Moving horizon based wavelet de-noising method of dual-observed geomagnetic signal for nonlinear high spin projectile roll positioning

Ting-ting Yin, Fang-xiu Jia, Xiao-ming Wang

ZNDY of Ministerial Key Laboratory, Nanjing University of Science and Technology, Nanjing, 210094, China

Keywords:High-spin projectile roll positioning Dual-observed geomagnetic signal Wavelet de-noising Discrete wavelet transform

ABSTRACT Phase-frequency characteristics of approximate sinusoidal geomagnetic signals can be used for projectile roll positioning and other high-precision trajectory correction applications. The sinusoidal geomagnetic signal deforms in the exposed and magnetically contaminated environment. In order to preciously recognize the roll information and effectively separate the noise component from the original geomagnetic sequence, based on the error source analysis, we propose a moving horizon based wavelet de-noising method for the dual-observed geomagnetic signal filtering where the captured rough roll frequency value provides reasonable wavelet decomposition and reconstruction level selection basis for sampled sequence; a moving horizon window guarantees real-time performance and non-cumulative calculation amount. The complete geomagnetic data in full ballistic range and three intercepted paragraphs are used for performance assessment. The positioning performance of the moving horizon wavelet de-noising method is compared with the band-pass filter. The results show that both noise reduction techniques improve the positioning accuracy while the wavelet de-noising method is always better than the band-pass filter. These results suggest that the proposed moving horizon based wavelet de-noising method of the dual-observed geomagnetic signal is more applicable for various launch conditions with better positioning performance.

1. Introduction

The 2-dimensional (2D) trajectory correction projectile (TCP),replacing the traditional fuse with trajectory correction module,provides an innovative approach for the traditional ammunition reconstruction.Having the advantages of high efficiency-cost ratio and precise ground attack capabilities, the TCP has become a research hotspot recently[1].The nonlinear roll attitude detection and control of canards is the foundation to generate the required correction force[2].Taking on the responsibility of forming a closed roll control loop, high spin projectile roll positioning has been an active research area due to the tough application environment and strong nonlinearity [3].

During the whole external ballistics, due to the energy consumption caused by air and friction, the actual projectile spin rate varies with time and space, That is, the amplitude and the frequency of the sinusoidal signal vary with time and different launching conditions [4,5]. Theoretically, the launching overload reaching up to 8000-20000 g,the spin frequency ranging from 150 to 400 r/s as well as the minimal allocation space, limit the applications of gyroscopes, accelerometers, optimal sensors and satellites. Without redundant reference system and weather restrictions, approaches based on geomagnetic sensors have been developed to address the problem of nonlinear roll state estimation.

Geomagnetic signal based positioning techniques can be classified into geomagnetic-independent, geomagnetic-gyroscopescombined and satellite-geomagnetic-combined (MR/GPS) techniques. Cao gives the solution based on the geomagneticindependent and MR/GPS techniques respectively with prior and real-time satellite provided ancillary angle information [6]. The former technique diverges in multiple trajectory correction conditions and the MR/GPS method described by Li is excluded due to error accumulation characteristics [7]. Regarding MR/GPS techniques, Yang [8] and Shi [9] describe a state estimation algorithm based on Kalman filters and quasi Newton Methods with feasibility and accuracy analysis. Cao describes the approximate roll positioning model where the low frequency satellite data are interpolated at sampling rates [10]. However, the sampled sequence is directly substituted where the noise is never considered in the above models.

During the trajectory correction process, a high-power electromagnetic actuator is controlled by the high-frequency pulses [11],which should be considered as the TCP platform-specific environmental noise which interferes with the sinusoidal signal to a certain degree as well as the other disturbing factors like the installation error, the measurement noise, the sensor noise and the magnetic noise from iron parts. Some corresponding methods to reduce the disturbance have to be taken during the process of electromagnetic actuator and measuring system design, such as spatial isolation of the sensor and the electromagnetic actuator system, electrical separation of the important measuring signals and control pulses as well as the power supply, the frequency isolation of the control pulses and the projectile rate,the metal shield,ground connection and so on. Meanwhile, consistency and good environmental stability of the permanent magnetics, electrical components and circuit layout have been guaranteed. Even so, the TCP platformspecific disturbance can never be avoided just reduced.

In order to conduct a successful attitude search procedure, it is desirable to decrease the specific noise before geomagnetic data substitution. The hardware-dependent filtering means, i.e.,resistance-capacitance filters, Butterworth or Chebyshev filters, is rolling period, the wavelet basis function is determined by previously acquired data. Based on the MR/GPS solution model, we compare the filtering and positioning performance based on the proposed method with the results of a Butterworth based bandpass filter.

2. Dual observed geomagnetic signal for projectile roll positioning

The MR/GPS positioning method is proved to be an effective solution for projectile roll attitude.The rolling sensitive axes of the geomagnetic sensor S1are consistent with a missile coordinate system,as shown in Fig.1,where ε,γ,φ denote the auxiliary angle,the magnetic measurement angle, and the desired roll angle,Bbx，Bby，BbzandBx，By，Bzrepresent the projection of geomagnetic vector B in missile and sensitive coordinates.

The theoretic model for projectile roll positioning is given as:

Taking the natural vector geomagnetic field as the reference coordinate system,the relationship of sensitive outputBbx，Bby，Bbzand projectile posture can be written as:one of the typical de-noising solutions [12]. However, the hardware-dependent filters cannot match different trajectory conditions due to constant parameters, additionally, severe phase shifts need to be compensated. Besides, the de-noising scheme based on Fourier transform is rejected for the time-varying spectrum.The Kalman based filter is another typical de-noising method with approximate projectile model in the roll channel[13].Among these, the unscented Kalman filter (UKF) is more adaptive to the nonlinear variance. However, the performance of the UKF heavily depends on the prior knowledge of system uncertainties, external disturbances and measurement noises,otherwise,probably leading to a suboptimal and even divergent result [14,15].

The wavelet based de-noising techniques have been widely applied in signal processing and data optimization, having advantages of multi-resolution analysis in both time and frequency domains.Wavelet transforms use various forms of mother wavelets to approximate or decompose the unknown signal.Kaloop et al.apply wavelet de-noising for global positioning system(GPS)monitoring observations[16].Lau describes a wavelet packets based de-noising method for repeat-time multipath filtering in GPS positioning[17].Sang et al. established a wavelet based noise reduction system for hydrologic series data analysis[18]and the performance of discrete wavelet transform is verified [19]. All proposals above are built on the known decomposition and reconstruction level which is unknown and nonlinearly varies in the projectile roll detection.

In this work,we present a dual-observer based moving horizon wavelet de-noising algorithm that may be applied to noise suppression. Instead of deviated prior information, the real-time dual observed geomagnetic signal strategy extracting rough roll rate extraction for wavelet parameter configuration is described.Considering the similarity of geomagnetic sequence shape for each where θ,ψ and φ are respectively the pitch,drift and roll angle,Bn，Be，Bddenote the projections of the reference geomagnetic vector B in navigation coordinate system.

In the condition that the signal is known, the magnetic measurement angle is computed as

Substituting φ=0 into Eq. (2), it becomes:

According to Fig.1, the auxiliary angle is obtained by:

Based on the geomagnetic field reference model, such as the World Magnetic Model 2015(WMM2015), the reference vector parameters can be derived with longitude, latitude, and altitude information provided by satellites, as shown in Eq. (6).

whereFis defined as the total field intensity of B,DandIare the corresponding declination and inclination angle of B in geomagnetic field, shown in Fig.1.

Fig.1. The coordinate system definitions in geomagnetic field and roll plane.

Then the solving model for ε can be rewritten as:

Based on the proved assumption of small angle of attack, the posture angle can be approximately substituted by ballistic inclination and deflection angle, proposed as:

Based on the proved uniaxial rotation hypothesis,differentiating Eq. (1) gives:

In theory, the solution for projectile roll positioning can be considered complete. However, judging from the experimental data, the original regulated circular curve has been severely distorted to a non-orthogonal, offset, and inconsistent-amplitude curve. The projectile roll rate value, as reference for the rudder control,is seriously disturbed where the noise vector is introduced and amplified by the differential operation, as shown in Fig. 2.

The whole schedule of dual-observed geomagnetic signal for projectile roll positioning, where the geomagnetic signal is simultaneously input to the frequency identification system and AD phase sampling system,is shown in Fig. 3.

The frequency identification system applies a zero-crossing comparison circuit and a timer capture interface, where timings such asandare recorded(see Fig. 4).

Therefore,

It can be seen that the trigger timing can be simply and reliably updated at every rising edge of Z-axis output by setting proper timer parameters without accumulation error, even though the system error based on the adjacent period roll rate assumption is introduced. In order to assess the feasibility of the timer captured frequency value as the reference for wavelet decomposition and reconstruction level selection, the systematic error δ ˙φ is modeled and analyzed by trajectory simulations, as shown in Fig. 5.

It is derived from Fig.5 that the system error is less than 0.7 rad/s and approximately linear with ˙φ, suggesting that the rough roll rate value can be applied as reference for subsequent noise elimination work.

Fig. 2. The experimentally sampled and solved geomagnetic signal.

Fig. 3. The dual-observed geomagnetic signal with wavelet de-noising module.

Fig. 4. The rough roll rate detection for wavelet de-noising parameter determination.

Fig. 5. The systematic error based on the adjacent period roll rate assumption.

3. De-noising with discrete wavelet transform (DWT)

The proposed wavelet de-noising approach consists of two steps. A selective wavelet basis function and level for signal decomposition is the first step.The second step is applying wavelet shrinkage to the wavelet coefficients of the sub-bands selected by a rough captured roll rate value and realizing wavelet reconstruction with the shrunken wavelet coefficients. A detailed description of the above steps is provided as follows.

The essence of the wavelet transform is to reveal the similarity degree between the decomposed signal and the wavelet basis function. The approximately sinusoidal shaped functions with positive symmetry and regularity, where no phase distortion and signal smoothness are promised,are preferential choices.The basic shape feature of geomagnetic signal is basically unchanged,which suggests that the basis wavelet function selection can be completed off-line with previous experimental data. The functions to be selected are numbered and the decomposition, data classification and reconstruction are repeated to find out the only function closest to the ideal signal.

Fig. 6 describes the three-level discrete wavelet de-noising procedure as well as the sub-band distribution. The input signal is a discrete time seriesXi=xi-N+1,xi-N+2, …,xi, with the moving horizon window of lengthNillustrated in Fig. 7.

Fig. 6. The discrete wavelet decomposition and reconstruction tree.

Fig. 7. Moving horizon window for wavelet de-noising.

The broad range of decomposition level, denotingNd,can be determined by the following rules withfmin andfmax provided by trajectory simulations.With the captured rough roll rate before,the decomposition level selection range can be further narrowed,where σ1represents the relatively small observational error which can be directly derived from Fig. 2.

Fig. 8. The experimental geomagnetic sensitive and detection modules.

Fig.9. The experimentally obtained rough projectile roll rate information and wavelet decomposition levels.

As an example, the sampling frequency for geomagnetic signalFs is 20 kHz and the timer captured roll rate value is 198 Hz.Therefore,fmin=188 Hz andfmax=208 Hz, and according to equation,Nd=7.

According to the frequency band distribution in Fig.6,the highfrequency sub-band at the highest decomposition level is the desired frequency band. Reserving coefficients of the corresponding band HNdand setting the rest coefficients to zero, the coefficients for reconstruction are prepared. The final step reconstructs the de-noised geomagnetic signal from the shrunken coefficients with the selected basis wavelet function.

4. Results and analysis

In order to assess the impact of the proposed projectile roll positioning based on moving horizon wavelet de-noising,the data acquisition test where the geomagnetic signal is dually observed in amplitude and frequency is conducted, as shown in Fig. 8. The photometrical module based on solar azimuth is also installed onboard. With corrected posture angle as well as accurate solar azimuth vector as input parameters, the roll angle information based on solar azimuth method is obtained as reference for detection performance evaluation [20], where the nonlinear moving horizon estimator and unscented estimator are applied respectively for roll rate and position measurement disturbance elimination.

After applying the proposed wavelet de-noising method to the experimental data series, we detect the projectile position information with the MR/GPS based solution model. The nonlinear projectile roll rate information, as reference for wavelet parameters,is plotted in Fig. 9.

In order to compare the conventional filtering method with the proposed one,the hardware band-pass filtered signal integrated in experimental circuits is collected at the same time with the noisy geomagnetic signal. The detailed filtering performance of three typical interrupted sequences is shown in Fig.10,where the noisy,the band-pass filtered, and the wavelet de-noised geomagnetic curves as well as the solved positioning results are displayed.

It is obvious that the proposed wavelet de-noising strategy and the band-pass filter can both effectively eliminate the noisy component from the original geomagnetic signal with approximate sinusoidal shape and acceptable positioning errors in the heavily disturbed conditions as shown in the first data set.However,in the slightly interfered sections in the latter two data series, the bandpass filter introduced more electrical noise and distortion than the original signal due to the weak signal in the filtering and amplification link.in addition, the resulting phase shifts also need to be compensated by the empirical model in real time.In order to objectively describe the filtering performance, we use the root mean square (RMS), maximum jerk value and the mean error as measures of the detection performance, where the respective amplitudes are processed in a dimensionless manner for comparison(see Tables 1-3).

The statistical information of the first data sequence proves that under the condition that the geomagnetic signal is exposed tostrong disturbances, the DWT and BPF methods can respectively increase the signal quality by nearly 70% and 50% while the BPF methods need extra phase compensation computation.The second and third data sequences indicate that when autocorrelation of the original geomagnetic signal is better, on the contrary, the SNR(signal-to-noise-ratio) of the hardware filtered signal is worse. On the whole, the DWT based de-noising strategy has much better adaptability without limitations of launching conditions and geomagnetic environments.

Table 2 Statistical comparison of de-noising performance for the second data sequence.

Table 3 Statistical comparison of de-noising performance for the third data sequence.

5. Conclusion

The geomagnetic signal can be dual observed in frequency and phase domains after simple signal conditioning circuit, where the rough projectile roll rate information can be obtained by integrated timers. With the captured information as reference where the feasibility is proved by trajectory simulations, the moving horizon wavelet de-noising parameters can be determined while the wavelet function selection is conducted off-line with previous experimental data.The experimentally collected original and bandpass filtered geomagnetic signals as well as the simulation wavelet de-noised signals are compared by wave parameters, positioning results shown as curves and statistical information. The noise reduction performance and the positioning stability and adaptability to different experimental conditions reflect the superiority of the proposed dual-observed wavelet de-noising and positioning strategy in width-fixed moving horizon window. Furthermore,more studies of projectile roll characteristics should be conducted in the future to improve the performance wavelet-based analyses,such as noise distribution characteristics and the decomposition level prediction.

Acknowledgement

This paper is funded by National Natural Science Foundation of China (61201391).