Comparison of calibration methods for accelerometers used in human motion analysis

ARTICLE IN PRESS

JID: JJBE

[m5G;August 30, 2016;15:13]

Medical Engineering and Physics 0 0 0 (2016) 1–11

Contents lists available at ScienceDirect

Medical Engineering and Physics journal homepage: www.elsevier.com/locate/medengphy

Comparison of calibration methods for accelerometers used in human motion analysis Alexis Nez∗, Laetitia Fradet, Pierre Laguillaumie, Tony Monnet, Patrick Lacouture PPrime Institute, CNRS – University of Poitiers – ENSMA, UPR 3346, Robotics, Biomechanics, Sport and Health, Futuroscope, France

a r t i c l e

i n f o

Article history: Received 11 January 2016 Revised 19 July 2016 Accepted 7 August 2016 Available online xxx Keywords: Microelectromechanical system (MEMS) Accelerometer Calibration Measurement model Human motion analysis

a b s t r a c t In the fields of medicine and biomechanics, MEMS accelerometers are increasingly used to perform activity recognition by directly measuring acceleration; to calculate speed and position by numerical integration of the signal; or to estimate the orientation of body parts in combination with gyroscopes. For some of these applications, a highly accurate estimation of the acceleration is required. Many authors suggest improving result accuracy by updating sensor calibration parameters. Yet navigating the vast array of published calibration methods can be confusing. In this context, this paper reviews and evaluates the main measurement models and calibration methods. It also gives useful recommendations for better selection of a calibration process with regard to a specific application, which boils down to a compromise between accuracy, required installation, algorithm complexity, and time. © 2016 IPEM. Published by Elsevier Ltd. All rights reserved.

1. Introduction Because microelectromechanical system (MEMS) development has resulted in sensors being cheap and small, accelerometers have been used in many applications. Recently, they have even invaded our daily lives with the influx of smartphones and connected objects. In practice, accelerometers are used for two main purposes: to directly obtain and interpret acceleration measure, to estimate another measure (such as orientation or position by integrating the acceleration), or by fusion with other measures (typically angular velocity). Acceleration is used directly for vibration measurement on a mechanical structure. For example, the analysis of vibration on rotating machinery can give information to diagnose defects and initiate maintenance operations. Another way to interpret vibration is modal analysis, which is the analysis of the dynamic response of a structure under vibrational excitation. For example, this enables us to check that a building’s natural frequency does not match that of earthquakes. The acceleration measure can also be directly interpreted in the fields of medicine and biomechanics especially for activity recognition [1]. The identification of known patterns over accelerations measured from body segments can identify a range of static and ∗

Corresponding author. E-mail addresses: [email protected] (A. Nez), [email protected] (L. Fradet), [email protected] (P. Laguillaumie), [email protected] (T. Monnet), [email protected] (P. Lacouture).

dynamic activities [2]. Typical applications are analysis of activities of daily living to improve rehabilitation treatment [3] and fall detection for the elderly [4]. Other less expected applications include the analysis of accelerometric patterns from gait for biometric user identification [5]. Regarding indirect measures based on acceleration, accelerometers are often used in static condition to estimate orientation. Indeed, 3D measured acceleration can be compared with g (vertical gravitational acceleration) and can provide two orientation angles, which are often described as roll and pitch angles. However, to estimate the full orientation parameters, that is to estimate the rotation along the vertical axis as well, accelerometers are not sufficient and are associated with magnetometers that provide an estimation of yaw angle [6,7]. In this way, Kemp et al. [8] measured 3D orientation of body parts for diagnosis of movement disorders. In non-static conditions however, accelerometers measure the combination of gravity and external acceleration, which means that orientation cannot be estimated properly [9]. Luinge et al. [10] succeeded in estimating body inclination during movements with large accelerations from a 3D accelerometer using Kalman filtering and making assumptions concerning the frequency of the movement measured. In fact, for 3D orientation measures, accelerometers are most often combined with gyroscopes to form an inertial measurement unit (IMU). Regarding these IMUs, algorithms were developed on the initiative of NASA, to integrate inertial data for spatial navigation [11]. Inertial navigation systems (INS) are now widely used in the spatial, aeronautic and military fields for spacecraft, plane and missile guidance. In the field of biomechanics IMUs are used to perform movement analysis and

http://dx.doi.org/10.1016/j.medengphy.2016.08.004 1350-4533/© 2016 IPEM. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

JID: JJBE 2

ARTICLE IN PRESS

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

postural evaluation. For example, El-Gohary et al. [12] use IMUs to detect and characterize turns during gait in patients with Parkinson’s disease. Al-Jawad et al. [13] use Kalman filtering, which is an excellent tool for data fusion coming from noisy signals, for static postural sway analysis from IMU data. The association of IMU and Kalman filtering enabled El-Gohary and McNames [14] to measure human joint angles, Sabatini [15] to perform gait analysis, and Zhang et al. [16] to apply pose estimation for cycling. Apart from orientation, IMU accelerometers can also estimate speed and position [17]. Indeed, using attitude estimated by the fusion of accelerometer and gyroscope data, acceleration can be expressed in the global frame. Then, by computing a double integration, the velocity and the position of the sensor from the starting point can be estimated. Following this method, IMU has become a promising device for human localization, and particularly for indoor pedestrian tracking [18]. Indeed it can be an alternative to global navigation satellite systems because the satellite signals are often too weak to penetrate buildings. For all of these applications, the relevance of results directly depends on the accuracy of the collected data. For example, in the particular case of position estimation, the accuracy of measured acceleration is a crucial point. Indeed, orientation is first estimated from the fusion of gyroscope and accelerometer data, after which acceleration can be expressed in the global frame. However, the second step is to double integrate acceleration to track changes in the IMU’s position. Due to the propagation of measurement errors through projection and integration calculations, errors rapidly accumulate in the tracked position. Such errors are collectively referred to as drift. In the literature, two categories of measurement errors are distinguished: stochastic errors (noise) and deterministic errors (calibration defects). Firstly, electronic sensors are disturbed by noise [19]. Thermal noise has the main influence on data collected from electronic sensors. As it is usually modeled with a Gaussian white noise, it has an impact over the entire frequency domain such that it cannot be filtered. Secondly, the definition of the link between raw output signals and estimated acceleration can be a factor of loss of accuracy. Unlike noise, this error can be corrected by defining an adapted measurement model and following an accurate calibration process. From the literature, measurement models with different levels of complexity can be selected, and several calibration methods have been described for accelerometers [20–24]. IMUs are commonly grouped into four performance categories: marine/navigation, tactical, industrial, and automotive/consumer grade. Marine/navigation grade sensors are the most precise, based on mechanical technology. Due to their high cost (from 100 000 to 1 million dollars) they are typically used in submarines, spacecraft, and military aircraft. Industrial and automotive categories are composed of sensors based on MEMS technology, and their main difference lies in the quality of sensor calibration. This paper focuses on calibration methods for MEMS accelerometers which are intended to be used in human motion analysis. As papers dealing with sensor calibration are flooding scientific literature, choosing an adapted approach for accelerometer calibration can be confusing. For clarification, the two following sections review the classical models and calibration procedures that are most commonly described in the literature. As one of the most important comparison criteria for these methods is their accuracy, the fourth section of this paper evaluates the magnitude of the errors resulting from each selected calibration method. Finally, the fifth section of this paper discusses these results and compares calibration methods, taking into account other criteria such as calculation complexity, consumed time, and required equipment. This paper concludes by offering some recommendations for the selection of a calibration method.

Fig. 1. Definition of the orthogonal frame built from the three accelerometer axes by three small rotations γyx , γzx and γzy (from Cai et al. [22]).

2. Measurement models Ideally, an accelerometer would have exactly the same sensitivity at any amplitude point within its specified amplitude range. It is generally agreed that the measurement model can be considered as linear when environmental conditions (temperature, for example) are sufficiently steady. The limit to how far the accelerometer’s output will differ from this perfect linearity is specified by the manufacturer in the datasheet. The measurement model for a perfectly linear accelerometer can be written as follows:

a = s.(u − b)

(1)

where a is the acceleration estimated from the electric potential u given by the sensor, by means of a scale factor s and an offset b. 2.1. First model For 3D acceleration measurements, the first model can be written in a matrix form:

a = S . (u − b) with

 

a=

ax ay ; S = az

(2)



sx 0 0

0 sy 0



0 0 ; u= sz

 

ux uy ; b = uz

  bx by bz

(3)

The link between sensor outputs and acceleration is defined via six calibration parameters (three scale factors and three offsets). This model is based on the restrictive assumption that the three axes of the accelerometers are perfectly orthogonal. Because of this limitation, this model is rarely used [20]. 2.2. Second model Due to the imprecise nature of the construction of a triaxial accelerometer, the three axes cannot be perfectly orthogonal. By defining three small rotations, starting from the first axis of accelerometers, an orthogonal frame can be defined (Fig. 1). Many authors define the link between real axes of accelerometers and the new orthogonal frame by a matrix T [21,22,25,26], such that the model becomes:

a = T . S . (u − b)

(4)

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

ARTICLE IN PRESS

JID: JJBE

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

with



1 −γyx

T =

0 1 −γzy

γzx

0 0 1

 (5)

The second version of the model (Eq. (4)) adds three angles (γyx , γzx , γzy ) to the set of calibration factors which is now composed of nine parameters. Although this model is certainly the most used in the literature [21,22,25,26], it is based on the assumption that a one-dimensional accelerometer is only sensitive to the acceleration along its axis. However, a transverse acceleration also has an influence on the measurement of a MEMS accelerometer [27]. Thus this model can be adequate for navigation and for some tactical grade sensors, but it is not fully adapted for lower grade sensors. In fact, most of the sensors used for human motion analysis are based on MEMS technology, so the model must be built to take into account cross-axis sensitivities. 2.3. Third model MEMS accelerometers are sensitive to transverse axes input. Therefore, the model defined by Eq. (4) must be built accordingly. The scale factor matrix becomes:



S=

sxx syx szx

sxy syy szy

sxz syz szz



(6)

where diagonal S elements represent the scale factors, and the offdiagonal S elements are the cross-axis factors. Moreover, misalignment between the accelerometer triad and the sensor box exists for the same reason as nonorthogonality within the triad. This error is sometimes taken into account in magnetometer sensor models [28], but surprisingly it is not considered for accelerometers. The link between the new orthogonal frame built on the triad and a frame defined by the sensor box can be defined by a misalignment matrix:



D=

dxx dyx dzx

dxy dyy dzy

dxz dyz dzz



(7)

Once cross-axis sensitivities and misalignment are included, the model becomes:

a = D . T . S . (u − b)

(8)

The product D . T . S can be replaced by a new matrix K whose factors depend on sensitivities, nonorthogonality, and misalignment. Thus the third version of the model can be written:

a = K . (u − b) with

K=



kxx kyx kzx

kxy kyy kzy

(9) kxz kyz kzz

 (10)

This final version of the model (Eq. (9)) depends on twelve calibration parameters (nine global factors ki j and three offsets bi ). Frosio et al. [23] used a similar model, but added the assumption that K is symmetric to reduce the number of calibration factors from twelve to nine. 3. Review of calibration processes Acceleration can be estimated from electric potential given by the sensor using the final measurement model described by Eq. (9). However, to obtain acceleration values, calibration factors (global factors ki j and offsets bi ) must be known. They are identified by means of a calibration process [29].

3

The choice of the calibration process should depend on the application. For example, to perform vibration measurements an accelerometer should be calibrated by controlled oscillations, as performed in the industrial field (norm ISO 16,063). So, more generally, calibration factors are obtained by comparing electric potential with applied accelerations close to the acceleration signal to be measured. In the case of an accelerometer used for attitude estimation, it can nevertheless be calibrated by controlled static orientations. Thus acceleration applied to the sensors can be deduced from the value of g (the gravitational acceleration) projected onto the sensor direction. In that case, the problem of applying controlled acceleration is replaced by the problem of positioning the sensor with regard to the vertical axis. This strategy is well adapted for accelerometers used in medicine and biomechanics. Using that principle, many calibration processes can be conducted. The following paragraphs review the main processes described in the literature, which can be divided into two parts: calibration from a reference and calibration without any reference. 3.1. Calibration from a reference Calibration from a reference is performed by comparing the electric potential given by the sensor with the acceleration deduced from the known position of the sensor. That means that the sensor orientation regarding the vertical must be controlled. 3.1.1. Direct identification from a leveled surface In the case of a rectangular-parallelepiped sensor box, calibration can be performed by putting the accelerometer on a horizontal table. The direct identification method is the easiest and fastest way to perform a calibration by considering the first linear version of measurement model (Eq. (2)). This method, called the “6-position method”, was first described by Titterton and Weston [30] and elsewhere [21,25]. The accelerometer is mounted on the table with each sensitive axis of the sensor pointing alternately up and down. From these six static acquisitions, scale factors and offsets can be directly calculated as follows:

b=

uup + udown 2

(11)

s=

uup − udown −1 2.g

(12)

where uup and udown are electric potentials given by the accelerometer when pointing upwards and downwards respectively. The accuracy of this calibration method depends on how well the accelerometer’s axes are aligned with the vertical axis. This means that the reference surface horizontality must be good and that the accelerometer box must be cube-shaped. If the shape of the accelerometer does not provide a good alignment of each axis with the table, a cube-shaped mounting frame can be used. As previously mentioned, MEMS output is known to be mainly affected by thermal noise which is usually modeled as a Gaussian white noise [19]. To reduce this impact on calibration, measured electric potentials must be averaged during the whole static phase. Even if this calibration method is relatively simple to implement, it has a major drawback. Because the identification of calibration factors is directly performed from the linear measurement model, cross-axis factors, misalignment and nonorthogonality cannot be estimated. Since these parameters are known to be significant for MEMS accelerometers, this can be a serious issue. 3.1.2. Optimized identification from a leveled surface To take into account cross-axis factors, misalignment and nonorthogonality, the previous method can be improved by estimating calibration factors from the least square criteria. In this

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

ARTICLE IN PRESS

JID: JJBE 4

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

case, the final version of the model (Eq. (9)) can be reorganized as follows:

3.2. Calibration without any reference

a = K . (u − b)

When calibration is performed from a series of known sensor orientations, the accuracy of calibration factors is directly dependent on how well the accelerometer is aligned with the vertical axis. Thus elaborated instrumentation can be required, which makes it mainly adapted for in-lab calibration. Calibration without any reference is more adapted for in-field calibration, because it does not require any alignment device and it can be performed with any type of sensors, whatever its shape. The original idea to perform in-use accelerometer calibration is presented by Lötters et al. [20]. This method is based on the fact that the modulus of acceleration in static conditions is equal to g:

(13)

u = K −1 . a + b u = N · a˜





with N = K −1 b

(14)

  and a˜ =

a 1

(15)

From the six static acquisitions described previously, the accelerometer outputs and the controlled accelerations from the sensor orientation can be stored in two matrices U and A:



U = u1

u2

u3

g

−g 0 0 1

0 g 0 1

⎡

⎢0 A=⎣ 0 1

u4 0 −g 0 1

u5 0 0 g 1

u6



(16)

⎤

0 0⎥ −g⎦ 1

(17)

Calibration factors stored in the matrix N can then be estimated using the following least square solution [21,31]:



N = U . AT . A . AT

−1

(18)

This equation returns the set of twelve calibration factors that minimize the error between estimated and controlled accelerations. As in the previous method, calibration accuracy depends on how well the accelerometer axes are aligned with the vertical axis. This calibration method could be improved by increasing the number of orientations to perform the optimization from more data. However, as a cube-shaped accelerometer has only six sides, the number of acquisitions can only be increased by redundant measures on the same sides. 3.1.3. Optimized identification from an orientation device A similar calibration method can be performed by changing the approach to controlling the direction of the accelerometer. In practice, more than six directions can be imposed by mounting the accelerometer on a turntable. Starting from the vertical axis, the accelerometer is rotated through a series of accurately known angles. As in the previous method, calibration factors are identified using the least square solution (Eq. (18)), where the size of matrices U and A could be greater than six. As in calibration on a horizontal surface, precise alignment between the turntable and the vertical axis is required. To obtain a direct link between the turntable and the accelerometer orientation, both must be accurately assembled. The fact that the accelerometer is fixed on the turntable is an advantage over previous methods because it avoids the negative effects arising from geometric defaults of the accelerometer housing. In fact, when calibration is performed by positioning the accelerometer on a horizontal surface, orthogonality defaults of the housing, and the fact that each side can be very narrow, induce misalignment between the internal accelerometers and the vertical axis. Because it avoids this issue, calibration on a turntable (or any device allowing accurate orientations) is known to give excellent results [21,30,32,33].

g=



ax 2 + ay 2 + az 2

(19)

The first step is to perform quasi-static phase detection during an acquisition. Next, the set of calibration factors is corrected by comparing the modulus of measured acceleration to g. This method has been adapted to calibrate accelerometers before an acquisition, and is called the “multi-position method” [21–26,32]. In the different methods proposed by many authors, calibration is always performed by positioning the accelerometer in n different static orientations. The resulting dataset is used to estimate calibration parameters that minimize the following least square objective function:

S (x ) =

n 

( a i ( x ) − g )2

(20)

i=1

where ai (x ) is the modulus of estimated acceleration at ith ith measurement, depending on calibration parameters stored in a vector x:

ai (x ) = |ai (x )| and ai (x ) are set by the model (Eq.(9 ) )

(21)

x = [kxx kxy kxz kyx kyy kyz kzx kzy kzz bx by bz ]

(22)

The added difficulty compared with the solution given by Eq. (18) is that the new function to optimize (Eq. (19)) is nonlinear. Calibration factors were calculated iteratively by minimizing the cost function (Eq. (20)) using Newton’s algorithm [23,26]. To converge on the global optimum, algorithms must be initiated with a rough estimation. Table 1 shows that the authors mainly used the manufacturer’s parameters, but these are not always available. Thus Syed et al. [21] used the “6-position method” described in Section 3.1.1 to initiate kxx , kyy , kzz , bx , by , bz . Cross-axis factors, nonorthogonality, and misalignment of MEMS accelerometers are known to be significant but still low, so they can be initialized from zero. Calibration parameters were obtained by solving the least square function by applying Cholesky’s decomposition to normal equations [21,25]. Zhang et al. [24,32] managed to keep the problem linear by writing it in matrix form. Thus the parameters can be estimated by a least square solution (as in Eq. (18)) without any

Table 1 Number of orientations, optimization algorithm and initial values used in seven papers. Ref.

Authors

Number of orientations

Optimization algorithm

Initial values

[25] [21] [26] [22] [23] [32] [24]

Shin et al. (2002) Syed et al. (2007) Skog et al. (2006) Cai et al. (2013) Frosio et al. (2009) Zhang et al. (2008) Zhang et al. (2010)

18 26 18 24 35, 42, 72 18 9, 18

Least-square solution Least-square solution Newton’s optimization Particle swarm Newton’s optimization Least-square solution Least-square solution

? 6-position method Manufacturer factors ? Manufacturer factors Method without initial value Method without initial value

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

ARTICLE IN PRESS

JID: JJBE

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

need for initial values. Unfortunately, a maximum of nine parameters can be identified by such a method. Therefore this method can be applied to the second model (Eq. (4)) but cannot estimate the twelve parameters of the third model (Eq. (9)). Finally, Cai et al. [22] added scale factor nonlinearities to the measurement model, such that the optimization was performed using the particle swarm algorithm which is more adapted for very non-linear problems. It can be seen that this problem is similar to the calibration of magnetometers by comparing the modulus of the measurement with Earth’s magnetic field. For this problem, the authors mainly used the Gauss–Newton algorithm to estimate calibration factors [34,35], rather than its second-order version (Newton algorithm). Indeed, Newton’s method requires us to write the Hessian matrix of the cost function with respect to each calibration parameter, in a literal form. Using the third model, it consists of a 12×12 matrix which is very time consuming. Gebre-Egziabher et al. also introduce a useful algorithm to establish initial conditions without the need for additional acquisition (for example, a 6-position procedure) [34]. This “two-step estimator” first estimates a list of combinations between calibration parameters, which are then extracted through algebraic manipulations. To enable convergence of the optimization process, we need at least as many equations as unknown. Thus from the third model (Eq. (9)), a minimum of n = 12 attitudes must be imposed on the accelerometer. However, there is clearly no consensus in literature with regard to choosing the number of imposed orientations (see Table 1). Moreover, the n equations must be uncorrelated, such as this is important to check that imposed orientations are all different from each other in terms of acceleration. For example, rotations around the vertical axis only do not give a different measurement in terms of acceleration. According to Zhang et al. [24], an optimal calibration scheme can be designed to maximize the numerical accuracy of calibration parameters. This optimal scheme is composed of attitudes with sensitive axes of accelerometers pointing up and down vertically, and attitudes with the gravity vector lying on the plane spanned by any two accelerometer sensitive axes and forming 45° or 135° angle with them. Also according to Zhang et al., adding more orientations leads to more accurate results.

5

4. Evaluation of calibration processes As explained previously, calibration is a very efficient way to improve the accuracy of data collected from a triaxial accelerometer. As detailed in the previous paragraph, the literature provides a quantity of calibration processes from which it can be difficult to find the “best method”. One of the important criteria to consider when selecting a calibration process is the acceptable magnitude of error made on the result. So there is a real need to evaluate each calibration process to identify which ones are the most and the least accurate. In addition, the choice of calibration method should take into account other parameters such as complexity, required time, and lab installation. Depending on requirements, a compromise should be found. 4.1. Presentation of the tested sensors In this study, eleven sensors coming from five different models were evaluated (Table 2). For completeness regarding the fields of medicine and biomechanics, we included two IMUs destined for movement analysis and two gyro-free models destined for actigraphy and movement analysis. Finally, we added a higher grade triaxial accelerometer used for vibration measurement, to highlight its differences regarding the sensors of interest. 4.2. Description of the evaluated calibration processes From the calibration processes described in the previous paragraph, Table 3 lists those that have been evaluated in this study, and the corresponding number of imposed orientations. The last column remains the references that are used for calibration: either controlled accelerations from known orientations, or the norm of the acceleration equal to gravitational acceleration. 4.2.1. Calibration A The method that is deemed to be the most accurate is optimized identification from an orientation device [21,30,32,33]. In this study, triaxial accelerometers were mounted on a motorized test bench (Fig. 2). A synchronous motor (B&R, Eggelsberg, Austria)

Table 2 Details of the eleven sensors used in the study. Reference

No. of tested sensors

Make-up

Field of use

APDM Opal

5

IMU (triaxial accelerometer–gyroscope–magnetometer)

Movement analysis

Cometa WaveTrack

2

IMU (triaxial accelerometer–gyroscope–magnetometer)

Movement analysis

Actigraph wGT3X-BT

1

Triaxial accelerometer

Actigraphy

Delsys Trigno IM Sensor

1

Triaxial accelerometer + EMG

Movement analysis

FGP FA3403-A9

2

Triaxial accelerometer

Vibration measurement

Picture

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

ARTICLE IN PRESS

JID: JJBE 6

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

Table 3 Identification and details of the seven calibration processes used in the study. Id.

Calibration process

Number of orientations

References

A

Optimized identification from an orientation device Optimized identification from a leveled surface Optimized identification from a leveled surface Direct identification from a leveled surface Multi-position method Multi-position method Multi-position method

9

Controlled accelerations

12

Controlled accelerations

6

Controlled accelerations

6

Controlled accelerations

12 30 54

a=g a=g a=g

B C D E F G

Fig. 3. 6-position scheme. Each axis is pointing upwards and downwards.

Fig. 4. Multi-position scheme for z axis pointing upwards from an approximately 45° inclined plan. The same procedure was followed for z axis pointing downwards, as well as x and y axes.

For sensors that were not cube-shaped, the mounting mechanism designed for assembly on the drive shaft was reused as an orthogonal casing. As for calibration A, the set of calibration parameters was estimated using the least square solution (Eq. (18)). Fig. 2. Triaxial accelerometer (Cometa WaveTrack) mounted on the motorized test bench.

was driven by a PID controller to obtain steady and accurate orientations. As described in Section 3.1.3, the orientation device must be aligned with the vertical axis and must be accurately assembled with the accelerometer. The horizontality of the bench was checked using a spirit level and the starting vertical orientation of the sensor was controlled with a plumb line. Sensors were accurately assembled on the drive shaft using specific mechanisms. As detailed in Appendix B, imposing three orientations over each sensor axis was sufficient to obtain accurate calibration parameters. In this paper, our starting position was always defined by the vertical (0°). Therefore, to perform three independent orientations over a complete rotation, each axis of the accelerometer was positioned at 0°, 120° and 240° which corresponds to a total of nine imposed orientations. From the set of nine acquisitions, calibration parameters were calculated using the least square solution (Eq. (18)) described in Section 3.1.2. 4.2.2. Calibration B and C For methods B and C, there is no need for a motorized test bench, but only a surface known to be horizontal. In this study, this was controlled using a spirit level. As described in Section 3.1.2, the sensor was positioned on the horizontal table over each face which led to six orientations for method C (see scheme Fig. 3). For method B, the number of orientations was increased to twelve by performing two acquisitions for each face.

4.2.3. Calibration D Direct identification from a leveled surface is the easiest and fastest way to estimate calibration parameters, because it does not require any optimization process. The orientation scheme is the same as for methods B and C (Fig. 3). As the method is based on the 6-parameter model (Eq. (2)), cross-axis factors cannot be estimated such that they are set to zero. The calculation is that described in Section 3.1.1. 4.2.4. Calibration E to G For calibration E to G, there is no need for any reference to control orientations. Triaxial accelerometers need only to be positioned in a series of different static orientations. In this study, we have performed calibration from 12, 30 and 54 orientations (respectively calibration E, F and G). As mentioned by Zhang et al. [24], an optimal scheme is composed of attitudes with sensitive axes of accelerometers pointing up and down and forming 45° or 135° angles with the vertical. That is why in this study the scheme was initiated with the 6-position attitudes (Fig. 3), and then with 8 orientations per sensor face on contact with an approximately 45° inclined plan (Fig. 4). This results in a total of 6 + 8 × 6 = 54 different orientations (calibration G). Calibration E and F was performed using the first 12 and 30 attitudes of this scheme. As explained in Section 3.2, the function to optimize is nonlinear. To take into account cross-axis sensitivities, the final version of the model (Eq. (9)) comprising twelve calibration parameters was considered, so the linear solution of Zhang et al. [24] could not be used. As is customary in solving this problem, we estimated the optimal set of calibration parameters iteratively [23,26,34,35]. To

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

ARTICLE IN PRESS

JID: JJBE

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

7

Fig. 5. RMS of the acceleration error (mean and standard deviation) induced by each calibration method. The grey shading highlights significant differences between calibration methods.

compensate for stability problems of the Gauss–Newton algorithm, we used the Levenberg–Marquardt algorithm. This algorithm was previously validated by comparison with results coming from the direct least square solution of Zhang et al. [24] when considering the second version of the model (Eq. (4)), which is only composed of nine parameters. We used the two-step estimator method of Gebre-Egziabher et al. [34] to estimate initial conditions. 4.3. Data analysis To evaluate the accuracy of the acceleration estimated from raw data and calibration parameters, sensors were inserted into a series of 36 controlled orientations in steps of 10° along their three axes, over the test bench described in Section 4.2.1. This resulted in a total of 3 × 36 = 108 control points by sensor. For each orientation, we calculated the error between estimated and controlled acceleration from the motor angle and g. Then we calculated the RMS of the error over the 108 imposed orientations. A Friedman test was used to determine a difference between the calibration methods ( p < 0.001). A Wilcoxon signed-rank test was used to compare calibration methods side-by-side and to detect significant differences (Fig. 5). In addition, the impact of the error of acceleration on the induced angle, which is a more eloquent quantity, was estimated. The angle α defining the incline from the horizontal of a onedegree-of-freedom object can be calculated from a uniaxial accelerometer as follows:

sin (α ) = a/g ⇒

α = arcsin(a/g)

(23)

By differentiating Eq. (23), the angle uncertainty Uα can be deduced from the acceleration uncertainty Ua and the angle α :

Uα =

Ua g.|cos(α )|

(24)

As shown by Eq. (24), this approach is only adapted when the sensor is close to the horizontal (α ∼0 ), otherwise angle uncertainty would become huge. The fact that Uα depends on the orientation is due to the derivate function of the sinus which is discussed in two technical notes [36,37]. From Eq. (24), the angle uncertainty in function of the incline can be plotted for each acceleration uncertainty resulting from the different calibration methods (see dashed lines in Fig. 7). A better way to calculate α is by using a triaxial accelerometer, which details measured acceleration along each component. In

this case, the incline of a one-degree-of-freedom object can be calculated as follows (see [36,37] for more details):

ax tan (α ) = ⇒ ay



α

ax = arctan ay



(25)

As previously discussed, the angle uncertainty Uα can be obtained by differentiating Eq. (25). The result (Eq. (26)) shows that angle uncertainty no longer depends on the incline (see solid lines on Fig. 7):

Uα =

Ua g

(26)

The calculation is detailed in Appendix A. Thanks to [37], in the global case of a 3-rotational-degrees-offreedom object, pitch and roll angles can be calculated in the same way by using the arctan function. In this way, angle uncertainty does not depend on the incline. 4.4. Results 4.4.1. Effect of the calibration methods on the acceleration error The mean acceleration errors and their standard deviations from all sensors of the study (except higher grade FGP accelerometers whose behavior is different) are given in Fig. 5. From results obtained from the Wilcoxon signed rank tests, non-significant differences (p > 0.05) between calibration methods appear on the graph. Thus, three calibration categories are identified and highlighted in different shades of grey. Results show that method A (calibration on the motorized bench) is the most accurate (RMSe ∼0.038 m/s2 ), and is significantly different from all other methods ( p < 0.01). Methods B and C (calibration over a horizontal surface) result in an intermediate precision (RMSe ∼0.061 m/s2 ) over the seven tested methods. The only difference between methods B and C is the number of imposed orientations, which does not induce any significant difference ( p > 0.05). Method D (direct identification) and methods E, F and G (multi-position) give less accurate estimations (RMSe ∼0.111 m/s2 ) of the acceleration, and their results are not statistically different. Finally, three calibration categories emerge from these results: method A (light shade), methods B–C (intermediate shade), and methods D–G (dark shade). Fig. 6a shows the RMS of the errors from each calibration method applied only to the five APDM Opal IMUs. Indeed, these sensors give both raw data (measured potential that we use to perform each calibration method), and calibrated data from the manufacturer’s software. Note that the measurement model used by

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

JID: JJBE 8

ARTICLE IN PRESS

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

a

b

Fig. 6. (a) RMS of the acceleration error on APDM Opals. (b) RMS of the acceleration error on FGP FA3403.

sensor orientation. In fact, the closer the accelerometer to the vertical axis (90°), the larger the induced measurement error. However, when the angle is in the 0–50° range, angle error is close to its minimal value. When computing the angle from a triaxial accelerometer, angle uncertainty becomes minimal and no longer depends on orientation. Indeed, large error induced by the accelerometer axis which is close to vertical is compensated by the other axis which is close to horizontal. Thus the minimal angle errors induced by method A, methods B–C, and methods D–G were about 0.22°, 0.36° and 0.65° respectively. 5. Discussion

Fig. 7. Angle uncertainties induced by the three calibration categories over the 0– 90° angular range, from a uniaxial accelerometer (dashed lines) and from a triaxial accelerometer (solid lines). 0° and 90° correspond to the horizontal and the vertical axes respectively.

the manufacturer is a black box and that the calibration method that has been followed is unknown. However, the comparison with acceleration obtained from manufacturer software is important to evaluate benefits that are provided by each calibration method. Therefore, when considering only APDM Opal IMUs, method A (calibration on the motorized bench) reduced the error by about 58% compared with manufacturer calibration. Methods B and C (calibration over a horizontal surface) enhance the accuracy of estimated acceleration slightly, while methods D–G give a result close to that obtained from manufacturer calibration. To highlight the difference in behavior between accelerometers that are typically used in biomechanics and those that are mainly destined for vibration measurement, Fig. 6b shows results obtained from the FGP FA3403 accelerometers. In this case, the method A once again gave the best result, while methods B–G were clearly equivalent. In particular, method D (direct identification), which is based on the 6-parameter version of the model, gave similar results to others that are based on the 12-parameter version of the model. 4.4.2. Effect of the calibration methods on angle error From Eqs. (24) and (26), Fig. 7 plots angle uncertainties converted from acceleration uncertainties which were induced by the three calibration method categories. Fig. 7 shows that acceleration uncertainty from a uniaxial accelerometer can induce different error angles, depending on the

The results that have been presented allow for a classification of calibration methods regarding their induced accuracy over measured acceleration or estimated orientation. The last section of this study offers some recommendations to better select a calibration method regarding a specific application, which boils down to a compromise between accuracy, required installation, algorithm complexity, and time. Results show that calibration over the motorized bench gave the most accurate acceleration estimation for the sensors used in the study (angular error of about 0.22°). This result is still valid for the higher grade FGP FA3403 accelerometers. Indeed, this method uses a series of very accurate orientations controlled by a servomotor. Moreover, the fact that the sensor stays mounted on the test bench during the overall process avoids the negative effects arising from geometric defaults of the accelerometer housing. Nevertheless, this calibration method requires specialized equipment to impose accurate orientations relative to a known reference. In this study, we built a test bench composed of a servomotor, but this could also be achieved by using a mechanical system such as an indexing head. This calibration method should be privileged for very accurate estimation of the orientation, but the required installation for calibration makes it difficult to implement outside of a lab. Optimized calibration over a horizontal surface (methods B– C) results in an angular error of about 0.36°, which is only 0.14° more than the previous one. Thus this calibration also gives accurate results and is relatively easy to implement. The only parameter that must be controlled is the horizontality of the surface on which the sensor must be positioned. This can be performed using a spirit level (accurate to the hundredth of a degree) or a laser level that ensures an accuracy over the table orientation in the order 10−9 degrees. However, the correct alignment of each side of the sensor can only be ensured if its box is cube-shaped. But the shape of accelerometers and IMU housings are sometimes too complex (see pictures in Table 2). In such cases, this calibration method can be performed by designing a cube-shaped mounting frame but, even if this becomes progressively easier thanks to 3D

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

JID: JJBE

ARTICLE IN PRESS

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

9

Table 4 Summary of mean acceleration and angle errors, required equipment and algorithm for the four identified calibration categories. Calibration method

Mean acceleration and angle errors

Required equipment

Algorithm

Optimized identification from an orientation device Optimized identification from a leveled surface Direct identification from a leveled surface Multi-position method

0.038 m/s2 – 0.22◦ 0.061 m/s2 – 0.36◦ 0.111 m/s2 – 0.65◦ 0.111 m/s2 – 0.65◦

Orientation device (motorized bench, indexing head) Leveled surface (+ mounting frame) Leveled surface (+ mounting frame) –

Simple (least-square solution) Simple (least-square solution) – Complex (iterative optimization)

printing, the process is more complicated. This study shows that increasing the number of orientations from six to twelve by redundant measurements has no impact on acceleration accuracy. Thus one measurement per sensor side is sufficient. This study highlights the great potential of this calibration method in the case of a cube-shaped sensor housing, because of its feasibility outside of the lab and its good level of accuracy. Direct identification of calibration parameters over a horizontal surface (method D) gave the lowest accuracy. This calibration method is based on the 6-parameter version of the measurement model, which does not take into account non-orthogonality, misalignment, and cross-axis sensitivities that are reputed to be non-negligible in MEMS sensors [21,23,38]. However, this calibration method gave equivalent accuracy compared to the others for the FGP FA3403 accelerometers. This illustrates the fact that these higher grade accelerometers are not highly sensitive to cross-axis accelerations, and that misalignment and non-orthogonalities due to manufacturing irregularities are insignificant. Even if this calibration method gave the worst results for MEMS sensors, the induced angular error was still less than one degree (about 0.65°). Moreover, it is the easiest to implement because there is no need for an optimization algorithm. As for the previous method, the sensor housing should be cube-shaped, otherwise a mounting frame must be used, which makes calibration too complex. Thus this method remains of great interest for applications that require frequent and fast calibration for cube-shaped sensors. Finally, the well-studied multi-position method gave results significantly less accurate than methods A–C (angular error of about 0.65°). In Fig. 5, multi-position (methods E–G) seems to be better than the direct identification (method D) but the difference is not significant (Fig. 5), which is consistent with the conclusions of Syed et al. [21]. However, Frosio et al. [23] concluded that the multiposition method was a little more accurate (error from −0.26° to +0.26°) than the so-called 6-position method (error from −1.54° to +1.15°). In any case, results obtained from these two methods are very close. As concluded in Zhang et al. [24], results seem to be more accurate when increasing the number of orientation (Figs. 5 and 6a), but this result is again not significant (Fig. 5). Therefore, this paper suggests that just putting an accelerometer in twelve different orientations should be sufficient. The main asset of this method is the fact that it does not require any controlled reference, which makes it appropriate for outdoor calibration and for all sensor shapes. However, a robust optimization algorithm must be implemented to produce an accurate result. This conclusion (summarized in Table 4) can be interpreted with regard to specific applications. First, using accelerometers for activity recognition by directly interpreting the acceleration signal does not require a high measurement accuracy. Indeed, in addition to acceleration values, this application analyzes the evolution of the signal to extract some key patterns. Thus in this case, manufacturing calibration should be sufficient. If default calibration is not available or is too mediocre, the direct identification (method D) could be envisaged for cube-shaped sensors, while multi-position is more adapted for other sensor shapes. Using accelerometers for estimating static or dynamic orientations requires a higher measurement accuracy. Each of the four methods summarized in

Table 4 could be selected, depending on desired accuracy and available time, equipment, and calculation tools. Finally, the use of IMUs for localization problems requires the highest accuracy over the measured acceleration. Indeed, acceleration must first be exploited to estimate orientation, and must then be integrated twice to get a position estimation. In this case, in-lab calibration using an accurate orientation device (method A) should be implemented. Some authors have studied human movements only in the sagittal or frontal plan [39]. In this case, segment orientation can be estimated from a uniaxial accelerometer. In this event, the angle error can be very large when the orientation is close to the vertical (Fig. 7). Thus the way to attach the accelerometer to the body segments should be selected with the aim of being close to the horizontal. However, when using a triaxial accelerometer its orientation relative to the body segments does not impact the angle error, which is always minimal. In this study, we tested five APDM Opal IMUs. Then we added two Cometa WaveTrack IMUs and two triaxial accelerometers (Actigraph and Delsys sensors) to assess inter-sensor brand variability among sensors that are intended to be used in human motion analysis. Finally, we added two FGP higher grade accelerometers to highlight their different calibration behavior arising from manufacturing irregularities and low cross-axis sensitivity. Thus our results are based on eleven sensors, which is not a huge number, but is sufficient to apply Friedman and Wilcoxon signed-rank tests. Adding more sensors to the study could improve statistical analysis, but it should be borne in mind that performing the seven calibration methods over one sensor is very time-consuming. Then, this paper does not integrate the temperature dependence of accelerometers. Temperature is known to affect measurement from an accelerometer [15] in a non-linear way. Therefore, the measurement models and calibration methods that are discussed in this paper can only be applied to experimentation performed at a steady temperature. To allow measurement when temperature is fluctuating, calibration parameter dependence with respect to the temperature should be identified. Thus measurement models and calibration methods become more complex and are not tackled in this paper. 6. Conclusions This study shows that in-lab calibration using a precise alignment installation offers the best accuracy. However, optimized calibration over a leveled surface is easier to implement and results in an only slightly less accurate acceleration which is acceptable for most of the biomechanical applications. This method nevertheless requires the design of a mounting frame when the sensor housing is not cube-shaped. To perform a calibration even faster, calibration parameters could be directly identified from the same scheme, but without an optimization process. This results in a less accurate measurement, even if the error over the estimated incline remains below 1°. Finally, the well-studied multi-position method gives equivalent results to the direct identification method, with no need of a controlled reference. However, in this case a robust optimization algorithm is required.

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

ARTICLE IN PRESS

JID: JJBE 10

[m5G;August 30, 2016;15:13]

A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

a

b

Fig. 8. (a) RMS of the acceleration error for different number of imposed orientations. (b) Zoom on the equivalent settings.

Conflict of interest

References

No conflict of interest. Acknowledgment This work has been sponsored by the French government research program “Investissements d’Avenir” through the Robotex Equipment of Excellence (ANR-10-EQPX-44). Authors would like to thank Biometrics France for lending IMUs. Appendix A Calculation of the angle uncertainty induced from a uniaxial accelerometer:

a α = arcsin

a ⇒ g da/g

sin (α ) = dα =



(27)

g da/g

da

 2 = 1 − sin α 2 = g · cos (α ) ( ) 1− a

(28)

g

Ua Uα = g.|cos (α )|

(29)

Calculation of the angle uncertainty induced from a triaxial accelerometers:

ax tan (α ) = ⇒ ay dα =

α

d ax /ay 1+

Uα =





 2 +

ax = arctan ay

−ax . day /ay 2

ax ay

ay . Ua g2

2

1+ +

 2



(30) =

ax ay

 −a . U 2 x a g2

ay . d ax −ax . day + g2 g2

=

Ua



ax 2 + ay 2 g2

=

Ua g

(31)

(32)

Appendix B Calibration over the motorized bench (method A) was tested by imposing different number of orientations over a complete rotation (from 2 orientations up to 36). Fig. 8 shows that imposing more than three orientations does not improve the resulting accuracy. However, two controlled orientations per axis are clearly not sufficient to perform an accurate calibration. Thus, our recommendation is to perform three independent orientations over a complete rotation.

[1] Godfrey A, Conway R, Meagher D, ÓLaighin G. Direct measurement of human movement by accelerometry. Med Eng Phys 2008;30:1364–86. doi:10.1016/j. medengphy.20 08.09.0 05. [2] Karantonis DM, Narayanan MR, Mathie M, Lovell NH, Celler BG. Implementation of a real-time human movement classifier using a triaxial accelerometer for ambulatory monitoring. IEEE Trans Inf Technol Biomed 2006;10:156–67. doi:10.1109/TITB.2005.856864. [3] Veltink PH, Bussmann HBJ, De Vries W, Martens WLJ, Van Lummel RC. Detection of static and dynamic activities using uniaxial accelerometers. IEEE Trans Rehabil Eng 1996;4:375–85. doi:10.1109/86.547939. [4] Li Q, Stankovic JA, Hanson MA, Barth AT, Lach J, Zhou G. Accurate, fast fall detection using gyroscopes and accelerometer-derived posture information. IEEE; 2009. p. 138–43. doi:10.1109/BSN.2009.46. [5] Gafurov D, Helkala K, Søndrol T. Biometric gait authentication using accelerometer sensor. J Comput 2006;1:51–9. [6] Yun Xiaoping, Bachmann ER, McGhee RB. A simplified quaternion-based algorithm for orientation estimation from earth gravity and magnetic field measurements. IEEE Trans Instrum Meas 2008;57:638–50. doi:10.1109/TIM.2007. 911646. [7] Lee JK, Park EJ. A fast quaternion-based orientation optimizer via virtual rotation for human motion tracking. IEEE Trans Biomed Eng 2009;56:1574–82. doi:10.1109/TBME.20 08.20 01285. [8] Kemp B, Janssen AJ, van der Kamp B. Body position can be monitored in 3D using miniature accelerometers and earth-magnetic field sensors. Electroencephalogr Clin Neurophysiol Mot Control 1998;109:484–8. [9] Bouten CV, Koekkoek K, Verduin M, Kodde R, Janssen JD. A triaxial accelerometer and portable data processing unit for the assessment of daily physical activity. Biomed Eng IEEE Trans 1997;44:136–47. [10] Luinge HJ, Veltink PH, Baten CTM. Ambulatory measurement of arm orientation. J Biomech 2007;40:78–85. doi:10.1016/j.jbiomech.2005.11.011. [11] Lefferts EJ, Markley FL, Shuster MD. Kalman filtering for spacecraft attitude estimation. J Guid Control Dyn 1982;5:417–29. doi:10.2514/3.56190. [12] El-Gohary M, Pearson S, McNames J, Mancini M, Horak F, Mellone S, et al. Continuous monitoring of turning in patients with movement disability. Sensors 2013;14:356–69. doi:10.3390/s140100356. [13] Al-Jawad A, Barlit A, Romanovas M, Traechtler M, Manoli Y. The use of an orientation Kalman filter for the static postural sway analysis. APCBEE Procedia 2013;7:93–102. doi:10.1016/j.apcbee.2013.08.018. [14] El-Gohary M, McNames J. Human joint angle estimation with inertial sensors and validation with a robot arm. IEEE Trans Biomed Eng 2015 pp. 1–1. doi:10. 1109/TBME.2015.2403368. [15] Sabatini AM. Quaternion-based extended Kalman filter for determining orientation by inertial and magnetic sensing. IEEE Trans Biomed Eng 2006;53:1346– 56. doi:10.1109/TBME.2006.875664. [16] Zhang Y, Chen K, Yi J, Liu L. Pose estimation in physical human-machine interactions with application to bicycle riding. Int Conf Intell Robots Syst IROS 2014:3333–8 2014. [17] Schepers HM, Roetenberg D, Veltink PH. Ambulatory human motion tracking by fusion of inertial and magnetic sensing with adaptive actuation. Med Biol Eng Comput 2010;48:27–37. doi:10.10 07/s11517-0 09-0562-9. [18] Woodman O, Harle R. Pedestrian localisation for indoor environments. In: Proc. 10th Int. Conf. Ubiquitous Comput. ACM; 2008. p. 114–23. [19] Woodman OJ. An introduction to inertial navigation. Tech Rep UCAMCL-TR-696, 14. University of Cambridge, Computer Laboratory; 2007. p. 15. [20] Lötters JC, Schipper J, Veltink PH, Olthuis W, Bergveld P. Procedure for in-use calibration of triaxial accelerometers in medical applications. Sens Actuators Phys 1998;68:221–8. doi:10.1016/S0924-4247(98)0 0 049-1.

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004

JID: JJBE

ARTICLE IN PRESS A. Nez et al. / Medical Engineering and Physics 000 (2016) 1–11

[21] Syed ZF, Aggarwal P, Goodall C, Niu X, El-Sheimy N. A new multi-position calibration method for MEMS inertial navigation systems. Meas Sci Technol 2007;18:1897–907. doi:10.1088/0957-0233/18/7/016. [22] Cai Q, Song N, Yang G, Liu Y. Accelerometer calibration with nonlinear scale factor based on multi-position observation. Meas Sci Technol 2013;24:105002. doi:10.1088/0957-0233/24/10/105002. [23] Frosio I, Pedersini F, Borghese NA. Autocalibration of MEMS accelerometers. IEEE Trans Instrum Meas 2009;58:2034–41. doi:10.1109/TIM.2008.2006137. [24] Zhang H, Wu Y, Wu W, Wu M, Hu X. Improved multi-position calibration for inertial measurement units. Meas Sci Technol 2010;21:015107. doi:10.1088/ 0957-0233/21/1/015107. [25] Shin E-H, El-Sheimy N. A new calibration method for strapdown inertial navigation systems. Z Für Vermess 2002;127:41–50. [26] Skog I, Händel P. Calibration of a MEMS inertial measurement unit. In: XVII IMEKO World Congr.; 2006. p. 1–6. [27] Bono RW, Seller EJ. The effect of high transverse inputs on accelerometer calibration. Cal Lab Int J Metrol 2011;18:31. [28] Liu Y, Li X, Zhang X, Feng Y. Novel calibration algorithm for a three-axis strapdown magnetometer. Sensors 2014;14:8485–504. doi:10.3390/s140508485. [29] Chatfield AB. Fundamentals of high accuracy inertial navigation. American Institute of Aeronautics and Astronautics; 1997. [30] Titterton D, Weston JL. Strapdown inertial navigation technology, vol. 17. 2nd edition. The Institution of Engineering and Technology; 2005. [31] Giansanti D, Maccioni G, Macellari V. Guidelines for calibration and drift compensation of a wearable device with rate-gyroscopes and accelerometers. In: Eng. Med. Biol. Soc. 2007 EMBS 2007 29th Annu. Int. Conf. IEEE. IEEE; 2007. p. 2342–5.

[m5G;August 30, 2016;15:13] 11

[32] Zhang H, Wu Y, Wu M, Hu X, Zha Y. A multi-position calibration algorithm for inertial measurement units. In: Proc. AIAA Guid. Navig. Control Conf. Exhib. Honol. HI USA; 2008. p. 18–21. [33] Hung JC, Thacher JR, White HV. Calibration of accelerometer triad of an IMU with drifting Z-accelerometer bias. In: Aerosp. Electron. Conf. 1989 NAECON 1989 Proc. IEEE 1989 Natl., vol. 1; 1989. p. 153–8. doi:10.1109/NAECON.1989. 40206. [34] Gebre-Egziabher D, Elkaim GH, David Powell J, Parkinson BW. Calibration of strapdown magnetometers in magnetic field domain. J Aerosp Eng 2006;19:87–102. [35] Vasconcelos JF, Elkaim G, Silvestre C, Oliveira P, Cardeira B. Geometric approach to strapdown magnetometer calibration in sensor frame. Aerosp Electron Syst IEEE Trans 2011;47:1293–306. [36] Fisher CJ. Using an accelerometer for inclination sensing. Appl Note 1057 Analog Devices; 2010. [37] Tilt measurement using a low-g 3-axis accelerometer. Appl Note 3182 STMicroelectronics 2010. [38] Mineta T, Kobayashi S, Watanabe Y, Kanauchi S, Nakagawa I, Suganuma E, et al. Three-axis capacitive accelerometer with uniform axial sensitivities. J Micromech Microeng 1996;6:431. [39] Tajdinan S, Afshari D, Mohammadi A, Tabesh H, others. Validation of triaxial accelerometer to continuous monitoring of back posture at sagittal and frontal planes in workplaces. Jundishapur J Health Sci 2014;6:319–26.

Please cite this article as: A. Nez et al., Comparison of calibration methods for accelerometers used in human motion analysis, Medical Engineering and Physics (2016), http://dx.doi.org/10.1016/j.medengphy.2016.08.004