Aided Navigation: GPS with High Rate Sensors Errata list. Jay A. Farrell University of California, Riverside

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1 Aided Navigation: GPS with High Rate Sensors Errata list Jay A. Farrell University of California, Riverside February 19, 16

2 Abstract. This document records and corrects errors in the book Aided Navigation: GPS with High Rate Sensors. The most up-to-date version of this document can be obtained from the authors website farrell. Thank you to the various readers who have requested clarifications or pointed out the errors corrected herein.

3 Chapter 1 Introduction p.8, eqn. (1.1) For consistency, the equation should read ˆv = â(t). The correction has no consequence in the remainder of the example since the discussion on the top of page 9 makes clear that (for this simple example) â = ã. 3

4 Chapter Reference Frames p.31, Table.1 The title for the second column should be Symbol. p. 4, text line 3 realized should be realize p.45, last eqn. of Example.4 The eqn. should read n e d t = R t e x y z e p.45 The sentence at the middle of the page should read Next, using eqns. (.78.79) it is straightforward... p.49, eqn. (.4) The eqn. should read u v w b = R b p.49 The penultimate sentence should read... rotations has singular points... p.5 The second sentence of Section.5.5 should read A small angle transformation is... p.5 The last sentence of Section.5.4 is poorly worded. The quantities being compared are x y z ω it = ω ie + ω et and ω ig = ω ie + ω eg. 4

5 In both cases, ω ie is constant. For the fixed tangent frame ω et is zero. For the geographic frame ω eg is dependent on the platform motion. The main point being that ω it ω ig, unless the platform is stationary. p.5 In the denominator of eqn. (.45) the R t b should be Rg b. p.5 Eqn. (.55) should read Ω g eg = λ sin(φ) φ λ sin(φ) λ cos(φ) φ λ cos(φ) The dot indicating the time derivative was misplaced on the (,3) element. p.55 The last of the three equations between eqns. (.6) and (.63) should read ( d t ) Ωdτ t e k 1 R a dt b (t) = where the dt has been corrected to dτ. p. 6 In Part 4 of Exercise.4, the last expression at the bottom of the page should be ( z + p (1 e ) ) 3 R M = b (1 e. ). 5

6 Chapter 3 Deterministic Systems p.74 In the penultimate sentence of Section 3.4, the eqn. reference should be to (3.41) (3.4). p.77 Eqn. (3.45) should read: x k+n = n 1 i= p.77 Eqn. (3.46) should read: n 1 Φ k+ix k + n 1 j= i=j y k+n = H k+nx k+n. Φ k+iγ k+ju k+j. p.79 In Example 3.1, the eigenvalues are at 1 ± j. The negative sign in the real part was missing in the original text. p.89 Near the middle of the page, the definition should be a i = u i Φ n p. The negative sign is missing in the text. p.89 The last matrix at the bottom of the page should be a a 1... UΦU 1 = a n a n... p.9 The role of the integer r o is confused in the discussion. The following changes should correct the confusion. 6

7 1. The dimensions of v 1 and v are reversed. They should be v 1 R ro and v R n ro.. The column dimensions for the matrices are backwards. They should be W 1 R n ro and W R n (n ro). p.93 In Example 3.18, in the state vector as shown in the right-hand side of the first equation at the middle of the page, the last two elements should be b a and b g, not ḃa and ḃg. p. 13 In the first line after eqn. (3.15), the definition should be δε g a = [ cos(θ) sin(θ) ] u p. 7

8 Chapter 4 Stochastic Processes p.111 The second equation on the page should read: P {W 1 < w W } = F w (W ) F w (W 1 ) = W W 1 p w (W )dw. p.117 Four lines below eqn. (4.) the the should be the. p.117 Five lines below eqn. (4.) small random affects should be small random effects. p.14 Eqn. (4.36) should read R v (τ) = σ vδ(τ). p.14 The last equation on the page should read: lim T 1 T v (t)dt = R v () = 1 T T π = 1 π S v (jω)dω σ vdω =. p.14 In the penultimate sentence of the penultimate paragraph (i.e., six lines from the bottom of the page), Gaussion should be Gaussian. p.18 Eqn. (4.5) should read S y (jω) = βσ w β + ω. The change is the ω in the denominator. 8

9 p.135 In the middle paragraph of the page, given that σ ω is the PSD should read given that σ ω is the PSD. p.141 In eqn. (15), readers have questioned the t in F 1 F 3 t t 1 t t 1 e F33s dsdt. This is correct as written. It can be more written as F 1 F 3 t t 1 τ t 1 e F33s dsdτ to indicate more clearly that τ is a dummy variable. p.143 In the last sentence of the first paragraph, Q d should be Qd. p.144 In Section 4.7.., all Q symbols should be replace by GQG. p.148 In the second line, principle should be principal. p.148 The second line of the main equation on the page should be ) 1 = exp ( v dv (π) 3 v c p.148 The phrase near the middle of the page should read... [39], but this value was... p.158 In the expression for Φ(t)P x ()Φ (t), the upper right element on the right-hand side should be P b t. p.158 The last sentence and a half should be: t Φ(τ)ΓQΓ Φ (τ)dτ = σ va t3 3 + σ ω t 5 b σ va t + σ ω t 4 b 8 σ va t + σ ω t 4 b 8 σv a t + σ ω t 3 b 3 σ ω b t 3 6 σ ω b t σ ω t 3 b 6 σ ω t b σ ω b t where we have used the fact that P() = diag(p p, P v, P b ). Therefore, the error variance of each of the three states is described by ( P p (t) = P p + P v t t 4 ) ( σ + P b + va t 3 + σ ω b t 5 ) 4 3 P v (t) = ( P v + P b t ) + (σ va t + σ ω b t 3 ) 3 P b (t) = P b + σ ω b t. p.16 The eigenvalues for the discrete-time system are.9 ±.7j, and.85. 9

10 Chapter 5 Optimal State Estimation p.171, eqn. (5.3) The hat is missing from the x + k 1 on the right-hand side. The equation should read ˆx k = Φˆx + k 1 + Gu k 1. p.18 Table 5.1 shows that, for large m, the computational and memory requirements of the batch algorithm are O(n m) and O(nm), respectively. Fig. 5.3, p. 188 The signs for the difference between y k and z k are reversed. Example 5.4, p.188 In the example, the notation P x (k) is used, when it should have been Pˆx (k). Example 5.4 Following is additional information regarding the derivation of Pˆx (k). Some preliminary statements are useful that aid the derivation. 1. The example goes through for two different sets of assumptions. First, x k and B can be considered deterministic, but unknown. Second, x k and B can be considered as random variables that are uncorrelated with each other and with n k and v k. Under either assumption, B is a constant. Those assumptions were not clearly stated, but either is reasonable. Under either of these assumptions: var(z k ) = E (z k E z k ) = E (x k + v k + B E x k + v k + B ) 1

11 = E (x k E x k + v k + B E B ) = E (x k E x k ) + E vk + E B E B ) = + (µσ) + (5.1). The estimate ˆx k is unbiased. This is easily shown: E x k ˆx k = E x k (z k ˆB ) k = E x k (x k + v k + B ˆB ) k ) = E ˆB k B + E v k = (5.) where the fact that E ˆB k B = is true from Example Note that z k and ˆB k are cross-correlated: E z k ˆBk = E = E = E ( z k ˆB k (r k k ˆB ) ) k 1 ( z k ˆB k (z k y k k ˆB ) ) k 1 ( z k ˆBk ) + 1 k k z k y kz k k = + µ σ +. k 4. Similarly, v k and ˆB k are cross-correlated: ( E v k ˆBk = E v k ˆB k (r k k ˆB ) ) k 1 ( = E v k ˆB k (z k y k k ˆB ) ) k 1 ( = E v k ˆBk ) + 1 k k z kv k y kv k k Derivation: = + µ σ +. k Pˆx (k) = E (x k ˆx k ) ( = E x k (z k ˆB ) k ) ( = E x k (x k + v k + B ˆB ) k ) 11

12 ( = E ( ˆB ) k B) v k = E ( ˆB k B) E ( ˆB k B)v k + E vk = = ( 1 + µ ) σ µ σ ( k ) k 1 µ + µ σ k + µ σ Checks: To investigate the validity of Pˆxk consider the following two checks on the result. 1. For k = 1, it is true that ˆx 1 = y 1. Therefore, Pˆxk = σ.. As k, Pˆxk (µσ). The result derived above satisfies both of these checks. 1

13 Chapter 6 Performance Analysis p.18, eqn. (6.4) The matrix ˆΦ should be ˆΦ k. p.18, Eqn. (6.11) The eqn. should be [ y x k = [H k, ] k ˆx k ] + ν k p., Step 3 The Qd k should be ˆQd k. p.5 In line 9, principal should be principle. p.33 Exercise should be Exercises. 13

14 Chapter 7 Navigation System Design p.41 The following paragraphs are meant to clarify the ideas of Section The discussion assumes that λ >. The text should state that the system is only observable over time intervals during which the acceleration u(t) is not constant. The following discussion supports this statement. Assuming that the acceleration measurement u(t) is constant, the observability matrix is λ y O = 1 λ y u λ 3 y λ 4 y as stated on page 41. At any time, given that u(t) is constant, rank(o) = 4, because the last two rows are linearly dependent. Given this observability matrix, using the methods of Section 3.6.3, the subspace spanned by the vectors: 1 q 1 =, q 1 =, q 3 = 1, q 4 = 1 u 14

15 is observable. Vectors in the subspace spanned by the vector: q 5 = u 1 are not observable. This mathematically shows the fact that for a constant acceleration, a constant bias of magnitude b u is indistinguishable from a scale factor error of magnitude α = bu u. For a constant acceleration, the estimation error converges to the subspace spanned by q 5. Note however that this subspace is different for different values of the acceleration (i.e., the vector q 5 (u) is a function of u). The only vector in the intersection of the subspace spanned by q 5 (u 1 ) and subspace spanned by q 5 (u ) for u 1 u is the zero vector. Therefore, over intervals of time where the acceleration u is changed, the system should be observable; however, the results in the Aided Navigation book only discuss observability for time invariant systems. For the following discussion of time varying systems, consider without loss of generality the time interval t [, t]. For a time varying system, we are interested in the rank of the observability grammian defined as M(t, ) = t Φ(τ, ) H HΦ(τ, )dτ where Φ(t, ) is the state transition matrix (see Section 3.5.3) for the time varying matrix F. For this example, with τ = and F as defined in eqn. (7.15), the state transition matrix is t 1 t t s u(τ)dτds t 1 t Φ(t, ) = u(τ)dτ 1 e λt. 1 Therefore, the observability grammian is M(t, ) = t 1 τ τ e λτ g(τ) τ τ τ 3 τe λτ τg(τ) τ τ 3 τ 4 τ 4 e λτ τ g(τ) e λτ τe λτ τ e λτ e λτ e λτ g(τ) g(τ) τg(τ) τ g(τ) g(τ) (g(τ)) dτ. 15

16 s where g(t) = t u(τ)dτds. This matrix has rank 5 unless u(t) is a constant vector, in which case the third and fifth columns are identical and the grammian has rank 4. Therefore, the system is observable over time intervals where the acceleration is not constant. p.46 In Table 7.1, the heading of the fourth column should be σ by. p.5 In eqn. (7.9), the left-hand side should be f(ˆx, ). 16

17 Chapter 8 Global Positioning System p.65 On line 14, course should be coarse. p.68 The second sentence in the last paragraph should read Therefore, the estimated value of x will be affected by the error ( p i ˆp i). p.74 Two lines after eqn. (8.18) the left-hand side of the equation should be R e a. p. 85 In the line after (8.48), h i = ˆp k ˆp i p k ˆp i should be h i = ˆp k ˆp i ˆp k ˆp i. p. 93 In the penultimate sentence of Section 8.4.5, (ĥ (p p )) s ĥ ) should be (ĥ (ˆp s p s ) ĥ p. 36 Following eqn. (8.84) the phrase should be where E cm is... than... p. 313 The equation above eqn. (118) should read as ρ i ro = (R(p r, ˆp i ) + c t rr + η i r + cδt i + cδt i a r + E i r) (R(p o, ˆp i ) + c t ro + η i o + cδt i + cδt i a o + E i o) The E i r and E i o were interchanged. p. 34 Eqn (8.147) has an extra opening parenthesis. 17

18 Chapter 9 GPS Aided Encoder-Based Dead-Reckoning p. 338 The sixth line should read... and two right increments.... p. 338 The phrase before eqn. (9.4) should be: Using the kinematic relationship v w = v b + ω b tb R where the subscript w denotes wheel and the superscript b denotes body, the body frame linear velocity of the center of each wheel is... p. 34 Eqn. (9.14) should read: p. 341 Eqn should read [ cos( ˆψk ) ˆp k = ˆp k 1 + sin( ˆψ k ) [ cos( ˆψk ) = ˆp k 1 + sin( ˆψ k ) ˆx = [ˆn, ê, ˆψ, ˆR L, ˆR R ], ] π C ] π C [ e L (k) ˆR L T k + ˆR e R (k) R T k [ ˆRL e L (k) + ˆR R e R (k)]. ] T k 18

19 Chapter 1 AHRS p. 354 In eqn. (1.1), the element in the fourth row, third column should be b 1. p. 357 The sentence after eqn. (1.15) should point to Section.5.4, not Section.5.3. p. 357 The sentence containing eqn. (1.18) should read: The initial yaw angle can be computed from the first two components of m w as ˆψ(T ) = atan( m w, m w 1 ). p. 358 In eqn. (1.3), the element in the fourth row, third column should be ˆb 1. p. 364 Eqn. (1.55) should read p.365 Eqn. (1.57) should read ρ = ˆR n b δω b ib ω n in. ρ = ˆR n b δx g ˆR n b ν g ω n in. p.365 The AHRS state space error model at the start of Section should read ρ δẋ g δẋ a = ˆR n b F g F a + I ˆR n b I I ρ δx g δx a ω n in ω g ν g ω a. 19

20 [p. 368 In eqn. (1.66) the element in the third row, third column should be P a, not P aa.

21 Chapter 11 Aided Inertial Navigation p. 381 The last phrase on the page should read In an inertial reference frame... p. 39 In eqn. (11.43), the rightmost factor in the right hand side should be v n e : v n e = R n b f b + g n (Ω n en + Ω n ie) v n e p. 39 In the line between (11.44) and (11.45), the factor R n b should be R b n. p. 395 In the first column, third row of F vp, the ˆv e should be v e. p. 396 In the sixth row, first column, the entry should be v e Ω D. p. 46 In the penultimate paragraph in the last sentence, the phrase Specific definitions for F vg and... should be Specific definitions for F ρg and... p. 41 Eqn. (11) is missing a transpose. p. 41 The eqn. for δnl a should read: k x1 fu + k x fv + k x3 fw + k x4 f u f v + k x5 f v f w + k x6 f w f u δnl a = k y1 fu + k y fv + k y3 fw + k y4 f u f v + k y5 f v f w + k y6 f w f u. k z1 fu + k z fv + k z3 fw + k z4 f u f v + k z5 f v f w + k z6 f w f u The f x at the right should be f u. p. 41 In eqn. (11.134), ν a should be ν g p. 41 In eqn. (11.134) and the subsequent equation, T p T p x bg. b g should be 1

22 p. 413 The R g b is eqn. (11.138) should be removed. p. 413 In the third row of δk g, the k p1 should be k r1. p. 419 After the definition of G, the sentence should state that ɛ D is unobservable. p. 419 At the end of the paragraph that defines G, the parenthetic comment should be (i.e., 1 milli-radian per 1 mg bias). p. 4 Eqn. (11.156) should be ˆR(ˆp e, ˆp i ) = ˆp e ˆp i. p. 45 The parenthese in eqn. (11.17) are incorrect. The second line should read: ( δy a = h δp n n [ˆr n ]ρ + ˆR n b δr b) + ν.

23 Chapter 1 LBL and Doppler Aided INS p. 447 In the second paragraph from the bottom, the end of the first sentence should read... it will always be the case that δˆx + (t ) will be zero. 3

24 Appendix A Appendix A. Notation No known errors. 4

25 Appendix B Appendix B. Linear Algebra Review p. 46 Between eqns. (B.15) and (B.16), the middle portion of the paragraph should read The equivalence expressed in eqn. (B.15) will be denoted... p. 467 Lemma B.5. eqn. (B.3) is missing an inverse. The equation should read (A + BCD) 1 = A 1 A 1 B ( DA 1 B + C 1) 1 DA 1. p. 47 In Section B.11, in the final result for P, the first column of the first row that reads d 11 + d u a + d 33 u 13 should read d 11 + d u 1 + d 33 u 13. 5

26 Appendix C Appendix C. Calculation of GPS Satellite Position & Velocity No known errors. 6

27 Appendix D Appendix D. Quaternions p. 5 The sentence containing eqn. (D.5) should read as follows: The norm of a quaternion is the square root of the scalar portion of b b b = b 1 + b + b 3 + b 4. (D.1) The vector portion of the quaternion b b is zero. p. 54 Section D..1 provides a single formula for the computation of a quaternion to represent a given rotation matrix. Alternative formulas may be preferred depending on the rotation matrix corresponding to a specific situation. Such formulas are available on the books website in the directory related to quaternions. 7