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MODELING OF THE DYNAMICS OF A GYROSCOPE THROUGHVOLTERRA SERIES

Oscar Scussel

Amanda Alves Silva

Samuel da Silva

[email protected]

[email protected]

[email protected]

UNESP - Univ Estadual Paulista, Faculdade de Engenharia de Ilha Solteira, Departamento de En-genharia Mecanica, Av. Brasil, n56, Centro, Zip-Code: 15385-000, Ilha Solteira, SP, Brasil

Abstract. The gyroscopes have been broadly discussed and studied due to their useful properties,mainly in atitute control of artificial satellites, naval and aircraft applications. Thus, to describethe nonlinear behavior of a gyroscope, this paper proposes to approximate the solutions of thenonlinear motion equations using Volterra series. The main advantage of Volterra series is thegeneralization of impulse response function (IRF) and allows to separate the response of the systemin linear and nonlinear components. To illustrate the approach a gyroscope in a Cardan suspensionwith two axes of freedom is considered, describing precession and nutation. The Volterra series areidentified based on time series data of torque, angular velocity and displacement of precession andnutation, considering cross-kernels contribution since the system has essentially multiple outputs.The results of this novelty method proposed here is discussed and are shown the advantages anddrawbacks of this approach with suggestions for further works.

Keywords: Nonlinear dynamical systems, Volterra series, Gyroscope

CILAMCE 2013Proceedings of the XXXIV Iberian Latin-American Congress on Computational Methods in Engineering

Z.J.G.N Del Prado (Editor), ABMEC, Pirenopolis, GO, Brazil, November 10-13, 2013

MODELING OF THE DYNAMICS OF A GYROSCOPE THROUGH VOLTERRA SERIES

1 INTRODUCTION

The gyroscope effect has enormous applications to understand rotor dynamics (Deimel, 1950;Scarborough, 1958). If a rigid body with a spin rotation has some disturbance that cause a changein the angular moment, a reaction appears in order to warrant the dynamic equilibrium of torque.This effect cause two motions very interesting: precession and nutation (Cannon, 1967). Severalpractical applications are found in the literature, as for example, aerospace navigation (Ni andZhang, 2011), ships and submarines stability (Braddon, 1960), sports (Chua et. al, 2011).

The equations that describe the nonlinear system are generally unknown a priori and is neces-sary to use a method that be able to identify a suitable nonlinear model from the experimental datameasured. In this sense, the Volterra models can be a good approach, because the parameters of thenonlinear model are linearly related to the output and this fact allows the extension of some resultsof linear systems to nonlinear ones (Mathlouthi et al., 2009). Volterra models were introduced bythe Italian Vito Volterra (1860-1940) by the end of the XIX century. These models are an alterna-tive way to describe the dynamical behavior of the nonlinear systems based on a functional powerseries (Rugh, 1981).

Volterra series have been extensively applied in many areas such as identification of nonlinearmechanical systems (da Silva et al., 2010), aeroelastic systems (Silva, 2005), viscoelastic systems(Zhang et. al, 1998), modeling of nonlinear circuits (Bojorsell et al., 2008), signal processing andadaptive filtering (Ogunfunmi, 2007; Kaiser, 1987), structural health monitoring (Chaterjee, 2009),biological systems (Marmarelis, 2004), etc.

The Volterra series represent a direct generalization of impulse response function (IRF) of thelinear systems. Moreover, it is possible to analyze the contribution of each component associatedin the total response, because the Volterra approach allows to separate the response of the system inlinear and nonlinear components (Schetzen, 1980). However, the use of Volterra series can impliesin some drawbacks.

The main drawbacks are that it requires a great numerical effort and problems of response con-vergence due to a large number of samples. Thus, the identification of Volterra kernels is difficult toachieve for complex analytical models due to overparametrization (Thouverez and Jezequel, 1998).One way to avoid and overcomes these drawbacks is the use of Wiener series. The Wiener serieswere introduced by Norbert Wiener and the idea proposed consists in to expand the Volterra seriesin terms of an orthonormal basis described for instance by Laguerre and Kautz filter (Lee, 1933;Kautz, 1954; Wiener, 1958). The orthonormality property of those filters often yield a simpler gen-eral model and allows to reduce the number of parameters to be estimated (Whalberg and Makila,1996). Among the filters mentioned, Kautz filters have been used in several applications involvingdynamical systems with oscillatory behavior because Kautz functions are composed by complexconjugate poles (Heuberger et. al, 2005; da Rosa et. al, 2007).

In this context, the goal of this work is to identify the Volterra kernels based on time seriesinput data of torque in order to describe the dynamic behavior of a gyroscope by predicting theangular outputs measured. Thus, in section 2 the identification of the Volterra kernels procedureusing the orthonormal Kautz basis with least squares method is briefly reviewed. In section 3, an

CILAMCE 2013Proceedings of the XXXIV Iberian Latin-American Congress on Computational Methods in EngineeringZ.J.G.N Del Prado (Editor), ABMEC, Pirenopolis, GO, Brazil, November 10-13, 2013

Oscar Scussel, Amanda Alves Silva, Samuel da Silva

experimental example of gyroscope in a Cardan suspension with two degree-of-freedom is consid-ered to illustrate the proposed methodology. Finally, in the section 4 are discussed the results andare proposed suggestions for further works.

2 SYSTEM IDENTIFICATION BASED ON VOLTERRA THEORY

This section present the identification of Volterra kernels procedure for single input - multipleoutput (SIMO) systems. The approach is based in the least squares method expanded in orthonor-mal basis described by Kautz functions with poles optimized through genetic algorithm.

2.1 Identification of Volterra kernels

The output yp(k) can be described using the following expression (Schetzen,1980):

yp(k) =+m=1

Hpm(k) (1)

where k = 1, 2, . . . , N are the samples collected for p = 1, 2, . . . , P outputs measured, Hpm(k) arethe Volterra functionals regarding to m th-order Volterra kernel hpm(n1, n2, . . . , nm) given by:

Hpm(k) =+

n1=

+n2=

. . .+

nm=

hpm(n1, n2, . . . , nm)mi=1

u(k ni) (2)

in wich u(k) is the in the input signal of the system. Thus, explicitly, the output yp(k) can be writtenby:

yp(k) =+m=1

+n1=

+n2=

. . .+

nm=

hpm(n1, n2, . . . , nm)mi=1

u(k ni) (3)

Furthermore, it is possible to separate the output yp(k) of the system by:

yp(k) = ylin(k) + yquad(k) + ycub(k) + . . . (4)

In the particular case when are considered three kernels h1(n1), h2(n1, n2) and h3(n1, n2, n3)given by:

ylin(k) =+

n1=

h1(n1)u(k n1) (5)

yquad(k) =+

n1=

+n2=

h2(n1, n2)u(k n1)u(k n2) (6)




ycub(k) =+

n1=

+n2=

+n3=

h3(n1, n2, n3)u(k n1)u(k n2)u(k n3) (7)

It is worth to note that Eq. (5) is the classical representation of the impulse response function(IRF) of linear systems. The Volterra approach is similar to the Taylor series represented by poly-nomials operators. However, Taylor series are limited to describe the nonlinear behaviour of systemwith memory, because the approximation of the response depends of the input only at that sametime. About this situation other advantage of the Volterra models are verified in Eq. (3) through thememory effect in the input data u(k ni).

Considering the following properties of the dynamical system:

Causality: hm(n1, . . . , nm) = 0 nk < 0, k = 1, 2, . . . ,m th-order kernel

Stability BIBO (Bounded-Input Bounded-Output): |hm(n1, . . . , nm)|


values N1, N2, N3, . . . , Nm. Thus, the Volterra functional described in Eq. (2) can be rewritten interms of an orthonormal representation:

Hpm(k) J1i1=1

. . .

Jmim=1

pi1,...,im

mj=1

ij(nj) (11)

where pi1,...,im are the coefficients of the orthonormal basis composed by functions ij whose theorthonormality property results from:

+k=0

q(k)r(k) =

1 where q = r0 where q 6= r (12)Thus, in the general case the output yp(k) of the system, described in Eq. (8), can be rewritten by:

yp(k) =

J1i1=1

. . .Jmim=1

pi1,...,im

mj=1

lij(k) (13)

where J1, J2, . . . , Jm are the number of functions considered in the orthonormal basis such thatJm


where p contains the values associated with the coefficients of the orthonormal base:

p = [p1(0)

p1(1) . . .

p1(J1)

p1,2(0, 0)

p1,2(1, 0)

p1,2(1, 1)

p1,2(2, 0)

p1,2(2, 1) . . .

p1,2(J2, J2) p1,2,3(0, 0, 0)

p1,2,3(1, 0, 0)

p1,2,3(1, 1, 0) . . .

p1,2,3(J3, J3, J3) . . .]

T

and the matrix is given by:

= [l1(k) l2(k) . . . lJ1(k) l1(k)2 l1(k)l2(k) l2(k)

2 l3(k)l1(k) l3(k)l2(k) l3(k)2 . . .

lJ2(k)2 . . . l1(k)

3 l1(k)2l2(k) l1(k)l2(k)l3(k) l2(k)

2l3(k) . . . lJ3(k)3 . . .]T

From the orthonormal expansion, it is possible to identify the Volterra kernels solving Eq. (11).Thus, the number of parameters to be estimated can be drastically reduced due to the number offunctions used in the orthonormal base is smaller than the numbers considered in the truncation ofthe multidimensional kernel in the physic basis, Jm


to use the transformation to the discrete domain using zm = em.(Fs)1 , where Fs is the sampling

rate, once Kautz functions are defined in the discrete-domain.

However, in the system identification context the parameters m and nm are, in some cases,totally unknown and it is need to employ a numerical procedure to obtain the optimal value of theKautz poles that minimize the prediction error function. Thus, follows the optimization procedure:

minimize Jp(, k) (20)

subject to

(low)nm nm (up)nm (21)(low)m m (up)m (22)J(low)m Jm J(up)m (23)nm, (up)m R+ (24)Jm 2Z+ (25)

where the p-th objective function Jp(, k) is obtained from the Euclidean norm of the predictionerror ep associated with the p-th output of the system:

Jp(, k) = ep(k) = yp(k) yp(, k) (26)

with = {J1, J2, . . . , Jm, 1, 2, . . . , m, n1, n2, . . . , nm} where J1, J2, . . . , Jm are the numberof functions and yp(, k) is the p-th output estimated using Votlerra models. In the set of constraints,the index up means the upper limitation and low is the lower value limitation defining the rangeof searching. This optimization problem can be solve by several approaches, for instance, geneticalgorithms. Figure 2 shows the flowchart of the approach for single input - multiple output (SIMO)systems.

3 APPLICATION EXAMPLE

In order to illustrate and exemplify the approach, a gyroscope in a Cardan suspension withtwo degrees-of-freedom was considered, to measure the angles of precession and nutation therespectives velocities. Figure 3 shows the Quanser gyroscope with 3 degrees-of-freedom used inthe experimental tests. The nutation was measured around of blue gimbal, the motion of precessionwas measured around the red gimbal and the silver gimbal was kept fixed. The gyroscope hassensors for measuring the angles and their angular rates. The three first Volterra kernels wereidentified using these angles, angular velocities and the torque applied in the axis y2 coincidentwith the axis y1 as showed in the Fig. 3(a) and Fig. 3(b).




Figure 2: Flowchart illustrating the identification of Volterra kernels procedure.

A sampling rate of Fs = 1 kHz and N = 4602 samples were used in the data acquisition. Thesignal u generated as input, showed in the Fig. 4, is a single step signal with amplitude 0.71 N.m,on a range of 0.38 seconds, activating the inner blue gimbal, as illustrated in the Fig. 3, such thatthe disk has 104.72 rad/s of spin velocity constant controlled by a motor.

From the input data generated and the four signals measured, displacement and angular veloc-ities of precession and nutation, was employed a genetic algorithm to estimate the optimal Kautzpoles, 2m1 e 2m, to build the Kautz filters described in Eq. (16) and Eq. (17) with m = 1, 2, 3kernels. Table 1 shows the optimization results from the p = 3 objective functions. The optimalvector = {J1, J2, J3, 1, 2, 3, n1, n2, n3} was obtained by genetic algorithm after 50 genera-tions with population size equals 300 using 0.8 as cross-over fraction.

The optimal values obtained in the optimization procedure led in the pairs of complex-polesin the z-domain 1 = 2 = 0.9821 + j0.0813, 3 = 4 = 0.9991 + j0.0266 and 5 = 6 =0.9987+j0.0193. Thus, the input signal was filtered by Kautz functions and was built the regressormatrix . From the regressors matrix, the vectors 1, 2, 3 and 4 were estimated solving Eq.(15) through classical least squares method. Therefore, the set of Volterra kernels Hpm(k) wereestimated solving Eq. (11) with p = 1, 2, 3, 4 outputs measured and m = 1, 2, 3 amounting to12 cross-kernels identified. Figure 5 shows the 1st order Volterra kernels, the impulse responsefunction (IRF) describing the linear contribution, associated to the outputs , , and .



(a) Schematic representation of the sys-tem considering the nutation.

(b) Schematic representation of the sys-tem considering the precession.

Figure 3: Quanser 3 DOF gyroscope with reference axis.

0 1 2 3 40

0.2

0.4

0.6

0.8

Time [s]

Tor

que

[N.m

]

Figure 4: Input signal u applied in the system.

The second and third kernels based on the outputs and are shown in the Fig. 6 and Fig.7, respectively. After that the Volterra kernels are identified, the outputs of the system, , , and were estimated by solving Eq. 8. In order to verify an accordance of the model identified usingVolterra series expanded in the orthonormal Kautz basis, Fig. 8 shows the comparison between theset of outputs data measured with the set of outputs data estimated. Thus, as shown in the Fig. 8, itis possible to verify a good estimation in the results of identification procedure.




Table 1: Parameters used in the optimization procedure using genetic algorithm.

kernel 1st order 2nd order 3rd order

Parameters n1 [Hz] 1 J1 n2 [Hz] 2 J2 n3 [Hz] 3 J3

Lower value 1 0.01 2 1 0.01 2 1 0.01 2

Upper value 80 0.2 10 80 0.2 10 80 0.2 10

Optimal value 13.35 0.17 6 4.23 0.017 8 3.08 0.05 8

0 100 200 300 4000.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

n1

h1 1(

n1)

(a) IRF related with the output .

0 100 200 300 4000.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

n1

h2 1(

n1)

(b) IRF related with the output .

0 100 200 300 4000.01

0.005

0

0.005

0.01

0.015

n1

h3 1(

n1)

(c) IRF related with the output .

0 100 200 300 400

0.04

0.03

0.02

0.01

0

0.01

0.02

0.03

n1

h4 1(

n1)

(d) IRF related with the output .

Figure 5: 1st order Volterra kernels estimated.

4 FINAL REMARKSIn the present paper the Volterra series expanded in the orthonormal Kautz basis were used

in order to obtain an approximation for the outputs displacement of nutation (), angular veloc-ity of nutation (), displacement of precession () and angular velocity of precession () from



(a) 2nd order. (b) 3rd order with cut surface in n3 = 100.

Figure 6: Higher order Volterra kernels estimated related with the displacement of nutation .

(a) 2nd order. (b) 3rd order with cut surface in n3 = 100.

Figure 7: Higher order Volterra kernels estimated related with the displacement of precession .

the gyroscope. The Kautz functions, with parameters optimized by genetic algorithm, allowed adrastic reduction in the number of parameters to be estimated in the identification procedure andavoided the overparametrization effect. The results showed the accuracy of the model identifiedand the useful features of Volterra series in nonlinear problems based by known only of the mea-sured data. Thus, the approach proposed here can be expanded considering the cross-kernels frommultiple-input/multiple-output (MIMO) systems and applied in problems involving modeling andidentification of nonlinear systems and control.




0 1 2 3 40.2

0.1

0

0.1

Time [s]

Dis

plac

emen

t of n

utat

ion

[rad

]

ReferenceEstimated

(a) .

0 1 2 3 4

0.6

0.4

0.2

0

0.2

0.4

0.6

Time [s]

Ang

ular

vel

ocity

of n

utat

ion

[rad

/s]

ReferenceEstimated

(b) .

0 1 2 3 40.3

0.2

0.1

0

0.1

Time [s]

Dis

palc

emen

t of p

rece

ssio

n [r

ad]

ReferenceEstimated

(c) .

0 1 2 3 40.9

0.25

0.4

Time [s]

Ang

ular

vel

ocity

of p

rece

ssio

n [r

ad/s

]

ReferenceEstimated

(d) .

Figure 8: Comparison between the outputs data measured with the outputs estimated by Volterra series.

ACKNOWLEDGEMENTS

The first author acknowledges his scholarship from the Coordination for the Improvement ofHigher Education Personnel (CAPES/Brazil). The second author acknowledges her scholarshipprovided by PROPe 20/2013 - Apoio a Jovens Talentos da Unesp. The authors would like tothank the financial support provided by the National Council for Scientific and Technological De-velopment (CNPq/Brazil) to the grant number 301582/2010-6 and 470582/2012-0 and Sao PauloResearch Foundation (FAPESP) to the grant number 12/09135-3.

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http://dx.doi.org/10.1016/j.proeng.2011.08.177http://dx.doi.org/10.1016/j.proeng.2011.08.177http://dx.doi.org/10.1007/s11071-005-1907-zhttp://dx.doi.org/10.1007/s11071-005-1907-zhttp://dx.doi.org/10.1016/S0020-7462(97)00050-4http://dx.doi.org/10.1016/S0020-7462(97)00050-4http://www.sciencedirect.com/science/article/pii/0005109895001980http://www.sciencedirect.com/science/article/pii/0005109895001980http://dx.doi.org/10.1114/1.82

INTRODUCTIONSYSTEM IDENTIFICATION BASED ON VOLTERRA THEORYIdentification of Volterra kernelsOptimization procedure of the Kautz filters

APPLICATION EXAMPLEFINAL REMARKS

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nonlinear systems

volterra models

nonlinear dynamical

nonlinear components

suitable nonlinear model

describethe nonlinear

bythe italian vito volterra

nonlinear ones mathlouthi