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# カルマンフィルタ講義資料

2018年度センシング工学１４回目のカルマンフィルタ講義資料。

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### カルマンフィルタ講義資料

1. 1. 2018 14
2. 2. (3)
3. 3. n x0 1 1 x0 (t,x) = (1, -1), (2, 1), (4, 4 x = t2 4 + x0
4. 4. n t=1,t=3,t=4 t=2 ¨t=1 t=3 t=4 n ¨ t=1 t=4 t=3 ¨
5. 5. 2 1 1 1 , i iN i i N i ii wav w w xw x s == å å = = Nixx ii ,,1 !=±= s
6. 6. x = t2 4 + x0 x0 = x − t2 4σx 2 =1, σt 2 =1 ∂x0 ∂x =1, ∂x0 ∂t = −t / 2 ∴σx0 2 = ∂x0 ∂x \$ % & ' ( ) 2 σx 2 + ∂x0 ∂t \$ % & ' ( ) 2 σt 2 =1+t2 / 4 x0 t →
7. 7. (t,x) = (1, -1), (2, 1), (4, 4) ˆx0 (1) = −1−12 / 4 = −5 / 4, σx0 2(1) = 5 / 4 ˆx0 (2) =1− 22 / 4 = 0, σx0 2(2) = 2 ˆx0 (1:2) = ((−5 / 4) / (5 / 4)+ 0 / 2) / (4 / 5+1/ 2) = −10 /13 σx0 2(1:2) =10 /13 ˆx0 (4) = 4− 42 / 4 = 0,σx0 2(4) = 5 ˆx0 (1,2,4) = (−10 /13/ (10 /13)+ 0 / 5) / (13/10 +1/ 5) = −2 / 3= −0.67(= −0.7) σx0 2(1,2,4) =10 /15 = 0.67(= 0.7) t=1 x=-1 t=2 x=1 t=4 x=4 σx0 2 =1+t2 / 4 x0 (1,2,4) = -0.7 0.8
8. 8. n t=3 ¨ t=1,2,4 ¨ t=2,4 t=1 → n t=3 x3 ¨ x3 = 32 4 + ˆx0 (1,2,4) ∂x3 ∂ˆx(1,2,4) 0 =1 x = t2 4 + x0 x3 = 9 4 + x0 ∴x0 = x3 − 9 4 ∴x3 = x − t2 4 + 9 4 2
9. 9. n ¨ !"(\$) &'(()) * ¨ !"(+:\$-+) &'((.:)/.) * n ¨
10. 10. n ¨ ¨ n ¨ n
11. 11. n x,y σx 2 σxy σxy σy 2 ! " # # \$ % & & (x,y) (determinant) σx 2 σxy σxy σy 2 =σx 2 σy 2 −σxy 2 ≥ 0 Schwarz (
12. 12. !" # "\$ %\$ & = 0: ) * "+,- = ."+ + 0+ 1 !" % & & &
13. 13. x2 x1
14. 14. n !"#\$ = &"!" + (" (" 0, *+, - &" *+, - k E[!0] = !0 V !0 = *34 - k k+1
15. 15. n k-1 !"#\$% &'()* + k ,"# -'( + ,"# = /#\$% !"#\$% -'( + = /#\$% + &'()* + + &1()* + "# 2# ,"# 2#
16. 16. n !" = \$"%" + '" '" 0, )*+ , \$" )*+ , k -%" = ."!" + /" V -%" =E -%" − %" , = )3+ , ." /"
17. 17. n !"# = %#&# + (# V !"# =E !"# − "# , = -./ , E[1#] = 0 → V[1#]→ min → 1# = !"# − "#
18. 18. !"# = %#&# + (# V !"# =E !"# − "# , = -./ , E[1#] = 3[!"# − "#] = 0 3[!"# − "#] = 3[%#&# + (# − "#] = 3 %#5#"# + %#6# + (# − "# = %#5# − 1 3 "# + 0 + (# = %#5# − 1 8"# + (# = 0 (# = 1 − %#5# 8"# &# = 5#"# + 6#
19. 19. !"# = %#&# + (# V !"# =E !"# − "# , = -./ , V[1#]→ min V 1# = 3[(1# − 3 1# ), ] = 3 ( %#6# − 1 "# + %#8# + (# − %#6# − 1 9"# − (#), = 3 ( %#6# − 1 ("#−9"#) + %#8#), = %#6# − 1 , : "# + %# , -;/ , = %#6# − 1 , <./ , + %# , -;/ , 2 "# 8# 1# "#
20. 20. !"# = %#&# + (# V !"# =E !"# − "# , = -./ , V[1#]→ min V 1# = %#3# − 1 , 5./ , + %# , -6/ , 7V 1# 7%# = 23# %#3# − 1 5./ , + 2%#-6/ , = 0 %# = 3#5./ , 3# , 5./ , + -6/ , , (# = 1 − 3# , 5./ , 3# , 5./ , + -6/ , ;"#
21. 21. !"# = %#&# + (# V !"# =E !"# − "# , = -./ , %# = 0#1./ , 0# , 1./ , + -2/ , , (# = 1 − 0# , 1./ , 0# , 1./ , + -2/ , 5"# !"# = 0#1./ , &# + -2/ , 5"# 0# , 1./ , + -2/ , = 0# , -2/ , &# 0# + 1 1./ , 5"# 0# , -2/ , + 1 1./ , 6/ 7/ 5"# &# = 0#"# + 8# 9["#] = &#/0# = "# = = &# 0# , = -2/ , 0# ,
22. 22. !"# = %#&# + (# V !"# =E !"# − "# , = -./ , !"# = 0#1./ , &# + -2/ , 3"# 0# , 1./ , + -2/ , = 0# , -2/ , &# 0# + 1 1./ , 3"# 0# , -2/ , + 1 1./ , 5 !"# = -!./ , = 6 !"# 6&# , -2/ , + 6 !"# 6 3"# , 1./ , = 1/ 0# , -2/ , + 1 1./ ,
23. 23. : n : "# , \$%& ' n (") = +),- ."),-, 0%1 ' = +),- ' \$%123 ' + \$5123 ' n ¨ .") = 61781 9 :1;<=1 9 (%1 61 9 781 9 ;<=1 9 = >1 9 ?=1 9 @1 >1 ; 3 A81 9 (%1 >1 9 ?=1 9 ; 3 A81 9 ¨\$%1 ' = B .%1 B:1 ' \$C1 ' + B .%1 B (%1 ' 0%1 ' = 1/ 61 9 <=1 9 + - 781 9
24. 24. 2 n n !=(\$%, \$') )[!] 2 2 + n → +,%
25. 25. 2 !"#\$ = &"!" + (" (" ), *" E[!-] = !- V !- = 0- !-2 1 0-2 2 !"#\$2 1 &"2 2 *"2 2 !"2 1 = + ("2 1
26. 26. 2 !"#\$ = &"!" + (" (" ), *" +!" = &",\$-!",\$ ." = &",\$/",\$&",\$ 0 + *",\$ &",\$2 2 *",\$2 2 = +."2 2 &",\$ 0 2 2 /",\$2 2
27. 27. !" = \$"%" + '" '" 0, )*+ , -%" = ."!" + /" V -%" =E -%" − %" , = 3" 2 4"1 1 \$"1 2 %" 2 1 = + '"1 1 4"1 1 ." 1 2 = + -%"2 1 /" 1 2
28. 28. 2 E[#\$] = '[()\$ − )\$] = 0 '[()\$ − )\$] = '[,\$-\$ + /\$ − )\$] = ' ,\$0\$)\$ + ,\$1\$ + /\$ − )\$ = ,\$0\$ − 1 3)\$ + /\$ = 0 /\$ = 4 − ,\$0\$ 3)\$ ()\$ = ,\$-\$ + /\$ V ()\$ =E ()\$ − )\$ 6 = 7\$ 0\$1 2 ,\$ 1 2/\$ 1 2 3)\$2 1 4 2 2= - -[ ]
29. 29. V[#\$] V #\$ = '[(#\$ − ' #\$ )(#\$ − ' #\$ )+ ] = (,\$-\$ − 1)/\$ ,\$-\$ − 1 + + ,\$123 4 ,\$ + 2 56\$ = ,\$7\$ + 8\$ V 56\$ =E 56\$ − 6\$ 4 = :\$ -\$ ,\$ ;[ ] /\$ -\$ < ,\$ T ;[ ] + ,\$ ,\$ T123 4 V #\$2 2 =
30. 30. V[#\$] V #\$ = '[(#\$ − ' #\$ )(#\$ − ' #\$ )+ ] = ,\$-\$ − 1 /\$ ,\$-\$ − 1 + + ,\$123 4 ,\$ + = ,\$ − 5\$ -\$/\$-\$ + + 123 4 ,\$ − 5\$ + + 6\$ 5\$= /\$-\$ + -\$/\$-\$ + + 123 4 78 6\$ = (9 − 5\$-\$)/\$ ,\$ = 5\$ ( ) 6\$ 2 :;\$ = ,\$<\$ + =\$ V :;\$ =E :;\$ − ;\$ 4 = ?\$
31. 31. !"= \$"%" & %"\$"%" & + ()* + ,- ." = (0 − !"%")\$" = \$" − \$"%" & %"\$"%" & + ()* + ,- %"\$" ." = (\$" ,- + %" & ()* ,+ %"),- (= 3") 2 45" = !"6" + 7" V 45" =E 45" − 5" + = 3"
32. 32. !"# = %# &' + )# * +,- &. )# &' %# &' /"# + )# * +,- &. 0# 1# = (%# &' + )# * +,- &. )#)&' 0# = )#"#(+4#: ) 0# )# * +,- &. )# 2 !"# = 6#0# + 7# V !"# =E !"# − "# . = 1# !"# %# &' )# )# ; +,- . /"# )# ; +,- &. <# )# )# ; +,- &. 1# %# &' %# &'
33. 33. 1-40 ! = cos 20° −sin 20° sin 20° cos 20° , = 1 0
34. 34. 1-10 ! = cos 20° − sin 20° sin 20° cos 20° , = 1 0 x y
35. 35. 10-20 ! = cos 20° − sin 20° sin 20° cos 20° , = 1 0
36. 36. 20-30 ! = cos 20° −sin 20° sin 20° cos 20° , = 1 0
37. 37. 30-40 ! = cos 20° − sin 20° sin 20° cos 20° , = 1 0
38. 38. y y
39. 39. ! = cos 20° − sin 20° sin 20° cos 20° , = 1 1 . = /0 + /2