GPS Navigation State-Space Models

This content delves into GPS navigation utilizing state-space models. It covers various scenarios such as Allan standard deviations of receiver clock time measurement error, GPS-aided 9-state INS with clock error models, differential carrier phase for positioning, and more. Explore the intricacies and applications of GPS technology in aerospace engineering.

• GPS navigation
• State-space models
• Aerospace engineering
• Allan standard deviations
• Clock errors

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1. B&H, CH. 11: GPS NAVIGATION [SEE THE NOTES FOR STATE-SPACE MODELS] PROB. 11.1 ALLAN STANDARD DEVIATIONS OF RECEIVER CLOCK TIME MEASUREMENT ERROR PROB. 11.2A: GPS AIDED 9-STATE INS + 2-STATE CLOCK ERROR MODEL: A) WITH RANGE MEASUREMENTS ; B) WITH FOUR RANGE AND THREE ATTITUDE MEASUREMENTS PROB. 11.6: RECEIVER HOST STATIONARY; 5-STATE KF: THREE POSITION STATES, TWO CLOCK STATES, WITH GPS MEASUREMENTS; PROB. 11.7: DIFFERENCED MEASUREMENTS 1-2,2-3,3-4, AND CLOCK ERROR ELIMINATED PROB. 11.8: DIFFERENTIAL CAREER PHASE FOR POSITIONING: 5-STATE MODEL (POSITION ERROR + CLOCK ERRORS): RECEIVER AT LAT=0, LON=0, A) 4 GPS SATELLITES; B) 3 SATELLITES Navigation and Guidance Course March 24, 2012 Department of Aerospace Engineering, IIT Bombay

2. Prob. 11.1 Allan Standard Deviations of Receiver Clock Time Measurement Error: Exact, and its Approximations from its Two-State Model Allan Variance0.5 and its approximations: Time Measurement Error: 1 (s) 10-8 Allan Allan approx Allan mod (h0/(2 ))0.5 2h-1 (2/3 2 h-2 )0.5 0.5 10-9 10-10 10-2 10-1 100 101 102 103 GPS Measurement Interval (s)

3. Prob. 11.2a: GPS aided 9-state INS + 2-state clock error model: a) with range measurements: Input Parameters Earth's angular speed: omegaE = 0.7292115e-4 rad/s [east: 0; north: omegaE*cos(lambda); up: omegaE*sin(lambda)]; lambda=latitude Power Spectral Density of a poor quality accelerometer : S_acc = 0.0036 (m/s^2)^2/Hz ; it introduces spurious acceleration 0.06 (m/s^2)/Hz^0.5 = 0.06/9.81 Vehicle with modest dynamics: ENU Geographic frame rate: [-lamda_dot; (omegaE+phi_dot)*cos(lamda); (omegaE+phi_dot)*sin(lamda)] = 0.0061 g/Hz^0.5 = 6100 micro g/Hz^0.5 Power Spectral Density of a high quality gyro: Sgyro = 2.35e-11 (rad/s)^2/Hz; It introduces spurious drift rate omegax=-.0000157 rad/s; omegay=.0000727 rad/s; 2.35e-11 ^ 0.5 = 4.85e-6 (rad/s)/Hz^0.5 = Identical accelerometers and gyros for three axes 2.78e-4 (deg/s)/Hz^0.5 :

4. Prob. 11.2a: GPS aided 9-state INS + 2-state clock error model: a) with pseudorange measurements ( =15 m) (1/4) Prob. 11.2a: GPS ranging aided-INS (1 Hz GPS Measurements) 10-1 North Tilt East tilt Azimuth Error Tilt about North and East Axis and azimuth error (rad) 10-2 10-3 10-4 0 100 200 300 400 500 600 700 800 900 1000 time (s)

5. GPS aided 9-state INS + 2-state clock error model: a) with pseudorange measurements ( =15 m) (2/4) Prob. 11.2a: GPS ranging aided-INS (1 Hz GPS Measurements) 102 rms east rms north rms altitude rms east, north, altitude position error (m) 101 100 0 100 200 300 400 500 600 700 800 900 1000 time (s)

6. GPS aided 9-state INS + 2-state clock error model: a) with pseudorange measurements ( =15 m) (3/4) Prob. 11.2a: GPS ranging aided-INS (1 Hz GPS Measurements) 101 rms east rms north rms vertical rms east, north, vertical vel error (m/s) 100 10-1 0 100 200 300 400 500 600 700 800 900 1000 time (s)

7. GPS aided 9-state INS + 2-state clock error model: a) with pseudorange measurements ( =15 m) (4/4) rms range bias error due to receiver clock (m) rms range rate error due to receiver clock (m/s) Prob. 11.2a: GPS ranging aided-INS (1 Hz GPS Measurements) 102 101 100 0 100 200 300 400 500 600 700 800 900 1000 time (s) Prob. 11.2a: GPS ranging aided-INS (1 Hz GPS Measurements) 102 101 100 10-1 0 100 200 300 400 500 600 700 800 900 1000 time (s)

8. Prob. 11.2b: GPS aided 9-state INS + 2-state clock error model: b) with four range and three attitude measurements; compare with case a) Prob. 11.2b: GPS clock and attitude measurements aiding 9-state INS 10-1 North Tilt Azimuth Error Tilt about North Axis and azimuth error (rad) 10-2 10-3 10-4 0 100 200 300 400 500 600 700 800 900 1000 time (s)

9. 10-1 North Tilt Azimuth Error Tilt about North Axis and azimuth error (rad) 10-2 b) With GPS range and attitude measurements: a multiantenna system 10-3 10-1 North Tilt Azimuth Error Tilt about North Axis and azimuth error (rad) 10-4 0 100 200 300 400 500 600 700 800 900 1000 time (s) 10-2 a) With GPS range measurements 10-3 10-4 0 100 200 300 400 500 600 700 800 900 1000 time (s) Brown and Hwang, Prob. 11.2: Integrated GPS/INS: 9-state INS + clock bias and rate model

10. Prob. 11.6: Receiver Host stationary; 5-state KF: three position states, two clock states, with GPS measurements; Q matrix % PSD of velocity white noise the integral of which is the position error in x, y, z % 5-state transition matrix: [x y z c*deltaT(range error) c*deltaTdot(range rate error)]' PHI=eye(state_dim); PHI(4,5)=dt; Sp=1.0; (m/s)^2/Hz % PSD of white noise inputs for two-state clock error % Measurement matrix (constant) H=[-0.3523 -0.0495 0.9346 1 0; 0.6199 0.7406 0.2593 1 0; -0.9506 -0.2553 0.1764 1 0; 0.9613 0.2129 0.1747 1 0]; % Measurement noise 1 sigma: 15 m, for 4 satellites GPS_sig = 15.0; % range measurement error, 1 sigma R = GPS_sig^2 *eye(4); Sf=0.4E-18*c^2 sec*(m/s)^2 Sg=1.58E-18*c^2 (1/sec)*(m/s)^2 Q=[Sp*dt 0 0 0 0; 0 Sp*dt 0 0 0; 0 0 Sp*dt 0 0; 0 0 0 Sf*dt+(Sg*dt^3)/3 (Sg*dt^2)/2; 0 0 0 (Sg*dt^2)/2 Sg*dt];

11. Prob. 11.6 Stationary host; Covariance Analysis of 5-state Position + clock error model with GPS measurements (1/2) 5-state Position + clock error Model, with GPS range measurements 35 x y z GPS 30 RMS Position error: x, y, z (m) 25 20 15 10 5 0 0 10 20 30 40 50 60 time (s)

12. Prob. 11.6 Stationary host; Covariance Analysis of 5-state Position + clock error model with GPS measurements (2/2) 5-state Position + clock error Model, with GPS range measurements 15 RMS clock range (m) range error Q(4,4)0.5 10 5 0 0 10 20 30 40 50 60 time (s) RMS clock range rate error (m/s) 102 range rate error Q(5,5)0.5 101 100 10-1 0 10 20 30 40 50 60 time (s)

13. Prob. 11.7: Differenced measurements 1-2,2-3,3-4, and clock error eliminated; position error sigma slightly greater than that with earlier 4 measurements (Prob. 11-6) Prob. 11.7: differencing meas. 1-2,2-3,3-4 to eliminate clock error 20 differenced and regular measurements: rms pos. error (m) differenced regular 18 16 14 12 10 8 6 4 2 0 0 20 40 60 80 100 120 140 160 180 200 Time(s)

14. Prob. 11.8a: Differential Career Phase for Positioning: 5-state model (position error + clock): receiver at Lat=0, Lon=0, 4 GPS satellites (1/2) Position error PSD % Error covariance matrix: 100 m; clock errors: 300 m range bias, 30 m/s range rate Sp=0.01; % m^2; How are the units consistent? Clock error sources: white noise PSDs Sf=0.4E-18*c^2 sec*(m/s)^2 Sg=1.58E-18*c^2 (1/sec)*(m/s)^2 PMINUS = [10000 0 0 0 0; % (100 m)^2 0 10000 0 0 0; 0 0 10000 0 0; 0 0 0 90000 0; % (300 m)^2 range bias equivalent to clock time bias Q=[Sp*dt 0 0 0 0; 0 Sp*dt 0 0 0; 0 0 Sp*dt 0 0; 0 0 0 0 900]; % (30 m/s)^2 range rate clock drift 0 0 0 Sf*dt+(Sg*dt^3)/3 (Sg*dt^2)/2; 0 0 0 (Sg*dt^2)/2 Sg*dt]; measurement noise: (0.005 m)^2 R=0.005^2 *eye(4);

15. Prob. 11.8a: Differential Career Phase for Positioning: 5-state model (position error + clock): receiver at Lat=0, Lon=0, 4 GPS satellites (1/2) Prob. 11.8a: 5-state model (position + clock) of Differential Carrier Phase Meas. 102 x (1 ) y (1 ) z (1 ) 101 x,y,z position change 1 (m) 100 10-1 10-2 0 100 200 300 400 500 600 700 800 900 1000 Time(s)

16. Prob. 11.8a: Differential Career Phase for Positioning: 5-state model (position error + clock): 4 satellites (2/2) Prob. 11.8a: Differential Carrier Phase Meas. 102 clock bias: range range rate RMS clock errors: range (m) and range rate (m/s) (1 ) 101 100 10-1 10-2 10-3 -100 0 100 200 300 400 Time(s) 500 600 700 800 900 1000

17. Prob. 11.8b: Differential Career Phase for Positioning: 5-state model (position error + clock): 3 satellites (1/2) Prob. 11.8a: 5-state model (position + clock) of Differential Carrier Phase Meas. 102 x (1 ) y (1 ) z (1 ) 101 x,y,z position change 1 (m) 100 Prob. 11.8: 5-state model (position + clock) of Differential Carrier Phase Meas. 102 10-1 x,y,z position change 1 (m) 101 x (1 ) y (1 ) z (1 ) 10-2 0 100 200 300 400 500 600 700 800 900 1000 100 Time(s) 10-1 10-2 0 200 400 600 800 1000 Time(s)

18. Prob. 11.8b: Differential Career Phase for Positioning: 5-state model (position error + clock): 3 satellites (2/2) Prob. 11.8a: Differential Carrier Phase Meas. 102 clock bias: range range rate RMS clock errors: range (m) and range rate (m/s) (1 ) 101 100 Prob. 11.8: Differential Carrier Phase Meas. 102 RMS clock errors: range (m) and range rate (m/s); 1 10-1 clock bias: range range rate 101 10-2 100 10-3 -100 0 100 200 300 400 Time(s) 500 600 700 800 900 1000 10-1 10-2 10-3 0 200 400 600 800 1000 Time(s)