Attitude estimation of a motorcycle via Unscented Kalman Filter

5th IFAC Symposium on Mechatronic Systems Marriott Boston Cambridge Cambridge, MA, USA, Sept 13-15, 2010

Attitude estimation of a motorcycle via Unscented Kalman Filter Stefano Corbetta*, Ivo Boniolo*, Sergio M. Savaresi* * Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza L. da Vinci, 32 20133 Milano, Italy, (e-mail: {corbetta,boniolo,savaresi}@elet.polimi.it). Abstract: In this work the problem of attitude estimation of a motorcycle is studied. A Kalman Filter

approach is used and the performance of the Extended Kalman Filter and the Unscented Kalman filter are compared. The estimation accuracy and robustness have been tested both with simulated and experimental data to show which are the advantages of the Unscented Transformation.

Keywords: attitude estimation, roll angle, data fusion, Extended Kalman Filter, Unscented Kalman Filter 1.

INTRODUCTION

The model is used in Section 4 to implement the EKF and UKF estimation process. In Section 5 a tuning process is applied to KF in order to obtain the best estimation accuracy of the roll angle of the vehicle. Section 6 and Section 7 are dedicated to the analysis of the estimation performance and the filters robustness. In Section 8 some conclusions are resumed.

In this paper, the Kalman filter approach is applied to solve the problem of attitude estimation of a motorcycle. In particular, a new description of the model is adopted and the Extended and Unscented Kalman Filter estimation methods are compared. This analysis is motivated by the fact that, especially in terms of robustness to the initial condition uncertainty, in many applications the UT (Unscented Transformation) guarantees better performance than the EKF which is based on linearization. In this paper the improvement of the performance due to the UT is tested estimating the attitude (roll angle and pitch angle) of a two wheeled vehicle.

2.

In this section, the simulation environment and the experimental set up used in this work are presented. All the algorithms presented in this paper are tested both on simulation and experimental data.

Reviewing the scientific literature, in (Gasbarro, Beghi, Frezza, Nori, & Spagnol, 2004) an interesting approach for estimating the whole vehicle trajectory is presented; it employs a vision system made by cameras complemented with MEMS accelerometers. In (Boniolo, Savaresi, & Tanelli, 2009) a method based on two angular rates is presented. This work is based on the application of the frequency separation principle and on the analysis presented in (Boniolo, Panzani, Savaresi, Scamozzi, & Testa, 2009).

The simulated data are provided by the MSC BikeSim® simulator and they are useful to test the implementation of the estimation algorithms. This tool is specifically designed to simulate the vehicle dynamics of motorcycles and it is based on a model developed and validated on the work described in (Sharp, Evangelou, & Limebeer, 2004).

Some contributions can be also found in the recent patent literature. For example, in (Hauser, Ohm, & Rol, 1995), (Schiffmann, 2003), (Schubert, 2005) some approaches are described, whose common purpose is to devise robust estimation methods with low cost (and small size) equipment. Specifically, (Schubert, 2005) focuses on roll-over detection mainly tailored to four-wheeled vehicles while (Hauser, Ohm, & Rol, 1995), (Schiffmann, 2003) focus on accelerometer-based estimation algorithms.

Fig. 1. Picture of the simulator environment. The principal signals provided by the simulator that are useful for the analysis are: •

The Kalman Filter method presented in this paper permits to obtain a lean angle error approximately of 2.6%. Moreover, implementing an UKF it is possible to achieve better robustness with respect to the initial condition uncertainty. The paper is organized as follow. In Section 2 the simulator and the experimental set up are presented. In Section 3 the process model is defined, with a state space representation. 978-3-902661-76-0/10/$20.00 © 2010 IFAC

SIMULATION ENVIROMENT AND EXPERIMENTAL SET-UP

• • • 511

Euler angles ZXY that describe the motorcycle orientation (see (Shuster, 1993), (Cocco, 1999), (Cossalter, 2002) and (Saccon, 2006)) Angular rates measured on the motorcycle COG (Centre Of Gravity) Accelerations measured on the motorcycle COG Velocity of the vehicle. 10.3182/20100913-3-US-2015.00029

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

The experimental data have been collected with a test conducted on the Santa Monica circuit in Misano Adriatico and on the Enzo Ferrari circuit in Imola (see Fig. 2).

the signals provided by the sensors mounted on the motorcycle. In avionic applications (e.g. (van der Merwe, Wan, & Julier, 2004), (Leffererts, Markley, & Shuster, 1982), (Crassidis & Landis Markley, 2003)), the vehicle dynamics can be fully measured and described combining positioning signals, angular rate signals, fixed-star measurements and magnetic field measurements. In four-wheeled vehicle applications different set of sensors are used to describe just a part of the vehicle dynamics. In a motorcycle application, the first problem that must be solved is the definition of a model that does not describe all the contributions that influence the inertial signals, but that is sufficiently accurate for estimating the attitude parameters of the vehicle.

Santa Monica

Enzo Ferrari

3.1. Measurements model The inertial sensors provide signals related with the linear accelerations and angular velocities measured on the body reference frame of the motorcycle. As depicted in Fig. 4, it is considered that the measurement axes are always aligned to the axes of the body reference frame, that is in accordance with the right hand rule with the x axis pointing forward and z axis pointing upward.

Fig. 2. Santa Monica circuit in Misano Adriatico and Enzo Ferrari circuit in Imola. The experimental vehicle has been instrumented with one strapdown Inertial Measurement Unit (IMU), that provides the angular velocities and the linear accelerations along three orthogonal axes, and two Hall-effect wheel encoders with 48 teeth, to measure the front and rear wheel rotational speed. To obtain the lean angle reference, a set of electro-optical sensors (Donati, 2004) have been used. As described in (Norgia, Boniolo, Tanelli, Savaresi, & Svelto, 2009), these sensors represent the unique solution to provide the attitude of the motorcycle respect to the asphalt. The reference roll angle is computed using two sensors mounted on each side of the motorcycle. Considering d1 and d2 as the distances measured by the left and the right sensor respectively and Ld as the mounting distance between the two sensors, the lean angle of the vehicle with respect to the asphalt is given by:  d1 − d 2  Ld

ϕ R = arctan

   

az, ωz

Z

z

Body Frame

ay, ωy Y X

ax, ωx

x

y

Inertial Frame

Fig. 4. Body reference frame definition and orientation of the measurement axes.

(1)

The motorcycle orientation is defined by the Roll-Pitch-Yaw rotational matrix, described in Equation (2). This representation describes the geometrical relation between the body reference frame and the inertial coordinate system where ϕ is the roll angle, ϑ is the pitch angle, ψ is the yaw angle of the vehicle and sε and cε are the sine and the cosine of the angle ε respectively.

Fig. 3. Instrumented vehicle with inertial measuring unit and electro-optical triangulators. The IMU has been mounted on the fuel tank while the electro-optical sensors have been fixed under the pedals, with a distance between them of 40 cm (Fig. 3). The signals are acquired on an ECU with a sampling frequency of 100 Hz and filtered with a second order filter at a frequency of 7 Hz. 3.

cϑ cψ − sϕ sϑ sψ cϑ sψ + sϕ sϑ cψ − cϕ sϑ    RZXY (ϕ ,ϑ ,ψ ) =  − cϕ sψ cϕ cψ sϕ   sϑ cψ + sϕ cϑ sψ sϑ sψ − sϕ cϑ cψ cϕ cϑ   

(2)

ω x  cϑ ϕ − sϑ cϕψ  ω  =  ϑ + s ψ  ϕ   y  ω z   sϑ ϕ + cϕ cϑψ   

(3)

In accordance with this description, the model of the gyroscope signals is defined as in (3) where ϕ is the roll angular rate, ϑ is the pitch angular rate and ψ is the yaw

PROCESS MODEL DEFINITION

angular rate (see (Shuster, 1993) and (Saccon, 2006)).

In a Kalman filter estimation process, the system description is related with the definition of the state variables and with 512

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

The model of the linear accelerations is very complex and it should describe all contributions due to linear translations and rotational movements around the centre of rotation of the vehicle. A simplified model can be obtained considering that: • •

The noises η x and η y are assumed to be uncorrelated white Gaussian noise with zero mean and covariance Q and R respectively as defined in (8).

[

center of rotation of the roll and pitch dynamics are the mounting position of the inertial unit; the lateral and vertical linear accelerations effects are neglected.

4.

After these simplifications the principal terms that affect the measured accelerations are the gravitational acceleration ( g

[

]

(4)

KALMAN FILTER APPLICATION TO NON LINEAR SYSTEM

The formulation of the EKF (see (Grewal & Andrews, 1993) and (Haykin, 2001)) is applied to (9) and the state vector can be estimated as:

1 0 P0 (+ ) =   0 1  xˆ k (− ) = xˆ k −1 (+ ) + f (xˆ k −1 (+ ), u k −1 )Ts yˆ k = g(xˆ k (− ), u k ) xˆ 0 (+ ) = [0 0]T

∂f (x, u ) ∂x x = xˆ k −1 (+ ),u =u k −1

Ac (xˆ , u, k − 1) =

x = [ϕ ϑ ]T

A(xˆ , u, k − 1) = e Ac (x,u,k −1)Ts ˆ

u = ω x ω y ω z g Vx V x

]T

]T ,

C (xˆ , u, k ) =

(5)

[

K k = Pk (− )C (xˆ ,u, k )T C (xˆ , u, k )Pk (− )C (xˆ , u, k )T + Rk , E

x = f (x, u ) + η x

xˆ k (+ ) = xˆ k (− ) + K k [y k − yˆ k ] Pk (+ ) = [I − K k C (xˆ , u, k )]Pk (− )

(6)

y = g(x, u ) + η y

where the functions f (x, u ) and g(x, u ) are non-linear functions defined as in (7) in which xi, i=1,2, and uj, j=1,…,6, are the i-th and j-th component of the vectors x and u respectively.

 c x 0 − s x 2 0 0 0 f (x, u ) =  2 u t x1 s x2 1 − t x1 cx2 0 0 0 − cx s x cx s x s x − s x cx u1 + cx cx u3  u4  2 1 2 1  1 2 2 1 2  0 cx1 − s x2 cx1 u1 + cx2 cx1 u3  u5  g(x, u ) =  s x1  c c s − s c − s c u + c c u  u   x1 x2 x2 x1 x2 x2 x1 1 x2 x1 3   6 

(

[

(

(

ηx = ηϕ ηϑ T ηy = η a x η a y ηa z

)

]

)

)

(10)

∂g(x, u ) ∂x x = xˆ k (− ),u =u k

Pk (− ) = A(xˆ , u, k − 1)Pk −1 (+ )A(xˆ , u, k − 1)T + Qk −1, E

the process is consequently described by:

]

(9)

y k = g(x k , u k ) + η y ,k

Define the state vector x, the input vector u and the output vector y as:

[

(8)

x k = x k −1 + f (x k −1 , u k −1 )Ts + η x,k

3.2. Process description

a y az

]

The Euler Forward (EF) method is applied to obtain the discrete time model of the non-linear process (6) as in (9), where Ts is the sampling time and the subscript k-1 and k indicate the sample at (k-1)-th and k-th instant respectively.

All terms that appear in (4) are measurable. In fact, inverting relation (3), the yaw rate ψ can be expressed as a function of the measured angular rates in the body coordinate system. Using simplified model (4) for the description of the linear accelerations the main error is committed in the representation of the acceleration acquired along the y axis of the vehicle.

[ y = [a x

[

In the field of attitude estimation, the description of the process is normally non-linear. The EKF and the UKF are widely used to solve the problem of state estimation. In this section, the two estimation methods are briefly presented.

), the longitudinal acceleration ( Vx ) and the centrifugal acceleration ( ψVx ). As a consequence, the acquired signals can be modelled as show in (4).  a x  − cϕ sϑ g + cϑ Vx + sϕ sϑψV x   a  =  sϕ g + cϕψV x   y   a z   cϕ cϑ g + sϑ Vx − sϕ cϑψV x   

]

E ηx ηxT = Q E ηy ηy T = R E ηx ηy T = 0 .

]

The linearized matrix Ac (xˆ , u, k − 1) is evaluated around the most recent estimation instant and can be approximated with the second order truncation of the Taylor series of

exponential matrix e Ac (xˆ ,u, k −1)Ts (Van Loan, 1978). The matrices Qk,E and Rk,E are the tuning parameters of the EKF and they represent the covariance of the noises η x and η y (7)

T

513

respectively. In the scientific literature, new non-linear implementations of the Kalman Filter are proposed (see (van der Merve & A. Wan, 2000) and (Grewal & Andrews, 1993)). The UKF presented in (van der Merwe, Wan, & Julier, 2004) is applied to the system in (9) to solve the problem of attitude estimation. The following parameters are defined:

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

•

α = 0.4 , β = 2 and k = 2 ;

•

the augmented state vector x ak = x Tk ηTx,k ηTy ,k

•

the weights Wi(m ) and Wi(c ) (where i = 0,...,2 L

5.

[

]

T

The definition of the noise covariance matrices acting on the state and output vectors is not a trivial problem. In accordance with the model description, the state transformation does not introduce any approximations, on the contrary, the output function is an approximated model, thus, the KF parameters can be constrained as Qk << Rk . Taking into account this constrain, to provide the best estimation performance via Kalman filter approach, a tuning process is applied to define the covariance matrices of the noise variables acting on the state and output vectors.

;

with L = 7 ) as defined in Equation (11).

λ

W0(m ) =

L+λ

λ

W0(c ) =

(11)

+1−α 2 + β

L+λ

1 , i = 1,...,2 L 2(L + λ )

Wi(m ) = Wi(c ) =

The tuning process is performed considering the MSE (Mean Square Error) of the roll angle estimation. This process is divided into two different steps.

where λ = α 2 (L + k ) − L . The UKF can been initialized as in (12).

[

xˆ 0a = xˆ T0 (+ ) 0 0

]

T

 P0 (+ ) 0  P0a =  0 Q0,U  0 0 

xˆ 0 (+ ) = [0 0]T

0   0  R0,U 

The first optimization is based on the definition of the covariance matrices as:

Qk = qI 2

(12)

1 0 P0 (+ ) =   0 1 

Rk = rI 3

{

xˆ ak −1 − γ Pka−1  

(

2L

∑W (

xˆ k (− ) =

i

i =0 2L

m)

i

(13)

X ix,k |k −1

c

x i , k |k −1

k

x i , k |k −1

0  1 Qk = q o   K 0 q  0 0 1 R k = ro 0 K r 0 0 0 1

)

− xˆ k (− )

T

i =0

Yk |k −1 yˆ k =

x k −1

2L

∑

v k −1

k

W (m ) Y i

i , k |k −1

2L

Pyk yk =

∑W ( ) (Y

Pxk yk =

∑W ( ) (X

i

i =0 2L

i

c

c

i , k |k −1

x i , k |k −1

)(

− yˆ k Yi ,k |k −1 − yˆ k

)(

)T

− xˆ k (− ) Yi , k |k −1 − yˆ −k

)

T

i =0

K k = Pxk yk Py−k1yk

By the analysis of the Fig. 5, it is clear that for both filers:

xˆ k (+ ) = xˆ k (− ) + K k [y k − yˆ k ] Pk (+ ) = Pk (− )

(17)

The KF parameters are tuned both in simulated and in real environment. In the experimental context, all the runs in Misano and Imola are considered, for each of them the estimation error and the corresponding MSE are calculated and the mean of the error variance is minimized. With this procedure it is ensured that the tuning of the filters is not biased by the considered run. In Fig. 5, the average MSE of the estimation error for the experimental test is depicted as a function of the parameters of the estimation process.

(14)

i =0

(16)

The second tuning step is based upon the focus of the application, that is the estimation of the lean angle and upon the model error of the accelerations description. In fact, the model of the acquired accelerations, better fits the signals measured along the x and z axis of the body reference frame, consequently, the variance of the noise signals can be differentiated as in (17), where the parameters qo and ro are fixed and the parameters Kq and Kr are defined so that MSE of the roll angle estimation is minimized.

)

∑W ( ) (X − xˆ (−))(X = g (X , u , X )

Pk (− ) =

}

qo , ro : min mse(ϕ − ϕˆ )

The state vector is estimated as in (14) in which the UT presented in (van der Merve & A. Wan, 2000) has been applied to the discrete non-linear dynamical system (9). X ak |k −1 = f X kx −1 , u k −1 , X kw−1

(15)

q = qo , r = ro

k ∈ [1,..., ∞] ,

calculating the sigma points as in (13) where γ = L + λ .

xˆ ak −1 + γ Pka−1

.

where q and r are such that:

The state variables can be estimated for

X ak −1 = xˆ ak −1 

KALMAN FILTER TUNING

•

− K k Pyk yk K kT

The best estimation accuracy is reached considering

[

]

q ∈ 10 −5 , 10 −7 and in this range the sensitivity to r

As for the EKF, the matrices Qk,U and Rk,U are the tuning parameters of the UKF and they represent the covariance of the noises η x and η y respectively.

•

is very low (see Fig. 5a); The best performances are reached considering

K q = 1 and K r = 1 (see Fig. 5b).

514

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

EKF 14

Analyzed Condition

UKF

mse(ε)

Simulation

Estimation Algorithm

12 10

Experimental test

EKF

JϕR

2.1%

JϕR

2.6%

UKF

JϕR

2.0%

JϕR

2.6%

8 6

4 1e-10

1e-9

1e-8

1e-7

1e-6

q

1e-5

1e-4

1e-3

50

150

250

350

450

550

650

750

850

Table 1: EKF and UKF estimation performance comparison.

r

(a)

ϕR ϕˆUKF ϕˆ EKF Est. EKF

Reference Est. UKF

60

EKF

7

40

6.5

Angle [°]

J mse(ε) MSE

20

6 5.5 5

4.5 1000

-20

100

10

1

UKF 1e-1

1e-2

KKq

1e-3

1

10

(b)

50

20

100

200

-40

500

-60

KKr

360

Fig. 5: Optimization of the KF parameters for experimental data (minimization of the MSE of the estimation error). 6.

SIMULATION AND EXPERIMENTAL RESULTS

380

7.

390

400 Time [s]

420

410

430

440

ROBUSTNESS ANALYSIS

To conclude the estimation performance comparison between the EKF and UKF, the filters robustness is taken into account. In particular, it is studied the effect of model errors and initialization uncertainty. To analyze the drop off of the performance both the experimental data collected on the Imola and Misano circuits are considered.

ϕ ϕ ϕˆ EKF Est. EKF

Reference R

50

370

Fig. 7: Estimation of the roll angle applying the EKF and UKF approaches to experimental data (one lap on the Imola circuit).

Now consider the performance comparison between the EKF and UKF approaches. In Fig. 6 the estimation performance related to simulation data are depicted showing the estimated ϕˆ EKF and ϕˆUKF and the reference roll angle φR. ˆUKF Est.UKF

20

EKF

0 15 10

J

/J

ESR ESR

0

Angle [°]

0

-50

5

0

50

Time [s]

100

150

0 30

Fig. 6: Estimation of the roll angle applying the EKF and UKF approaches to simulation data (one lap on simulated track).

20 15

20

10

15

10

∆ϕE 0 φ

5

5 0

0

∆E ϑ0 θ

Fig. 8: Robustness to mounting attitude estimation error of the EKF and the UKF.

In Fig. 7, an experimental test on the Imola circuit is considered and φR is compared to ϕˆ EKF and ϕˆUKF . It is shown that the EKF and the UKF have the same estimation accuracy.

First of all, it is considered that the IMU has been aligned to the body coordinate system, with a simulated mounting attitude error (∆ϕ o , ∆ϑo ,0 ) . The reduction of the roll angle

This conclusion is also confirmed by the results in Table 1 where the Error to Signal Ratio (ESR) defined in (18) is compared.

J ϕ R = mse(ϕˆ − ϕ R ) / mse(ϕ R )

UKF 25

estimation accuracy is evaluated as ESR ESR0 , where ESR and ESR0 are the estimation performance with and without model errors respectively. In Fig. 8 the loss function trend is depicted both for the EKF and the UKF and it can be deduced that the two observers have the same robustness degree and

(18) 515

Mechatronics'10 Cambridge, MA, USA, Sept 13-15, 2010

that the roll angle error ∆ϕ o is much more critical then the error ∆ϑo . It can be concluded that the proposed approaches based on Kalman filtering are robust with respect to uncertainty of the pitch angle mounting estimation.

Cocco, G. (1999). Motorcycle Design and Technology. Milan: Giorgio Nada Editore.

Now consider the robustness to the initial conditions. In Fig. 9 the estimation error of the EKF and of the UKF are compared for different values of the estimation initialization ϕˆ (0) . It is highlighted that the UKF has a better robustness to initial conditions than the EKF. In this framework, a realistic value of the lean angle initialization error is ±55° and, in this range, it is clear that the less robustness of the EKF implementation is not a critical aspect for the roll angle estimation performance.

Crassidis, J. L., & Landis Markley, F. (2003). Unscented filtering for spacecraft attitude estimation. AIAA Guidance, Navigation and COntrol Conference and Exhibit, (p. 1-9). Austin.

Cossalter, V. (2002). Motorcycle Dynamics. Milwakee: Race Dyanmics.

Donati, S. (2004). Electro-Optical Instrumentation - Sensing and Measuring with Lasers. Prentice Hall. Gasbarro, L., Beghi, A., Frezza, R., Nori, F., & Spagnol, C. (2004). Motorcycle trajectory reconstruction by integration of vision and MEMS accelerometers. Proceedings of the 43th Conference on Decision and Control, (p. 779–783). Nassau.

20

Grewal, M. S., & Andrews, A. P. (1993). Kalman filtering: theory and Practice. Prantice-Hall.

Estimation error [°]

0 -20 -40

Hauser, B., Ohm, F. H., & Rol, G. (1995). Motorcycle ABS using horizontal and vertical acceleration sensors. US Patent 5,445,44.

ϕˆ (0)0°= 0° ϕˆ (0)30° = 30° ϕˆ (0)60° = 60° ϕˆ (0)89° = 89°

-60

Haykin, S. (2001). Kalman Filtering and Neural Networks. John Wiley & Sons Inc.

-80 0

10

20

30

40

50 Time [s]

60

70

80

90

100

(a)

Leffererts, E. J., Markley, F. L., & Shuster, M. D. (1982). Kalman Filtering for Spacecraft Attitude Estimation. Journal of Guidance, Control and Dynamics , 5 (5), 417-428.

20

Estimation error [°]

0

Norgia, M., Boniolo, I., Tanelli, M., Savaresi, S. M., & Svelto, C. (2009). Optical Sensors for Real-Time Measurement of Motorcycle Tilt Angle. IEEE Transactions on Instrumentation & Measurement , 58 (5), 1640-1649.

-20

ϕˆ (0)0°= 0° ϕˆ (0)30° = 30° ϕˆ (0)60° = 60° ϕˆ (0)89° = 89°

-40 -60 -80 0

10

20

30

40

50 Time [s]

60

70

80

90

Saccon, A. (2006). Maneuver Regulation of Non-Linear Systems: The Challenge of Motorcycle Control. Ph.D Thesis, Università degli studi di Padova, Padova.

100

(b)

Schiffmann, J. K. (2003). Vehicle roll angle estimation and method. European Patent EP1,346,883.

Fig. 9: Robustness to the initial conditions. (a) EKF roll angle estimation error. (b) UKF roll angle estimation error. 8.

Schubert, P. J. (2005). Vehicle rollover sensing using angular accelerometer. US Patent US 1,139,83.

CONCLUSIONS

Sharp, R. S., Evangelou, S., & Limebeer, D. J. (2004). Advances in the modelling of motorcycle dynamics. Multibody System Dynamics , 12, 251–283.

In this paper the problem of estimating the attitude angles of a motorcycle is presented using a Kalman filter approach. In detail, the EKF and UKF implementation are applied. The estimation performance comparison highlights that the UKF and the EKF have the same estimation accuracy. The main advantage of the UKF is a major robustness to the initialization uncertainty that is due to the non-linear computation of the covariance matrix of the state estimation error. 9.

Shuster, M. D. (1993). A Survey of Attitude Representations. The Journal of the Astronautical Sciences , 41 (4), 439–517. van der Merve, R., & A. Wan, E. (2000). The Unscented Kalman Filter for Nonlinear Estimation. Symposium 2000 on Adaptive Systems for Signal Processing, Communication and Control (AS-SPCC). Alberta.

REFERENCE

van der Merwe, R., Wan, E. A., & Julier, S. I. (2004). Sigmapoint Kalman filters for nonlinear estimation and sensorfusion: Applications to Integrated Navigation. Proceedings of the AIAA Guidance, Navigation & Control Conference.

Boniolo, I., Panzani, G., Savaresi, S. M., Scamozzi, A., & Testa, L. (2009). On the Roll Angle Estimate via Inertila Sensors: Analysis of the Principal Measurament Axes. DSCC 2009.

Van Loan, C. (1978). Computing integrals involving the matrix exponential. IEEE Transactions on Automatic Control , 23 (3), 395-404.

Boniolo, I., Savaresi, S. M., & Tanelli, M. (2009). Roll angle estimation in two-wheeled vehicle. Control Theory & Applications, IET , 3 (1), 20-32. 516