EE 570: Location and Navigation
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1 EE 570: Locatio ad Navigatio Error Mechaizatio (NAV) Aly El-Osery Kevi Wedeward Electrical Egieerig Departmet, New Mexico Tech Socorro, New Mexico, USA I Collaboratio with Stephe Bruder Electrical ad Computer Egieerig Departmet Embry-Riddle Aeroautical Uivesity Prescott, Arizoa, USA April 4, 2016 Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
2 NAV Attitude Error C b = C b (Ωb ib Ωb i) = d dt [ (I + [δψ ] b ])Ĉ b = (I + [δ ψ b ])Ĉ b Ωb b = [δ ψ b ]Ĉ b + (I + [δ ψ b ]) Ĉ b = (I + [δ ψ b ])Ĉ b (ˆΩ b b + δωb ib δωb i) (I + [δ ψ b ])Ĉ b ˆΩ b b + Ĉ b (δωb ib δωb i) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
3 NAV Attitude Error C b = C b (Ωb ib Ωb i) = d dt [ (I + [δψ ] b ])Ĉ b = (I + [δ ψ b ])Ĉ b Ωb b = [δ ψ b ]Ĉ b + (I + [δ ψ b ]) Ĉ b = (I + [δ ψ b ])Ĉ b (ˆΩ b b + δωb ib δωb i) (I + [δ ψ b ])Ĉ b ˆΩ b b + Ĉ b (δωb ib δωb i) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
4 NAV Attitude Error C b = C b (Ωb ib Ωb i) = d dt [ (I + [δψ ] b ])Ĉ b = (I + [δ ψ b ])Ĉ b Ωb b = [δ ψ b ]Ĉ b + (I + [δ ψ b ]) Ĉ b = (I + [δ ψ b ])Ĉ b (ˆΩ b b + δωb ib δωb i) (I + [δ ψ b ])Ĉ b ˆΩ b b + Ĉ b (δωb ib δωb i) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
5 NAV Attitude Error C b = C b (Ωb ib Ωb i) = d dt [ (I + [δψ ] b ])Ĉ b = (I + [δψ b ])Ĉ b Ωb b = [δ ψ b ]Ĉb + (I + [δ ψ b ]) Ĉ b = (I + [δψ b ])Ĉ b (ˆΩ b b + δωb ib δωb i) (I + [δψ b ])Ĉ b ˆΩ b b + Ĉ b (δωb ib δωb i) [δψ b ]δωb b 0 Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
6 NAV Attitude Error C b = C b (Ωb ib Ωb i) = d dt [ (I + [δψ ] b ])Ĉ b = (I + [δψ b ])Ĉ b Ωb b = [δ ψ b ]Ĉb + (I + [δ ψ b ]) Ĉ b = (I + [δψ b ])Ĉ b (ˆΩ b b + δωb ib δωb i) = (I + [δψ b ])Ĉ b ˆΩ b b + Ĉ b (δωb ib δωb i) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
7 NAV Attitude Error C b = C b (Ωb ib Ωb i) = d dt [ (I + [δψ ] b ])Ĉ b = (I + [δ ψ b ])Ĉ b Ωb b = [δ ψ b ]Ĉ b + (I + [δ ψ b ]) Ĉ b = (I + [δ ψ b ])Ĉ b (ˆΩ b b + δωb ib δωb i) (I + [δ ψ b ])Ĉ b ˆΩ b b + Ĉ b (δωb ib δωb i) [δ ψ b ] = Ĉ b (δωb b δωb i)ĉ b = [Ĉ b (δ ω b ib δ ω b i) ] (1) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
8 NAV Attitude Error C b = C b (Ωb ib Ωb i) = d dt [ (I + [δψ ] b ])Ĉ b = (I + [δ ψ b ])Ĉ b Ωb b = [δ ψ b ]Ĉ b + (I + [δ ψ b ]) Ĉ b = (I + [δ ψ b ])Ĉ b (ˆΩ b b + δωb ib δωb i) (I + [δ ψ b ])Ĉ b ˆΩ b b + Ĉ b (δωb ib δωb i) [δ ψ b ] = Ĉ b (δωb b δωb i)ĉ b = [Ĉ b (δ ω b ib δ ω b i) ] (1) δ ψ b = Ĉ b (δωb ib δ ω b i) (2) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
9 NAV Attitude Error (cot.) Computig δ ω b i Recall that ω b i = C b ω i Expressig the above equatio i terms of estimates we get ˆ ω i b + δ ω b i = Ĉ b (I [δψ b ])(ˆ ω i + δ ω i) δ ω b i Ĉ b (δ ω i [δψ b ]ˆ ω i) Substitutig this result i Equatio 2 δ ψ b = ˆ Ω i δψ b + Ĉ b δ ω b ib δ ω i (3) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
10 NAV Attitude Error (cot.) Computig δ ω i Usig Taylor series ad retaiig the first order terms where r eb = [L b, λ b, h b ] T ad δ ω i = ˆ ω i δ r ˆ r eb + ˆ ω i δ v eb ˆ v eb eb ω ie cos ˆL b + ˆ v eb,e R ˆ ω i = ˆ ω ie + ˆ ω E (ˆL b )+ĥ b e = ˆ v eb,n R N (ˆL b )+ĥb ω ie si ˆL b ˆ v eb,e ta ˆL b R E (ˆL b )+ĥ b Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
11 NAV Attitude Error (cot.) Defie terms for simplicity ˆ ω i ˆ r eb ˆ ω i ˆ v eb ˆ Ω i = [(ˆ ω ie + ˆ ω e) ] = F ψψ (4) RE (ˆLb)+ĥb = = F ψv (5) RN(ˆLb)+ĥb ta 0 ˆLb 0 RE (ˆLb)+ĥb ˆ v eb,e ω ie si ˆL b 0 (RE (ˆLb)+ĥb)2 = ˆ v 0 0 eb,n = F ψr (6) (RN(ˆLb)+ĥb)2 ω ie cos ˆL ˆ v b + eb,e 0 ˆ v eb,e ta ˆLb (RE (ˆLb)+ĥb) cos 2 ˆLb (RE (ˆLb)+ĥb) 2 Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
12 NAV Attitude Error (cot.) Fial Expressio δ ψ b = F ψψ δ ψ b + F ψv δ v eb + F ψr δ r eb + Ĉ b δ ω b ib (7) where F ψψ, F ψv ad F ψr are defied by Equatios 4, 5 ad 6, respectively. Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
13 Velocity v eb = C b f b ib + g b (Ω e + 2Ω ie) v e eb (8) ˆ v eb = Ĉ b ˆ f b ib + ˆ g b (ˆΩ e + 2ˆΩ ie)ˆ v eb = (I [δ ψ b ])C b ( f b ib δ f b ib ) + ˆ g b (ˆΩ e + 2ˆΩ ie)ˆ v eb (9) Computig the δ v eb we obtai δ v eb = v eb ˆ v eb = [δ ψ b ]C b f b ib + Ĉ b δ f b ib + δ g b + (ˆΩ e + 2ˆΩ ie)δ v eb (δω e + 2δΩ ie)ˆ v eb Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
14 Velocity Cot. (ˆΩ e + 2ˆΩ ie) The term (ˆΩ e + 2ˆΩ ie ) is dervied as 0 2ω ie si ˆL b ˆ v eb,e ta ˆLb RE (ˆLb)+ĥb RN(ˆLb)+ĥb [(ˆ ω e + 2ˆ ω ie) ] = 2ω ie si ˆL b + ˆ v eb,e ta ˆLb 0 2ω ie cos ˆL b + ˆ v eb,e = F ΩΩ (10) RE (ˆLb)+ĥb RE (ˆLb)+ĥb ˆ v eb,n 2ω ie cos ˆL b ˆ v eb,e 0 RN(ˆLb)+ĥb RE (ˆLb)+ĥb ˆ v eb,n Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
15 Velocity Cot. (δ ˆΩ e + 2δ ˆΩ ie)ˆ v eb The term (δ ˆΩ e + 2δ ˆΩ ie )ˆ v eb is dervied as (δ ˆΩ e + 2δ ˆΩ ie)ˆ v eb = F ΩΩ ˆ r eb δ r ebˆ v eb + F ΩΩ δ v ebˆ v ˆ v eb (11) eb Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
16 Velocity Cot. Gravity Error δ g b 2g 0(ˆL b ) r e es (ˆL b ) δh b (12) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
17 Velocity Error Cot. After tos of algebra δ v eb = F vψδ ψ b + F vvδ v eb + F vr δ r eb + Ĉ b δ f b ib (13) where F vψ, F vv ad F vr are defied by Equatios 14, 15 ad 16, respectively. Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
18 Velocity Cot. F vψ, F vv ad F vr F vr = ˆv eb,d (RN(ˆLb)+ĥb) F vv = ˆv eb,e ta ˆLb + 2ω (RE (ˆLb)+ĥb) ie si ˆL b 2ˆv eb,n RN(ˆLb)+ĥb [ ] F vψ = (Ĉ b ˆ f b ib ) (ˆv eb,e )2 sec 2 ˆLb 2ˆv ω ie cos ˆL b 0 RE (ˆLb)+ĥb eb,e 2ˆv eb,e ta ˆLb 2ω (RE (ˆLb)+ĥb) ie si ˆL ˆv eb,n b RN(ˆLb)+ĥb ˆv eb,n ta ˆLb+ˆv eb,d ˆv eb,e + 2ω RN(ˆLb)+ĥb (RE (ˆLb)+ĥb) ie cos ˆL b 2ˆv eb,e 2ω (RE (ˆLb)+ĥb) ie cos ˆL b 0 ˆv eb,n ˆv eb,e sec2 ˆLb + 2ˆv RE (ˆLb)+ĥb eb,n ω ie cos ˆL b 2ˆv eb,d ω ie si ˆL b 0 ˆv eb,n ˆv eb,e ta ˆLb+ˆv eb,e ˆv eb,d (RE (ˆLb)+ĥb)2 2ˆv eb,e ω ie si ˆL b 0 (ˆv eb,e )2 ta ˆLb ˆv eb,n ˆv eb,d (RE (ˆLb)+ĥb)2 (RN(ˆLb)+ĥb)2 ( ) (ˆv eb,e )2 + (ˆv eb,n )2 2g0(ˆLb) (RE (ˆLb)+ĥb) 2 (RN(ˆLb)+ĥb) 2 res e (ˆLb) (14) (15) (16) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
19 Positio ˆ r eb = ˆ v eb,n R N (ˆL b )+ĥb ˆ v eb,e cos ˆL b (R E ( ˆL ) b )+ĥ b ) ˆ v eb,d (17) Computig δ r eb usig taylor series expasio ad retaiig oly the first order terms δ r eb = F rψδ ψ b + F rv δ v eb + F rr δ r eb (18) where F rψ, F rv ad F rr are defied by Equatios 19, 20 ad 21, respectively. Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
20 Positio Cot. F rψ, F rv ad F rr F rψ = (19) R N(ˆL b)+ĥb F rv = (R E (ˆL b)+ĥ b) cos ˆL b (20) 0 0 F rr = ˆv eb,e ta ˆL b 0 (R N(ˆL b)+ĥ b) cos ˆL b ˆv eb,n (R N(ˆL b)+ĥ b) 2 ˆv eb,e (R N(ˆL b)+ĥ b) 2 cos ˆL b (21) Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
21 Summary - i terms of δ f b ib, δ ω b ib δ ψ b F δ v ψψ F ψv F ψr δ ψb 0 Ĉ b b = F vψ F vv F vr δ v b + Ĉ b 0 δ f b ib (22) δ ω b δ r b F rv F rr δ r b ib 0 0 Attitude NAV Velocity Error Nav Positio Error Summary Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
22 Notatio Used Truth value Measured value x x Estimated or computed value ˆ x Error δ x = x ˆ x Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
23 Notatio Used Truth value Measured value Estimated or computed value x x Nothig above ˆ x Error δ x = x ˆ x Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
24 Notatio Used Truth value Measured value Estimated or computed value x x ˆ x Use tilde Error δ x = x ˆ x Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
25 Notatio Used Truth value Measured value Estimated or computed value x x ˆ x Error δ x = x ˆ x Use hat Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
26 Notatio Used Truth value Measured value x x Estimated or computed value ˆ x Error δ x δ x = x ˆ x ˆ x Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
27 Liearizatio usig Taylor Series Expasio Give a o-liear system x = f ( x, t) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
28 Liearizatio usig Taylor Series Expasio Give a o-liear system x = f ( x, t) Let s assume we have a estimate of x, i.e., ˆ x such that x = ˆ x + δ x x = ˆ x + δ x = f (ˆ x + δ x, t) (23) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
29 Liearizatio usig Taylor Series Expasio Give a o-liear system x = f ( x, t) Let s assume we have a estimate of x, i.e., ˆ x such that x = ˆ x + δ x x = ˆ x + δ x = f (ˆ x + δ x, t) (23) Usig Taylor series expasio f (ˆ x + δ x, t) = ˆ x f ( x, t) + δ x = f (ˆ x, t) + x δ x + H.O.T x=ˆ x ˆ x f ( x, t) + x δ x x=ˆ x Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
30 Liearizatio usig Taylor Series Expasio Give a o-liear system x = f ( x, t) Let s assume we have a estimate of x, i.e., ˆ x such that x = ˆ x + δ x x = ˆ x + δ x = f (ˆ x + δ x, t) (23) Usig Taylor series expasio f (ˆ x + δ x, t) = ˆ x f ( x, t) + δ x = f (ˆ x, t) + x δ x + H.O.T = x=ˆ x ˆ x f ( x, t) + x δ x x=ˆ x Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
31 Liearizatio usig Taylor Series Expasio Give a o-liear system x = f ( x, t) Let s assume we have a estimate of x, i.e., ˆ x such that x = ˆ x + δ x x = ˆ x + δ x = f (ˆ x + δ x, t) (23) Usig Taylor series expasio f (ˆ x + δ x, t) = ˆ x f ( x, t) + δ x = f (ˆ x, t) + x δ x + H.O.T x=ˆ x ˆ x f ( x, t) + x δ x x=ˆ x δ x f ( x, t) x δ x (24) x=ˆ x Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
32 Actual Measuremets Iitially the accelerometer ad gyroscope measuremets, f b ib ad ω b ib, respectively, will be modeled as f b ib = f b ib + f b ib (25) ω b ib = ω b ib + ω b ib (26) where f b ib ad ω b ib are the specific force ad agular rates, respectively; ad f b ib ad ω b ib represets the errors. I later lectures we will discuss more detailed descriptio of these errors. Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
33 Actual Measuremets Iitially the accelerometer ad gyroscope measuremets, f b ib ad ω b ib, respectively, will be modeled as f b ib = f b ib + f b ib (25) these terms may ω b ib = ω b ib + ω b be expaded further ib (26) where f b ib ad ω b ib are the specific force ad agular rates, respectively; ad f b ib ad ω b ib represets the errors. I later lectures we will discuss more detailed descriptio of these errors. Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
34 Error Modelig Example Accelerometers f b ib = b a + (I + M a ) f b ib + l a + w a (27) Gyroscopes ω b ib = b g + (I + M g ) ω b ib + G g f b ib + w g (28) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
35 Error Modelig Example Accelerometers f b ib = b a + (I + M a ) f b ib + l a + w a (27) Biases Gyroscopes ω b ib = b g + (I + M g ) ω b ib + G g f b ib + w g (28) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
36 Error Modelig Example Accelerometers f b ib = b a + (I + M a ) f b ib + l a + w a (27) Misaligmet ad SF Errors Gyroscopes ω b ib = b g + (I + M g ) ω b ib + G g f b ib + w g (28) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
37 Error Modelig Example Accelerometers f b ib = b a + (I + M a ) f b ib + l a + w a (27) No-liearity Gyroscopes ω b ib = b g + (I + M g ) ω b ib + G g f b ib + w g (28) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
38 Error Modelig Example Accelerometers f b ib = b a + (I + M a ) f b ib + l a + w a (27) Gyroscopes G-Sesitivity ω b ib = b g + (I + M g ) ω b ib + G g f b ib + w g (28) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
39 Error Modelig Example Accelerometers f b ib = b a + (I + M a ) f b ib + l a + w a (27) Noise Gyroscopes ω b ib = b g + (I + M g ) ω b ib + G g f b ib + w g (28) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
40 Pos, Vel, Force ad Agular Rate Errors Positio error Velocity error Specific force errors δ r γ βb = r γ βb ˆ r γ βb (29) δ v γ βb = v γ βb ˆ v γ βb (30) δ f b ib = f b ib ˆ f b ib (31) ef b ib = f b ib ˆ f b ib = δf b ib (32) Agular rate errors δ ω b ib = ω b ib ˆ ω b ib (33) e ω b ib = ω b ib ˆ ω b ib = δ ω b ib (34) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
41 Attitude Error Defiitio Defie δc γ b = C γ b Ĉ γ b = e [δ ψ γ γb ] I + [δψ γ γb ] (35) This is the error i attitude resultig from errors i estimatig the agular rates. Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
42 Attitude Error Properties The attitude error is a multiplicative small agle trasformatio from the actual frame to the computed frame Ĉ γ b = (I [δ ψ γ γb ])C γ b (36) Similarly, C γ b = (I + [δ ψ γ γb ])Ĉ γ b (37) Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
43 Specific Force ad Agular Rates Similarly we ca attempt to estimate the specific force ad agular rate by applyig correctio based o our estimate of the error. ˆ f b ib = f b ib ˆ f b ib (38) ˆ ω b ib = ω b ib ˆ ω b ib (39) where ˆ f b ib ad ˆ ω b ib are the accelerometer ad gyroscope estimated calibratio values, respectively. Basic Defiitios Liearizatio Iertial Measuremets Attitude Error Estimates of Sesor Measuremets Aly El-Osery, Kevi Wedeward (NMT) EE 570: Locatio ad Navigatio April 4, / 23
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