Mobile Robotics, Mathematics, Models, and Methods

Transcription

1 Mobile Robotics, Mathematics, Models, and Methods Errata in Initial Revision Prepared by Professor Alonzo Kelly Rev 10, June 0, 015 Locations in the first column are given in several formats: 1 x,y,z means page x, para y, line z x, Fig yz means figure yz on page x 3 x, Eq yz means Equation yz on page x 4 etc Generally, errors in cross references will make it more difficult to follow derivations With a few exceptions, errors in math (both in-line and otherwise) are typically notational A few severe issues are noted in the comments Look for the word severe If your time is limited, I suggest that you use a pencil in your text to fix the severe issues and the cross reference errors Everything else is minor able 1: ypo and In-line Math Edits 8, 8,5 n RA n RA is is Add space 113,6, we have: we have the matrix: clarify quantity is a matrix 118,8,3 Let the set of solutions to the above constraint be parameterized locally by a parameter s to produce the surface Each element of the vector level curve is a 119,1,1 We can conclude that the gradient g x 1467,1, composite vector of unknowns x xs xs Let the set of solutions to the above constraint be parameterized locally in one direction by a parameter s to produce the curve xs xs is called a level curve We can conclude that the gradient g x composite vector of unknowns x his discussion is confusing when applied to n dimensions remove underline move delta symbol

2 able 1: ypo and In-line Math Edits 151 Form the matrix (math font) V whose rows are two arbitrary vectors (math font, underlined) v 1 and (math font, underlined) v in the planeuse constrained optimization to show that the unit vector that maximizes the sum of its squared projections onto both vectors is an eigenvector of (math font) V V 176, 7,1 Consider next two frames of reference that are rotating with respect to each one another 19,4,1 411 Wheel Wheel Steering Control 195,,1 Box 4 WMR Forward Kinematics: Offset Wheels V V Form the matrix whose rows are two arbitrary vectors v 1 and v in the plane Use constrained optimization to show that the unit vector that maximizes the sum of its squared projections onto both vectors is an eigenvector of V Consider next two frames of reference that are rotating with respect to one another 411 Wheel Steering Control Box 4 WMR Forward and Inverse Kinematics: Offset Wheels remove format spec, take V down fromfinal superscript remove word each delete extra word add words 13,8,3 projection that occurs at line 11 projection that occurs at line 1 17,3,8 the eigenvectors associated with the eigenvectors associated with the n 1 largest eigenvalues the n 1 largest eigenvalues of of C C C C move second C down from superscript 17,4, However, later content However, earlier content 4,6,1 he state vector is xt and xt yt t xt he state vector is and xt yt t change theta to psi 51,3,3 counteract the increasing tendency increasing tendency to roll over counteract the increasing tendency to roll over delete double phrase 61,3,1 using the above Jacobian using the Jacobian delete word 6,3,1 the parameters may only be obtainable p the parameters obtainable may only be change letter p to rho 68, Papers see comment move Raol and Wong up to Books

3 able 1: ypo and In-line Math Edits 68, 7 Proceedings of International symposium on Reading Review, 011 Proceedings of International Symposium on Robotics Research, ,3 Spot 51 he exponent is called the Mahalanobis distance (MHD) he exponent x C 1 x x C 1 x is called the (squared) Mahalanobis distance (MHD) add word squared 78,1, when the exponent (the squared Mahalanobis distance) is a constant 90,4, in terms of it evolution over time; 308, 7 It is not possible to invert this relationship to determine R from xx 309 he weighted least squares solution from Equation 579 for H 11 and when the exponent (the squared Mahalanobis distance) is a constant in terms of its evolution over time; It is not possible to invert this relationship to determine R from xx Also note that: xˆ * xx H R 1 z In other words, the solution covariance appears in the formula for the solution itself he weighted least squares solution from Equation 579 for H' 11 and add word squared change it to its Add sentence to clarify add primes to H and R R diag z x is R' diag z x is 310,, Now, suppose that another measurement z 6 was produced at the same time as the first with variance z ,11,5 By substituting Equation 585 into Equation 580, we immediately get: Now, suppose that another measurement z 6 was produced at the same time as the first with variance z 15 We already know that the covariance of the MLE estimate is the first matrix in the solution herefore, from Equation 585 and Equation 584, we can immediately write: Change variance subscript to z Add sentence and reuse earlier result

4 able 1: ypo and In-line Math Edits 313, 11,1 Subsequent measurements have covariances of R 3 diag 1 01 and Subsequent measurements have covariances of R 4 diag R diag 1 01 R 4 diag 01 1 and Change 01 to 1 35,4,3 he complement of the linearized filter is an system dynamics model he complement of the linearized filter is a system dynamics model Change an to a 331,1,1 Let be the position vector of the wheel relative to the body frame and let denote the angular velocity which, is assumed Let be the position vector of the wheel relative to the body frame and let denote the angular velocity, which is assumed move comma 346,3,5 the propo- did not We will denote by sition that event that occur A A the propo- did not We will denote by sition that event occur A A delete second that 353, 5, Bayes, filter [17] Bayes filter [17] remove comma 361,,3 (Figure 548) Figure 548 remove brackets 375, 3,3 However, there may no solution However, there may be no solution add word be 76,, imagine placing the vehicle at the origin imagine placing the vehicle on the hyperbola at y0 401, measure the angle measures the angle add s 401,3,1 Inexpensive devices have bandwidths on the order or Hz 401,3, a rebalance loop that prevent the sensor 403,4,1 neither constant nor entirely a real force Inexpensive devices have bandwidths on the order of Hz a rebalance loop that prevents the sensor neither constant nor entirely associated with a real force or -> of add s add two words 407,5, is fixed with a lasing medium is filled with a lasing medium fixed->filled 414,6, but the above assumes a sequence zyx but the above assumes a sequence zxy 419,3,3 n n Note that R v and R v are very n n Note that v and R v are very different fonts different matrices different matrices

5 able 1: ypo and In-line Math Edits 43,5,6 three to five meter accuracy hree to five meter accuracy capitalize 438,3,1 he trajectory control level considers and entire trajectory 453,1 Provide the details of the transfer function derivation for the cascade controller shown in Figure 714 he trajectory control level considers an entire trajectory Provide the details of the transfer function s (before assuming K vi 0 ) derivation for the cascade controller shown in Figure 714 Its tricky and->an 469,4,6 and an entire terminal state and an entire terminal state xs f xs f x f y f f f x f y f f f add brackets 484,5,1 Much of the forth coming discussion 491,3,1 Of course, there are situations, such as operation in cluttered environments where Much of the forthcoming discussion Of course, there are situations, such as operation in cluttered environments, where join words add comma 503,,3 Most sensors measure anyway Most sensors measure surfaces anyway add word surfaces 503,4, Sampled versus Continuum 503,4,6 In such a representation computational complexity Sampled versus Continuum In such a representation, computational complexity remove space add comma 515,1,1 Although states estimation Although state estimation states->state 530,7, similarity measures 9 similarity measures [9] add [] 536,5,3 same color it is not possible same color, it is not possible add comma 539,4,4 defined by the Bessel function Figure ,5, In the latter case the corners can be very small defined by the Bessel function (Figure 86) In the latter case, the corners can be very small add () add comma 560,5,5 Hence, a camara that is sensitive Hence, a camera that is sensitive camara->camera 56,4,1 identical basic principles to those or sonar identical basic principles to those of sonar or->of 569,3, Section 1310 of Chapter 5 Section of Chapter 5 570,4,4 optics are also relative inexpensive optics are also relatively inexpensive relative->relatively

6 able 1: ypo and In-line Math Edits 69,7,1 Conversely, the residual Conversely, the residual zx b cx can be zx b gx can be added added c -> g 635,4,1 Let the measurement project onto a a few states 637, 3 to locate the landmark Figure ,5 Soundness Feasible Admissible Completeness If any If not Optimality Let the measurement project onto a few states to locate the landmark (Figure 948) Soundness Feasible Admissible Completeness If any If not Optimality delete redundant word add () Indent nd,3rd,4th,5th bullets 651, 3, set of roads and vehicles set of roads, and vehicles 655,5,1 to encode arbitrary gradient to encode arbitrary strategies gradient->strategies 659,1, described in terms of a traversal of a tree that restrict traversal is discovered during the search process described in terms of a traversal of a tree that is discovered during the search process delete restrict traversal 66,,1 the g values nodes the g values 664,,1 is com- he cost estimate puted fx is com- he cost estimate puted fx add space 666,3,6 total costs any new nodes added to OPEN 668,6,6 order to process child nodes is unknown means this total costs of any new nodes added to OPEN order to process child nodes is unknown, means this insert of add comma 669, 8,4 In difficult terrain the details In difficult terrain, the details add comma 671,4,3 (only when headed) (only when needed) headed->needed 679,5,5 Node costs refer to the estimated total cost Node costs refer to the estimated total cost estimate->estimated

7 able : Math and Algorithm Edits 40 Add p subscript f x p + f f x p + f p p x f x p p x f x p 77,Eq severe error, replace diagonal with zeros 95, Eq 96 R1 R B B replace superscript B R BP 1 R B 1 R1 R B1 B R BP 1 R B 1 with B1 97,Eq 97 c k i s k i i i j i j i s k c jak s jbk + a j c k i s j i ak j i j i + c jbk +b j c k i s k i i i j i j i s k c jak s jbk + a j c k i s j i ak j i j i + c jbk +b j move column separators right 14, Eq 37 remove letters mp subject to:c'mpx 0 subject to: cx 0 17, Eq 31 1 no underline on c fx --x Q xx x b 1 + x+ c fx --x Q xx x+ b x+ c 163, Eq 3119 reciprocal k p ---- t k p ---- s r c t 199, Eq 46 s1 s1 severe error, change r c1 d s 1 c 1 r c1 d s 1 c 1 sign of all y coordinates s3 s3 r c3 d s 3 c 3 r c3 c 3 s4 r c4 d s c d s 4 c 4 s r c s4 r c4 d s 3 d s c d s 4 c 4 08,1 severe error, add minus i v v 00cx i + sy i v v i 00cx i + sy i sign to 3rd expression only 10, Eq 494 c x s c L c x s c Lc s c L s c Lc change subscript to x

8 able : Math and Algorithm Edits 4, Eq 4111 v v v w switch right super and v z v z subscript, leave left alone cx 0 cx x x 0 y w y v 7,7 change k+1 to k ---- y t k + 1 y y k + k y t k + 1 y y k + k 7,8 change k+1 to k t y k ---- t + u k + 1 y k + 1 y k + 1 y k u k 1 + y k 4, Eq 415 change theta to psi c t s c t s everywhere s t c s t c 4, 7,1 change theta to psi 00 sv t x t + cv t y t 00 sv t x t + cv t y t everywhere (including 00 cv t x t sv t y t 00 cv t x t sv t y t inline lines above) , 8,1 change theta to psi c t s c t s everywhere s t c s t c 43, Eq 4154 change theta to psi c t s t yt c t s t yt everywhere s t c t xt s t c t xt 43, Eq 4155 change theta to psi c t s t yt c t s t yt everywhere (just like line above) s t c t xt s t c t xt 63,7 Vt cost Vt sint t c 0 s Vt cost Vt sint t,, c 0 s change theta to psi everywhere

9 able : Math and Algorithm Edits 64,1 change theta to psi c c c c everywhere s 1 W s 1 W s 1 W s 1 W 64, change theta to psi v r + v l c v r + v l c everywhere v r + v l s v r + v l s v r v l W v r v l W 65, Eq 41 x t Fx t + Gu t x t Fx t + Gu t missing dot (time derivative) 74, 1 sigma is not inside 1 px x exp square root px x exp 98,10 change k subscript to k- k 1 k k k k 1 k k k k 1 k k k ,3,3 delete first col of nd x i s i x i l i s i matrix c i l + i y i s i l i c i 304,7 add transpose Expt xt 0 xt 0 t Expt xt 0 xt 0 tt y i + l i c i 311,1, he measurements relationship is: z x z H I x' x he measurement relationship is: H I x' Remove s at end of word measurements Add second equality 311, Eq 584 Add primes to H and R x' H R 1 H 1 H R 1 z x' H' R' H' H' R' z 38, Eq 5111 Change theta to psi x x everywhere (3rd col) y y and thetadot to psidot everywhere (3rd row) v v

10 able : Math and Algorithm Edits 38, Eq 511 Change subscripts of last two elements to k+1 k + 1 v k k k + 1 v k + 1 k , Eq 68 change theta to psi x y 380, Writing total differentials: Writing total differentials: change theta to psi x y + + F F G G F F G G 0 391,6 change theta to psi xx 0 xy 0 x 0 xx 0 xy 0 x 0 xy 0 yy 0 y 0 x 0 y 0 0 xy 0 yy 0 y 0 x 0 y ,4 W W W W W O severe, superscript W - S *S O S *S O > O 601,4 601,5 argmin * argmin f 1 --r Zr Z f argmax * argmax f zpred Z f z obs 1 --r Zr Z change * z obs zpred Z change * to to 608, argmin * argmin change to * and Z f 1 --r Zr Z f 1 --r Xr X to X 69, Eq 943 c -> g cx b gx b

11 able : Math and Algorithm Edits 66, algorithm expand- NodeDijkstra, line , algorithm expand- NodeAstar, line 05 gnew xg gnew x next g else if ( ) else if ( ) x -> xnext, severe bug if( x next O && fnew xf ) if( x next O && fnew x next f ) x -> xnext, severe bug 665, algorithm expand- NodeAstar, line 08 else else if( x next C && fnew xf ) if( x next C && fnew x next f ) x -> xnext, severe bug able 3: Cross Reference Edits 41,1, Figure 7 Figure 8 139,4, section Figure 314 section 315 Figure -> Section 314-> ,1 Solve Equation 3104 Solve Equation > 5 170,6,1 Substituting this into Equation 3139 above Substituting this into Equation 3140 above 180, 1, (Box 41) Equation 411, Equation 41, and Equation 413 Equation 41, Equation 413, and Equation ,4 Rewrite Equation 440 Rewrite Equation , 6,1 Equation 491 Equation ,10,1 Algorithm 4 Algorithm 41 4,3 Based on Equation 474 Based on Equation > 5, 3 constraint from Figure 46 constraint from Figure > 17 8,1,1 Equation 4114 Equation ,, he transform of our first-order system (Equation 4113) he transform of our first-order system (Equation 4114) 58,1 Section 3 of Chapter 3 Section 33 of Chapter 3 clarify reference 58, 5 the functional in Equation 31 in Chapter 3 the functional in Equation 3133 in Chapter 3 64,, 1 Equation 414 Equation 416

12 able 3: Cross Reference Edits 65,6, Jacobian of the left-hand side of Equation 41 Jacobian of the left-hand side of Equation 4 96,8,3 Equation 558 Equation ,4,3 Equation 561 Equation ,3, Equation 571 Equation ,5,1 Equation 569 Equation ,, Equation 4138 Equation ,3, Equation 580 Equation ,7,1 he result of Equation 587 he result of Equation ,10, It can be derived by returning to Equation 583 and including the above 350,6,4 the computation of the denominator in Equation 514 requires that 361,5,1 Recall Equation 5151 and Equation 515 It can be derived by returning to Equation 584 and including the above the computation of the denominator in Equation 5143 requires that Recall Equation 5150 and Equation Section 456 of Chapter 4 Section 4566 of Chapter 4 more precise xref 415,,3 based on Equation 76 in Chapter 415,3,1 Equivalently, based on Equation 13 in Chapter based on Equation 81 in Chapter Equivalently, based on Equation 133 in Chapter 44,7 first two lines of Equation 657 first two lines of Equation ,3,5 Figure 68 illustrates the equivalent Figure 63 illustrates the equivalent 596,1,1 then we can use Equation 55 then we can use Equation ,3,1 For the camera, we can use Equation 81 For the camera, we can use Equation 8 615,6,3 Figure 538 Figure ,4,1 his is of the form of equation Equation ,1,3 varies along the line of the laser (Figure 948) his is of the form of Equation 937 varies along the line of the laser (Figure 949) delete redundant word

13 able 4: Figure Edits 65, Fig as drawn see comment x,y axes should not be partially erased 416, Fig 630 i i change n sub/sup script n 001 i i e 001 to e n i n c 0 s e i e c 0 s 48, Fig 633 caption two antenna two antennae 503,Fig 744 as drawn see comment increase contrast in left image 598, Fig 916 as drawn see comment increase contrast in right figure to see highlighted fork holes 618, Fig 935 caption Line are approximately 1 pixel wide Lines are approximately 1 pixel wide line -> lines 61, Fig 939 as drawn see comment increase contrast to see feature traces 663, Fig 1019 as drawn see comment increase contrast to see left image