UKACC Control 2012 D t Determination of Dynamic Flexure i ti f - PowerPoint PPT Presentation

UKACC Control 2012 D t Determination of Dynamic Flexure i ti f D i Fl Model Parameters for Ship Angular p g Deformation Measurement Wei Wu 1, 2 , Sheng Chen 2, 3 , Shiqiao Qin 1 1 School of Opto-Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China 2 School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, UK 3 King Abdulaziz University, Jeddah 21589, Saudi Arabia

O tli Outline 1 Background Parameters Estimation Approach 2 Simulation System 3 Results and Analysis 4 Conclusions 5

B Background k d � Ship Angular Deformation Ship angular deformation refers to Ship angular deformation refers to the two frames angle displacement Pitching: cross x -axis Pitching: cross x axis Rolling: cross y -axis Yawing: cross z -axis rolling pitching deformation deformation

B Background k d LGU 2 � Measurement Approach LGU 1 Ship deformation: i i φ ( t )= φ 0 + θ ( t ) Measurement system where φ is time-invariant component where φ 0 is time-invariant component, θ ( t ) is dynamic component, which is usually modeled as a second-order Gauss-Markov process, the correlation function is second order Gauss Markov process, the correlation function is ⎛ ⎞ α ( ( ) ) τ = σ − α τ βτ + β τ = 2 ⎜ i ⎟ R ( ) exp cos sin , i x y z , , θ β β i i i i ⎝ ⎝ ⎠ ⎠ i i i in which σ 2 is the variance, α is the damping factor, β is the circular frequency circular frequency.

B Background k d Schematic diagram of ship deformation measurement system Schematic diagram of ship deformation measurement system Kalman Filter ( ( ) ) θ θ φ φ ψ ψ ˆ ˆ Measurement function: M t f ti m m s s Z Z = B - A B A + B C B C -C C dcm 0 i m i s � State function: X = FX +w � � X = φ θ θ ψ ψ ε ε � � T T T T T T T T [ ] State vector: 0 m s m s

B Background k d Specifically, the measurement vector is given by and the matrices A and B are given by d th t i A d B i b where C ij and C' ij are the components of DCMs of LGU 1 and LGU 2 , respectively.

B Background k d The state transition matrix is given by e s e s o s g ve by in which The state noise covariance is The state noise covariance is

B Background k d � Determine the Dynamic Flexure Model Parameters • Empirical method The parameters are determined according to experience p g p • Statistical method The parameters are obtained from previously recorded Th t bt i d f i l d d measurement data In actual condition, the parameters depend on sea condition, ship velocity and ship structure, etc. It requires to estimate the parameters on line requires to estimate the parameters on-line.

B Background k d � Our Novelty The dynamic flexure information is existing in attitude difference measured by LGU 1 and LGU 2 . ( ( ) ) θ φ ψ ψ ˆ ˆ m s Z = B - A + B C -C dcm 0 i m i s Assume the dynamic flexure can be depicted as a second- A th d i fl b d i t d d order Gauss-Markov process, we developed an on-line dynamic flexure parameters estimation method by dynamic flexure parameters estimation method by utilising the attitude difference measured by two LGUs, and Tufts-Kumaresan (T-K) method was applied to obtain a robust and accuracy estimates.

Parameters Estimation Approach P t E ti ti A h The attitude matching function can be written as ( ( ) ) ( ( ) ) θ φ φ ψ ψ ψ ψ ˆ ˆ m s Z = B + B - A + B C -C dcm dcm 0 0 i i m m i i s s � ≈ B θ Z dcm ( ( ) ) Remove the second term : for ( B A ) is a small, and φ 0 : for ( B - A ) is a small, and φ 0 B - A φ φ Remove the second term 0 0 can be compensated to several mrads using the course estimate results, so the multiply results are small and can be removed. ( ) ψ ˆ ψ ˆ Remove the third term : for the frequency of m s B C - C i m i s attitude error caused by gyro bias and random walk noise is far less than θ , this term can be removed through a high-pass filter.

Parameters Estimation Approach P t E ti ti A h � The correlation function of is given by Z dcm � � τ τ = = + + τ τ = = θ θ θ θ + + τ τ R R ( ) ( ) Z Z ( ), ( ) t Z t Z ( ( t t ) ) ( ) ( ), ( t t ( t t ) ) Z dcm dcm Recall that the correlation function of dynamic flexure θ ( t ), based on the second-order Gauss-Markov process assumption is th d d G M k ti i ⎛ ⎞ α ( ( ) ) τ = σ − α τ βτ + β τ 2 ⎜ ⎜ ⎟ ⎟ R ( ) exp cos sin θ β β ⎝ ⎝ ⎠ ⎠ Therefore, the parameters σ 2 , α and β can be obtained from R Z ( τ ) R Z ( τ ).

Parameters Estimation Approach P t E ti ti A h � T-K Method The T-K methods is widely applied in estimation of The T-K methods is widely applied in estimation of parameters for closely spaced sinusoidal signals in noise M ∑ ∑ ( ( ) ) = ⎡ ⎡ − α + β β ⎤ ⎤ + = � y n ( ) ( ) a exp ⎣ j j n ⎦ q n n ( ), ( ) 1,2, 1 2 , N N l l l = l 1 where M is the number of sinusoidal signals, N is the g sample length, a l is the amplitude, α l is the damping factor and β l is circular frequency. The parameters α l and β l can be resolved by using T- K method. Then, substitute the estimate results α l and β to above equation and the magnitude a can be β l to above equation, and the magnitude a l can be resolved by using the least square methods.

Parameters Estimation Approach P t E ti ti A h � Parameters Estimation Procedure � Initialization � Initialization ˆ i ˆ i • Calculate the DCMs of and , derive Z dcm C C s m φ φ ˆ • Compensate the Compensate the using course estimation results using course estimation results 0 0 • Remove gyro errors through high-pass filter � T-K Based Parameters Estimation • Calculate the correlation function R Z ( τ ) of Z dcm • Construct the T-K prediction function and evaluate the frequency β /2 π and damping factor α • Calculate the variance σ 2 using the least square algorithm � KF Based Angular Deformation Measurement

Si Simulation System l ti S t Gyro samples Parameters estimation and KF based g generation deformation measurement T-K method Schematic diagram of gyro samples generation and g gy p g dynamic flexure parameters estimation

Si Simulation System l ti S t The ship attitude can also be modeled as a second-order Gauss- Markov process, whose correlation function takes the form p , ⎛ ⎞ α ( ) ξ τ = σ − α τ β τ + β τ ⎜ ⎟ 2 ( ) xp cos sin R e i ⎜ ⎟ ξ ξ ξ ξ ξ β β i i i i i ⎝ ⎝ ⎠ ⎠ ξ ξ i Ship attitude parameters Magnitude Magnitude Frequency Frequency Damping factor Damping factor α σ β π / 2 (s -1 ) (deg) (Hz) ξ ξ ξ i i i Pitch Pitch 2.2 2.2 0.18 0.18 0.10 0.10 Roll 3.4 0.07 0.06 Yaw 0.8 0.05 0.12 Set according Identified from to experience experiment data

Si Simulation System l ti S t T True dynamic flexure parameters d i fl t Magnitude Frequency Damping factor Set according σ σ α α β β π π to experience to experience / 2 / 2 (s 1 ) (s -1 ) (mrad) (mrad) (Hz) (Hz) i i i i i Identified from Pitch 0.40 0.19 0.13 experiment data Roll 0.68 0.17 0.11 Yaw 0.50 0.18 0.10 In order to reflect actual measurement environment we In order to reflect actual measurement environment, we σ 2 add Gaussian white noise with variance in dynamic ς i flexure signal. The SNR is defined by g y σ 2 = i SNR 10log σ i 10 2 ς ς i i

Results and Analysis i l d A lt R

R Results and Analysis lt d A l i Mean estimate errors for dynamic flexure parameters magnitude σ 2 , frequency β /2 π and damping factor α as well as measurement error with different SNR

C Conclusions l i � we have developed an on-line dynamic flexure parameters estimation approach based on T-K p pp method for KF based ship angular deformation measurement � Compared with previous methods, the proposed method offers: method offers: • on-line estimation (not require a priori knowledge) • accurate estimation i i • robust to noise and work conditions

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