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Vehicle System Dynamics: InternationalJournal of Vehicle Mechanics andMobilityPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/nvsd20

Estimation of land vehicle roll andpitch anglesH. Eric Tseng a , Li Xu a & Davor Hrovat aa Research and Advanced Engineering, Ford Motor Company ,Dearborn, MI, 48124, USAPublished online: 11 Apr 2007.

To cite this article: H. Eric Tseng , Li Xu & Davor Hrovat (2007) Estimation of land vehicle roll andpitch angles, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility,45:5, 433-443, DOI: 10.1080/00423110601169713

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Vehicle System DynamicsVol. 45, No. 5, May 2007, 433–443

Estimation of land vehicle roll and pitch angles

H. ERIC TSENG, LI XU* and DAVOR HROVAT

Research and Advanced Engineering, Ford Motor Company, Dearborn, MI 48124, USA

In this article, two kinematics-based observers are proposed to estimate the vehicle roll and pitchangles by using an inertial measurement unit. The observers are mathematically proven to be stable ifthe vehicle yaw rate is not zero. With a design variation of the observer gains, the estimated roll or pitchangle is shown to further asymptotically converge to the true value, eliminating possible errors causedby the biases of the acceleration signals. Simulation results show that accurate estimation of bothpitch and roll angles can be achieved without the help of external sensors such as global positioningsystems, either by using the accelerometer-based reference pitch or roll angle as the maneuver varies,or by using an observer with zero steady-state error property.

Keywords: Vehicle dynamics control; State estimation; State observer

1. Introduction

In recent years, many vehicle chassis control systems have been developed to enhance vehiclestability and handling performance in critical dynamic situations. Such control systems includeyaw stability control systems [1] (also known as electronic stability control [2]), roll stabilitycontrol systems [3], integrated vehicle dynamic control systems [4, 5], etc. In these systems,knowledge of the vehicle roll and pitch attitude is very important for satisfactory controlperformance. For example, in yaw stability control systems, the effect of vehicle body roll andpitch, as well as the dynamically changing road bank and road grade is significant, becausethey directly lead to the gravity components measured by the accelerometers [6, 7], which inturn affect the accuracy of vehicle state estimation such as longitudinal velocity and sideslipangle. In roll stability control systems, roll angle is one of the most important variablesused to construct feedback pressure command to combat the detected roll instability [3].Hence a successful vehicle dynamics control system must involve an accurate determinationof the vehicle roll and pitch attitude. However, for production vehicle applications, the vehicleattitude must be estimated, as cost and availability issues often exclude the possibility of directmeasurements.

Over the last few years, a lot of effort has been devoted to the estimation of vehicle rollor road bank angle. Fukada [7] addressed the problem in his work on vehicle sideslip angleestimation. He derived the lateral tire force from the lateral acceleration measurement andfrom a tire model independently. Then the two versions of lateral tire force are compared toobtain the road bank angle. However, with this approach, the sideslip angle and road bank

*Corresponding author. Email: lxu9@ford.com

Vehicle System DynamicsISSN 0042-3114 print/ISSN 1744-5159 online © 2007 Ford Motor Company (FORD)

http://www.tandf.co.uk/journalsDOI: 10.1080/00423110601169713

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434 H. Eric Tseng et al.

estimation affect each other, and the stability of the algorithm has not been addressed. Inref. [8], Nishio et al. calculated the road bank angle by comparing the lateral accelerationmeasurement to the centripetal acceleration (i.e. the product of yaw rate and vehicle speed).Since the lateral accelerometer also measures the derivative of the lateral velocity, this approachis subject to large error. To overcome this technical difficulty, Tseng [6] proposed a methodwhich practically decouples the road bank induced gravity from the derivative of the lateralvelocity. In ref. [9], the road bank estimation algorithm proposed in ref. [6] was simplified, andthe fidelity of the estimate was further improved by decomposing the estimate into multiplefrequency layers and applying a different weighting factor at each layer. In refs. [10, 11], aglobal positioning system (GPS) was utilized to aid the road bank angle estimation, by meansof Kalman filter and disturbance observer, respectively. In ref. [12], a method which furtherseparates road bank from vehicle body roll was presented. While GPS helps to improve theestimation accuracy in an open sky environment, the performance generally deteriorates whenthe satellite signals bounce off reflective surfaces, or fewer than three or four satellites can be‘seen’ (i.e. driving through a tunnel). Another, perhaps more fundamental, limitation is thatGPS devices are not at all common and/or cost effective in current production vehicles.

This article investigates the use of inertial measurement units (IMU) in estimating the vehicleroll and pitch angles. A typical IMU consists of three accelerometers and three gyroscopesmounted in a set of three orthogonal axes. The IMU measures the acceleration and the rotationrate of the vehicle in all three dimensions at a high sampling rate, typically at frequencieshigher than 100 Hz. Recent progress in the development of micro-electro mechanical systemshas made it possible to put IMU on production vehicles because of their small size, low cost,and ruggedness.

Two kinematics-based observers are proposed to estimate vehicle roll and pitch angles.A reference roll/pitch angle is first obtained from longitudinal and lateral accelerations, yawrate, and vehicle longitudinal velocity. This reference angle captures the low-frequency com-ponent of the vehicle attitude. To captures the high frequency component, a kinematics modelbased state observer is applied to integrate the reference angles with additional roll and pitchrate signals. The observer is mathematically proven to be stable if the yaw rate is non-zero.However, the estimates at this stage may be biased due to the offset of the accelerometer signals.A variation of the state observer design further allows the estimate to asymptotically convergeto the true roll or pitch angle and eliminates any possible bias contained in the reference angles.

2. Motion of a land vehicle

Without loss of generality, it is assumed that the IMU is placed at the vehicle center of gravity,and there is no misalignment with respect to the vehicle body frame. Using the kinematicrelationship between IMU output and the derivatives of the Euler angles, and assuming thatthe rotation rate of the earth is negligible, the equations of vehicle motion can be written as [13]

φ̇ = p + (q · sin φ + r · cos φ) · tan θ (1)

θ̇ = q · cos φ − r · sin φ (2)

ψ̇ = (q · sin φ + r · cos φ) · sec θ (3)

ν̇x = ax + r · νy − q · νz + g · sin θ (4)

ν̇y = ay − r · νx + p · νz − g · sin φ · cos θ (5)

ν̇z = az + q · νx − p · νy − g · cos φ · cos θ (6)

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Land vehicle roll and pitch angles 435

in which �ν = [νx, νy, νz]T represent linear velocities, �ω = [p, q, r]T represent angular rates,and �a = [ax, ay, az]T represent linear accelerations, all in body frame; �θ = [φ, θ, ψ]T

represent the three Euler angles, roll, pitch, and yaw, respectively; g is the gravitational constantwhich is assumed to be known.

For vehicle dynamics control purposes, the Euler yaw angle ψ (or the heading angle)is not required. Furthermore, since the vehicle is constrained to move on the road surface,the vertical velocity νz is normally very small and can be neglected. Thus the estimationalgorithms proposed in the following are based on equations (1), (2), (4) and (5) only, withνz ≈ 0. Theoretically, the vehicle attitude can be computed via mathematical integration, ifthe initial condition is known and �ω is measured by the gyro sensors. In practice, however,direct integration tends to drift due to sensor bias and inevitable numerical errors. Absolutesensors such as GPS are often required to eliminate errors constantly due to gyro integration.

3. Reference roll and pitch angles

In this section, rough estimates of the vehicle pitch and roll angles are presented by utilizingthe sensors typically available on vehicles equipped with electronic stability control or yawstability control. These sensor signals include longitudinal and lateral accelerations, yaw rate,wheel speeds, and steering wheel angle.

As seen from equations (4) and (5), vehicle pitch and roll angles can be calculated if νx , ν̇x ,νy and ν̇y were available:

θ = arcsin

(ν̇x − ax − r · νy

g

)(7)

φ = arcsin

(ay − r · νx − ν̇y

g · cos θ

)(8)

Although it is possible to obtain fairly accurate νx and thus ν̇x from wheel speed sensors whenwheel slip is small, νy and ν̇y are generally not available on current production vehicles. Thusequations (7) and (8) cannot be implemented. Fortunately, during many maneuvers, νy or ν̇y isfairly small and can be neglected. In such cases, the so-called reference pitch and roll angles,θref and φref , respectively, are given by the following equations:

θref = arcsin

( ˙̂νw − ax

g

)(9)

φref = arcsin

(ay − r · ν̂w

g · cos θref

)(10)

where ν̂w represents the wheel speed based longitudinal velocity calculation, and ˙̂νw itsderivative. In practice, in addition to the approximation errors, the reference signals fromequations (9) and (10) are also affected by the low frequency biases in acceleration measure-ments ax and ay . Similar to the method in ref. [9], by exploiting the information from a simplevehicle model, the results of equations (9) and (10) may be further improved and made robustto vehicle maneuvers, road disturbances, and road/tire friction coefficient. The details areomitted due to space limitations.

In the next two sections, two state observers and their variations are proposed to integratethe reference angles with the roll and pitch rate signals.

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4. Attitude observer utilizing rate sensors

Equations (1) and (2) can be rewritten as

φ̇ = p + r · θ + q · sin φ · tan θ + r · (cos φ · tan θ − θ)

θ̇ = q − r · φ + q · (cos φ − 1) + r · (φ − sin φ)(11)

In matrix form, the above equation becomes[φ̇

θ̇

]=

[0 r

−r 0

]·[φ

θ

]+

[p

q

]+

[�φ

�θ

](12)

where �φ = q · sin φ · tan θ + r · (cos φ · tan θ − θ) and �θ = q · (cos φ − 1) + r · (φ −sin φ). Assuming �φ ≈ 0 and �θ ≈ 0, equation (12) is simplified as[

φ̇

θ̇

]≈

[0 r

−r 0

]·[φ

θ

]+

[p

q

](13)

This simplified relation among yaw rate, pitch rate, roll rate, Euler roll, and pitch angle allowsus to formulate the problem as state estimation of a linear time-varying system and constructobservers using modern linear control theory.

In the following, the reference roll angle described in section 3 is used as a pseudo-measurement, and an observer is designed as [14]:⎡

⎣ ˙̂φ

˙̂θ

⎤⎦ =

[0 r

−r 0

]·[φ̂

θ̂

]+

[p

q

]+

[2α|r|

(α2 − 1)r

]· (φref − φ̂) (14)

where φ̂ and θ̂ are the estimated roll angle and pitch angle, respectively, and α is a positiveconstant. Suppose the reference signal is fairly accurate, i.e. φ ≈ φref . The corresponding errordynamics are thus given by[ ˙̃

φ

˙̃θ

]=

[0 r

−r 0

]·[φ̃

θ̃

]−

[2α|r|

(α2 − 1)r

]· φ̃ (15)

where φ̃ = φ − φ̂, θ̃ = θ − θ̂ . Consider a positive definite function

V = 1

2α2φ̃2 + 1

2θ̃2 (16)

Its derivative along the trajectory of equation (15) is

V̇ = α2φ̃ · ˙̃φ + θ̃ · ˙̃

θ = −2α3|r|φ̃2 (17)

If yaw rate r is a non-zero constant, then φ̃ → 0 and θ̃ → 0 as t → ∞ by LaSalle’s invariantset theory. In case the assumption φ ≈ φref is not valid but instead θ ≈ θref , one can use thefollowing complementary observer as an alternative:⎡

⎣ ˙̂φ

˙̂θ

⎤⎦ =

[0 r

−r 0

]·[φ̂

θ̂

]+

[p

q

]+

[2α|r|

(α2 − 1)r

]· (θref − θ̂ ) (18)

and the analysis can be done in similar fashion as in equations (15)–(17). In practice, it isusually possible to assess/predict the quality of reference roll/pitch angle based on vehicle

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Land vehicle roll and pitch angles 437

maneuvers [9]. If one of the reference angles is considered to be credible, then both roll andpitch angles can be estimated based on the above approaches to take advantage of the ratesensor signals and reference angle information.

However, if there is a constant bias φ̃ref in the reference roll angle, i.e. φref = φ − φ̃ref , theerror dynamics then become

[ ˙̃φ

˙̃θ

]=

[−2α|r| r

−α2r 0

]·[φ̃

θ̃

]+

[2α|r|

(α2 − 1)r

]· φ̃ref (19)

and the conclusion drawn from equation (17) no longer holds. Assume ˙̃φ ≈ 0 and ˙̃

θ ≈ 0 inequation (19) give the steady-state estimation errors:

φ̃ =(

1 − 1

α2

)· φ̃ref (20)

θ̃ =(−2|r|

α

)· φ̃ref (21)

The bound of φ̃ and θ̃ can therefore be derived since the conservative bound of φ̃ref canusually be obtained based on vehicle maneuvers and sensor specifications. Similar analysisalso applies to equation (18).

In summary, the proposed observer provides the value and uncertainty bound of the pitchangle estimate if those of the reference roll angle are available, and vice versa.

5. Attitude observer with no steady-state error

As seen from equation (19), the estimation of the steady-state roll and pitch angles is affected bythe errors of the reference angles – the ‘measurement’used for observer feedback. Consideringa special case of equation (14) by assigning α = 1 gives⎡

⎣ ˙̂φ

˙̂θ

⎤⎦ =

[0 r

−r 0

]·[φ̂

θ̂

]+

[p

q

]+

[2|r|

0

]· (φref − φ̂) (22)

Equations (14) and (22) provide the corresponding error dynamics

[ ˙̃φ

˙̃θ

]=

[−2|r| r

−r 0

]·[φ̃

θ̃

]+

[2|r|

0

]· φ̃ref (23)

It is easy to check the steady-state roll angle estimation error φ̃ → 0. That is, the accuracyof the steady-state roll angle estimate is no longer affected by error of the reference rollangle. A similar observer can be used to improve the pitch angle estimation and remove thesteady-state error as well:⎡

⎣ ˙̂φ

˙̂θ

⎤⎦ =

[0 r

−r 0

]·[φ̂

θ̂

]+

[p

q

]+

[2|r|

0

]· (θref − θ̂ ) (24)

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438 H. Eric Tseng et al.

6. Simulation results

In order to illustrate the proposed estimation algorithms, simulation results are obtained basedon the kinematics described by equations (1) and (2). In the simulation, a simple scenario inwhich a vehicle is driven down a spiral ramp at a constant speed is used (yaw rate r = 0.5 rad/s).

0 5 10 15 20−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time (s)

road bank (rad)estimate (rad)

Figure 1. Road bank estimation (φref = φ).

0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

road grade (rad)estimate (rad)

Figure 2. Road grade estimation (φref = φ).

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0 5 10 15 20−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time (s)

road bank (rad)estimate (rad)

Figure 3. Road bank estimation (φref �= φ).

The ramp is assumed to have a constant grade (θ = 0.1 rad), and its bank angle is zero,i.e. φ = 0.

The observer given by equation (14) in which α = 1.2 is first used to estimate the road bankand grade. The initial estimation errors are assumed to be φ̃(0) = −0.1 rad and θ̃ (0) = 0.2 rad.When the reference roll angle is accurate (φref = 0), it is seen from figures 1 and 2 that both

0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time (s)

road grade (rad)estimate (rad)

Figure 4. Road grade estimation (φref �= φ).

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440 H. Eric Tseng et al.

0 5 10 15 20−4

−2

0

2

4

6

Time (s)

Rol

l Ang

le (r

ad)

0 5 10 15 20−10

−5

0

5

10

Time (s)

Pitc

h A

ngle

(rad

) road gradefrom eqn. (22)from eqn. (14)

road bankfrom eqn. (22)from eqn. (14)

Figure 5. Steady-state estimation (φref �= φ).

0 2 4 6 8 10 12 14 16 18−250

−200

−150

−100

−50

0

50

Ste

erin

g W

heel

Ang

le (d

eg)

Time (s)

0 2 4 6 8 10 12 14 16 185

10

15

20

25

Veh

icle

Spe

ed (m

/s)

Time (s)

Figure 6. Steering wheel angle and vehicle speed.

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Land vehicle roll and pitch angles 441

0 5 10 15 20−20

−10

0

10p

(deg

/s)

0 5 10 15 20−5

0

5

10

q (d

eg/s

)

0 5 10 15 20−40

−20

0

20

r (de

g/s)

0 5 10 15 20−5

0

5

a x (m/s

2 )

0 5 10 15 20−10

−5

0

5

a y (m/s

2 )

Time (s)0 5 10 15 20

5

10

15a z (m

/s2 )

Time (s)

Figure 7. Sensor signals.

0 2 4 6 8 10 12 14 16 18−8

−6

−4

−2

0

2

Rol

l Ang

le (d

eg)

Time (s)

0 2 4 6 8 10 12 14 16 180

0.5

1

1.5

2

2.5

Pitc

h A

ngle

(deg

)

Time (s)

MeasurementEstimation

MeasurementEstimation

Figure 8. Estimation versus measurement.

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442 H. Eric Tseng et al.

φ̂ and θ̂ converge to their true values, respectively. However, when the reference angle is notaccurate (φref = 0.05 rad is used in the simulation), the estimation errors no longer convergeto zero (see figures 3 and 4).

To reduce the roll angle estimation error when the roll angle reference is not accurate, thesteady-state observer (22) is applied and the results are given in figure 5. It shows that whilethe pitch estimation error remains large, the roll angle estimate converges to the true value.That is, the accuracy of the steady-state roll angle estimate is no longer affected by error ofthe reference roll angle, which agrees with the analysis in section 5. The estimates generatedby observer (14) are also plotted out (dotted lines) for comparison.

Post-processing results using vehicle testing data are also obtained. A J-turn maneuver ona flat surface is given as an example. The observer is based on equation (18), and the initialvalues are computed using reference signals (9) and (10). Figure 6 shows the steering wheelangle input and the vehicle speed, and figure 7 shows the inertia sensor signals. In figure 8,the estimated roll and pitch angles are compared to the measurements from an integratedGPS/INS system, which indicates fairly small estimation errors during the maneuver.

7. Conclusions

This article investigates the estimation of a land vehicle’s roll and pitch attitude by using anIMU and the kinematics model. It is observed from the study that accurate estimates of bothpitch and roll angles can be achieved, either by switching the utilization of accelerometer-based reference pitch or roll angle as the maneuver varies, or by using an observer with zerosteady-state error property to incorporate the rate sensor information with the accelerometerinformation.

References

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[2] Van Zanten, A., Erhardt, R., Pfaff, G., Kost, F., Hartmann, U. and Ehret, T., 1996, Control aspects ofthe Bosch-VDC. Paper presented at the Proceedings of the 3rd International Symposium on Advanced VehicleControl (AVEC), Aachen, Germany, pp. 573–608.

[3] Brown, T. and Rhode, D., 2001, Roll over stability control for an automotive vehicle. United States Patent.Patent Number: 6324446.

[4] Abe, M., 1996, On advanced chassis control technology for vehicle handling and active safety. Paper presented atthe Proceedings of the 3rd International Symposium on Advanced Vehicle Control (AVEC), Aachen, Germany,pp. 1–12.

[5] Nagai, M. and Yamanaka, S., 1996, Integrated control law of active rear wheel steering and direct yaw momentcontrol. Paper presented at the Proceedings of the 3rd International Symposium on Advanced Vehicle Control(AVEC), Aachen, Germany, pp. 451–470.

[6] Tseng, H.E., 2000, Dynamic estimation of road bank angle. Paper presented at the Proceedings ofthe 5th International Symposium on Advanced Vehicle Control (AVEC), Ann Arbor, Michigan, USA,pp. 421–428.

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[9] Tseng, H.E. and Xu, L., 2003, Robust model-based fault detection for roll rate sensor. Paper presented at theProceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, pp. 1968–1973.

[10] Ryu, J., Rossetter, E. and Gerdes, J., 2002, Vehicle sideslip and roll parameter estimation using GPS. Paperpresented at the Proceedings of the 6th International Symposium on Advanced Vehicle Control (AVEC),Hiroshima, Japan, pp. 373–380.

[11] Hahn, J., Rajamani, R., You, S. and Lee, K., 2002, Road bank angle estimation using disturbance observer.Paper presented at the Proceedings of the 6th International Symposium on Advanced Vehicle Control (AVEC),Hiroshima, Japan, pp. 381–386.

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[13] Greenwood, D., 1988, Principles of Dynamics (2nd ed) (Englewood Cliffs, NJ: Prentice-Hall).[14] Ungoren, A.Y., Peng, H. and Tseng, H.E., 2002, Experimental verification of lateral speed estimation methods.

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