EE 570: Location and Navigation
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1 EE 570: Location and Navigation Gyro and Accel Noise Characteristics Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA February 20, 2013 Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
2 Gyro Constant Bias ( /h) A constant in the output of a gyro in the absence of rotation, in /h. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
3 Gyro Constant Bias ( /h) A constant in the output of a gyro in the absence of rotation, in /h. Error Growth Linearly growing error in the angle domain of ǫt. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
4 Gyro Constant Bias ( /h) A constant in the output of a gyro in the absence of rotation, in /h. Error Growth Linearly growing error in the angle domain of ǫt. Model Random constant. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
5 Gyro Integrated White Noise Assuming the rectangular rule is used for integration, a sampling period of T s and a time span of nt s. t 0 ǫ(τ)dτ = T s n ǫ(t i ) (1) i=1 since E[ǫ(t i )] = 0 and Cov(ǫ(t i ), ǫ(t j )) = 0 for all i = j, Var[ǫ(t i )] = σ 2 Var [ t ] E ǫ(τ)d τ = T s ne[ǫ(t i )] = 0, i (2) 0 [ t ] ǫ(τ)d τ = Ts 2 nvar[ǫ(t i )] = T s tσ 2, i (3) 0 Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
6 Gyro Integrated White Noise 80 Integrated Angle Noise (deg) Time (sec) Gyro Noise Characteristics Aly El-Osery (NMT) Accel Noise Characteristics Allan Variance EE 570: Location and Navigation Using PSD and Allan Variance February 20, / 14
7 Angle Random Walk ( / h) Integrated noise resulted in zero-mean random walk with standard deviation that grows with time as We define ARW as σ θ = σ T s t (4) ARW = σ θ (1) ( / h) (5) In terms of PSD ARW( / h) = 1 PSD(( 60 /h) 2 /Hz) (6) Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
8 Angle Random Walk ( / h) Integrated noise resulted in zero-mean random walk with standard deviation that grows with time as We define ARW as σ θ = σ T s t (4) ARW = σ θ (1) ( / h) (5) In terms of PSD ARW( / h) = 1 PSD(( 60 /h) 2 /Hz) (6) Error Growth ARW times root of the time in hours. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
9 Angle Random Walk ( / h) Integrated noise resulted in zero-mean random walk with standard deviation that grows with time as We define ARW as σ θ = σ T s t (4) ARW = σ θ (1) ( / h) (5) In terms of PSD ARW( / h) = 1 PSD(( 60 /h) 2 /Hz) (6) Error Growth ARW times root of the time in hours. Model White noise. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
10 Gyro Bias Instability ( /h) Due to flicker noise with spectrum 1/F. Results in random variation in the bias. Normally more noticeable at low frequencies. At high frequencies, white noise is more dominant. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
11 Gyro Bias Instability ( /h) Due to flicker noise with spectrum 1/F. Results in random variation in the bias. Normally more noticeable at low frequencies. At high frequencies, white noise is more dominant. Error Growth Variance grows over time. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
12 Gyro Bias Instability ( /h) Due to flicker noise with spectrum 1/F. Results in random variation in the bias. Normally more noticeable at low frequencies. At high frequencies, white noise is more dominant. Error Growth Variance grows over time. Model First order Gauss-Markov. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
13 Accel Constant Bias (µg) A constant deviation in the accelerometer from the true value, in m/s 2. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
14 Accel Constant Bias (µg) A constant deviation in the accelerometer from the true value, in m/s 2. Error growth Double integrating a constant bias error of ǫ results in a quadratically growing error in position of ǫt 2 /2. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
15 Accel Constant Bias (µg) A constant deviation in the accelerometer from the true value, in m/s 2. Error growth Double integrating a constant bias error of ǫ results in a quadratically growing error in position of ǫt 2 /2. Model Random constant. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
16 Velocity Random Walk (m/s/ h) Integrating accelerometer output containing white noise results in velocity random walk (VRW) (m/s/ h). Similar to development of ARW, if we double integrate white noise we get t 0 ǫ(τ)dτdτ = T 2 s,sensor n i=1 i j=1 ǫ(t j ) (7) Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
17 Velocity Random Walk (m/s/ h) Integrating accelerometer output containing white noise results in velocity random walk (VRW) (m/s/ h). Similar to development of ARW, if we double integrate white noise we get t Error Growth 0 ǫ(τ)dτdτ = T 2 s,sensor Computing the variance results in n i=1 i j=1 ǫ(t j ) (7) σ p σt (3/2) Ts 3 (8) Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
18 Velocity Random Walk (m/s/ h) Integrating accelerometer output containing white noise results in velocity random walk (VRW) (m/s/ h). Similar to development of ARW, if we double integrate white noise we get t Error Growth 0 ǫ(τ)dτdτ = T 2 s,sensor Computing the variance results in n i=1 i j=1 ǫ(t j ) (7) σ p σt (3/2) Ts 3 (8) Model White noise. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
19 Accel Bias Stability (µg) Error growth Grows as t 5/2. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
20 Accel Bias Stability (µg) Error growth Grows as t 5/2. Model First order Gauss-Markov. Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
21 Allan Variance Introduction It is a time domain analysis techniques designed originally for characterizing noise in clocks. It was first proposed by David Allan in Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
22 Allan Variance Computation 1 Divide your N-point data sequence into adjacent windows of size n = 1, 2, 4, 8,...,M N/2. 2 For every n generate the sequence y j (n) = x [ ] nj + x nj+1 + +x nj+n 1 N, j = 0, 1,..., 1 (9) n n 3 Plot log-log of the Allan deviation which is square root of σ 2 Allan (nt s) = versus averaging time τ = nt s N 1 1 2(N 1) j=1 (y j y j 1 ) 2 (10) Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
23 One-sided PSD - Typical Slopes PSD(( /h) 2 /Hz) -2 RRW AWN +2 PSD((m/s 2 ) 2 /Hz) -2 Acceleration RW VWN +2-1 BI ARW 0-1 BI VRW 0 FREQUENCY(HZ) FREQUENCY(HZ) Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
24 Allan Deviation - Typical Slopes ALLAN DEVIATION( /h) AWN -1/2 ARW rate trend +1 RRW +1/2 ALLAN DEVIATION((m/s/h)) VWN -1/2 acceleration trend +1 acceleration RW +1/2 VRW 10 4 BI BI 0 AVERAGE TIME (S) AVERAGE TIME (S) Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
25 Noise Parameters Noise Type AV σ 2 (τ) PSD (2-sided) Quantization Noise 3 α2 τ 2 (2πf) 2 α 2 T s Angle/Velocity Random Walk α 2 τ α 2 Flicker Noise Angular Rate/Accel Random Walk 2α 2 ln(2) π α 2 τ 3 α 2 2πf α 2 (2πf) 2 Ramp Noise α 2 τ 2 2 α 2 (2πf) 3 Aly El-Osery (NMT) EE 570: Location and Navigation February 20, / 14
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