A Traceable Inertial Calibration ProcedureSuited for MEMS Sensing

Surya SinghAustralian Centre for Field Robotics

University of SydneyNew South Wales 2006, Australia

Email: spns@acfr.usyd.edu.au

Abstract—The wide availability of MEMS inertial sensors haslead to a diversity of applications for compact IMUs in mobilerobotics. Determination of the sensitivity, bias, noise, and non-linear effects of these units is important for robust estimation andaccurate operation. A traceable and dynamic off-line calibrationprocedure is presented based around MEMS device charac-terization equipment and/or equipment available in roboticslaboratories. As with a full inertial calibration configuration, thisprocedure provides a traceable sensor model and a means ofdetermining system alignment and cross-coupling. However, thisis more accessible as the calibration can be performed locally.The method was tested with a custom, high-frequency, MEMStransducer based IMU with results different from nominal values,yet within manufacturer specifications.

I. INTRODUCTION

The combination of micro-electromechanical systems(MEMS) accelerometers and gyroscopes to form an inertialmeasurement unit (IMU) for the entire system is an increas-ingly prevalent aspect of mobile robotics, particularly fortrajectory control and mapping problems. Such units are notonly compact localization sensors, but also allow for increasedperformance on dynamic platforms even in the case of self-stabilizing, compliant designs.

The precise operating parameters governing the inertialtransducers are not initially known as the values given by themanufacturer are typically against a linear sensor model andare specified within a significant range. Further, these valuesare for the individual parts and not for the IMU, for which theaxis alignment and coupling for the orthogonal triad [1] (orrelated redundant configuration [2]) needs to be determined.Hence, for high accuracy operation, it is necessary to calibratethe sensor, even if the measurements are compensated bysmoothing or filtering as later processing adds delay and doesnot directly address scaling [3].

Calibration is an integral part of the sensing process asit relates measurements back to a reference standard. Whilecommercial calibration is possible, values can vary due toexact operating conditions (e.g., temperature and voltages),especially for compact, low-cost MEMS inertial transducers[4]. Online calibration affords a means for performing checksunder operating conditions and is useful for tuning noise andcross coupling variances that might be used in later estimationprocesses [1].

In the case of inertial sensors, calibration checks measuredaccelerations and angular rate values against those from ref-erence forces and motions, such as against the accelerationgenerated in a gravity oriented ground reference frame. Thiscan be performed using a mechanical platform in which theIMU is precisely oriented while spinning at a known rate [1];however, such platforms are complex and expensive. Varioussimplifications have been proposed. For example, the inertialsensor can be moved while being tracked optically [5] oroptimization techniques can be applied while the IMU isspinning so as to determine the misalignment of its sensors toagainst gravity [6], [7]. The issue with the fist approach is thatit requires access to an optical tracking apparatus, while thelatter approach, while allowing for simultaneous calibrations,can only provide acceleration calibrations against a presumedknown gravity reference (i.e., ± ≈ 1g) and thus provides nomeasure of accelerometer nonlinearity.

To get dynamic acceleration calibration and skewing, aprocedure using MEMS resonator characterization equipmentis proposed. The central idea is to integrate the inertialmeasurements and thus make comparison against a referencevelocity measurement. The accelerometer calibration and axisalignment, which if presumed stable for the IMU, can then beused to simply the turntable calibration of the gyroscopes. Incomparison to online estimation techniques [7], this methodproduces a traceable result and does not make assumptionsabout the functional representation of the noise or bias, whichcan lead to over-fitting of parameters.

Experiments were performed using a custom assembly builtfrom best-of-breed MEMS transducers. Referred to as theSensor Cube (shown in Fig. 1), it uses an orthogonal (cu-bic) configuration. Redundant sets of sensors were added foroperation across multiple sensitivities and for part robustness,but were not aligned in using an optimal strategy. Indirectlyinferable measurements, such as obtaining angular rates fromthe six accelerometers, were not computed and checked [8]for these experiments.

Traceability is provided by comparing the INS output signalto synchronized recordings made by a reference instrumentwith a NIST traceable calibration standard. Thus, the finalSensor Cube measurements relate to standard units within thevariance of the instruments, the operating assumptions made,and the limits of uncertainty associated with this procedure.

Fig. 1. The Sensor Cube features a redundant sets of gyros and accelerome-ters in an orthogonal configuration to sense motions of up to 10g acceleration,572◦/sec angular rate, at rates up to 400-Hz, in an ≈ 0.1kg package.

II. ACCELERATION CALIBRATION

Calibration of inertial measurements is complicated by boththe coupling of gravity and the IMU dynamics as the sensingaxis are not necessarily aligned, oriented, and intersecting.This results in the output signal being coupled to variousfactors. In the case of an accelerometer, the output signal,sa, absent of noise is

sa = soR ((a− o

nRg) + α× r + ω × (ω × r)) (1)

where soR is the rotation of the sensor relative to the calibration

frame, onR is the rotation of the gravity (g) frame relative to

the calibration frame, and ω and α are the angular rates andaccelerations respectively.

If care is taken to prevent the frame from rotating, the sec-ondary tangential and centripetal accelerations can be assumedto be negligible. Thus the object becomes to estimate the actualacceleration and the frame orientations. A controlled sinusoidtesting configuration allows for filtering and extended testing.

Based on this notion, the this accelerometer calibrationprocedure builds on techniques for checking the resonantproperties of MEMS cantilevers [9]. The setup as shown inFigures 2 and 3 consists of: a Laser Doppler Vibrometer(LDV: Polytech OFV 3001), a two-channel signal analyzer(HP 89410A), an oscilloscope (HP 54542A), a piezoelectricshaker, and a linear signal filter (Krohn-Hite Model 3750R). These parts are supported on a vibration isolated table(Newport I-2000 series isolator).

The signal analyzer is programmed to generate a sinusoid ofvarious frequencies with the amplitudes adjusted to maintain aconsistent power level. This signal is then amplified and sentto the piezoelectric shaker. The expected accelerations are:

Vsignal = A sin(ωt) (2)xdrive = kxsignal = kA sin(ωt) (3)

adrive =d2xdrive

dt2= −ω2xdrive (4)

where Vsignal is the signal voltage, xdrive is the drivenposition of the shaker, adrive is the driven acceleration, andthe terms A and k are proportionality constants reflectingapparatus settings. Assuming that the accelerometer acts asa linear transducer gives:

Fig. 2. General acceleration calibration setup. A vector signal analyzerfor precision signal generation, which results in shaker vibration of knownfrequency that is captured simultaneously by both the LDV and the sensor

Fig. 3. Side view of acceleration calibration setup used for sensor setcalibration showing the laser alignment and reflective target.

asensor = S1Vsensor + S0 (5)

vsensor =asensor

2πf(6)

where Vsensor is the transducer voltage, S1 is the sensitivity,S0 is the offset, and v is the measured velocity. Since thesensor and LDV are sensing the same motion, the calibrationequation becomes (for a known LDV gain, G):

v(LDV ) = vsensor (7)S1Vout + S0 = 2πfGLDV VLDV (8)

The technique for solving for the online noise model in-volved static and dynamic operations of the sensor elementsduring which the signal was taken. The procedure followedstandard noise quantification protocols except for modifica-tions that removed the extensive shielding and associatedprecautions typically in place as the goal is to obtain apragmatic noise model rather than determine performancelimits. As expected, at lower frequencies the signal has highernoise densities. It is suspected that this is due to the flickernoise effects common in transistorized integrated circuits suchas those integrated with the sensor elements.

This leads to a procedure where by a sweep of frequenciesallows for calibration of various acceleration values. Thefrequency needs to be selected so as not to exceed the ac-celerometer bandwidth while being large enough to be drivenby the shaker. Typical ranges used were from 20 to 500 Hz.Although it is may be measured by a monitoring oscilloscope,the frequency, f , is obtained from the signal generator (byassuming that damping effects are negligible). Example outputis shown in Figure 4 and 5. Although the vibration isolationtable and careful setup minimized axis cross-coupling duringactuation, data was recorded from all three accelerometer axes.

Fig. 4. Typical output from reference LDV velocity measurement

0 0.05 0.1 0.15 0.2 0.25 0.32.3

2.35

2.4

2.45

2.5

2.55

2.6

2.65

Time (sec)

Acc

eler

om

eter

Vo

ltag

e (V

)

x-axisy axisz-axis

Fig. 5. Uncalibrated accelerometer measurements. Axis cross-coupling isvisible (taken during a 67 Hz experiment, x is forward, y is aligned withgravity, and z is to the side)

Analysis of the resulting data via linear least-squares al-lowed for the determination of the sensitivity and offset valuesfor all directions on each of three accelerometer units. Thisalso allowed for a determination of the noise bandwidth andverified that the noise from the MEMS units was non-trivialand colored.

III. GYROSCOPE CALIBRATION

In a similar motivation as with the accelerometer procedure,calibration of the gyroscopic sensors (gyros) was undertakenusing an encoder instrumented turntable. This provides areference measure of the angular position, from which standardrotational rates were obtained by differentiation. Processingof this standard against sensor output gives a mean sensitivityspecification and offset result, which is used to form the linearsensor model relating the transducer signal to measured rates.As with the acceleration procedure, care is taken to align theaxis of the sensor to all three axes.

The calibration apparatus is shown in Fig.6 and consists ofa phonograph turntable (Technics SL-BD10) to which a 1000-count traceable, calibrated, reference encoder (US DigitalUSD#611-2”-1000-9040-B00 disk with Agilent HEDS-9040module) is installed on the platen shaft. Sensor output is thenpassed through a trimmed unity gain buffer circuit (basedon Texas Instruments TLC2274 amplifiers) so as to reducesignal propagation losses and multiplexing effects. The datafrom both sources is then simultaneously captured using a16-channel multiplexed PC-104-based data acquisition card(Measurement Computing PC104-DAS16JR/16). Each signalis digitized at 3125 Hz using a 16-bit analog to digitalconverter (Texas Instruments ADS7805) and then logged usingcustom software.

As expected, the turntable operates at two nominal rates:33 1

3 and 45 revolutions per minute (rpm) (or 198◦/sec and270◦/sec respectively). However, these rates are not guaran-teed or regulated. Further, the additional weight of the sensingapparatus results in a slight imbalance and in additionalfriction so that the manufacturer’s tuning is no longer valid.Thus, the unit is operated primarily at these two rates but withadditional values observed due to various perturbations present(e.g., friction). The reference rotational rates (or simple angu-lar velocities) are found from the encoder data by measuringthe elapsed interval time, te, between one encoder unit (i.e.,one transparent and opaque patch). This gives in radians:

θreference =2πC · te

(9)

and in degrees:

θreference =360o

C · te(10)

where C is the total number of encoder units on the disk(C = 1000 for this setup). Quadrature is treated as an increasein this count. This definition does introduce the possibilityof a digitization error as the encoder signal measurement issynchronized with the sensor readings, but not necessarily the

Fig. 6. The gyroscope calibration has the sensor was placed level with theplaten surface of an instrumented turntable. Simultaneous measurements wererecorded by the data acquisition PC, which was controlled using an externallaptop

rising and falling edge of the encoder pulse train. With thisparticular setup, the highest speed will result in 750 encoderunits per second, which is near the sampling rate. Underthese conditions, the maximum error for a single particularmeasurement is 10%. This is treated through the use ofa moving average as this error only effects instantaneousmeasurements not the average. An alternative solution is to usea data acquisition system providing for simultaneous readingsof multiple channels (i.e., non-multiplexed) and data types(i.e., digital and analog sources).

The sensitivity of the gyros is found via least squarestechniques. In order to encode direction (i.e., rotational sense)the output of the gyros is biased to a half of the inputvoltage (i.e., nominally 2.5 V). For simplicity the term VR isintroduced to represent this voltage. In the case of the SiliconSensing CRS 3-11 gyros this is given by:

V CRSR ≡ 5

2

[2Vo − Vs

Vs

](11)

where Vo is the output and Vs is the supply voltage. Thusthe rate output of the gyros, which is given in Equation 11,becomes:

RATE · S = V CRSR (12)

where the sensitivity (S) is given in units relative to that ofthe rate (typically Volts

◦/sec ).The sensitivity maybe found directly by solving Eq. 12

directly with:

S =VR

RATE(13)

A linear approximation of the cross-coupling effects mayalso be obtained by examining these side channels via theaforementioned procedure. Under non-impact conditions theywere found to be over an order of magnitude less than principalaxis signals (∼ 0.1 mV

◦/sec ). The results of the calibrationthough are not completely sufficient to cover all expectedoperating conditions. Some of these variations include the lackof large angular accelerations, a greater range of angular rates,and gyro variations due to temperature.

As shown in Fig. 7, signal noise leads to an uncertainmeasurement of the sensitivity. This may be treated by takingthe mean of this calculation for the various cases.

0 0.5 1 1.5 2 2.5 3-3.8

-3.75

-3.7

-3.65

-3.6

-3.55

Time (sec)

Sca

le F

acto

r (m

V/(

o/s

ec))

Fig. 7. Average sensitivity found by application of Eq. 13. The meansensitivity is shown as a dashed line. Negative value since reference sensoris CW positive (shown for the roll axis).

An alternative solution is to apply a least squares estimatebetween the reference value and the measured signal (as VR).This approach has the advantage of reducing the computationby providing a least-squares value for the sensitivity andaccounts for minor biases present in the transduction of thesignal. It does, however, assume that the noise is symmetricand thus may be colored by outliers. For the sensors underconsideration the difference in sensitivities by both methodsis typically less than 1%. A result of this calculation is givenin Fig. 8.

0 50 100 150 200 250 300-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

Measured Rate (o/sec)

CR

S R

atio

met

ric

Vo

ltag

e

RefernceGyro model (3.7 mV/(o/sec))

Fig. 8. Least-squares sensitivity estimate (for the roll axis)

IV. EXAMPLE CALIBRATION RESULTS

The calibration procedure was applied to the aforemen-tioned Sensor Cube (see also Fig. 1). The result of thecalibration process is a series of transducer coefficients. Atabulation for the accelerometers is give in Table I. A similartabulation of the values for the three primary (Silicon Sensing)gyros and an average value for the secondary (Analog Devices)gyros on the sensor cube is given in Table II.

AccelerometerSensitivity

(mVg

)Deviation(mili− g)

Calibration Reference Calibration Reference2g Unit 650 660 1.5 0.710g Unit 97 100 20 10

TABLE ISENSOR CUBE ACCELEROMETER CALIBRATION VALUES

AxisSensitivity( mV◦/sec

)Deviation(◦/sec )

Calibration Reference Calibration ReferencePitch 3.61 3.49 1.5 1.1Roll 3.67 3.49 0.9 1.1Yaw 3.71 3.49 1.7 1.1

BackupGyros 5.19 5.00 0.8 0.6

TABLE IISENSOR CUBE GYROSCOPE CALIBRATION VALUES

0 0.5 1 1.5 2 2.5 3120

140

160

180

200

220

240

260

280

300

Time (sec)

Rat

e (o

/sec

)

ReferenceCalibrated GyroManufacturer specification

Fig. 9. Overlaying the measurement based on the calibrated model andthe reference values shows that sensor transduction is generally linear. Forcomparison, calculation based on the default sensitivity is included

V. CONCLUSIONS

A calibration procedure is presented for MEMS IMUsbased on techniques for resonator characterization. By usinga driven, but not necessarily precision controlled, shaker andLaser Doppler Vibrometer it is possible to calibrate an custominertial suite at a variety of dynamic modes. Processing canbe performed at the bench using a vector signal analyzer, or

off-line using a digital signal processing software. While theequipment used is not common in robotics laboratory circles,it is available in MEMS and clean-room characterizationfacilities.

Future work is considering means of relating on-line es-timated to the calibrated values. Thus calibration from oneunit can be used to qualify additional units on the mobilerobot. Future work is also considering a simpler means oftraceable calibration for the accelerometers as MEMS charac-terization equipment, while available in university clean-roomsand MEMS research, is too specialized for general roboticslaboratories.

VI. ACKNOWLEDGMENTS

The author acknowledges the support of the Rio Tinto Cen-tre for Mine Automation and the ARC Centres of ExcellenceProgramme funded by the Australia Research Council andNew South Wales State Government. Experimental testing waswith the technical assistance of the Stanford Microsystemsand the Stanford Microstructures and Sensors Laboratories.This material is based upon work supported by the NationalScience Foundation under Grant No. IIS-0208664.

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