IMU: Generic Model Development Approach - Semantic Scholar

V Simpósio Brasileiro de Engenharia Inercial __________________________________________

IMU: Generic Model Development Approach G. Baldo Carvalho, S. Theil, H. Koiti Kuga

Resumo— Este trabalho apresenta um modelamento genérico para IMU (Inertial Measurement Unity). Através de ajustes de parâmetros de desempenho de sensor tais como bias, ru´ıdo e resposta em frequência além de outros, o modelamento oferece capacidade para simular desempenho de sensor desejado. possibilitando testar requisitos de sensor e desempenho. O modelo e´ comparado com uma simualaça˜ o de sensor IMU provida pelo Instituto ZARM da Universidade de Bremen. Os dados de simulaça˜ o de trajetória são referentes ao protótipo PHOENIXEADS. Palavras-Chave— IMU, Simulaça˜ o, Sensor de navegaça˜ o inercial. Abstract— In this work a generic a modeling for IMU (Inertial Measurement Unity) is presented. By setting sensor performance parameters such as bias, noise and frequency response among others, the modeling offers the capability to simulate required sensor performance enabling testing sensor requirements and performance. The model is compared against IMU sensor simulation provided by ZARM Institute of University of Bremen. The trajectory simulation data is related to the PHOENIX-EADS prototype. Keywords— IMU, Simulation, Inertial Navigation Sensor

I. D EVICE OVERVIEW The Inertial Measurement Unity (IMU) is an integrated sensor unity constituted by three accelerometers and three gyroscopes mounted in three orthogonal axes capable of providing measurements of three-dimensional specific force and attitude rate with respect to the inertial reference frame. These measurements, when integrated in time allow to find navigation solution for inertial navigation systems. However, since sensors are not perfect, their measurements are affected by various errors and then affecting navigation solution computations. That is the reason why the knowledge about sensor performance and its internals is important for determining its impacts on a system.

Sensor Dynamic - the process by which inputs viewed by the sensor are modified by the sensor dynamic. • Measurement - the process by which the resulting inputs are translated to the sensor output (Displacements or Volts instead of force or attitude rates). Besides that, some model parameters are modeled in 3 parts as follow: • Nominal - regarded as the mean value, sometimes function of temperature • Stability day-day - regarded as the standard deviation that change the Nominal value from operation to operation, but remaining fixed during one operation time. • Stability in run - regarded as the standard deviation that change the Nominal value during the operation time. •

III. ACCELEROMETER T RIAD GENERIC MODEL A. Gravitational gradient effect When the sensor is not under the calibration reference conditions a gravitational force difference effect appears in the specific force f p as function of the inertial position ri , attitude q pi and references ri0 and q pi0 . f pG = f p +∆g p = f p +(T pi (q pi )·g i (ri )−T pi (q0 pi )·g i (ri0 )) (1)

Gravitational gradient effects gi (ri) Tip(qip) Tip(q0ip)

g

p

-

x x

g0

∆g

p

+

fp

p

fG p

gi (ri0)

II. I NERTIAL M EASUREMENT U NITY M ODEL Since there are a lot of types of accelerometers and gyros, the IMU can be assembled in many different ways using different integration strategies and sensor types. This results on different device performances, which can nonetheless be modeled by a common mathematical model by adjusting parameters ( [3], [4], [5], [6] and [9]). To ease, the model is analyzed in 3 different levels: • Input Change - the process by which inputs are changed and generate the input actually viewed by the sensor. G. Baldo Carvalho, Flight Test Division, EMBRAER S.A., Gavião PeixotoSP, Brazil. S. Theil, Center of Applied Space Technology and Microgravity ZARM, University of Bremen, Bremen, Germany. H. Koiti Kuga, Division of Space Mechanics and Control, INPE, São José dos Campos-SP, Brazil. Emails:[email protected], [email protected], [email protected]

Fig. 1.

Gravitational gradient model.

B. Input Change In the accelerometer, the main sources of signal change are Misalignments, Cross-Coupling and Rotation Sensitivity. 1) Misalignment: The misalignment consist of a nonorthogonal transformation represented by the accelerometer cluster misalignment matrix T pa (α(t)) with respect to platform reference axes causing that the specific force components are not aligned to each input axis:

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f aM = T pa (α(t))−1 · f p

(2)


fG p α

Misalignment

αD xc

αR

x-1

xcD xc

x

σD α = αD · α σR α = αR · α

+

x

Cross Coupling

a

Tpa

δD term is the stability day-to-day applied to the random values γi and δR term is stability-in-run applied to the random values µi (t). The nominal value is αN . The standard deviations are function of the nominal values α and given parameters for stability day-to-day αD and in-run αR :

KXCa

a

rsa

Tap

ωipp

x

a R

KRSa

2) Cross Coupling: The cross coupling effect is due the cross influence of orthogonal components to the input of a given axis regarded as a scaling matrix

fa

+

x

Rot Sensitivity

rsDa rsRa Input Change Fig. 2.

Accelerometer input change model.

where a and p denote non-orthogonal accelerometer triad and platform reference frame. Since the transformation is nonorthogonal the inverse transformation matrix must be used.   cαxy cαxz −cαyx sαyz cαzx sαzy −sαzx  (3) T pa (α(t)) = cαxy sαxz cαyx cαyz −sαxy sαyx cαzx cαzy

where c = cos, s = sin. The misalignment angles α(t) are shown in Figure 3.

Z

p

X

αxy

αyz Yp

(10)

rsaxy rsayy rsazy

 rsaxz rsayz  rsazz

X

The main sources of signal change are Frequency Response, Noise, Bias, Sensor Resolution and Input Saturation.

Accelerometer misalignments.

α(t) = αN + δD αN + δR αN (t)

(4)

Sensor Dynamic T

a

H (s)

where

αN

  αxy αxz    αyx  ,  =  αyz  αzx  αzy

  σD αxy · γ1 σD αxz · γ2    σD αyx · γ3    δD α N =   σD αyz · γ4  σD αzx · γ5  σD αzy · γ6   σR αxy · µ1 (t) σR αxz · µ2 (t)   σR αyx · µ3 (t)   δR αN (t) =   σR αyz · µ4 (t) σR αzx · µ5 (t) σR αzy · µ6 (t)

(11)

C. Sensor Dynamic

a

Fig. 3.

f aRS = f a + K aRS (rsa (t)) (T ap · ω pip )

(9)

where rsa (t) are the Rotation Sensitivity factors modeled similarly as given by equation 4 by using the standard deviations as function of the nominal value rsaN and given parameters for stability day-to-day rsaD and in-run rsaR .

Ya αyx

(8)

 a rsxx K aRS (rsa (t)) = rsayx rsazx

αzy

αxz

f aXC = f a + K aXC (xca (t)) f a   0 xcxy a xcxz a 0 xcyz a  K aXC (xca (t)) = xczx a a a xczx xczy 0

where xca (t) are the cross coupling factors modeled similarly as given by equation 4 by using the standard deviations as function of the nominal value xca and given parameters for stability day-to-day xcaD and in-run xcaR . 3) Rotation Sensitivity: The Rotation Sensitivity effect is due the linear influence of the actuating angular rate ω pip as a scaling matrix to the input of a given accelerometer axis.

p αzx Z

a

(7)

bTa pol

ba Bias

fa Freq. Response

bDa bRa

(5)

+

+ n

Resolution

fsa

Saturation

a

Noise

nEqa

rRa

fωa fm a

(6) Fig. 4.

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Accelerometer sensor dynamic model.


1) Frequency Response: Every sensor as a dynamic system, has its frequency response that affects the spectrum of the signal. To model this effect, an analog filter H a (s) can be used where its dynamics is described by its poles pan , zeros a zm and gain K a . QMZ a (s + zm ) fã (s) H a (s) = FaR = K a · Qm=1 (12) N P a) f (s) (s + p n n=1 However, since the model treats with digital data, this filter must be then translated to a digital filter H a (z) ( [1], [10]): PM a −m f˜Fa R (z) a m=0 bm z (13) H (z) = a = PN f (z) 1 − n=1 can z −n

where f˜Fa R (z) and f a (z) are the z transform of the filter input and output. Note that the bam coefficients integrate the gain and that the number of analog poles and zeros are not necessarily identical to the number of discrete poles and zeros. For each accelerometer: f˜Fa R (k) = PM PN a a a ã   m=0 bm · f (k − m) + n=1 cn · fF R (k − n), k ≥ max(M, N )   ã f , k < max(M, N )

(14)

0.02s, the change is not detectable and the digital filter will be not able to apply the desired dynamic effect. Although the Nyquist law states the presented relation, in the practice a minimum relation of ten times would be desirable to better describe the dynamic change through samples, what means fωa ≥ 10ωc . 2) Noise: Noise is regarded as standard deviation due unmodeled internal effects or due non desirable actuating specific forces. a a = f˜ + na (t) f˜ (16) N oise

na (t) = naEQ ·

p

fωa · µ(t) = σnaEQ · µ(t)

where µ(t) is a random value variable during the operation, fωa is the sensor sampling rate in [Hz] and naEQ is the random walk parameter. Note that it is driven by a standard deviation equivalent random walk, which must be multiplied by the square root of the sampling rate to obtain a real white noise standard deviation. 3) Bias: Bias is the persistent offset of the output with respect to the input due construction imperfections and temperature variations. a a f˜B = f˜ + ba (t, T )

a

where the delayed filter outputs f˜F R (k−n) are put in feedback with the delayed inputs f a (k − m) through the digital filter coefficients bam and can . Note that at the very beginning, when the number of samples is less then max(M, N ), there is not enough data to apply the filter. In this case the filter is not applied and the measurement does not suffer any effect. Lets say that the desired dynamic is modeled by a second order filter: 1 H a (s) = K a · 2 (15) s + 2ξ a ωca s + (ωca )2 Here (ωca )2 is the cutoff frequency (rad/s) and ξ a is the system dumping, which defines at which magnitude the cutoff will take place: M agca = 1/(2 ξ a ). This defines a low pass filter and as such it will allow low frequencies while cutting off high frequencies causing a phase delay in the steady state of the signal. Note that the Magnitude response is not normalized. If a normalized response is desired instead, the gain can be adjusted as K a = (ωca )2 . There is yet an important problem in the discrete domain to be observed. Depending on which is the relation between the speed of dynamic response, translated by system time constant τ a = 1/(ξ a ωca ) and the system sampling rate fωa , that can happen that the system dynamic is much faster than the sampling rate. That means the response of the system is too fast for the sampling process to map enough data to describe the state change of the signal, which means that the application of a digital filtering will not simulate properly the desired dynamic effect. To illustrate, consider a sampling rate of 100Hz ( 0.01s). By the Nyquist sampling law the sampling rate should be at least twice faster than the desired changing rate. If the system dynamic response is such that a response happens faster than

(17)

(18)

where ba (t, T ) is the time variable bias as function of temperature T modeled similarly as given by equation 4 by using the standard deviations as function of the nominal value baN and given parameters for stability day-to-day baD and in-run baR . The nominal values baN (T ) in function of temperature are modeled as order n temperature polynomial coefficients: baN (T ) = ban · T n + ban−1 · T n−1 ... + ba0

(19)

As an example, the temperature dependency can be modeled as a second order polynomial ( [8]): ba (T ) = ba2 · T 2 + ba1 · T + ba0 db bmastd = dT T =Tstd a bm ba (0) − ba (Tstd ) std ba2 = + 2 Tstd Tstd a a b (Tstd ) − b (0) ba1 = 2 − bmastd Tstd ba0 = ba (0)

(20)

where the basic values ba (0), ba (Tstd ), bmastd are given and Tstd represents an adopted standard temperature. a 4) Resolution: Resolution rR is regarded as the minimal value that can be sensed by the sensor represented by means of a quantization process. ! fã a a ˜ fR = rR · int (21) a rR where int is a function that takes the nearest integer towards zero.

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5) Input Saturation: Saturation is regarded as the maximal input value that the sensor can stand, outside of which every input saturates to a maximum value represented by the full scale f sa . a f˜sat =

( f sa · fã /|fã |, fã ,

fã ≥ f sa fã < f sa

where the terms knla are deduced from non-linearity factors nla and full scale parameter f sa as: v a (max(fin )) = sf a · max(fin ) v a (max(fin ))nl = sf a · max(fin ) + knla · max(fin )2 nla = v a (max(fin )) − v a (max(fin ))nl

(22)

= −knla · max(fin )2 a

f s = max(fin ) nla knla = − (f sa )2

D. Measurement

(26)

The main sources of signal change are Scale Factor, Scale Factor Non-linearity, Digital Resolution and Delay. fm a T

Measurement sfTa pol δv0a

∆ta va(fm a)

KSFa

x

x2

+

x

sfDa sfRa

-

Delay

Scale Factor

Digitization

fs

rDa

a

Non-Linerarity

nla nl

Fig. 5.

a D

nl

KNLa

a R

Accelerometer measurement model.

1) Scale Factor: Scale Factor is represented by a linear gain scaling matrix, which relates the input (specific force) to the output (Displacement or Volts for example). a a a v aSF (f˜ ) = K aSF (t, T ) · f˜ + K anl (t) · (f˜ )2

K aSF

 a sfx = 0 0

0 sfya 0

 0 0  sfza

(23)

(24)

a

where sf are the scale factors modeled similarly as for bias given the standard deviations as function of the nominal value a as function of the temperature and given parameters for sfN a stability day-to-day sfD and in-run sfRa . K anl (t) is the nonlinearity in the scale factor treated in next section

a

Note that the f s parameter must be given in the same unit as the input fin and not in the output unit. However, nla is given in the output unit. The nla are the non linearity modeled similarly as given by a equation 4 as function of the nominal value nlN and given a a parameters for stability day-to-day nlD and in-run nlR . a 3) Digitization: Digital Resolution rD is regarded as the precision with which the digital output is digitized. a v˜ a a v˜D = rD · int (27) a rD where int is a function that takes the nearest integer towards zero. 4) Delay: Delay is the representation of the amount of time that the sensor takes to deliver an output once a given input is presented. a It can be represented by a memory function f∆ that buffers a the current measurement, to only deliver it ∆t seconds later: ( a a f∆ (v a (f˜ ), ∆ta ), t ≥ ∆ta a ã v ∆ (f , t) = (28) 0, t < ∆ta Note that the output is null till the time is greater than ∆ta . IV. G YRO T RIAD GENERIC MODEL The gyros modeling is similar to the one presented for the accelerometers with some differences in the input change part of the model. A. Input Change The difference with respect to accelerometers modeling is the absence of rotation sensitivity and the inclusion of GSensitivity and G2-Sensitivity. 1) G-Sensitivity: The G-Sensitivity effect is due the linear influence of the actuating specific force as a scaling matrix to the input of a given gyro axis

2) Non-linearity: The non-linearity of the scale factor effect is due the difference of the sensor scale factor from the modeled linear behavior, which cause distortions on the scaling of the input. As shown by equation 23, it can be regarded as a scaling matrix

K anl

 a knlx = 0 0

0 knlya 0



0 0  knlza

(25)

ωGS gip = ω gip + K gGS (t) (T gp · f p )  g  gsxx gsgxy gsgxz K gGS (gsg (t)) = gsgyx gsgyy gsgyz  gsgzx gsgzy gsgzz

(29) (30)

where gsg (t) are the G-Sensitivity factors modeled similarly as given by equation 4 by using the standard deviations as function of the nominal value gsgN and given parameters for stability day-to-day gsgD and in-run gsgR .

--- 65 ---


ωipp

β

Tgp

Misalignment

Tpg

x-1

spectrum, it can be noted that the proposed IMU modeling does not reshape the base spectrum of the original input while adding some changes around it.

fG p

x

VI. C ONCLUSIONS βD xcg

βR

KXCg

Cross Coupling

This paper presented a generic IMU modeling where an example was implemented and compared against IMU modeling simulation provided by ZARM(University of Bremen) for the Phoenix prototype (EADS). The result showed that the proposed model was very similar and resemble the original input despite of sensor disturbances, showing then the model validity. Some features where however noted which present the capability of the model to simulate some behavior of real sensors more accurately than the ZARM-IMU model. The model is then left to the community for further improvement, correction, adaptations as well as verification for other performance parameters.

+

x

xcDg xcRg

x fg KGSg

gsg

G Sensitivity

gsDg gsRg g

g2s

+

ωipg

xT

KG2Sg

G2 Sensitivity

g2sDg g2sRg

Fig. 6.

+

x

x

x Input Change

ACKNOWLEDGEMENT

Gyro input change model.

2) G2-Sensitivity: The G2-Sensitivity effect is due the square influence of the actuating specific force as scaling matrices to the input of a given gyro axis.  g p T  (T p · f ) · KxgG2S (t) · (T gp · f p ) p  g g  g p T ωG2S gip = ω gip + (T p · f ) · Ky G2S (t) · (T p · f ) (31) (T gp · f p )T · Kz gG2S (t) · (T gp · f p )   g2sxgxx g2sxgxy g2sxgxz g2sxgyy g2sxgyz  KxgG2S (g2sgi (t)) =  0 0 0 g2sxgzz   g g g g2syxx g2syxy g2syxz g g  g2syyy g2syyz Ky g (g2sgi (t)) =  0 (32) G2S g 0 0 g2syzz   g g g g2szxx g2szxy g2szxz g g  g2szyy g2szyz Kz gG2S (g2sgi (t)) =  0 g 0 0 g2szzz

where g2sgi (t) are the G2-Sensitivity factors modeled similarly as given by equation 4 by using the standard deviations as function of the nominal value g2sgN and given parameters for stability day-to-day g2sgD and in-run g2sgR . V. S IMULATION A. System modeling

The authors would like to thank Embraer S.A. and CNPq/DAAD cooperation for supporting the development of this work. R EFERENCES [1] A. Antoniou, editor. Dgital Filters: Anaylsis, Design and Applications. McGraw-Hill Inc., 1993. [2] P.F. Beer and E.R. Johnston Jr. Mecânica Vetorial para Engenheiros: Dinâmica. McGraw-Hill, 1981. [3] K. R. Britting. Inertial Navigation Systems Analysis. John Willey And Sons Inc., 1971. [4] A.B. Chatfield. Fundamentals of High Accuracy Inertial Navigation, volume 174 of Progress in Astronautics and Aeronautics. American Institute of Astronautics and Aeronautics, 1997. [5] J. L. Farrell. Integrated Aircraft Navigation. Academic Press, 1976. [6] A. Lawrence. Modern Inertial Technology: Navigation, Guidance and Control. Springer Verlag, 1992. [7] R.H. Parvin and G. Merrill. Inertial Navigation. D. Van Norstrand Company Ltd., 1962. [8] C. Pearce. The Performance and Future Developent of a MEMS SIVSG and its Application to The SiIMU. In AIAA Guidance, Navigation, and Control Conference and Exhibit. AIAA, August 2001. [9] R. M. Rogers. Applied Mathematics in Integrated Navigation Systems. AIAA Education Series, 2000. [10] C. B. Rorabaugh, editor. Dgital Filter Designer’s Handbook: Featuring C Routines. McGraw-Hill Inc., 1993. [11] J.B. Scarborough. The Gyroscope: Theory and Applications. Interscience Publishers Inc., 1958. [12] S.H. Stovall. Basic Inertial Navigation. Technical Report NAWCWPNS TM 8128, Naval-Air Warfare Center/Weapons Division, September 1997. [13] J.R. Wertz, editor. Spacecraft Attitude Determination and Control. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1978.

The Simulated System is analyzed with inputs from a trajectory simulation provided for Phoenix-EADS while an IMU simulation from ZARM Institute is also provided as comparison. The first conclusion from the presented pictures is that the proposed model resembles the inputs as expected. It also adds noise and other tendencies to the measured input such as an real sensor. The more clear tendency we can note is that the proposed model allowed the simulation of more realistic bias which has no constant mean over the time. If the power frequency domain is analyzed through the power

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Power fft(fp),fft(imu−fa)

fp,imu−fa

2.5

0.2 imu−fx imu−fy imu−fz fx fy fz

0

−0.2

pimu−fx pimu−fy pimu−fz pfx pfy pfz

2

−0.4

Power fp

1.5

fp (g)

−0.6

−0.8

1

−1

0.5

−1.2

−1.4 0

−1.6

Fig. 7.

0

50

100

150 Time (s)

200

250

0

0.5

1

1.5

300

Fig. 11. forces.

IMU model and reference trajectory for specific forces.

2.5

3

3.5 4

x 10

IMU model and reference trajectory for power spectrum specific

−6

fp − imu−fa

Power fft(wipp),fft(imu−wipg)

x 10

dax day daz

−0.05

2 freq (Hz)

pimu−rollrate−wx pimu−pitchrate−wy pimu−yawrate−wz prollrate−wx ppitchrate−wy pyawrate−wz

1.2

−0.06 1

−0.07 Power wipp

Dif acc (g)

0.8

−0.08

−0.09

0.6

0.4

−0.1

−0.11

0.2

−0.12

50

100

150 Time (s)

200

250

0

0

5

10

15

freq (Hz)

Fig. 8. Difference between IMU model and reference trajectory for specific forces.

Fig. 12. rates.

4

x 10

IMU model and reference trajectory power spectrum for angular

Power fft(fp),fft(zarm−imu−fa)

fp,zarm−imu−fa

2.5

0.2 zarm−imu−fx zarm−imu−fy zarm−imu−fz fx fy fz

0

−0.2

pzarm−imu−fx pzarm−imu−fy pzarm−imu−fz pfx pfy pfz

2

−0.4 Power fp

1.5

fp (g)

−0.6

−0.8

1

−1

0.5

−1.2

−1.4 0

−1.6

Fig. 9.

0

50

100

150 Time (s)

200

250

−6

x 10

dax day daz

2

1

0

0.8

−2

0.4

−6

0.2

50

100

150 Time (s)

200

2

2.5

3

3.5 4

x 10

Power fft(wipp),fft(zarm−imu−wipg) pzarm−imu−rollrate−wx pzarm−imu−pitchrate−wy pzarm−imu−yawrate−wz prollrate−wx ppitchrate−wy pyawrate−wz

0.6

−4

0

1.5

1.2

Power wipp

Dif acc (g)

4

1

Fig. 13. ZARM IMU model and reference trajectory for power spectrum specific forces.

fp − zarm−imu−fa

x 10

0.5

freq (Hz)

ZARM IMU model and reference trajectory for specific forces. −3

0

300

250

0

0

5

10 freq (Hz)

Fig. 10. Difference between ZARM IMU model and reference trajectory for specific forces.

15 4

x 10

Fig. 14. ZARM IMU model and reference trajectory power spectrum for angular rates.

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