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`Dependence of inertial measurements of distance on accelerometer noise,”
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`Article  in  Measurement Science and Technology · July 2002
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`DOI: 10.1088/0957-0233/13/8/301
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`APPLE 1045
`
`1
`
`

`

`INSTITUTE OF PHYSICS PUBLISHING
`
`Meas. Sci. Technol. 13 (2002) 1163–1172
`
`MEASUREMENT SCIENCE AND TECHNOLOGY
`
`PII: S0957-0233(02)35079-3
`
`Dependence of inertial measurements of
`distance on accelerometer noise
`
`Y K Thong, M S Woolfson1, J A Crowe, B R Hayes-Gill and
`R E Challis
`
`School of Electrical and Electronic Engineering, University of Nottingham,
`Nottingham NG7 2RD, UK
`
`E-mail: Malcolm.Woolfson@Nottingham.ac.uk
`
`Received 20 March 2002, in final form 16 May 2002, accepted for
`publication 24 May 2002
`Published 1 July 2002
`Online at stacks.iop.org/MST/13/1163
`
`Abstract
`An investigation is made into the errors in estimated position that are caused
`by noise and drift effects in stationary accelerometers. An analytical study is
`made into the effects of biases in the accelerometer data and the effects of
`changing the cut-off frequency in the anti-aliasing filter. The root mean
`square errors in position are calculated as a function of time and sampling
`frequency. A comparison is made between the theoretical results and
`experimental data taken from two commercial accelerometers.
`Recommendations are made regarding the calibration of accelerometers
`prior to their use in practical situations.
`
`Keywords: accelerometers, noise, micro-electro-mechanical systems, inertial
`navigation systems
`
`1. Introduction
`
`Accelerometers are widely used in many applications to
`determine position. These devices can be used either on their
`own or in combination with other navigation equipment, for
`example gyroscopes [1, 2] or velocity meters [3]. Application
`areas are numerous varying from measurement of forces on a
`car that is turning or accelerating [4] to the ‘smart pen’ which
`can store what it writes for the future [5]. Another application
`is the investigation of structures under impact load [6]. The
`authors have been looking, in particular, at the application of an
`accelerometer-only inertial navigation system (INS) to various
`desktop applications, for example its use as a computer mouse.
`The basic principle of the accelerometer as an inertial
`sensor is very straightforward:
`the accelerometer measures
`acceleration and displacement
`is determined by double
`integrating the data. The integration could be carried out using
`analogue methods [7, 8] or it could be performed numerically
`after the data have been digitized [9].
`However, there is the problem of measurement noise and
`drift [10–12]. It is shown in [1] that the standard deviation of
`the measured position due to acceleration noise, in the absence
`of drift and initialization errors, increases as t 1.5 where t is the
`
`1 Author to whom any correspondence should be addressed.
`
`integration time. This result is derived by using the continuous
`Kalman filter. In [13], it is suggested that the standard deviation
`of the error in position increases as t. In [14], it is assumed
`that if ε represents the accelerometer error, then the measured
`position would have an error that is proportional to εt 2. This
`last assumption would only be true if the error concerned were a
`bias rather than white noise. What this prior work demonstrates
`is a lack of consensus regarding how noise affects the rms errors
`in the estimated displacement.
`The accelerometer data has already been filtered by an
`in-built anti-aliasing filter.
`It is found that further filtering
`reduces the absolute value of the error in position, but there
`is still a tendency for the variation in the positional error to
`increase with time. Another problem with this additional
`filtering of the input to the accelerometer is that one would
`be reducing the bandwidth of measurable accelerations. As
`examples, acceleration data taken from two commercially
`available accelerometers are shown in figures 1(a) and (b).
`The accelerometers are both at rest on an optical bench. The
`sampling frequency is 3 kHz. For each accelerometer, the data
`have been filtered using a moving average of 1000 samples so
`that the effects of drift can be brought out. It can be seen that
`there is noise and drift in the data, for both accelerometers,
`which will contribute to errors in the estimated position.
`
`0957-0233/02/081163+10$30.00 © 2002 IOP Publishing Ltd Printed in the UK
`
`1163
`
`2
`
`

`

`(a)
`
`0
`
`200
`
`400
`
`600
`
`800
`
`1000
`
`1200
`
`1400
`
`1600
`
`1800
`
`Time (seconds)
`
`(b)
`
`01234
`
`-1
`
`-2
`
`-3
`
`-4
`
`01234
`
`-1
`
`-2
`
`-3
`
`-4
`
`milli g
`
`milli g
`
`0
`
`200
`
`400
`
`600
`
`800
`
`1000
`
`1200
`
`1400
`
`1600
`
`1800
`
`2000
`
`Time (seconds)
`
`Figure 1. Output signals from (a) ADXL250 and (b) Crossbow
`CXL01F3 accelerometers.
`
`It can be seen from equation (2) that
`(i) RMS(s(T )) varies as T 2 and
`(ii) for a particular time, RMS(s(T )) for a dc signal is
`independent of the sampling frequency.
`
`2.3. Rms error in double integration of coloured noise from a
`stationary accelerometer
`In this section, an expression for the rms errors in position as a
`function of time, arising from double integration, are derived
`for the two cases of white noise and coloured noise formed by
`passing white noise through a single-pole filter.
`Let the data segment be T seconds long, the number of
`samples be N and the sampling frequency be fs Hz.
`Now the time between samples is 1/fs seconds, hence, for
`N samples,
`T = N
`fs
`Let σd be the standard deviation of noise for each data
`point. We model
`the data as coloured noise with no
`‘signal’ component, i.e. an absolutely stationary horizontal
`accelerometer.
`Let {a[n]} represent the noisy acceleration measurements,
`with a[n] signifying the acceleration at sample point n. It is
`assumed that the data are stationary. In this case, we can define
`the pth lag of the autocorrelation function as
`r[p] = E[a[i] · a[i + p]]
`where E[·] signifies expectation.
`
`.
`
`(3)
`
`(4)
`
`Y K Thong et al
`
`For the ADXL250 [15] accelerometer, figure 1(a), the main
`contribution to the output from the accelerometer is noise
`with a relatively small amount of drift. For the Crossbow
`CXL01F3 accelerometer [16], figure 1(b), there is less noise
`than for the ADXL but the contribution from drift effects
`is more significant. The question to be asked is how the
`aforementioned increase in positional error depends on the
`parameters of the accelerometer, the sampling frequency, the
`filter parameters and the level of noise.
`The aim of this paper is to provide a theoretical study
`of the errors caused in the measurements of position by the
`noise in the accelerometer. This investigation would be of
`use in deciding whether to use a particular accelerometer in
`a particular application. We shall be using as our model
`a stationary accelerometer so that any errors are due to
`the noise and not due to any contributions from motion of
`the accelerometer.
`In this way, the errors in the estimated
`displacement arising from the double integration of noise
`are isolated from the corresponding errors from specific
`acceleration signals.
`Firstly, an expression will be derived for the rms errors as
`a function of time, for the case of ideal double integration of
`coloured noise. The specific cases of white noise and filtering
`using an analogue single-pole filter are discussed subsequently.
`In particular, we address the following two questions.
`(i) Given a set value of the sampling frequency, how do
`the root mean square (rms) errors in position vary with
`integration time?
`(ii) Given a set value of the integration time and bandwidth,
`how do the rms errors vary with sampling frequency?
`The theoretical work will be assessed by comparison with
`the analysis of experimental data taken for an accelerometer
`on an optical bench, where intrinsic vibration amplitude
`is minimal and well below the noise amplitude of the
`accelerometer that is used.
`
`2. Theory
`
`2.1. Introduction
`
`A theoretical analysis is now made of the dependence with
`integration time and sampling frequency of the rms error of the
`measured displacement from accelerometer measurements. In
`section 2.2, the case when the measurements are represented
`by a dc bias is described. In section 2.3, the double integration
`of coloured noise is first described and the particular cases of
`white noise and noise filtered by a single-pole filter will be
`analysed.
`
`2.2. Double integration of a dc bias
`
`We first look at the case where a dc signal is being double
`integrated.
`If the acceleration is a constant, A, then the
`estimated displacement assuming zero initial displacement and
`velocity is given by
`
`s(T ) = 1
`2 AT 2.
`As A is constant, then the rms value of A is equal to A and the
`rms value of displacement is hence given by
`RMS(s(T )) = 1
`2 AT 2.
`
`(1)
`
`(2)
`
`1164
`
`3
`
`

`

`Dependence of inertial measurements of distance on accelerometer noise
`
`Substituting for C[0], C[1], . . . , C[N − 1] into equation (11)
`the following expression can be derived for the mean square
`value of correlated noise:
`E[(DN )2] = R1 + R2
`
`(16)
`
`where
`
`and
`
`N
`
`R2 = 2
`N 2
`
`r[j](N − j ).
`
`(17)
`
`(18)
`
`R1 = r[0]
`N−1(cid:1)
`j=1
`R1 is the variance of the acceleration in the absence of
`correlations between samples which would be E[(DN )2] for
`white noise. R2 is the contribution from the correlations
`between samples of the noise.
`In the derivation of equation (16), the effects of bias, for
`example from the acceleration due to gravity or dc offset, have
`been ignored.
`
`2.3.1. White noise. For the white noise model, it is assumed
`that the noise is uncorrelated from sample value to sample
`value. In this case R2 in equation (16) is zero. Hence, from
`equations (16) and (17),
`E[(DN )2] = r[0]
`
`(19)
`
`.
`
`Now the dc value of the acceleration at the Nth sample
`point is given by an average of the noise values over the first
`N sample points:
`
`DN = 1
`
`N
`
`N(cid:1)
`i=1
`
`a[i].
`
`(5)
`
`This summation approaches zero as the number of samples
`becomes infinite:
`
`N→∞ DN = 0.
`
`lim
`
`(6)
`
`However, for a finite number of samples, DN will in general
`be non-zero.
`It is now assumed that the underlying acceleration is a
`constant over the whole data interval. The average of the
`acceleration measurements over N samples, equation (5), is
`taken as the estimate of this constant acceleration. Thus, the
`displacement at time T can be determined from the following
`analytical double integration:
`s(T ) =
`
`DN dt
`
`(cid:3)
`
`dt = 1
`2 DN T 2
`
`(7)
`
`(cid:2)
`
`T
`
`(cid:2)
`
`t
`
`0
`
`0
`
`where it is assumed that DN is a constant over the time
`interval T and that the initial velocity and displacement of the
`accelerometer are zero, consistent with the assumption that the
`accelerometer is stationary. It should be pointed out that, in
`practice, the data would be numerically integrated from sample
`to sample. Hence, equation (7) is a simplified approximation
`to the estimate of displacement found in practice.
`As the accelerometer is stationary, s(T ) in equation (7)
`can be considered to be an error in the measured displacement.
`Now DN will depend on the particular sequence of noise
`values up to time T . Taking rms values of both sides of
`equation (7),
`
`RMS(s(T )) = 1
`2 T 2RMS(DN ).
`
`(8)
`
`From equation (5), the expectation value of the square of the
`mean of the acceleration is given by
`E[(DN )2] = 1N 2 E[(a[1] + a[2] + ··· + a[N])
`
`× (a[1] + a[2] + ···a [N])].
`Expanding the brackets, using the symmetry condition
`E[a[i] · a[j]] = E[a[j] · a[i]]
`
`(10)
`
`(9)
`
`and using the stationary property, equation (4), we may rewrite
`equation (9) as
`
`E[(DN )2] = C[0] + C[1] + C[2] + C[3] + ··· + C[N − 1]
`
`N 2
`
`where
`
`C[0] = N r[0]
`C[1] = 2(N − 1)r[1]
`C[2] = 2(N − 2)r[2] . . .
`C[N − 1] = 2r[N − 1].
`
`N
`Let σd be the standard deviation of the noise, so that
`r[0] = σ 2
`d .
`Taking square roots of both sides of equation (19) and
`substituting for r[0] from equation (20),
`RMS(DN ) = σd√
`
`(20)
`
`(21)
`
`.
`
`N
`
`Substituting for RMS(DN ) from equation (21) into equa-
`tion (8), we obtain the following expression for the rms errors
`in position:
`RMS(s(T )) = 1
`T 2 σd√
`2
`N
`Substituting for N from equation (3) above,
`= 1
`RMS(s(T )) = 1
`T 2 σd√
`σd√
`2
`2
`Tfs
`fs
`
`.
`
`(22)
`
`T 1.5.
`
`(23)
`
`the rms error in
`
`Hence, for a fixed sampling frequency,
`estimated displacement increases as T 1.5.
`It is also of interest to investigate the effect of increasing
`the sampling frequency on RMS(s(T )) keeping the integration
`time, T , constant. Intuitively, we would expect RMS(s(T ))
`to go to zero, as we are averaging over more samples N
`(see equation (21)); note that this is a consideration only for
`discrete, rather than continuous, processes.
`From equation (23),
`RMS(s(T )) = C√
`(24)
`where C = 0.5σd T 1.5 is, in this case, a constant. It should
`be noted that equation (24) is appropriate for the simplified
`
`fs
`
`1165
`
`(11)
`
`(12)
`
`(13)
`
`(14)
`
`(15)
`
`4
`
`

`

`where ωc = 2πfc is the 3 dB cut-off frequency in rad s
`−1. This
`type of filter is built into the two accelerometers under study.
`Using Parseval’s theorem, if noise with power spectral
`−4 Hz
`−1 is input to the single-pole filter with
`density 1
`2 σ 2c cm2 s
`
`frequency response given by equation (28), then the energy of
`(cid:2) ∞
`(cid:2) ∞
`the output signal is given by
`Eout = σ 2
`|H (jω)|2 dω = σ 2
`c
`2π
`
`0
`
`1 π
`
`c2
`
`Hence
`
`ω2
`c
`c + ω2 dω.
`ω2
`(29)
`
`0
`
`(cid:6)(cid:7)∞
`
`0
`
`ω ω
`
`c
`
`(cid:4)
`
`(cid:5)
`
`−1
`
`1 ω
`
`c
`
`c
`
`.
`
`c
`
`= σ 2
`Eout = σ 2
`c ω2
`ωc
`(30)
`tan
`2π
`4
`Substituting ωc = 2πfc into equation (30) and taking square
`roots of both sides of this equation, it can be shown that the
`rms value, σf , of the filtered noise is given by
`(cid:7)0.5
`(cid:3)
`σf =
`Eout =
`
`(cid:4)
`
`πfc
`2
`
`σc.
`
`(31)
`
`(cid:4)
`
`In the appendix, it is shown that the rms error in displacement
`using the filter model in equation (28) is given by
`RMS(s(T )) = T 2
`ασ 2
`c fs
`2N 2
`2
`−2α) − e
`2 (1 − e
`−α + e
`× N
`(1 − e−α)2
`where N = Tfs is the number of samples processed and
`α = 2πfc
`fs
`
`(cid:9)(cid:7)0.5
`
`−(N+1)α
`
`.
`
`(32)
`
`(33)
`
`It should be noted that a similar analysis can be made by using
`the continuous-time version of equations (16)–(18):
`E[d(T )2] = 2
`(T − t )r(t ) dt
`T 2
`
`T
`
`0
`
`(cid:2)
`
`where r(t ) is given by equation (A.4) and d(T ) is the time-
`averaged filtered accelerometer signal, analogous to DN in
`equation (16). Further details are contained in [19].
`It is of interest to investigate the time dependence of the
`rms errors for doubly integrated filtered noise for small and
`large integration times.
`
`(34)
`
`Small time approximation. For small enough α, the following
`approximations can be made:
`−α ≈ 1 − α +
`α2
`e
`2
`−2α ≈ 1 − 2α + 2α2.
`(35)
`e
`These approximations would be valid for fc (cid:6) fs. Let the
`number of samples N be small enough so that the last two
`terms in the numerator of equation (32) cannot be neglected.
`In addition, we make the approximation valid for small
`enough α and N:
`−(N+1)α = 1 − (N + 1)α +
`
`e
`
`(N + 1)2
`2
`
`α2.
`
`(36)
`
`Y K Thong et al
`
`model used in equation (7). This result will be tested later
`when experimental data are analysed.
`Hence, equation (24) predicts that if we keep T constant
`but change fs then RMS(s(T )) is proportional to the square
`root of the inverse of the sampling frequency.
`To summarize, for the case of white noise,
`
`(i) for a particular sampling frequency, RMS(s(T )) varies
`with integration time as T 1.5;
`(ii) for a particular integration time, RMS(s(T )) varies as the
`square root of the inverse of the sampling frequency.
`
`Equation (23) is in good agreement with the result in [1],
`where it is shown that, in the absence of initialization and
`drift errors,
`the rms error in estimated position from an
`accelerometer is given by
`RMS(s(T )) = 1√
`3
`
`(cid:3)
`RvT 1.5
`
`(25)
`
`where Rv is the variance of continuous noise. This expression
`has been derived using state-space analysis. In [17], it is shown
`that Rv is related to the variance, σ 2
`d , of discrete noise by
`Rv = σ 2
`
`.
`
`d
`fs
`
`(26)
`
`(27)
`
`T 1.5.
`
`Substituting for Rv from equation (26) into equation (25), we
`find that
`RMS(s(T )) = 1√
`σd√
`3
`fs
`Apart from the constant factor pre-multiplying the expressions,
`equations (23) and (27) are in agreement with each other.
`Equation (27) can also be derived by considering the
`double integration of acceleration as an integrated Wiener
`process [18]. In the approach used in [1], double integration
`is carried out continuously up to the time of interest. In the
`simplified approach used in this paper, the rms acceleration
`at
`the time point of interest
`is found first.
`Then, an
`analytical double integration is carried out, assuming that this
`acceleration is a constant over the interval of integration, to
`obtain an estimate of the displacement. Unlike the approach
`in [1], this latter analysis is retrospective in nature leading to
`a different constant prefactor in equations (23) and (27).
`The advantage of the analysis presented in this section is
`that it is easier to understand physically the factors that have
`lead to the dependence of the rms error in displacement on both
`the sampling frequency, fs, and time, T , in equation (23).
`
`The
`Noise filtered with a single-pole filter.
`2.3.2.
`accelerometer data will,
`in practice, be filtered prior to
`processing. An anti-aliasing filter will have a finite cut-off
`frequency and, even after conversion to digital form, it may
`be required to filter the digital signal further prior to double
`integration.
`Equations (16)–(18) apply to the general case where no
`particular filter is specified.
`In this discussion, we model
`the anti-aliasing filter as a single-pole filter, with frequency
`response
`H (jω) = ωc
`ωc + jω
`
`(28)
`
`1166
`
`5
`
`

`

`Dependence of inertial measurements of distance on accelerometer noise
`
`the variance of the mean value of the acceleration, E[(DN )2],
`in equation (16): R1 from the variance of each acceleration
`data value and R2 from the cross-correlation between the
`acceleration values of different sample points. As N increases
`in equation (18), then cross-correlations between pairs of
`samples more distantly positioned in time will contribute
`the more distantly
`to R2. According to equation (38),
`related in time are two samples, the less will be the cross-
`correlation value. Hence, as N increases, the contributions
`from strongly correlated sample pairs to R2 will become less
`significant and eventually R2 (cid:6) R1 which is the case for white
`noise. Therefore, as N, and hence T , increases, the filtered
`acceleration data can be approximated as white noise, and the
`dependence of the rms error in position estimate on T and fs
`would be as for white noise.
`
`General case. Between the limits of small and large time,
`the dependence of the rms position error for accelerometers
`on time will be more complicated than a simple power law, as
`indicated by equations (37) and (42). However, we may define
`a ‘local power law’ as follows. Define the rms error in s(T ) at
`time T as RMS(s(T )). We write this as
`RMS(s(T )) = A(T )T p(T )
`
`(43)
`
`where A(T ) is a function of time and p(T ) is a time varying
`index. For the two cases of uncorrelated white noise and dc
`bias p(T ) would be a constant at 1.5 and 2 respectively.
`The variation of p(T ) with T has been calculated for filter
`cut-off frequencies of 50, 200 and 500 Hz, and a sampling
`frequency of 3 kHz. The log to base 10 of RMS(s(T )) in
`equation (32) is computed as a function of the log to base 10
`of integration time and the local slope is computed between
`adjacent samples; this slope is p(T ) in equation (43) above.
`The results are shown in figure 2.
`It can be seen that, for
`each cut-off frequency, the p-index starts off at a value of 2
`and decreases monotonically with time to 1.5 as the effects of
`correlation become less, and the value for p(T ) approximates
`that for white noise. As expected, the smaller the cut-off
`frequency, the more the correlation between adjacent samples,
`and hence the slower is the decrease with time of p(T ) from 2
`to 1.5.
`
`3. Analysis of experimental data
`
`For the purpose of examining the application of the above
`theory, we have used two different accelerometers, which are
`(1) an Analog Devices ADXL250 [15] and (2) a Crossbow
`CXL01F3 [16]. The ADXL250 has a noise density, σc, rated
`−0.5. This value is signficantly higher
`at around 500 µg Hz
`than the corresponding value for the CXL01F3, which is
`−0.5. It will be of interest to investigate how this
`100 µg Hz
`difference in noise densities for these two accelerometers is
`reflected in the differences in the rms position errors, as a
`function of time and sampling frequency.
`The filtered noise model, equation (32), will be used as a
`comparison with experiment. Any discrepancies between this
`model and the experimental data will also be discussed.
`
`1167
`
`If we use the approximations (34)–(36) in equation (32),
`substitute for α from equation (33) and use equation (31) it
`can be shown that for small integration times
`RMS(s(T )) = 1
`2 σf T 2.
`
`(37)
`
`Hence, for small enough times, the rms error in measured
`position is proportional to T 2 which is similar to the case for
`the rms position error for a double integrated bias, equation (2).
`In addition, in this limit, the rms error is independent of the
`sampling frequency used.
`This dependence on T can be explained as follows. For
`filtered noise, the nth autocorrelation lag, r[n], is given by
`equation (A.6) in the appendix:
`r[n] = πfcσ 2
`2
`
`(cid:5)
`
`c
`
`exp
`
`(cid:6)
`
`.
`
`− 2πfcn
`fs
`
`(38)
`
`Now, r[n] decreases to zero as n increases. However, if N in
`equation (18) is small enough, then most of the correlation lags
`r[n] will be significant and of comparable magnitude: r[0] ≈
`r[1] ≈ r[2] ≈ ··· ≈ r[N − 1] and E[(DN )2] in equation (16)
`would approximate the result for a dc bias. Hence, in the limit
`of small integration time, the time dependence of the rms error
`in position is the same as for a dc bias. The difference between
`the cases of coloured noise and dc bias is that the rms error
`for the latter will be independent of the cut-off frequency, fc.
`Coloured noise is relatively broadband and the associated rms
`error will reduce with fc as indicated in equations (31) and (37).
`
`Large time approximation. Now let us look at the limit of
`large time, where N is large enough that in the numerator of
`equation (32) the following approximation can be made:
`−2α) (cid:7) e
`−α − e
`(1 − e
`−(N+1)α.
`
`(39)
`
`N 2
`
`Using this approximation, substituting for α from equation (33)
`and N from equation (3), and substituting for σf from
`equation (31), it can be shown that
`RMS(s(T )) = 1
`2
`
`(cid:8)
`
`T 1.5σf
`f 0.5
`s
`
`(cid:9)
`
`.
`
`(40)
`
`(1 − e
`−2α)0.5
`1 − e−α
`The theoretical limit to fc allowed in the context of the
`sampling theorem is given by
`fc = fs
`2
`
`(41)
`
`although in practice fc would have to be less than this value.
`In this case, α in equation (33) is given by π. Making the
`−π ≈ 0, equation (40) simplifies to
`approximation e
`RMS(s(T )) = 1
`σf√
`2
`fs
`
`T 1.5.
`
`(42)
`
`Comparing this with equation (23), we can see that for large
`times, the rms errors in displacement for coloured noise have
`the same relation with T and fs as for white noise.
`In this
`case, the rms value of the coloured noise, σf , replaces the
`white noise standard deviation, σd. This can be explained with
`equations (16)–(18). There are two types of contribution to
`
`6
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`Y K Thong et al
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`0
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`p-index
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`Time (seconds)
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`Time (seconds)
`
`Figure 2. p-index versus integration time for filter cut-off
`frequencies of 500 Hz (full curve), 200 Hz (dashed curve) and 50 Hz
`(dotted curve).
`
`3.1. Rms errors versus time for accelerometers
`
`Measurements were made with the accelerometer at rest on
`an optical bench.
`In order to perform Monte Carlo type
`estimations of the rms errors as a function of time and sampling
`frequency, we have recorded 30 min of data at a sampling
`frequency of 3 kHz, resulting in 5.4 × 106 samples of data.
`Now suppose that we wish to compute the rms error after
`a time corresponding to N samples. To do this, the data set
`is divided up into M blocks of N samples. Within each block
`of data, double integration is carried out using the trapezoidal
`rule, and the square of the error (taking the actual position as
`zero) at the Nth or last sample in each block is computed.
`Suppose that the square of the error in position at the Nth
`sample in the kth block is given by ε2(k, N ). The mean square
`error at the Nth sample point is computed by averaging this
`value over all blocks as follows:
`ε2(N ) = 1
`
`ε2(k, N ).
`
`(44)
`
`M(cid:1)
`k=1
`
`M
`
`The rms error in position at sample point N is then estimated as
`[ε2(N )]0.5. This figure of merit is computed for block lengths
`of N = 100 200 300 up to 10 000 samples corresponding to
`block time increments of 0.0333 s.
`The output of the analogue to digital converter has a
`nominal zero level of 2.5 V. However, it has been found, in
`practice, that the dc value is not exactly equal to 2.5 V and
`subtracting this value from the whole data segment could result
`in a bias that would significantly affect the doubly integrated
`results. To avoid this, the mean value over each block of N
`samples is computed and is subtracted off each sample within
`that block.
`The 3 dB frequency for the in-built low pass filter for
`the Crossbow is set equal to 125 Hz and this parameter is set
`initially to 50 Hz for the ADXL.
`To compare experimental data with the theory, we need
`to know σc in equation (32), which is the spectral density for
`unfiltered continuous noise. The procedure that is adopted is
`as follows. Let the experimentally estimated rms position in
`displacement at time T be RMS(s(T )). From equation (32),
`
`1168
`
`fs
`
`(cid:11)0.5
`
`(45)
`
`Figure 3. Rms position error versus integration time for ADXL
`with filter cut-off of 50 Hz and sampling frequency of 3000 Hz. Full
`curve, theory for analogue filter; dashed curve, experimental data.
`an estimate of σc can be found from
`√
`2fs (1 − e
`−α)RMS(s(T ))
`σc =
`2
`2 (1 − e−2α) − e−α + e−(N+1)α
`T (2πfc)0.5 N
`with α = 2πfc
`.
`For both accelerometers, fs = 3 kHz. The value of
`RMS(s(T )) is taken at T = 3 s. Note that N = Tfs = 9000.
`For the ADXL accelerometer, fc = 50 Hz, and it is found
`that RMS(s(3)) = 0.613 cm. Substituting for RMS(s(3)) into
`equation (45), it is found that σc = 0.334 cm s
`−2 Hz
`−0.5. The
`−0.5,
`corresponding value given in the data sheets is 500 µg Hz
`−2 Hz
`−0.5.
`which is 0.491 cm s
`For the Crossbow accelerometer, fc = 125 Hz and
`it is found that RMS(s(3)) = 0.111 cm.
`Substituting
`is found that σc =
`these values into equation (45),
`it
`−2 Hz
`−0.5. The corresponding value given in the
`0.0603 cm s
`−0.5 which is 0.0981 cm s
`−2 Hz
`−0.5.
`data sheets is 100 µg Hz
`It should be noted that this estimation procedure for σc
`effectively fits the model in equation (32) to the data at T = 3 s.
`The test of the suitability of this model is whether it has a good
`fit to the experimental data for all integration times.
`
`the theoretical rms
`In figure 3,
`ADXL accelerometer.
`estimation errors in position are plotted as a full curve and
`the experimental rms values are plotted as a dashed curve.
`There is excellent agreement between theory and experiment.
`The slope of the theoretical log:log plot, computed between
`integration times 0.033 and 3.333 s, is found to be 1.512
`and the corresponding value for the experimental curve is
`1.596, which is 5.6% larger. The slopes of the theoretical
`and experimental curves are slightly above 1.5 because of
`the effects of correlation between acceleration data points due
`to the filtering, (section 2.3.2). In addition, drift effects, for
`example due to temperature, may be another significant factor
`contributing to the index for the experimental curve being
`above 1.5.
`
`Crossbow accelerometer. The experimental and theoretical
`rms errors in position as a function of time are shown in
`figure 4(a). The filtered noise model, equation (32), has been
`used to derive the theoretical curve.
`
`7
`
`

`

`Dependence of inertial measurements of distance on accelerometer noise
`
`3.2. Rms errors at a particular time as a function of sampling
`frequency
`
`Next we investigate the variation of rms error in position on
`the sampling frequency for a fixed integration time. The basic
`sampling frequency used is 3 kHz. The data are divided up
`into M = 1000 blocks of N = 5040 samples, each block
`corresponding to a duration of 1.68 s. The rms error after
`1.68 s is achieved by double integrating the data over each
`block and working out the rms error according to equation (44)
`with N = 5040.
`Lower sampling frequencies are simulated by decimation
`of the acceleration data in the double integration process. In
`this way, rms errors are computed over an integration time of
`1.68 s for sampling frequencies of 3, 1.5, 1 kHz, 750, 600, 500,
`428.6, 375, 333.3 and 300 Hz.
`These results at different sampling frequencies will be
`compared with the dc bias model equation (2) and the filtered
`noise model, equation (32).
`
`For the ADXL accelerometer, data
`ADXL accelerometer.
`have been taken for the following cut-off frequencies for the
`accelerometers’ built-in low pass filter: 500, 200 and 50 Hz.
`The variation of rms error in position at time 1.68 s as
`a function of sampling frequency is shown in figure 5 for
`the three cut-off frequencies used. The experimental results
`are shown as dashed curves; the theoretical predictions as full
`curves.
`Looking at the experimental curves, it can be seen that
`the smaller the cut-off frequency of the filter, the less the rms
`errors in position vary with sampling frequency. This can be
`explained qualitatively, by noting that the lower the cut-off
`frequency, then the more significant will be the correlations
`between the samples of noise. In this case, the data segment
`will behave more like a dc bias where the rms errors are
`independent of the sampling frequency, and the theoretical
`slope of the log:log plot of rms error versus sampling frequency
`will be zero. On the other hand, as one increases the cut-off
`frequency, then the filtered data segment will behave more
`like pure white noise and the rms errors will have a variation
`with sampling frequency as described in equation (24). The
`corresponding slope of the log:log plot would then be −0.5.
`Experimentally, the slopes of the log:log plots are −0.502
`for a cut-off frequency (fc) of 500 Hz, −0.36 for fc = 200 Hz
`and −0.07 for fc = 50 Hz, illustrating the trend from white
`noise to bias behaviour as the cut-off frequency is decreased.
`The results in figure 5 demonstrate that the model in
`equation (32) is in good agreement with the experimental data,
`particularly for sampling frequencies larger than 1 kHz.
`In practice only sampling frequencies greater than the
`Nyquist frequency, taken as 2fc, would be used.
`In this
`case, it can be seen that for frequencies greater than the
`Nyquist frequency, the errors decrease relatively slowly with
`increasing sampling frequency. Hence, beyond a certain
`sampling frequency there is little benefit, from the point of
`view of rms error in noise, in increasing this value further.
`
`Crossbow accelerometer. For the Crossbow accelerometer,
`data are taken for a cut-off frequency of 125 Hz only, as this
`value is fixed for this particular device.
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`1169
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`RMS Error in Position (cm)
`
`Figure 4. (a) Rms position error versus integration time for
`Crossbow with filter cut-off of 125 Hz and sampling frequency of
`3000 Hz. Full curve, theory for analogue filter; dashed curve,
`experimental data. (b) Rms position error versus integration time for
`Crossbow with filter cut-off of 125 Hz and sampling frequency of
`3000 Hz. Full curve, theory for bias; dashed curve, experimental
`data.
`
`Agreement between theory and experiment is much poorer
`for this accelerometer. When plotted on a log:log scale,
`the overall slope of the theoretical curve is 1.505 compared
`with the experimental value of 1.926. The closeness of the
`slope to 2 for the experimental curve suggests that the filtered
`noise model, equation (32), is inappropriate to describe the
`dependence with integration t