mems imu carouseling for ground vehicles · mems imu carouseling for ground vehicles (post-print) 1...

MEMS IMU CAROUSELING FOR GROUND VEHICLES (POST-PRINT) 1

MEMS IMU Carouseling for Ground VehiclesJussi Collin Member, IEEE

Abstract—MEMS gyroscopes have advantageous properties fororientation sensing and navigation as they are small, low-costand consume little power. However, the significant noise at lowfrequencies results in large orientation errors as a function oftime. Controlled physical rotation of the gyroscope can be usedto remove the constant part of the gyro errors and reduce low-frequency noise. As adding motors for this would increase thesystem cost, it would be advantageous to attach gyros to a rotatingplatform that is already built in the vehicle. In this paper wepresent theory and results for novel navigation system whereinertial measurement unit is attached to the wheel of a groundvehicle. The results show that a low-cost MEMS IMU can providevery accurate navigation solution using this placement option.

Index Terms—MEMS gyroscopes, Dead Reckoning

NOMENCLATURE

ω angular rate vectorω magnitude of angular rate vectorωB

IB angular rate of coordinate frame B relativeto coordinate frame I, expressed in coordinateframe B coordinates

ωrot angular rates after the carousel transformationωrot\3 angular rates after the carousel transformation;

2 × 1 vector omitting the carousel axis mea-surement

ωY scalar heading (yaw) rate of a vehicleωR scalar roll rate of a vehicleCA1

A2 direction cosine matrix that transforms vectorsfrom coordinate frame A2 to A1

(ω×) 3× 3 skew symmetric form of 3× 1 vector ω()i desired signal in the absence of noise()b constant bias, independent of desired signal()n noise process, independent of desired signalE(x) expected value of random variable xV(x) variance of x, V(x) = E(xxT )− µxµ

Tx

xL vector x expressed in local coordinate framewhere the first axis is defined by plumb-bobgravity direction

xB vector x expressed in body frame (wheel)coordinates

xV vector x expressed in vehicle frame coordinatesxI vector x expressed in inertial frame coordinatesϕ wheel phase angle, cumulative rotation about

B-frame z-axis

The author is with the Department of Pervasive Computing, TampereUniversity of Technology, Finland. E-mail: [email protected]

c⃝ 2014 IEEE. This is a post-print of publication in the IEEE Transactionson Vehicular Technology, doi: 10.1109/TVT.2014.2345847 . Personal use ofthis material is permitted. Permission from IEEE must be obtained for allother users, including reprinting/ republishing this material for advertising orpromotional purposes, creating new collective works for resale or redistribu-tion to servers or lists, or reuse of any copyrighted components of this workin other works.

aBSF specific force acceleration measured by ac-celerometers

g, g gravity vector and its magnitudeR wheel radiusp(k) vector describing the vehicle position at time

k

I. INTRODUCTION

ALTHOUGH Global Navigation Satellite Systems (GNSS)provide accurate positioning service for ground vehicles

in line of sight conditions, the availability and accuracydrops in urban canyons and underground [1], [2]. Inertialsensors can be used to complement the GNSS service, butmaintaing the accuracy of GNSS over long outages requiresquite expensive sensor system [3], [4]. The drift caused bydouble-integration of accelerometer data can be avoided byusing the odometer of the vehicle, but the connecting toexternal system is often difficult due to different standards,data transfer speeds, and difficulties in obtaining accuracyinformation along with the data [5], [6]. Another problem isthe drift in heading information caused by the gyro noise [7].Today’s microelectromechanical system (MEMS) gyros aresuitable for vehicular applications in terms of size and cost,but the noise properties of typical MEMS gyroscope (largebias and significant 1/f noise) lead to large cumulated errorsafter integration from angular rates to heading, roll and tiltangles [8]. In this paper we show that inertial measurementunit attached in the vehicle wheel can actually solve the abovementioned problems.

Mitigation of MEMS gyro noise is an actively studied topic,and solutions vary from using external updates [7], [9] toadvanced statistical signal processing methods for filteringthe gyro noise while retaining the signal [10]–[13]. However,in the absence of external updates it is quite impossible totransform the low-cost MEMS gyro to high-performance low-noise gyro with signal processing methods alone. This is dueto the fact that the 1/f noise heavily overlaps with the usefulsignal frequency bands. To separate the problematic noisecomponents (bias, 1/f noise) from the signal, methods such asindexing and carouseling have been studied [14]–[19], evenmore actively recently as the MEMS gyro applications havegrown rapidly [20], [21]. It can be shown that the carouselingmethod can remove the individual gyro biases from 3D inertialmeasurement unit (IMU) completely [17], but this methodrequires a complex motor system wherein the IMU shouldbe attached. In many applications controlled rotation aboutone axis is sufficient [22], [23], but still the cost and sizeof motor and control system can take away the advantages ofMEMS technology [24]. In this paper we apply the carouselingmethod, but without the controlled rotation. Instead, we place

2 MEMS IMU CAROUSELING FOR GROUND VEHICLES (POST-PRINT)

the IMU to the wheel of the vehicle and estimate the rotationindirectly using accelerometer data. As a side-product theaccelerometer data can be used as virtual odometer, and thusthe system can provide a complete 2D dead-reckoning solutionwithout need to access the onboard odometer of the vehicle.

II. INTENTIONAL ROTATION OF INERTIAL SENSORS

Intentional rotation of angular rate sensors with known inputsignal is a common procedure for calibrating the scale factorsof the sensors [25]. Intentional, controlled rotation duringmeasurement missions is also known method to reduce themeasurement errors [14], [26]–[29]. Two common methodsare indexing (or maytagging) and carouseling [23], [30]. Inthe former method the sensor sensitivity axis is turned ± 180◦

between the measurements and the sign of the measurementdata is changed accordingly. As additive bias remains constantlarge part of angular rate sensor error disappears when thetwo measurements are combined. However, the measuredsignal should not change during the rotation process andthus indexing is mainly used for applications where constantangular rotation signal is observed, such as surveying orNorth finding [31]–[33]. In carouseling such restriction isremoved by rotating the sensor system continuously, at aconstant angular rate, for example. In this case there is no deadtime between the measurements and varying angular rotationsignals can be extracted from the data. A wheel of a groundvehicle would fulfill the requirement for carouseling motion if

• wheel rotation speed is constant• the rotation angle of the wheel is known.

Such system would include intentional rotation, but in practicethe first requirement does not hold and thus the rotation isnot controlled. The second requirement is not a problem tomodern vehicles with anti-lock braking systems [34], [35], butthis information is often difficult to obtain. In the followingsections we address these issued and introduce a frameworkfor a dead reckoning system that is completely based oninertial sensors data but does not suffer from gyroscopebiases and does not require error-prone double-integration ofaccelerometer data.

III. COORDINATE FRAME TRANSFORMATION FORWHEEL-IMU

Using general attitude update equation [36]

CA1A2 = CA1

A2(ωA2IA2×)− (ωA1

IA1×)CA1A2, (1)

we define A2 = B, e.g. the wheel-frame and A1 = V , e.g.the vehicle chassis frame to obtain

CVB = CV

B(ωBIB×)− (ωV

IV×)CVB . (2)

Assuming that CVB is known, for dead reckoning (DR) solution

we need to solve ωVIV. We right-multiply both sides by CB

V toobtain

(ωVIV×) = CV

B(ωBIB×)CB

V − CVBC

BV. (3)

We construct the frames so that B and V share a commonz-axis, and the basis vectors are orthogonal (Fig.1). Thus,

CVB =

a b 0−b a 00 0 1

, (4)

and furthermore a2 + b2 = 1. This result is derived using thedirection cosine matrix property AT = A−1. We define thegyro measurements as ωB

IB = [p q r]T , and thus

(ωBIB×) =

0 −r qr 0 −p−q p 0

. (5)

The second term in the right side of (3) is in this definition

CVBC

BV = CV

B(ωBVB×)CB

V

=

a b 0−b a 00 0 1

0 −x 0x 0 00 0 0

a −b 0b a 00 0 1

=

0 −x 0x 0 00 0 0

.

(6)

We will combine (4), (5), and (6) to the (3) to obtain

(ωVIV×) = 0 −r a2 − r b2 + x a q − b p

r a2 + r b2 − x 0 −a p− b qb p− a q a p+ b q 0

. (7)

Simplification by using the identity a2 + b2 = 1 yields

(ωVIV×) =

0 −r + x a q − b pr − x 0 −a p− b q

b p− a q a p+ b q 0

. (8)

If the vehicle travels on a flat (forward direction) surface, andwe neglect the Earth rotation, then r = x and we obtain aresult:

(ωVIV×) =

0 0 a q − b p0 0 −a p− b q


. (9)

On the other hand, if we set the z-gyro to zero (that is,we do not include z-axis gyro at all), and rotate the gyromeasurements directly, we obtain

ωrot ≡ CVB [p q 0]T =

[a p+ b q a q − b p 0

]T. (10)

Skew-symmetric form of this vector is

(ωrot×) =

0 0 a q − b p0 0 −a p− b q


, (11)

exactly the same as (9). Thus, the coordinate transformationto gyro data can be done by using external angle informationthat does not suffer from drift as low-cost gyroscope would. Tokeep the system independent from the host vehicle, it is advan-tageous to estimate the required CV

B using two accelerometerswith sensitivity axes orthogonal to axis of rotation of thewheel. It should be noted that in traditional inertial navigationequations the gyro data is not transformed to another framebut the gyro data is used to build transformation for theaccelerometer data. However, the above analysis shows thatin the wheel-attached IMU the result will be the same in theabsence of noise. The advantages for realistic situation withMEMS gyros suffering from significant noise processes willbe shown in the following sections.

3

B

V

Fig. 1. Axes definitions for wheel-fixed frame (B) and chassis-fixed frame(V)

A. Effect of Carouseling to the Gyro Bias

With the rotation model explained above, the matrix CVB is

a function of the wheel phase angle ϕ:

CVB =

cos(ϕ) − sin(ϕ) 0sin(ϕ) cos(ϕ) 0

0 0 1

. (12)

Then, by using (10), and adding time t, we get

ωrot(t) =

p(t) cos(ϕ(t))− q(t) sin(ϕ(t))q(t) cos(ϕ(t)) + p(t) sin(ϕ(t))

0

(13)

As a simple error model, we add constant biases to the gyrooutputs, e.g. p(t) = pi(t) + pb, q(t) = qi(t) + qb, pb = qb.Then ωrot(t) can be separated to

ωrot(t) = ωroti(t) + ωrotb(t), (14)

wherein

ωroti(t) =

pi(t) cos(ϕ(t))− qi(t) sin(ϕ(t))qi(t) cos(ϕ(t)) + pi(t) sin(ϕ(t))

0

, (15)

and

ωrotb(t) =

pb cos(ϕ(t))− qb sin(ϕ(t))qb cos(ϕ(t)) + pb sin(ϕ(t))

0

. (16)

Assuming the wheel rotates at a constant speed, ϕ(t) = rt,the elementary integral rules show that cosine and sine termswill be zero once the gyro data is being integrated over full

revolution (T = 2πr ),∫ T

0

ωrotb(t)dt =

∫ T

0pb cos(rt) dt−

∫ T

0qb sin(rt) dt∫ T

0qb cos(rt) dt+

∫ T

0pb sin(rt) dt

0

=

000

,

(17)

and thus the biases disappear. If the wheel rotation speedchanges, there will be small residual angle in the resultingdelta-heading.

B. Mixing of Yaw Rate and Roll RateTo further analyze how heading rate (ωY ) and roll rate (ωR)

are mixed in the wheel IMU, assume first that ωR = 0 andapply (10) and (12) to obtain

p1 = ωY cos(ϕ)

q1 = −ωY sin(ϕ) .(18)

Similarly, with ωY = 0

p2 = ωR sin(ϕ)

q2 = ωR cos(ϕ) .(19)

and adding these linearly yields the gyro signals observed bythe wheel-mounted IMU,

p = ωY cos(ϕ) + ωR sin(ϕ)

q = −ωY sin(ϕ) + ωR cos(ϕ) .(20)

By omitting the 3rd axis, (10) can be written as

ωrot\3 =

[cos(ϕ) − sin(ϕ)sin(ϕ) cos(ϕ)

] [pq

]=

[cos(ϕ) p− sin(ϕ) qsin(ϕ) p+ cos(ϕ) q

].

(21)

Combining (20) and (21) yields the heading rate (first elementof 2× 1 vector ωrot\3)

cos(ϕ) (ωY cos(ϕ) + ωR sin(ϕ))

− sin(ϕ) (−ωY sin(ϕ) + ωR cos(ϕ)) , (22)

and roll rate (the second element of ωrot\3)

sin(ϕ) (ωY cos(ϕ) + ωR sin(ϕ))

+ cos(ϕ) (−ωY sin(ϕ) + ωR cos(ϕ)) . (23)

Multiplying out brackets yields

ωY cos2 (ϕ) + ωY sin2 (ϕ)

+ ωR cos(ϕ) sin(ϕ)− ωR cos(ϕ) sin(ϕ)

= ωY (24)

and

ωR sin2 (ϕ) + ωR cos2 (ϕ)

+ ωY sin(ϕ) cos(ϕ)− ωY sin(ϕ) cos(ϕ)

= ωR, (25)

showing that the original signals remain unaltered in the noise-free case.


C. Effect of Carouseling to Scale Factor Errors

Assuming that the vehicle is negotiating a turn with constantheading rate ωY the gyro measurements can be expressed asp = ωY cos(ϕ) and q = −ωY sin(ϕ). By dividing the gyromeasurement to true value plus scale factor error we obtainp = ωp + αpωp. Then, after the transformation to the vehicleframe the scale factor errors (along the V-frame x axis, similaranalysis holds for the y-axis) appear as:

αV x = ωY (αp cos(ϕ) cos(ϕ) + αq sin(ϕ) sin(ϕ)). (26)

The integral of (26) is zero over full rotation only if the scalefactor errors are equal but of opposite sign, but this wouldbe a very unlikely condition for MEMS gyros. There existsCarouseling maneuvers that remove the scale factor errorsas well [17], but for the uncontrolled rotation in the wheelthese errors remain. However, as the main error component(gyro bias) is mitigated efficiently the bias state vector can bereplaced with scale-factor state of the ’virtual’ V-frame x-axisgyro. Keeping the error state vector low-dimensional meansthat methods such as map matching with particle filteringremain practical for implementation [37].

IV. WHEEL-IMU AND CONING MOTION

Obviously, the rotation rates with passenger cars are quitefast, and attitude processing must be optimized for fast motion.It is interesting to compare the motion of the wheel to thetraditional coning motion. From [38] we get angular rates forconing motion:

ω =

ω sin θ cosωt−ω sin θ sinωtω(1− cos θ)

. (27)

As noted in the previous section, the angular rates during aturn would be:

ωBIB =

ωY cos(rt)−ωY sin(rt)

r

, (28)

where the z-rotation rate r accumulates as rt = ϕ(t). Settingθ = π/2 in (27) yields

ω =

ω cosωt−ω sinωt

ω

. (29)

Furthermore, setting r = ωY in (28) yields

ωBIB =

ωY cos(ωY t)−ωY sin(ωY t)

ωY

, (30)

i.e. the wheel-gyro unit is on pure coning motion if thevehicle makes a very sharp turn. This is useful fact fororientation computation performance analysis, as we know thatthe rotation vector for this motion is ( [38], assuming suitableinitial orientation):

φ(t) =

π2 sin(ωY t)π2 cos(ωY t)

0

. (31)

For other turn rates the similar analysis can also be done, andby knowing the exact rotation vector enables the estimation oferrors caused by infrequent sampling rate, for example.

V. WHEEL PHASE ANGLE AND 2D TRAJECTORY

To observe the wheel phase angle ϕ(t) various methods canbe used. For example, switches or magnets can be used to findthe time for full cycle and ϕ(t) can then be estimated via in-terpolation. To keep the system self-contained and fully basedon inertial sensing, accelerometers can be used as follows. Theaccelerometer triad measures specific force acceleration [39]

aBSF = pB − gB, (32)

and assuming pB is known or a noise term one obtains

aBSF = −gB + n

= CBL

−g00

+ n.(33)

Furthermore, assuming the L frame coincides with the vehicleframe V

1

gaBSF =

(CV

B

)T 100

+ n,

(34)

and thus the first row of CVB in (12) is observable, and phase

angle ϕ(t) can be estimated. In addition to accelerometernoise the term n includes nuisance factors such as vehicleacceleration and vibration and shocks caused by the roadsurface. If the IMU is not perfectly mounted on the rotationaxis at least one accelerometer will also measure centripetalacceleration. This adds speed-dependent bias to term n. Tomaintain good estimate of ϕ extended Kalman filter [40] orsimilar filter should be implemented in a way that the effect ofvehicle acceleration is taken into account. For example, withϕ and ϕ as states, the measurement equation for the x and yaxes accelerometers would be

zk =

[axay

]=

[cos(ϕ) /g− sin(ϕ) /g

](35)

and Jacobian H for the filter would be

Hk =

[− sin(ϕ) /g 0− cos(ϕ) /g 0

]. (36)

Then constant velocity or similar model can be used to reducenoise [41]. As ϕ is a filter state, centripetal accelerationcorrection can be performed for the noise term n. Obviously,the lower the dynamics of the vehicle the more accuratelyϕ can be estimated. The inaccuracy in estimation of ϕ leadsto misalignment error in the V-frame which in turn acts asa scale factor error in navigation [36]. In practice there is atradeoff when the car is accelerating on braking: inaccuracyin estimation of ϕ would lead to scale factor errors but on theother hand accurate estimate would mean that gyro biases arenot completely cancelled in the transformation as (17) doesnot hold. Transformations that keep (17) valid on the expense

5

of scale factor errors might be preferable in the cases wherebias errors are large. One such transformation is cosine seriesfit to the accelerometer data over full wheel cycle [42].

The state ϕ should be treated as freely growing continu-ous state variable to handle situations where the vehicle isreversing and to carry information about distance traveled. Tocomplete the 2D trajectory, the position can be updated indiscrete steps (per full wheel rotation, for example) using

CLV = CL

V (ωrot×) (37)

and

p(k) = p(k − 1) + CLV

0ϕ(k − 1

2 )R0

. (38)

As in traditional DR, the initial position and heading areobtained from another navigation system.

VI. ERROR PROPAGATION

In this section we show that Allan variance, originallydeveloped for characterizing the stability of frequency stan-dards [43], is a convenient tool for error propagation analysisof the wheel IMU as well. The Allan variance is a functionof averaging time, computed as

σ2A(τ) =

1

2 (N − 1)

N−1∑j=1

(y(τ)j+1 − y(τ)j)2 (39)

where σ2A(τ) is the Allan variance for averaging time τ . The

values of y(τ)j and N are obtained by dividing the data yinto disjoint bins of length τ , y(τ)j is the average value ofthe jth bin, and N is the total number of bins [8], [44]. Asthe wheel IMU measurements are conveniently propagated asaggregated data blocks (accumulated to angles after each 2πwheel rotation), from (16) we can see that instead of squarewave function used in Allan variance [45] the averages areobtained using sine wave function:∑

ωrotb(t)∆t = ∑pn(t) cos(ϕ(t))∆t−

∑qn(t) sin(ϕ(t))∆t∑

qn(t) cos(ϕ(t))∆t+∑

pn(t) sin(ϕ(t))∆t0

.(40)

In (40) the bias terms of (16) are changed to time varyingprocesses pn(t) and qn(t), and the sum is taken over samplescollected during one full rotation of the wheel, u ≤ ϕ <u+ 2π. From statistical point of view (40) is quite close to

P = ∑pn(t)S (ϕ(t))∆t−

∑qn(t)S (ϕ(t))∆t∑

qn(t)S (ϕ(t))∆t+∑

pn(t)S (ϕ(t))∆t0

=

τ (pj − pj+1)− τ (qj − qj+1)τ (qj − qj+1) + τ (pj − pj+1)

0

.

(41)

where S(x) = sgn sin(x), n is the number of samples, 1∆t

is the sampling rate and time for full rotation is 2τ = n∆t.Assuming P is random vector (statistic of the observed gyro

noise sample), constant wheel speed and statistically indepen-dent and identical gyro noises for the two gyros we obtain

diag(V(P )) =

τ2[2V (pj − pj+1) 2V (pj − pj+1) 0

]T.

(42)

On the other hand, taking expected value of (39) for p-gyroyields

E(σ2A) =

1

2 (N − 1)

N−1∑j=1

E((pj − pj+1)

2), (43)

and furthermore assuming the gyro noise process is meanstationary we see that the variance term in (42) is also inthe expectation of Allan variance:

E(σ2A) =

1

2 (N − 1)

N−1∑j=1

V (pj − pj+1) . (44)

This connection between Allan variance and error propagationunder carouseling is very useful fact for error propagationanalysis, because MEMS gyro manufacturers typically provideAllan variance data in the specifications instead of detaileddescriptions of the underlying stochastic processes. Usinglinear algebra tools (supporting material, [46]) we can collectthe following observations for carouseling:

• effect of white noise component in the gyro errors (inrad/s) remain unaltered;

• effect of random walk in the gyro errors will be reducedsignificantly and the errors become uncorrelated in time;

• effect of 1/f noise (as defined in [8]) will be reducedand errors become negatively correlated between fullrotations;

• random constants disappear completely.The fact that error components to be integrated become

uncorrelated or negatively correlated between consecutivesamples means that carouseling obviously is very powerfultool in MEMS gyro error mitigation in navigation applications,where angular rates itself are not of interest but the integratedangles. To illustrate this, Fig.2 shows data from low-costMEMS gyro [47]. The unit was stationary on a table for onehour right after the turning the unit on. Directly integratingthe biased angular rates to angles would lead to very largeerrors, as shown in Fig. 3a. Fig.3b shows the angular errorsfor same gyro data, assuming the data has been collectedfrom constantly rotating wheel, processed using (13) andthen integrated to yaw and roll angles. This shows clearlythe strength of carouseling, errors due to biases disappearcompletely.

VII. TEST SETUP AND RESULTS

In the previous section example the gyro was not actuallyrotated and thus this result applies only for noise terms that areindependent of the signal. To observe how vibration of wheeland inaccuracies in CV

B transformation affect the result we per-formed a test run with passenger car with inertial measurementunit attached to the back (non-steerable axle) wheel. The test


0 10 20 30 40 50 60

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (minutes)

Ang

ular

Rat

e (d

eg/s

)

X−GyroY−Gyro

Fig. 2. Angular rates measured by L3GD20 gyroscope.

0 10 20 30 40 50 60

0

100

200

300

400

500

600

700

Time (minutes)

Ang

le (

deg)

X−GyroY−Gyro

(a)

0 10 20 30 40 50 60

−1.5

−1

−0.5

0

0.5

1

1.5

Time (minutes)

Ang

le (

deg)

YawRoll

(b)

Fig. 3. Gyro data integrated to angles, without (a) and with carouseling (b).

Fig. 4. Test vehicle

vehicle, shown in Fig. 4, included the following measurementunits:

• dual-frequency GPS receiver [48] with roof antenna [49]• inertial measurement unit with 3D accelerometer [50] and

3D gyroscope [47] fixed to the left rear wheel center. Thecentripetal acceleration effect to the IMU was verified tobe less than 1 m/s2 during the test.

• identical inertial measurement unit fixed to the centerconsole of the vehicle

GPS reference was computed with post-prosessing soft-ware [51] and with 3 km baseline the reported positionaccuracy was below 10 cm (2σ) during the test at 20 Hzoutput rate. IMU measurements were recorded to a memorycard and post-processed in Matlab [52]. As the sampling rateof the IMU was limited to 100 Hz, the car speed was keptbelow 20 km/h during the 10 minute run. The horizontal speedduring the test is shown in Fig. 5. Forward acceleration wasbelow 0.5 m/s2 during the test. The length of the driventrajectory was 1680 m. GPS course over ground solution andits smoothed derivative was used as heading and heading ratereference. The test area was relatively flat parking lot and thuszero was used as reference roll rate.

A. IMU data

Raw accelerometer data (Fig. 6) show cosine and sine of ϕin accordance with (35). As true ϕ reference was not avail-able, EKF measurement prediction is shown. The EKF errorestimate of the state was 4 deg (2σ). Raw gyro data in Fig. 7shows the cosine-modulated yaw rate. Shown GPS yaw rateestimate draws envelope curve as (20) explains. Zero crossingsof predicted measurements were used for cycle counting andCV

B was constructed after each cycle using sine and cosine fitsof the predicted measurements. This step ensures that constantbias is isolated from the heading computations. The cost ofnot using optimal estimate of ϕ is increased scale factor error.Balancing between these options should be based on vehicledynamics and selected sensor set.

After the coordinate frame transformation CVB , the signal

and bias switch roles as (24) and (16) suggest. In Fig. 8 the

7

0 100 200 300 400 5000

2

4

6

8

10

12

14

Time (s)

Hor

izon

tal s

peed

(km

/h)

DGPS

Fig. 5. Speed profile

−1 0 1 2 3

−1

−0.5

0

0.5

1

1.5

Time (s)

Spe

cific

For

ce (

g)

X−AccY−Acccos(φ)−sin(φ)

Fig. 6. Accelerometer data measured from the wheel and EKF predictionsof the measurements

150 160 170 180 190 200−20

−10

0

10

20

30

Time (s)

Ang

ular

Rat

e (d

eg/s

)

X−GyroY−GyroDGPS

Fig. 7. Gyro data measured from the wheel and DGPS yaw rate estimate

140 150 160 170 180 190 200−20

−15

−10

−5

0

5

10

15

20

Time (s)

Ang

ular

Rat

e (d

eg/s

)

Carouseled Yaw RateDGPS

Fig. 8. Gyro data after CVB transformation and DGPS yaw rate estimate

0 100 200 300 400 500 6000

20

40

60

80

100

120

140

160

Time (s)

Rol

l ang

le e

rror

(de

g)

Wheel InertialGyro+Odometer

Fig. 9. Errors in roll angle estimate

original bias is now oscillation around the yaw rate signal. Asthe heading update is done once per full rotation this zero-mean oscillation has no effect in the final result.

B. Heading and Roll Estimates

Expected scale factor errors are very small in roll anglesas the test surface was relatively flat. In this direction theadvantages of bias removal are clearly visible, as Fig. 9 shows.The cabin fixed gyroscope error accumulates to 154 degreeswhile the wheel-IMU error accumulates to 10 degrees. Bothvalues are larger than Fig. 3 suggests, probably due to vibrationand other environmental differences between laboratory andfield tests. In wheel-IMU heading estimation the scale factorerrors are clearly visible as Fig. 10 shows, the wheel-IMUerror accumulates to 20 degrees. This result is obtained usingnominal scale factors obtained form laboratory turn table tests.To separate the scale factor errors we calibrated the virtualgyro (the transformed gyro data shown in Fig. 8) scale factorusing DPGS heading up to time t = 300 s. This correction


100 200 300 400 500

0

10

20

30

40

50

60

70

Time (s)

Hea

ding

ang

le e

rror

(de

g)


Fig. 10. Errors in heading angle estimate, nominal scale factors

100 200 300 400 500

0

5

10

15

20

25

Time (s)

Hea

ding

ang

le e

rror

(de

g)

NominalCalibrated

Fig. 11. Errors in heading angle estimate

(α = 0.02) based on single scalar parameter improves theheading accuracy clearly as shown in Fig. 11. To estimatehow vehicle dynamics affect the heading estimate we droveone (curved) part of the trajectory with more accelerationsand braking (1-2 m/s2) and compared the heading estimateto constant speed result driven in the same curve. The results(Fig. 12 indicate that high dynamics indeed have strong effecton heading errors and adaptive data processing would berequired (accept more bias error but limit scale factor errors,for example).

C. Position Estimates

The results with nominal scale factors (Figs.13,14,15) showthat the wheel-IMU provides clearly more accurate solutionthan the conventional DR (result obtained replacing wheel-IMU heading estimates with cabin gyro heading estimates).The effect of gyro bias is clearly visible in the conventionalDR-positioning solution. After the scale factor calibration thewheel-IMU shows very accurate results, not exceeding 8 m in

0 10 20 30 40 50 600

2

4

6

8

10

12

Time (s)

Hea

ding

Err

or (

deg)

FastSlow

Fig. 12. Heading errors with high and low dynamics

−40 −20 0 20 40 60 80 100 120 140−60

−40

−20

0

20

40

60

80

Easting (m)

Nor

thin

g (m

)

DGPSWheel Inertial

Fig. 13. Uncalibrated wheel-IMU position results

the results shown in Fig. 16. In this case a fair comparison toconventional DR is difficult, as scale factor correction alonewould not fix the bias errors. Thus this kind of positioncomparison is omitted, but it is clear that for low-dynamicground vehicles requiring long unaided navigation solutionsthe wheel-IMU has many benefits over traditional DR.

It should be noted that the trajectory computation was per-formed without estimation of the sideslip angle, a factor thatbecomes relevant at high speeds [53, Ch.4.7]. However, as biaserror component is being removed, bringing in the necessaryscale factor correction information from aiding systems suchas GNSS or street maps [54], [55] is more straightforward.Obviously more tests are needed to verify the allowed speedlevels and applicability to general car navigation. Nevertheless,for slow-speed low-dynamic ground vehicles it is clear thatwheel-IMU can provide more accurate position solutions inlong missions than conventional DR-system equipped withsimilar gyros.

9

−20 0 20 40 60 80 100 120 140−50

0

50

100

Easting (m)

Nor

thin

g (m

)

DGPSGyro+Odometer

Fig. 14. Conventional DR position results

0 100 200 300 400 500 6000

10

20

30

40

50

60

70

80

Time (s)

2D E

rror

(m

)


Fig. 15. 2D position errors, nominal scale factors

0 20 40 60 80 100 120

−20

−10

0

10

20

30

40

50

60

70

80

Easting (m)

Nor

thin

g (m

)

DGPSWheel Inertial

Fig. 16. Calibrated wheel-IMU position results

VIII. CONCLUSIONS

In this article we have shown that placing the inertialmeasurement unit to the wheel of a ground vehicle hastwo clear advantages: the motion of the wheel removes theconstant bias of the gyroscopes and travelled distance canbe estimated from the accelerometer data. Especially for low-dynamic ground vehicles this approach is shown to be superiorto conventional dead reckoning with odometer when a low-cost MEMS-gyro is used to provide the heading information.The test results are obtained using vehicle driving slowly andon a relatively smooth surface and the use of accelerometerfor wheel phase angle tracking was fairly accurate for thispurpose. We acknowledge that for higher dynamics of thevehicle and when driving on gravel roads the accelerometerdata will be contaminated with significant centripetal andmotion-caused accelerations. For that purpose, use high-rangegyro with the sensitivity axis perpendicular to the wheel planeshould be considered to complement the accelerometer-based(bias-free) observations. To apply this method to passengercars at highway speeds would require an IMU with quite widebandwidth, and solving the challenges at high speeds remains afuture research topics. In addition, there is requirement to bringelectricity to the wheel and need for wireless data transfer, butthese are engineering challenges that can be solved and areactually studied actively [56]–[64]. As the major error sourceof MEMS gyros, angular errors due to bias instability areeliminated, the described method opens new applications forinertial navigation systems. In addition, there is a very largepotential for wheel-based sensing in general, not restricted toEarth surface [65] or navigation applications [66].


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Jussi Collin (M’11) received the M.Sc. and Dr.Tech.degrees from the Tampere University of Technology,Tampere, Finland, in 2001 and 2006, respectively,specializing in sensor-aided personal navigation. Heis currently a Senior Research Fellow with theDepartment of Pervasive Computing, Tampere Uni-versity of Technology. His research interests includestatistical signal processing and novel sensor-basednavigation applications. Dr. Collin is a Vice Chairof the IEEE Finland Section Signal Processing andCircuits & Systems Chapter.

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