Study on a Ship with 6 Degrees of Freedom Inertial Measurement System and Related Technologies

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1 Oen Access Library Journal Study on a Shi with 6 Degrees of Freedom Inertial Measurement System and Related echnologies Jianhui Lu,, Yachao Li, Shaonan Chen, Yunxia Wu Shandong rovince Key Laboratory of Ocean Engineering, Ocean University of China, Qingdao, China College of Engineering, Ocean University of China, Qingdao, China Received 5 March 6; acceted March 6; ublished 5 March 6 Coyright 6 by authors and OALib. his wor is licensed under the Creative Commons Attribution International License (CC Y). htt://creativecommons.org/licenses/by/4./ Abstract We research shi 6 DOF inertial measurement method and dislacement reconstruction form acceleration signal to establish the stra-down inertial navigation measurement system through sensors configuration technology. he inertial measurement system consists of three grous of vertically mounted angular rate and acceleration sensors. hrough designing an FIR integration filter and studying the shi dislacement signal reconstruction from the acceleration, the filter arameters design method and imlementation way were determined. y means of the beam vibration to analog shi heave motion reconstructs linear dislacement form acceleration signals. he root mean square error of the reconstructed line dislacement is.4 mm, and the mean value of the dislacement ea value is 5.9%, which is close to the design recision of the filter. he test shows that the measurement system and the FIR filter to achieve the redetermined recision with noise suression. Keywords Ocean Engineering, SINS, Shi Attitude Detection, Dislacement Reconstruction echnology Subject Areas: Oceanology. Introduction he 6 DOF motion measurement technology has been a widesread concern in the industry. Due to the difficulty in finding a fixed oint of reference, the real shi motion measurement is difficult in engineering, and the establishment of the line dislacement is esecially difficult. Feasible methods mainly include GS and inertial measurement system, due to the current GS accuracy is hard to meet the requirements. he research focused on ow to cite this aer: Lu, J.., Li, Y.C., Chen, S.N. and Wu, Y.X. (6) Study on a Shi with 6 Degrees of Freedom Inertial Measurement System and Related echnologies. Oen Access Library Journal, 3: e56. htt://dx.doi.org/.436/oalib.56

2 the measured acceleration by inertial measurement system to reconstruct the shi dislacement (linear dislacement and angular dislacement) [] []. Inertial measurement system can directly establish the shi heading and attitude, and in fact often needs the sway, surge and heave motion data, such as offshore suly, construction oerations, aircraft etc. Stradown inertial navigation system solves the roblem of the traditional measurement latform system, and the measuring system has the advantage of light quality, small volume, low cost, long life and easy maintenance etc., and has been alied in ractice [3]-[6]. In this aer, a new tye of stradown inertial navigation system is established, and an integral filtering algorithm is designed, which can reconstruct the shi line motion from the line acceleration and angular velocity measurement signals.. Coordinate System and the osture arameters of the Shi he shi has 6 degrees of freedom movement, so first establish shi motion attitude coordinate system as shown in Figure, therefore, exort exress shi osture arameters, and then study the measuring method. In the shi coordinate system, O - XYZ is coordinate system for inertial reference system, O-xyz is shi carrier coordinates, O is a fixed oint on the shi, Ox is arallel axis ointing to bow, oint C is the article of shi, the shi view as a rigid body, is any oint on the shi. r vector is from O oint to O oint, S vector is from O oint to oint, r vector is from O oint to oint. According to the definition of euler Angle, introducing coordinates O-xyz with new one to the attitude Angle O - XYZ coordinates, and ϕθσ,, resectively for the roll Angle, itching Angle and bow wave Angle, the shi swaying, surge and heave is the oint C s dislacement in O - XYZ. On the measuring oint at the shi, to establish the relationshi between shi 6 degrees of freedom motion arameters and the measurement signal. According to Figure, r = r + S () where r, r and S were comared with the vector inertial coordinate system O - XYZ, this equation can also be described as, r = r + () AS he matrix A is the coordinate transformation matrix of coordinate system O - XYZ to coordinate system O-xyz. S vector in the vector coordinate O-xyz reresented as guide, by the nature of the matrix, S = AS, this equation can also be described as, r = r + AA S S vector. he formula () for one order (3) he vector ω is the carrier coordinates with resect to the rotational angular velocity of O-xyz inertial coordinate system O - XYZ, according to the rincile of rigid body inematics, Figure. Shis coordinate system. OALibJ DOI:.436/oalib.56 March 6 Volume 3 e56

3 where this Equation (4) can also be described as r = r + ω S (4) ωz ω y ω = ωz ωx ωy ωx = + S (5) r r ω where ω is rotation angular velocity which is the carrier relative to the coordinate system O-xyz, according to the equation where, then ω = Aω A A = Aω = ω A (6) A = A ω + Aω = Aω ω + Aω (7) ϕ sinθ ω x θ cosϕ cosθsinϕ ω = y σ sinϕ cosϕcosθ ω z he Euler angles can be obtained by the formula of numerical integration. he seventh formula into formula (3) of the two order derivative, r = r + Aω S + Aω ω S he Formulas (5) and Formulas (9) rojects onto the carrier coordinates, (9) () r = r + ω S r = r + ω S + ω ω S () 3. he Measurement Framewor and Sensor Arrangement In order to obtain the shi angular acceleration and sway, surge and heave arameter, taing the sensor layout as shown in Figure the measurement system consists of 3 vertically mounted sensor grou, each consisting of angular velocity and acceleration sensor. (8) Figure. he sensor configuration for shi motion measurement. OALibJ DOI:.436/oalib.56 3 March 6 Volume 3 e56

4 According to the measurement scheme, firstly we can calculate the shi angular acceleration, taing into account the b in y and z direction of acceleration is not measured, so can be obtained, the same way for oint a and c, ( a a ) ω = ρ + ωω () z x bx by x y ( a a ) ω = ρ + ωω (3) y z az ax x z ( a a ) ω = ρ + ωω (4) x y cy cz y z herefore, the acceleration of the shi motion is obtained by the acceleration sensor and the angular rate gyro sensor. For shi swaying, surging and heaving motion arameters, it can be obtained by formula 9, r = r + Aω S + Aω ω S C C C (5) r = Ar + Aω S + Aω ω S C C C (6) herefore, as long as the acceleration angular velocity, angular acceleration of the shi and three axis are measured, we can calculate the shi sway, surge and heave acceleration. 4. Integral Filter Design Method o design a second-order integration filter is the ey to rebuilt dislacement from the acceleration, and then determine the filter arameters. he ideal frequency resonse is, E = (7) ω ( ω) wo order integral filter design method includes the digital filter method and the frequency domain integral method. he finite imulse resonse (FIR) filter and infinite imulse resonse filter (IIR) digital filter has been widely used [7] [8]. Since the initial information of the IIR filter is needed to be measured, it is difficult to obtain the real shi, on the other hand, the FIR filter does not need the initial information of the measured object, but by choosing the aroriate filter arameters for two-order integral. 4.. he Design of FIR Integration Filter he design method of FIR filter includes maximum flatness and variational aroximation. he main characteristic of maximum smoothness method is system filtering having best effect when near the angular frequency ω = π, belonging to the band-ass filter, but it needs to estimate frequency in advance. he variational aroximation method is transforming the two-order integral roblem into an extremum roblem, d u t a( t) = a, (8) < t <, u( ) = u, u( ) = u where a is measured acceleration. It needs to solve a dislacement function when it is transformed into functional roblem, d u t d t (9) Min ( u) = a t u d Obviously this is a boundary value roblem, because it is difficult to measure the initial dislacement and seed of the carrier, therefore, this function has no definite solution. For the ill osed roblem, the regularization technique is used to estimate the solution in advance, and the reconstructed dislacement is close to the given static dislacement, because only the dynamic dislacement can be reconstructed, the static equilibrium oint can be set to zero. hen R = u r < () OALibJ DOI:.436/oalib.56 4 March 6 Volume 3 e56

5 R is a regular function, r is determined solution domain, the regular conditional Equation () is introduced into the functional extremum roblem in the form of enalty function. d u( t) β () Min ( u) = d d u a t u t + Functional extreme value roblem formula () is called the ihonov regularization method, β is a non negative regularization arameter. he larger the value of β is, the regular function have the greater the influence on the dislacement solution, when β is zero, the Equation () become a boundary value roblem. In the absence of the initial value, the equation is not solvable, the variation of functional roblem Equation () is d δu d u( t) d d () δ u = a t uu t β δ + = he solution of this variational roblem includes frequency domain and finite element method, etc. his aer uses the frequency domain method. According to the division of integral method, 4 3 d u d a dδu d u d u da δu β u 4 t a δu 3 + d + = (3) hen, functional extremum roblem for control equations and boundary conditions are, 4 d u d a + β u = (4) 4 3 d d d 3 u u a a,, t = = < < Due to the dislacement and velocity are unnowable at the boundary, so this aer uses the Newman boundary condition. Fourier transform of the Equation (4), and the transfer function of the control equation is ( ω) ( πf ) ( πf ) ω = = 4 4 ω + β + β where f is the frequency, reresents the control equation of the transfer function, the dislacement can be reconstructed by inverse Fourier transform, ω = = 4 ( ) ω + β ( ω) ( ) u t F F a t F F a t From the Equation (6) we can see that the low resistance filter does not need to be introduced in the dislacement reconstruction, because the transfer function of the governing equation has the ability to suress the low frequency noise. In order to determine the transfer function and the regularization arameter β, the introduction of the normalized frequency f = f f, f is the target frequency. he accuracy of the target frequency α is defined as the control equation of the ratio of the transfer function and the ideal transfer function, then Equation (5) is related to the target frequency for normalization of transfer function ( ω) ( ) = = (7) 4 4 πf + β πf he ratio between the transfer function of the control equation and the ideal transfer function 4 4 acc ω ( ω) = = = ω + β + β πf At the time of f = the recision function acc E E is (5) (6) (8), recision to, when the frequency close to the target OALibJ DOI:.436/oalib.56 5 March 6 Volume 3 e56

6 acc frequency, the recision function raid convergence to. At the time of < f, recision function mainly deends on the regularization arameter values β. In the target frequency f, mae f =, the accuracy at the target frequency is α = + β 4 πf So get regularization arameter values β under certain recision where λ ( α ) 4 α α β = = λ α α ( πf ) ( πf ) α =, then into Equation (7) and Equation (8), acc ( ) ( ) = = 4 4 f + λ α f + λ α For different target accuracy α, control equation of the transfer function and the difference between ideal transfer function and accuracy function is shown in Figure 3 and Figure 4. When frequency is greater than the target, the control equation of the transfer function and the ideal transfer function almost unanimously, has (9) (3) Figure 3. Comarison of transfer functions under different recision. Figure 4. Comarison of accuracy functions under different recision. OALibJ DOI:.436/oalib.56 6 March 6 Volume 3 e56

7 nothing to do with the target accuracy, So the control equation of transfer function for signal reconstruction dislacement; At the same time, the control equation of the transfer function can inhibit below target frequency acceleration signal, the signal is generally to measure noise and disturbance of constant value, the jamming signal frequency is lower, the transfer function of the noise suression caability. he Figure 4 shows that the higher the accuracy of target, the transfer function of the lower noise suression caability. As a result, recision of target and noise suression ability between the two filter design index. he choice of target accuracy deends on the secific roblem. 4.. he Realization of FIR Integration Filter he traditional FIR filter is similar to a given transfer function in frequency domain. Set the acceleration measurement time, every time the FIR filter can be exressed as where c is FIR filter coefficient, wih is ( ) C + (3) u = ut = t ca = t c a t+ t = = u + is dislacement through the filter and the moment of t, time window + t, the tye of Fourier transform to Equation (3), f is the transfer function of the FIR filter, hen, C = e + + f t (3) = iπ f t c e + + f t (33) = iπ f t c ( f ) f c = cos 4 4 ( π f + + ) d π + λ where fs = t is frequency, f = f f is the ratio of target frequency to samling frequency. Control s equation of the transfer function in frequency domain is even function. ecause the control equation of the transfer function with increase raidly declining, as shown in Figure 3, therefore, Equation (34) oints in almost has nothing to do with the integral uer limit f.. So, it can be set = f, c + + = c+ + f,thus, the relation between, c + + and is indeendence. Although the number of FIR filter coefficients can be changed with the change of the filter order, when f is under certain circumstances, the filter coefficient c + + constant. Gibbs henomenon of high frequency oscillation occurs in the transfer function of FIR filter, in order to reduce the oscillation amlitude, filter order time should be enough big, smooth filter coefficients in tends to converge to zero. FIR filter, therefore, should be the last item on the coefficient of corresonds to the zero oint, = is = In general, select the rectangular window. Refactoring dislacement can be used when no overla or overlaing time window, corresonding to different dislacement calculation strategy. In overlaing time window, will be calculated for each moment corresonding dislacement, only will be a time window in central calculation result as the outut, not the effect of overlaing time window boundary value estimation error will be reduced. hrough the target cycle and the size of the window wih defined filter dw = t = = Nw (36) f f (34) (35) OALibJ DOI:.436/oalib.56 7 March 6 Volume 3 e56

8 where d w is exressed as a unit of time the filter size, Nw = is exressed as the filter size target cycle. Numerical analysis of Equation (34), and c + + the relation curve between the zero return for.687 eriod; he filter size relative to the initial three zero were.846, 4. and times the target cycle; Due to the filter coefficient of zero in the first lace not convergence, when N w =.846, a significant oscillating filter transfer function is; when N w = 4. and N w = 7.594, recision function of oscillation amlitude in the accetable range. he larger the filter size, the smaller the oscillation amlitude of the recision function. Due to the symmetry of FIR filter coefficients, FIR filter transfer function can be exressed as ( f ) = ( t) c + c cos ( πf t) (37) C = FIR filter corresonding normalization transfer function and accuracy function is 5. Exerimental Verification C( ) = ( π ) c+ + c+ + cos ( π f ) (38) = acc C ( ) = ( π f ) c+ + c+ + cos ( π f ) (39) = he ey of the roblem is to reconstruct the line dislacement from the acceleration measurement signal, the Oriental institute of vibration and noise control INV6 vibration test system is adoted, the acceleration sensor is installed in the middle of the simle beam, the vibration acceleration of the beam is measured, and the dislacement of the vibration of the beam is installed at the lower art of the measuring oint. Reference sensitivity of iezoelectric accelerometer is C/m s. he test system shows in Figure 5. he middle of the beam is subjected to harmonic excitation 5 z to 45 z, the acceleration sensor of laser dislacement sensor samling frequency and samling frequency is z. According to the accelerometer measurement signal, through FIR filter reconstruction beam dislacement of measuring oints, and comared with the laser dislacement sensor measured dislacement. Vibration acceleration signal of a simly suorted beam under z excitation shows in Figure 6. he target frequency of the selected FIR filter is z, the ratio between target frequency and samling frequency is., the acceleration resonse reconstruction beam at the measuring oint dislacement. Reconstruction after artial details shows in Figure 7, there is no obvious hase lag between the outut dislacement and the actual dislacement of the FIR filter, FIR filter reconstructed outut dislacement relative to the laser dislacement sensor measurement more accurate the root mean square error (RMSE) is.4 mm, the mean value of the dislacement ea value is 5.9%, close to the filter design accuracy is 97%, it shows that the FIR filter is designed to realize the second integral, reconstruct dislacement from the acceleration signal, to the exected target design recision, and has certain ability of noise suression. Figure 5. Exerimental modal. OALibJ DOI:.436/oalib.56 8 March 6 Volume 3 e56

9 Figure 6. he acceleration measurement signal of beam. 6. Conclusions Figure 7. FIR filter reconstruction dislacement. A stra down inertial navigation attitude measurement system is established, and the integral filtering algorithm is used to measure the shi attitude and the three axis acceleration resonse; with the vibration of the beam simulation shi heaving, inverse dislacement is measured by acceleration. he results of the study are as follows: ) A stra down inertial measurement system for shi 6 degrees of freedom is established, sensor configuration technology is exorted, measurement system is comosed of three grous of vertical installation of sensors, and each grou is made u of an angular rate and acceleration sensor. ) he FIR integral filter is designed, and the design method of the filter arameters is studied to realize the goal of reconstructing the line dislacement and angular dislacement of the acceleration signal from the shi. 3) With the vibration of the beam simulation shi heaving, adot the design of FIR filter, linear dislacement from the accelerometer signal reconstruction, reconstruct dislacement relative to the laser dislacement sensor to measure the dislacement of root-mean-square deviation of.4 mm, the dislacement ea average error is 5.9%, close to the filter design recision being 97%. 4) he exerimental results show that the 6 degree of freedom inertial measurement system and the FIR filter can effectively realize the reconstruction of the dislacement from the acceleration signal and achieve the goal of the redetermined design recision. And it has certain ability of noise suression. 5) Analysis shows that, in order to achieve a certain recision, the order of the filter is often high, the measurement system has a large amount of comutation and memory cost. Fund his study is suorted by the national science and technology major secial sub tas (ZX556-3). OALibJ DOI:.436/oalib.56 9 March 6 Volume 3 e56

10 References [] ong, Y.., Kim,.-K. and Lee,.S. () Reconstruction of Dynamic Dislacement and Velocity from Measured Accelerations Using the Variational Statement of an Inverse roblem. Journal of Sound and Vibration, 39, htt://dx.doi.org/.6/j.jsv..5.6 [] Lee,.S., ong, Y.. and ar,.w. () Design of an FIR Filter for the Dislacement Reconstruction Using Measured Acceleration in Low-Frequency Dominant Structures. International Journal for Numerical Methods in Engineering, 8, [3] Wang, F. (8) Research on Marine Mini-ye Attitude Measurement System and Design. arbin Engineering University, arbin. [4] Cardou,. and Angeles, J. (8) Estimating the Angular Velocity of a Rigid ody Moving in the lane from angential and Centrietal Acceleration Measurements. Multibody System Dynamics, 9, [5] Cardou,. and Angeles, J. (9) Linear Estimation of the Rigid-ody Acceleration Field From oint-acceleration Measurements. Journal of Dynamic Systems Measurement and Control, 3, -. htt://dx.doi.org/.5/.379 [6] Angeles, J. (999) he Angular-Acceleration ensor of Rigid-ody Kinematics and Its roerties. Archive of Alied Mechanics, 69, 4-4. htt://dx.doi.org/.7/s4954 [7] seng, C.-C. and Lee, S.-L. (9) Design of Digital IIR Integrator Using Discrete artley ransform Interolation Method. IEEE International Symosium on Circuits and Systems, aiei, 4-7 May 9, htt://dx.doi.org/.9/iscas [8] seng, C. C. and Lee, S.L. () Design of Digital Integrator Using Stancu olynomial and Numerical Integration Rules. th International Symosium on Communications and Information echnologies, oyo, 6-9 October, htt://dx.doi.org/.9/iscit OALibJ DOI:.436/oalib.56 March 6 Volume 3 e56