ERRATA TO: STRAPDOWN ANALYTICS, FIRST EDITION PAUL G. SAVAGE PARTS 1 AND 2 STRAPDOWN ASSOCIATES, INC. 2000

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1 ERRATA TO: STRAPDOWN ANAYTICS, FIRST EDITION PAU G. SAVAGE PARTS 1 AND 2 STRAPDOWN ASSOCIATES, INC Corrections Reported From April 2004 To Present Pg Section , first paragraph - Frequency folding is when the oscillation frequency is a multiple of the sample frequency, not the inverse as in the book. Sections and The section titles are interchanged. The Section title should be "E FRAME DEFINED ERROR PARAMETER RATE EQUATIONS FROM N FRAME DEFINED PARAMETER RATE EQUATIONS". The Section title should be "N FRAME DEFINED ERROR PARAMETER RATE EQUATIONS FROM E FRAME DEFINED PARAMETER RATE EQUATIONS". Pg The equation in the last sentence should be ( ). Corrections Reported From January 2004 To April 2004 Pg Delete the minus sign in front of 1 2 g in the μ zy equation. Corrections Reported From November 2003 To January 2004 Pg. B-2 (Volumes 1 and 2) - Add following Attitude: "Average of Averages filter " Pg. B-7 (Volumes 1 and 2) - Add to Quantization error (on inertial sensor outputs), Defined: " " -1-

2 Corrections Reported From June 2003 To November 2003 Pg , Table Axis Along Outer Rotation Fixture Axis for SEQUENCE NUMBER 20 should be -Y. Pg Equation ( ) should be: Conditions For Zero x Coupling Into Δx, Δy: x = I x * Φ xx or H x * = Hx = Φ xx Φ yx = 0 H x * = Hx ( ) or x = Diagonal H x * = Hx * If x (i=j, j) 1 Then Φ xx(i,j) = Φxx (i,j) And Φ yx (i,j) = 0 For All i Corrections Reported From 2000 To June 2003 Pg. 3-44, last line - Should be: "From Equation ( ), the conclusion following ( ), and the above definitions we have:" Pgs. 7-33, 7-34, 7-58, 7-59, 7-60, and Correction To The Navigation Frame Rotation Compensation Terms In The Velocity And Position Integration Algorithms The following is a revision to pages dealing with the small correction applied during acceleration transformation operations to account for rotation of the Frame during the m cycle. I(m) The term of concern is C I(n-1) I(n-1) - I Δv SFm in Equation ( ) which is incorrect in the form shown. The correct term is developed by rederiving Equation ( ). Using the matrix chain rule and integration by parts: -2-

3 Δv SFm I(m) = Δv SFm (t) = C I(m-1) I(m-1) I(n-1) C CB C B(t) asf dt I(m) = C I(m-1) (t) - C I(m-1) I(m-1) I(n-1) C CB I(m-1) I(n-1) C CB t C B(τ) asf dτ dt ( ) (t) Assuming C I(m-1) is approximately constant and linearly over the m cycle we then get: t C B(τ) asf dτ changes approximately Δv SFm I(m) C I(m-1) I(m-1) I(n-1) C CB - 1 I(m) C T - I m I(m-1) I(m-1) I(n-1) C CB t m t - t C B(τ) asf dτ m-1 T m dt I(m) = C I(m-1) I(m-1) I(n-1) C CB C I(m) I(m-1) = 1 2 C I(m) I(m-1) = 1 2 C I(m) I(n-1) I(m-1) I(n-1) - I C CB I(m-1) I(n-1) + I C CB I(m-1) I(n-1) + C CB ( a) With this result, Equations ( ) and ( ) are unchanged but Equation ( ) becomes: Δv SFm = 1 2 C I(m) I(n-1) I(m-1) I(n-1) + C ΔvSFm I(n-1) I(n-1) 1 = Δv SFm + 2 C I(m) I(m-1) I(n-1) - I + C I(n-1) - I ΔvSFm I(n-1) ( ) -3-

4 I(m) In the fifth line of Section , the analytical term C I(n-1) 1 2 C I(m) I(m-1) I(n-1) - I + C I(n-1) - I ΔvSFm I(n-1) I(m) Also change C I(n-1) I(n-1) - I Δv SFm should then be: - I in the sixth line of Section to the previous expression. For the position update in Section 7.3.3, the Δv SF(t) equation in ( ) should be: and the ΔR SFm 1 Δv SF(t) = 2 C (t) + C (m-1) (n-1)(t) I(n-1) (n-1) ΔvSF = 1 2 C (t) (m-1) = 1 2 C (t) (m-1) = 1 2 C (m) - I t - t m-1 (m-1) T m = 1 2 C (m) (m-1) = 1 2 C (m) (n-1) + I C (m-1) (n-1)(t) (n-1) ΔvSF - I C (m-1) (n-1)(t) (n-1) ΔvSF + (m-1) (n-1)(t) C(n-1) ΔvSF C (m-1) (n-1) t - t m-1 ΔvSFm (n-1) T m + C (m-1) (n-1)(t) (n-1) ΔvSF - I C (m-1) (n-1) t - t 2 m-1 ΔvSFm (n-1) + C (m-1) (n-1)(t) 2 (n-1) ΔvSF T m - C (m-1) (n-1) t - t 2 m-1 ΔvSFm (n-1) + C (m-1) (n-1)(t) 2 (n-1) ΔvSF T m equation in ( ) becomes: ΔR SFm = ζ (n-1) n-1,m - ζ n-1,m-1 Δv SFm Tm + C (m-1) (n-1) B CB(m-1) (n-1) ΔR SFm The effect on the Chapter 17 trajectory generator is: C m 1 m-1 I - 2 ζ v m ( ) ΔR SFm = ζ m-1 v m Δv SFm Tm + C B m-1 ΔR B SF m ( ) The previous corrections should also be applied to the equivalent terms in Equations ( ) and ( ). -4-

5 Pg. 8-6, last line - Should be replaced by: "δω Bias is determined by first writing from ( ):" Pg In the first sentence add "then" before "yields". The line preceding Equation ( ) should read: "Applying ( ) in ( ) also obtains for K Bias in alternate compensation Equations ( ):" Pg. 8-8, paragraph following Equation ( ) - Add at the end: "Equations ( ) and ( ) can then be used to obtain Ω Wt and K Mis in alternate compensation Equations ( )." Pg. 8-30, first definition following Equation ( ) - Delete: "(or pulse count)". Pg Include the F(Ω) equation from ( ) in the ( ) summary equation set. Pg , Equations ( ) - The following should replace the equivalent terms in the original version: Iω YR = ω c sin β 1 φ cos φ - cos α Iω ZR = - ω c sin β 1 φ sin φ - sin α Pg Add to the end of the first sentence following the Equation ( ) N definitions: " premultiplied by C." Pg , definition at top of page - Delete: "(positive in the plus φ direction)". Pgs and Change ( ) to: I ψ TMis I I = C B - CB0 δk TMis ( ) I Delete the Δψ TMis definition following ( ) and add the following in place of the last sentence beginning on page 13-24: "The same effect holds for any coordinate frame in which ψ is being evaluated, not only the I Frame, provided that rotation of the frame is properly taken into account. For example, for the local level Frame, Equation ( ) becomes: ψ TMis I I = C I CB - CB0 δk TMis = C B - I CB I0 δk TMis = C B - I I0 I C C I0 CB I I0 δk TMis = C B - I I0 C C I0 B δktmis I0 ( a) = C B - I C C 0 I0 B0 δktmis -5-

6 where 0 = Frame orientation in inertial space at time t = 0. and for more specificity, I = Discrete attitude of the Frame in non-rotating inertial space at the current time. BI0, I0 = Discrete attitudes of the B and Frames in non-rotating inertial space at time t = 0. I The previous notation is introduced to clearly denote C as being a relative attitude matrix I0 between two time points evaluated in the I Frame. Without the I notation (e.g., C 0) the matrix would have different values depending on the coordinate frame in which it was evaluated. For example, contrast Equations ( ) and ( ) with ( ) and ( ) (substituting for N) while recognizing that ρ ω EN = ω E." Pg , Paragraph following table - Should read: "We see from the table that the maximum for the error function occurs near q 2 / q 3 = 3 for which the error is % (i.e., the approximate sum solution P ΩSum is 10.9% smaller than the true solution P ΩSimult ). The standard deviation of P ΩSum the Ω Sum error at q 2 / q 3 = 3 (i.e., the square root ratio compared to one) is - 5.6%. P ΩSimult For large or small values of q 2 / q 3 the error in the P ΩSum solution is zero. Thus, we see that use of P ΩSum as an approximation to P ΩSimult has little error over the full range of q 2 / q 3." Pg Delete the last paragraph. Pg Add the following at the beginning of the first sentence following Equation ( ): "Under horizontal maneuvering, " Pg Add the following at the end of the second paragraph: "Also note that by E N continuously updating C N during the alignment process, ε is implicitly controlled to zero, thereby validating the ε N = 0 approximation of the previous paragraph." Pg to The first equation in ( ) and Equation ( ) should be: ψ N = C N B B ωib δ α ψ Quant + δ α ψ VQuant + G PψV = 1 2 C N B 1 B ωib 2 C N B B ωib N N 0 - a SF CB ( ) -6-

7 The 1/2 factor in the previous expressions is the ratio between the velocity and attitude update frequencies and is needed because there is 1 contribution of δ α ψ Quant and 1 contribution of δ α ψ VQuant (i.e., = 2 quantization noise contributions) to the attitude update cycle for each velocity update cycle. Equation ( ) provided in the book is consistent with this change. The same effect also applies to Equation ( ) which becomes: G PVR = - 1 r N B C B ωib CB N - 1 r N B C B ωib 0 C B N CB N ( ) The 1/r multiplier in ( ) arises because there are r-1 contributions of δυ VQuant and 1 contribution of δυ VRQuant (i.e., r = r quantization noise contributions) to the velocity update cycle for each position update cycle. Equation ( ) provided in the book is consistent with the previous change. Move the r definition following Equation ( ) up to the definition group following Equation ( ). Pg Delete the fourth bullet line. Pg At the end of the second sentence following the top of page definition, change "divided by the velocity." to "divided by the great circle angular velocity. The angular velocity equals the velocity along the great circle divided by the great circle radius." Change Equation ( ) to: T S/GC = R Avg θ GC/Range ( ) V GC Add the following definition at the bottom of the page: R Avg = Average distance to earth's center over the θ GC/Range angle (See Table R calculation). Can be approximated as R at the start of the trajectory segment. Pg In the second sentence in the second paragraph, change "velocity" to "velocity magnitude". Pg Add the following at the start of the first sentence: "Recognizing that during the turn V V V V u XV is horizontal, perpendicular to the vertical u Z, hence u Z u XV is a unit vector," Pg Add a closing bracket to the first equation in ( ). Pg. B-1 (Volumes 1 and 2) - Add to Accelerometer error characteristics: "12.4" -7-

8 Pg. B-2 (Volumes 1 and 2) - Add to Angular rate sensor error characteristics: "12.4" Pg. B-4 (Volumes 1 and 2) - Under Free azimuth coordinates (defined), change "2.2" to "4.5" Pg. B-7 (Volumes 1 and 2) - Add to Schuler, Dr. Maximilian: "13.2.2" Pg. B-9 (Volumes 1 and 2) - Under Wander azimuth coordinates (defined), change "2.2" to "4.5" Page (Part 2): The first IF statement under the DO loop should be changed to: IF j is less than or equal to the specified Φ λy expansion order, THEN: The indentation for the last two lines in the DO loop should be the same as for the preceding line: T IF j is equal to the specified Φ λy expansion order: B = Φ λy Φ λλ j M λλ j Φ λλ j-1 = M λλ ΔΦλλm j j! M j λλ = Φ λλ -8-