% p34 = plot(1:10:Tplot, 1000*error(kk, b).int_sh_timeave(1:10:Tplot), 'r-'); % % p35 = plot(1:10:Tplot, 1000*error(kk, b).state_est_timeave(1:10:Tplot), 'g-'); % title('Mean Error of target with and without bias correction'); % xlabel('Time (seconds)'); % ylabel('Mean Error (m)'); % legend('Without bias correction','Bias EKF correction','Bias sample and hold correction'); %, 'Bias EKF and State LKF' % figure(5); hold on; p55 = plot(1:Tplot, bsaz(b,1:Tplot), 'b-.'); P56 = plot(1:Tplot, bsel(b,1:Tplot), 'r-.'); p51 = plot(1:TS:Tplot, Sensor(1).bias_sh(2,1:TS:Tplot), 'k:'); p52 = plot(1:TS:Tplot, Sensor(1).bias_est(2,1:TS:Tplot), 'k-'); % p53 = plot(1:TS:Tplot, Sensor(2).bias_sh(1,1:TS:Tplot), 'b:'); % p54 = plot(1:TS:Tplot, Sensor(2).bias_sh(2,1:TS:Tplot), 'k:'); p57 = plot(1:TS:Tplot, Sensor(1).bias_sh(1,1:TS:Tplot), 'k:'); p58 = plot(1:TS:Tplot, Sensor(1).bias_est(1,1:TS:Tplot), 'k-'); xlabel('Time (secs)'); ylabel('Bias (rads)'); % legend('Without bias filter', 'With bias filter', 'True bias'); title('Sensor 1 Bias measurements without and with Kalman Filtering'); figure(4); hold on; p45 = plot(1:Tplot, bsaz(b,1:Tplot), 'b-.'); P46 = plot(1:Tplot, bsel(b,1:Tplot), 'r-.'); p41 = plot(1:TS:Tplot, Sensor(2).bias_sh(2,1:TS:Tplot), 'k:'); p42 = plot(1:TS:Tplot, Sensor(2).bias_est(2,1:TS:Tplot), 'k-'); % p43 = plot(1:TS:Tplot, Sensor(2).bias_sh(1,1:TS:Tplot), 'b:'); % p44 = plot(1:TS:Tplot, Sensor(2).bias_sh(2,1:TS:Tplot), 'k:'); p47 = plot(1:TS:Tplot, Sensor(2).bias_sh(1,1:TS:Tplot), 'k:'); p48 = plot(1:TS:Tplot, Sensor(2).bias_est(1,1:TS:Tplot), 'k-'); title('Sensor 2 Bias measurements without and with Kalman Filtering'); xlabel('Time (secs)'); ylabel('Bias (rads)'); end % b==3 % Print results results.error(b+1,1) = biasvect(b)'; results.error(b+1,2) = error(kk, b).int_nocorr_runave; results.error(b+1,3) = error(kk, b).int_nocorr_RMS; results.error(b+1,4) = error(kk, b).int_EKF_runave; results.error(b+1,5) = error(kk, b).int_EKF_RMS; results.error(b+1,6) = improv(kk, b).int_EKF; results.error(b+1,7) = error(kk, b).int_sh_runave; results.error(b+1,8) = error(kk, b).int_sh_RMS; results.error(b+1,9) = improv(kk, b).int_sh; results.error(b+1,10) = error(kk, b).state_est_runave; results.error(b+1,11) = error(kk, b).state_est_RMS; results.error(b+1,12) = improv(kk, b).state_est; 169
data = results.error(b+1,:) stars = [nstars_1 nstars_2] fprintf(savefile, '%12.6e\t', data); fprintf(savefile, '\r Number of stars: Sensor 1: %6.0f\t, Sensor 2: %6.0f\n', stars); fprintf(savefile, 'MSE: \t'); fprintf(savefile, '%12.6e\t', MSE(kk,:)); fprintf(savefile, 'RMS: \t'); fprintf(savefile, '%12.6e\t', RMS(kk,:)); fprintf(savefile, 'Mean Error: \t'); fprintf(savefile, '%12.6e\t', Mean_error(kk,:)); if b == 3 figure(6); hold on; p64 = plot(kstart:10:kend, error(kk, b).state_est_timeave(1:10:kend-kstart+1), lk); title('Mean Error of target with EKF bias correction'); xlabel('Time (seconds)'); ylabel('Mean Error Km'); axis([0 1800 0 1]); end end end % end bias % end kk % ------------------------------ Calculate and plot PCRLB ----------------------- tcrb = 600; Qn = [1/3 1/2 0 0 0 0; 1/2 1 0 0 0 0; 0 0 1/3 1/2 0 0; 0 0 1/2 1 0 0; 0 0 0 0 1/3 1/2; 0 0 0 0 1/2 1]; Qcrb = process_error*Qn/1000; for t = 1:tcrb Jz_1 = PCRLB(target.posit(:,t), Sat(1).posit(:,t), oNoise); Jz_2 = PCRLB(target.posit(:,t), Sat(2).posit(:,t), oNoise); if t == 1 J(:,:,t) = eye(6); else J(:,:,t) = inv(Qcrb+ A*inv(J(:,:,t-1))*A') + .5*Jz_1 + .5*Jz_2; end % if t==0 CRLB(:,:,t) = inv(J(:,:,t)); CRLBave(t) = CRLB(1,1,t)^2 + CRLB(3,3,t)^2 + CRLB(5,5,t)^2; end % end t 1-50 figure(8) 170
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Improved Space Target Tracking Thro
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DEDICATION I dedicate this disserta
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TABLE OF CONTENTS Page LIST OF TABL
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LIST OF TABLES Table Page Table 1.
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Figure 29. Mean error of target pos
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u Deterministic input g Gravitation
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The system then utilizes a separate
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Using higher altitude to gain advan
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Associate tracks from other element
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For the nominal scenario Table 1 pr
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1.3. Purpose of the Study The curre
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section. In a missile defense scena
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As mentioned above, errors in satel
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2. STATE OF THE ART From the early
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2.1.1. The Kalman Filter The Kalman
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2.1.4. The Unscented Kalman Filter
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y accuracy of an initial guess of t
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the state vector in some additive m
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. Orbital Coordination Frame (O-Fra
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e. Telescope Pointing Frame (T-fram
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an object sighting message (OSM). T
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vector until a new measurement is m
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ias correction and tracking algorit
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3.2.2. Target model For this study
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4. BIAS MEASUREMENT AND CORRECTION
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Table 4. Mean error in target posit
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otation axis and measures azimuth d
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For this study, the Aura catalog wa
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position of the satellite. This ang
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As target tracking proceeds there w
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Latitude about a 0.25 and 0.5 Hertz
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4.4. Bias modeling and measurement
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β az = tan −1 Δz ex − Δz act
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and β k = β k−1 + K β z β k
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Bias (rads) Bias measurements witho
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Using a test scenario we studied th
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Mean Error (m) 800 700 Mean Error o
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z X S 2 v 2 y v 1 i S 1 i Figure
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independent estimate from each sens
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z k = (x k ) + v k (5-12) where h(x
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the = 90 degrees, the ellipse beco
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to be white and independent with re
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where (L + λ)P x is the ith row or
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5.2.6. Sensor Fusion Following the
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Mean Error Km Table 11. Mean error
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Mean Error Km Mean Error Km 0.8 0.7
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Mean Error Km Mean Error Km 1 0.9 0
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Mean Error Km Mean Error Km 1 0.9 0
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6. FILTER PERFORMANCE MEASURES In t
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J z k = E M J z k: M = M p M J z (k
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Error variance (km)2 Error variance
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epsilon epsilon 6000 5000 4000 3000
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Each scenario test, with the except
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Furthermore, the bias correction ha
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sampling, we tested two different b
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spacecraft attitude, or the measure
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first was using the intercept point
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also be tested on maneuvering targe
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z APPENDIX A: SIMULATION RESULTS Sc
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Mean Error Km Mean Error Km 0.8 0.7
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Bias (rads) Bias (rads) Sensor 1 Bi
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Mean Error Km Bias (rads) 3 x 10-4
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Bias (rads) Mean Error Km 1 0.9 Mea
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114 NEA-SF-DEVELOP Case 1 Filter ty
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116 NEA-SF-DEVELOP Time step = 5 se
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- Page 167 and 168: APPENDIX B: SELECTED MATLAB PROGRAM
- Page 169 and 170: end % t end % bias_senario_case = 2
- Page 171 and 172: Tcount2 = 0; % --------------------
- Page 173 and 174: nearest_1 = NNtable_1(1,Q(11)); end
- Page 175 and 176: Tfirst2 = 0; Tcount2 = 0; end % end
- Page 177 and 178: else errorstatement = 'star_case er
- Page 179 and 180: % observation and Satellite positio
- Page 181: error(kk, b).int_EKF_MSE = sum(erro
- Page 185 and 186: Bias Extended Kalman Filter functio
- Page 187 and 188: Linearized Kalman Filter Function f
- Page 189 and 190: APPENDIX C: ORBITOLOGY BASICS The f
- Page 191 and 192: REFERENCES 178
- Page 193 and 194: [13] Chang, C. B. "Ballistic Trajec
- Page 195 and 196: [40] Sande, C., and N. Ottenstein.
- Page 197 and 198: Hue, Carine, Jean-Pierre le Cadre,