s. To complete the P-I procedure and to entirely performP-II, five dynamic trials were carried out.

Calibration P-I can be articulated as follows. First,by postprocessing a static trial, matrices NP-I and SP-I areevaluated by means of equation (10). Second, by elaboratinga dynamic trial, matrix RP-I is computed using NP-I and SP-I inequation (12). Thus, calibration P-I is based on thepostprocessing of one static trial and five dynamic ones andgenerated five global matrices MP-I. Figure 2 illustrates theprocess graphically.

In calibration P-II, in turn, only the dynamic data areelaborated to determine matrices NP-II, SP-II and RP-II by meansof equation (15) and then equation (12). In the presentcase calibration P-II produces five global matrices MP−II (seefigure 2).

Figure 3. Errors in the AC position rP-I and rP-II with respect to rOS,�rP-I, P-II; errors in the AC orientation nP-II with respect to nP-I, αP-II.

Finally, the procedure P-REF consists in the computationof MP-REF from the set of matrices NP-I, SP-I and ROS,which is obtained by the direct measurement of AC positionsusing an optoelectronic system. P-REF is assumed asthe absolute reference calibration for the following reasons:(i) the optoelectronic system assures better metrologicalperformances in terms of 3D position measurement, ROS, incomparison with the double-pendulum apparatus and (ii) theangular encoders equipping the double pendulum yield, inthe static calibration, highly accurate measurements for thedetermination of sensitivity and unity vectors, SP-I and NP-I.

The data postprocessing, which focused on the assessmentof metrological performances of P-I and P-II, was executedin two steps: (i) calibration tests finalized to the evaluationof the error associated in location, orientation and sensitivityfor each AC and (ii) validation tests for the estimation of theerror associated with the linear and angular acceleration valuesprovided by the GF-IMU.

3.2.1. Calibration tests. The validation of P-I with respect toP-REF only consisted in the comparison with the AC positionsdetermined by OS (i.e., rP-I versus rOS). The validation ofP-II with respect to P-REF, instead, always included both thecomparison with the AC positions obtained by OS and thecomparison with the directions and sensitivities obtained bythe static trials (i.e. rP-II versus rOS, SP-II versus SP-I and nP-II

versus nP-I).The differences between each AC position obtained with

P-I and P-II and the position measured by the optoelectronicsystem (see figure 3) are

�rP-I,P-II = |rP-I,P-II − rOS|. (16)

The error in the AC sensitivity is

�SP-II = |SP-II − SP-I|. (17)

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Meas. Sci. Technol. 19 (2008) 055204 P Cappa et al

(a) (b)

Figure 4. Locations of the ten bi-axial ACs (a) and the six bi-axial ACs (b) over the lightweight foam bicycle helmets.

The error in terms of axis directions is given by the angle αP-IIdefined by nP-II and nP-I (see figure 3):

αP-II = arccos(nP-II · nP-I). (18)

Finally, the 100 values (20 ACs and five nominallyidentical trials) were elaborated to obtain the mean valuesand the standard deviations of the above-mentioned fourvariables.

3.2.2. Validation tests. For the evaluation of the calibrationprocedures, five dynamic motion sequences were monitoredwith the GF-IMU always imposed by means of the double-pendulum apparatus. Based on the experimental data, wecalculated the acceleration components (a0 − g) and ω byusing the 11 different global matrices (see figure 2) in fivevalidation tests. So, to estimate the accuracy inherent to eachcalibration procedure, we calculated the following variables:

ε|a0−g|P-I,P-II,P-REF

=√∑Nv

i=1 (|(a0 − g)|P-I,P-II,P-REF − |(a0 − g)|E)2i

Nv

(19)

ε|ω|P-I,P-II,P-REF

=√∑Nv

i=1 (|ω|P-I,P-II,P-REF − |ω|E)2i

Nv

(20)

where Nv is the number of collected samples in a validationtest and (a0 − g)E and ωE are the accelerations evaluatedwith the encoders. Consequently, we obtained the followingsets of data: 25 values (five global matrices and five validationtests) for ε|a0− g|P-I, ε|ω|P-I and ε|a0− g|P-II,ε|ω|P-II and 5 values(one global matrix and five validation tests) for ε|a0−g|P-REF

and ε|ω|P-REF.To exploit the effects induced by sensor redundancy,

we also decided to analyse a cluster constituted by aminimal number of ACs by computing the same variablesindicated in equations (19) and (20); previous studies [19, 20]demonstrated that 12 linear ACs is the minimal number toevaluate the complete acceleration state of a rigid body andto ensure a unique solution. The new cluster was obtainedby means of the same data acquired in the validation tests byvirtually ‘removing’ four bi-axial ACs. Among all the 210

Table 1. Errors in the estimation of unity vector and sensitivitydetermined by means of P-II with respect to P-REF and in theevaluation of the AC position computed by P-I and P-II with respectto P-REF.

Indices Mean SD

αP-II (deg) 6 4�SP-II (mV V−1/m s−2) 0.4 0.3�rP-I (mm) 7 3�rP-II (mm) 32 18

possible subsets of six of the ten bi-axial ACs, we chose thespecific configuration (see figure 4) that maximizes index cof the pseudoinversion condition of global matrix M. Thatcriterion was already used in [24].

The differences among the three calibration proceduresfor both the array configurations were evaluated by meansof a one-way analysis of variance ANOVA. The lowerthreshold for the ANOVA test was chosen equal to 0.01,i.e., when p < 0.01 occurred then at least one variableresulted significantly different from the others. To state ifthe differences between pairs of variables are significant, theTukey multiple comparison Honestly Significant Differencetest was performed, assuming a threshold of 0.05.

4. Results and discussion

4.1. Calibration tests

The results relative to the discrepancies among the calibrationprocedures in the evaluation of positions, attitudes andsensitivities of the entire set of ten bi-axial ACs are summarizedin table 1.

The error values inherent to P-II, in terms of αP-II and�SP-II, are noticeable and can significantly compromise theoverall metrological performances of the GF-IMU. The errorscan be ascribed to the following reason. In P-I and P-REF,equation (10) implies a number of coefficients to be determinedequal to four, which is equal to the number of the degreesof freedom describing the AC model and which are givenby O and S, and the unity vector n. In P-II, however,equation (15) is solved as a linear system with 13 coefficientswhile the actual number of the degrees of freedom of the AC

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Meas. Sci. Technol. 19 (2008) 055204 P Cappa et al

(a) (b)

Figure 5. Root mean square errors in the evaluation of translational (a) and rotational (b) acceleration components as a function of thechosen calibration procedure and the AC number that sensorize the GF-IMU.

is equal to 7, which is given by the parameters O, S, n andr. In synthesis, if one solves equation (15) as a linear system,neglecting six constraints among the coefficients, it impliesP-II to be more sensitive to experimental errors such as thenumerical postprocessing of the encoders and the noise levelof the AC outputs.

As concerns the error in the estimation of the AC positions(�rP-I), P-I produces a value of 7 mm, so the procedure exhibitsan inaccuracy greater than that of the optoelectronic system,which, as mentioned in section 3.1, is ∼1 mm. The value of�rP-I represents the intrinsic limit of the novel theoreticalapproach used in P-I. Moreover, P-II exhibits a sensibleincrease in the error associated with the estimation of the ACposition with respect to P-I; in fact, �rP-II is about four/fivetimes �rP-I. It is clear that the further increase of the observeddiscrepancies is caused by the mismatches between actual andestimated orientation (αP-II) and sensitivities (�SP-II) of eachAC that caused the error in position to go unbounded.

In general, from a comparative examination of theobtained data, it emerges that a feasible calibration proceduremust include a static trial before the dynamic one to properlycompute the AC parameters. Hence, the procedure P-I givesbetter metrological results than the procedure P-II.

4.2. Validation tests

The results relative to the validation tests are summarized intable 2 for both the ten bi-axial and six bi-axial AC clusters.

As regards the (a0 − g) component and ω evaluated withthe entire set of ten bi-axial ACs, see table 2(a), it emergesthat P-I provided values that are close to the ones calculatedby means of P-REF. Hence, the metrological performances ofP-I (see table 1) are worse than the ones of an optoelectronicsystem in the determination of the AC position and do notsignificantly affect the RMSE of translational and rotationalaccelerations. Conversely, the comparative examination of P-II with P-REF shows a sensible worsening of the metrological

Table 2. Root mean square errors in the estimation of translationaland rotational accelerations: (a) GF-IMU equipped with ten bi-axialACs; (b) GF-IMU equipped with six bi-axial ACs. Statisticaldifferences found among groups (p < 0.05) are indicated by asuperscript.

Calibration procedure

P-REF P-I P-II

(a)ε|a0− g| (mm s−2) 225P-II 231P-II 463

% FS 2.2P-II 2.3P-II 4.6ε|ω| (deg s−2) 54P-II 57P-II 220

% FS 1.8P-II 1.9P-II 7.7

(b)ε|a0− g| (mm s−2) 230P-II 237P-II 736

% FS 2.3P-II 2.4P-II 7.4ε|ω| (deg s−2) 69P-II 75P-II 467

% FS 2.3P-II 2.5P-II 15.6

performances; in fact, the choice of P-II causes a doubling ofε|a0−g| and a quadrupling of ε|ω|.

The observed differences in the metrological quality of thecomputed parameters can be ascribed to the following reason.The evaluation of the translational acceleration component isbased on the postprocessing of the outputs of at least threelinear ACs, while the estimation of rotational accelerationneeds at least 12 linear ACs. Thus, taking into account thatwe elaborated ten bi-axial ACs, the estimation of ω is, incomparison with (a0 − g), intrinsically affected by a highererror due to a reduced redundancy. The ANOVA and theTukey tests confirm for both acceleration components that nosignificant differences are observed between P-I and P-REF,while P-II is statistically different to P-REF and, consequently,to P-I.

The similar behaviour observed for P-I and P-REF witha ten bi-axial linear AC cluster is confirmed when the GF-IMU is constituted by six bi-axial linear ACs (see table 2(b)).

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Meas. Sci. Technol. 19 (2008) 055204 P Cappa et al

Moreover, the significant worsening of metrologicalperformances when P-II is used is also confirmed; in factε|a0− g| and ε|ω| are three and seven times the correspondingerrors obtained with P-REF. The differences observed in theestimated errors can always be ascribed to the level of sensorredundancy. The differences that are statistically significantare always between P-II and P-REF.

Finally, the data obtained for ε|a0− g| and ε|ω| by means ofthe two GF-IMUs that differ in the sensor number are depictedin figure 5.

Several issues are worth considering in this figure. First,by reducing the number of sensors to 12 and focusing attentionon P-I and P-REF, an unnoticeable variation of ε|a0− g| and asmall increase of ε|ω| are observed. Second, the standarddeviations associated with the two parameters, always limitedto P-I and P-REF, do not increase by reducing the sensornumber. Third, conversely, P-II is highly dependent on thenumber of ACs used; in fact, the reduction of the number ofsensing axes to 12 causes an increase of 60% and 100% forε|a0− g|and ε|ω|, respectively. Fourth, the standard deviationvalues also significantly increase for a six bi-axial AC cluster.In general, the advantage of obtaining all the parameters forthe AC cluster with the conduction of only dynamic trials hasa major drawback because of the poor metrological quality ofthe obtained data.

5. Conclusions

In this paper we proposed a novel algorithm to entirelycalibrate an AC cluster—i.e. in terms of position, orientationand sensitivity of the sensor—and examined two calibrationprocedures (P-I and P-II) that use an ad hoc instrumenteddouble-pendulum apparatus; the former is characterized bythe conduction of static and dynamic trials, while the latter isbased on only dynamic trials. The calibration procedures wereused on two instrumented helmets equipped with ten and sixbi-axial linear ACs.

We compared P-I and P-II with a procedure (P-REF)based on an optoelectronic system characterized by bettermetrological performances of the test bench used here.A global analysis of the obtained data demonstrated thatP-I, in conjunction with redundant AC configurations, iscomparable to the reference calibration procedure P-REF inthe measurement of translational and angular accelerations.P-II, instead, caused a doubling of ε|a0− g| and a quadrupling ofε|ω| with respect to P-REF and then it showed a non-negligibledecrease in performance.

Lastly, a similar analysis could be used to determine theerror in velocity and displacement; this will be examined inthe ongoing research phase.

Acknowledgments

This work has been supported in part by a grant from theItalian Ministry of University and Research (PRIN-2006) andby a grant from ‘Sapienza’ University of Rome (ex-60%—year 2006). The authors are also deeply grateful to theChildren’s Hospital ‘Bambino Gesu’ IRCCS for the use of theoptoelectronic system installed at the ‘Movement Laboratory’.

Appendix

A.1. For each AC

Linear acceleration of the origin of the generic AC:

a = a0 + ω × r + ω × (ω × r) (A.1)

a = a0 + F(ω)r + F(ω2)r (A.2)

a = a0 + Wr (A.3)

where F is the cross product matrix:

F(m) =⎡⎣ 0 −mz my

mz 0 −mx

−my mx 0

⎤⎦

W = W(ω, ω) = F(ω) + F(ω)2.

The output voltage of the AC is

V = S(nT (a − g)) + O. (A.4)

Combining equations (A.3) and (A.4)V − O

S= nT (a0 − g) + nT Wr = nT (a0 − g) + kTw (A.5)

where the terms of the last addendum of equation (A.5) are

k =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−F(n)r⎡⎣−n2r2 − n3r3

−n3r3 − n1r1

−n1r1 − n2r2

⎤⎦

⎡⎣n2r3 + n3r2

n3r1 + n1r3

n1r2 + n2r1

⎤⎦

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, w =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ω1

ω2

ω3

ω21

ω22

ω23

ω2ω3

ω3ω1

ω1ω2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

A.2. Indexing for N axes:

Equation (A.5), written for all axes, may be written as follows:

S−1(V − O) = NT (a0 − g) + K · w (A.6)

V = M[

wa0 − g

]+ O, (A.7)

with

N = [n1 . . . ni . . . nn ] : orientation matrix;

O = [O1 . . . Oi . . . On ]T : offset vector;

V = [V1 . . . Vi . . . Vn ]T : output vector;

S =

⎛⎜⎝

S1 0. . .

0 Sn

⎞⎟⎠ : sensitivity matrix of the ACs;

K =

⎡⎢⎢⎢⎢⎢⎢⎣

k1...

ki

...

kN

⎤⎥⎥⎥⎥⎥⎥⎦

: rotational sensitivity matrix;

M = S[K NT ] : global sensitivity matrix.

8

Meas. Sci. Technol. 19 (2008) 055204 P Cappa et al

The vector of angular acceleration and translational (a0 − g)component may be calculated from equation (A.7):[

w(a0 − g)

]= M∗(V − O) (A.8)

where M∗ is the pseudoinverse of M; the angular accelerationcomponents are given by the corresponding parts of w:

ω =⎡⎣w1

w2

w3

⎤⎦ . (A.9)

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