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Chapter 7 The Unscented Kalman Filter

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"                             %&' && &()         #  "    $  *     +  %,' ,( '- ,.)   /         #         0    012                   3    "4   5                  012          3      4                                012                012                  7                         "   #                    012                    &        3    "4    6 3 $   # 1      3   #           $   #           #    3        8                4            3                    "               #9  "4              #     8            #      Æ      %'-)          "  #        4     3    "                3       #                         0                        3                   +35 +          "       #            8               !      3                    4            : :                             <  ;< ;&    3                       process noise measurement noise vk nk input output uk x k +1 xk Δ F yk H xk state  ;<=           ;<          6                      8                              3  34                          9  ;&           #     9  ;(   >                 "                      :   ;(              3       5                               "      ?                                 :               "                      5  3           *     5   3                : @  :                                   4     ;' @          3                              ;'      ;,                     Æ  A 4 B                              9  * %';)  >      %'<)       C  &    3     #          9  ;'    8                        &                                  4                         : : ;D ;;                                        3              4      "  4    8                       C  ,    3 8 3 4+35        C  D 8     5  #   4     9  ;,    3          3     !           #          ! 8          #     9  ;(                            *      9  ;D            #         + C   "    8 3 8         Æ        # 8 3 E         #            0             *                        4   ++9           = F     :                     ++9               3     0             A B      "                       ;G               E             :             ;-    :     5            ;<.            :                               ( ;<<                         "                           4                                       9       9 "      ;<&  3         0            4     4  :             :  .               :            :  ;<(                             :          #   4        ;- ;<.  ;<<                    + 4C                      4         "               3                  0    E         "        F  *          F      :                  :    :     @ %         ) ;<' ;<,                                 F :  : F :     )     % ) %      H            :   F ;<; ;   1 %(-)  %,.)    +   *    #  CK   "                 3     #             x − γ P x ⎤⎦                      #   x +γ Px        2    #          i Weighted sample covariance  "      8  3     ;(   &4    =              + 4C M             5        M          #   ,              #    % &  (         #    3    #     4    ;<'     12    "             =    )  : %  #         ;(.            12       3   #       ;(< 6   3   $  7                             ½ 12 Ô  ¾   È  1 2      +     / 0    *    ½   "     1 2 " *    , 3             *        *      14 35 "  2 %  *       / 0         "  "                "   %     .     3     ,           / 0      Ê   ;       Linearized (EKF) Actual (sampling) UT sigma points covariance mean      Ý  Ì Ü    weighted sample mean and covariance  transformed sigma points true mean true covariance Ì UT mean Ü UT covariance                 !   " #      $  "  !  ;(= )  *                            3   #                          12N                                                          ;(&   3   3 3                 K                                 "        9  ;'   3     4            8               3  3              C  5          D 7        3          4   4         C                      5 K    Æ     4 #        8 3 E G       5 = F  : F : %  )  : %   F    F   ) % ) : %F  + +)   :   <      F     F   ) : % ;(, ;(D ;(;  + +  + + + +  ;(G  C  =  : F   F  @     F       ;(-   =      : %     F :           :   %        : %     F :       ) ;'.  ;'<   F  )%     F )  ;'& ) ;'(  ;'' +     =   :  :            %    %       F )%     F )    F )%     F ) ;',  ;'D     F : F @    F   :       :          ) : % :   :  : %            ;'; ;'G ;'-    )  :   @ :     :      ;('  !,!#$           -   :            5 = F  : %  )  : %   F     F  )   <    ;,. ;,<  C  =   : F  F  @     F       ;,&   =   )   : %   F :           :   %     F     : %   )   F :        ;,( ;,' )%     F  @ ) ;,, ;,D ;,; +     =   :  :           %     %    @     F )%     F )    F )%     F ) ;,G  ;,-     F : F @    F   :            :   @  : :     ;D. ;D< ;D& :      :       :       ;('  !,!"$     $      <.  :         θ2 θ1 u l2 , m2 l1 , m1 M x % #        ;'=       #                  4   %&' && &() *   3                     3      4              )   ! 8       9  ;'           4             : %  O  O  O )                             %       )  :       @  @  P   @ & P      P     :  @  @ &  O   @   O   ;D(   ;D'    P @ &  P       : < @ &    @ &   O       @ &  P   @ '    ( @      @ &   P      @ (    P  :      &   O      '   P ;D, ;DD ;D; ;DG   4     5        ..&       5                1  9K1     5             8      ' 467#    8   *   "  *             1 2 9 1 2                  "   "             "  :              *     ;<7         1 21 2 1 2   1 2     "     7  #    *        1 2   1 2                               "    8      "    "     ;<7                           !          <<                    #                                                                             3       5    Q 4   @&,R4&'       /   ,              ;, C                 #                  #        5            *                            observed observed 10 5 2 cart velocity cart position 4 0 −2 −4 0 −5 −10 0 50 100 −15 150 0 50 un−observed 0.5 0 −0.5 −1 −1.5 0 50 100 5 0 −5 −10 −15 150 0 50 observed 150 10 pendulum 2 velocity pendulum 2 angle 100 un−observed 0.5 0 true state noisy observation EKF estimate UKF estimate −0.5 −1 150 10 pendulum 1 velocity pendulum 1 angle 1 −2 100 un−observed 0 50 100 5 0 −5 −10 −15 −20 150 time 0 50 100 150 time &      #        #  ! '       #  (     "               ! )*+  &,-.  /*" # /+"   /+0  ;,=   "  =             "   "  #$%   .$%      <&       Estimation of Mackey−Glass time series : EKF 5 x(k) clean noisy EKF 0 −5 200 210 220 230 240 250 260 270 280 290 300 k Estimation of Mackey−Glass time series : UKF 5 x(k) clean noisy UKF 0 −5 200 210 220 230 240 250 260 270 280 290 300 k Estimation Error : EKF vs UKF on Mackey−Glass normalized MSE 1 EKF UKF 0.8 0.6 0.4 0.2 0 0 100 200 300 400 500 600 700 800 900 1000 k    1 2    3       3  !      3               !  ;D= - .    3  #             4       0      4       + 404(.     %(, &G)     4    "                 :   5                 @   3   ;D-                             6 3  0         + 40           4    :  @       4         =     :          <    :  .     . .   . . @    . .     < . <(         <  .     @     .         : <  . .        4    @     ;;.            #   5          ;D   4                 #    961        #     4     (E            8        "          3 4                                        "       Q    +48 %'D <&)            "      8                   %&-)          "                     F  8    "                  4        F                          =    :    @     F :  %     F  @    F ) ;;< ;;&             "          "    7          "           5   4     :   @    ;;( ;;'           5              8    "       S         #9        #                       "                               ;;                #       5             + 40  4        4                                            4                              3        5      9<  3                9&         #9      E    9       8      "          <'     xk time series xˆk − L−1  ;;=  3  4# 3   3       ! -! /   #$  01  xˆk +1 -! /   %  &'(( 01 9< .&. .;. .&; 9< .(, .(& .&G 9& .&. .(< .<- 9& .(, .&& .&( #9 .<. .&' ..G #9 .&( .&< .  &4<.4<.4'                : %< <) @.G               %< <)         .G     ;<.    "        #     " "              <..                       <. ...  8          # <-     True Mapping Learning Curves on Test Set 0.7 averaged RMSE −1 x2 −0.5 0 0.5 −0.5 0 0.5 0.6 0.5 0.4 0.3 0.2 1 0 50 100 150 200 x1 epochs NN Classification : EKF trained NN Classification : UKF trained −1 −1 −0.5 −0.5 x2 x2 1 −1 EKF UKF 0 0.5 1 −1 0 0.5 −0.5 0 0.5 1 −1 1 x1 −0.5 0 0.5 1 x1 &  =9   -     #  !   ( )   . /      . 7*+*+> 185 . - ## 1  . *   / *++    0!  ;<.= &.         Inverted Double Pendulum : parameter estimation 0 10 model MSE EKF UKF −5 10 5 10 15 20 25 30 35 40 iteration        . :  #          !    ) /      . - ## 1  0  ;<<=     )    1          9  ;(<          4    : %     )                5   <. : % O  O  O )      ;<<      +9            #             /              "             =             .,. .;, .;, .,. <,. #  .,. .;, .;, .,. <'-   .,. .;, .DG .', <(,                            &35    ' &        3      5    E  %'()       &4       4        9 "    "                  : <..     @ <      : <   : <     5   ;-(  E                        5          #               4    5  4+9          4     4   %'&)=   @ .: @  : &< ;-' ;-,          A B  "3   5          7      :           5    +9   5     4      5    :         4              C  ,     1    #        /        5    3      : %<.      <   )             4 5         N  N            5   5       ;-& S &       #      #             *                       5  +9   #           #    4       ;<& S         "              5    K4  4 >  ?   4+  9 A B  +8?8E      3         #     5   Function Value 20 10 EKF UKF 0 f(X) 10 −20 10 −40 10 1 2 3 4 5 6 7 k Model Error 20 10 EKF UKF 0 MSE 10 −20 10 −40 10 1 2 3 4 5 6 7 k -  # 9 9 9  $   #  !   ( )      .  0  ;<&=        . /     " >   3    &   "  "              * * "                     *        ?4#        *  "               *     * "        Ö  &&       !     1                                ;; 8     4   3      $  K    1  4> 4   +34     3 4+35 +   8            C  ,  D    3             K #  4   $ #             %,&)     4                     8          4               4"         4       4"         4   4      #   *        %(D)  4          *    = %   )            4       Q  =        :    : <    . .  @       ;-D  @ ;-;       Q  4             8       #              - .  *         4              #      "       + 404(.        961  (E     4                        0         (E 8  D4<.4< +?>  !"   0  961                  "    + 40   8 ,4(4< +?>                                 6        3           4       8               3      /     Q           ;<( 8 "    + 40         K #        #       &(     0.55 Chaotic AR neural network 0.5 Dual UKF Dual EKF Joint UKF Joint EKF normalized MSE 0.45 0.4 0.35 0.3 0.25 epoch 0.2 0 5 10 15 20 25 30 0.7 Mackey−Glass chaotic time series Dual EKF Dual UKF Joint EKF Joint UKF 0.6 normalized MSE 0.5 0.4 0.3 0.2 0.1 0 epoch 5 10 15 20 25 30 35 40 45 50 55 60 Estimation of Mackey−Glass time series : Dual UKF 5 x(k) clean noisy Dual UKF 0 −5 200 210 220 230 240 250 260 270 280 290 300 k ?                     ! ?      *+ @        <    3! A B          C     ! A6   B         3        !  ;<(= &'     /   3       Q #            4     ;<'         #   9  #  Q  *  + C  E  *                       # 6               3        961  3         #             ;<,           D.  961     /         8 "3     #  <.. E                        Q  9    6       #  5   /    5                 #     u1 = N (0,1) u2 = N (0,1) 1 0 0 ⎤ ⎡ x1 ⎤ ⎡ x1 ⎤ ⎡ 0 2 ⎢  ⎥ ⎢ x −ω −2ζ1ω1 0 0 ⎥ ⎢ x1 ⎥ ⎢ 1⎥ = ⎢ 1 ⎥⎢ ⎥ 0 0 1 ⎥ ⎢ x2 ⎥ ⎢ x2 ⎥ ⎢ 0 ⎢  ⎥ ⎢ ⎥⎢ ⎥ 0 −ω22 −2ζ 2ω2 ⎦ ⎣ x2 ⎦ ⎣ x2 ⎦ ⎣ 0 y1 (pos) y2 (pos)  ;<'= 1  &   ! 20 ω1 (rad/s) 19.5 19 EKF UKF Actual 18.5 18 0 1 2 3 4 5 6 7 8 9 10 54 EKF UKF Actual 53.5 ω2 (rad/s) 53 52.5 52 51.5 51 50.5 0 1 2 3 4  ;<,= 5 Time (s) 6 7       &, 8 9 10     #6 701     3       #   * Q      <,    %')                                    4   8              * >  8E 8              C9           %<.) 8           #              "      + C        9 9                 E     ;- ;<.  ;<< 3           + C                              +       "      3         :                   9     M : <     3       0           Æ      K              0    ;-G     : 0     <     0   ;--     ++9  F :  ) %          3      "         3      9   "           + C                 "   9                Æ      :             8   Æ 3       3      C      <  $ +         7                     *              % 9      3                             0             &G          0 9   :  :  : 9   % 9  9 % 9     9  9  0 9    % 9  9   % 9   & 9  0 9  % 9  9    0 9    9      5       9  9   & : % 9   *         5             &   0 9  ;<..  < 0 9 & 9 % 9  9    0 9 & 9 % 9  9   ¼  ¼   9  9   ¼  ¼ 9  0  9 & 9 % 9  9 & 9 % 9  9     ¼ & 9 0 9      ¼ & 9  :  : : :       ¼       5   3          %  8                 3    9   : % 9    % 9                 %                          +                            = & : &          9     ;<.<                  9    9                                ;<.< % %                                   3                 <   0  &   : 0 9   0  &   ;<.&    < &       : &   &        N     5     &     Q       &- % 9               0 9   &      3     3                         "         3          4  9   :              Æ 9      &    ;<.(       ;<.'     "               Q       :                 &         3         ;-G  ;--       3   Æ        0 9  : 0    4                                                     0 3      3   4              ) 0   // )       9 9          =                        4     5        <          5  8         /                        4 "                                                 /       8                +  C + 4C +C+C           A B      :              %-) 9                     ,)'- %<< '' 'G)    %<, (<)  94   9 1     K                             <            M : <      &   M : <           /                &           ; 1           8 ;D<       "   "        =   :<       %               #  =   :<  = ;       <                     :    I    I  @       I                       :          I   :                   :   %      I   )%      I   )            I   :1       :          ;<<< ;<<& ;<<( ;<<'                :  :            %         I   )%     I   ) ;<<,  %      I  )     I   )%      ;<            "                            "   ('         1&  )    3     4          : < @ $  #'       0  ! ( 8 4   : .&  ( : ., ( .....< 0              @ (   & @              45      @ ;<<-   # : ..'      (  :       @( &           : .,      (  .        (. ) (.  ;<&.     0   /   "         : <    D.  3        <..    8    "    &..       #         : <  : .  : &              ;< 5       /   "              4 4  +9        ;&.                /    "               #>      & 0 4  & )  K    "                                    3   %             %() E  9            * : , :    +              @ +       01 ;<&< ;<&& /     3    "   .(;' ..<, #      # .&G. ..<& >   =    .'&' ..,( >   = +C+C    .'<; ..,, >   =   .(<. ..   =    +C+C    .(.; ..<, >   = #  A ..;. ...D ..;' ...G      B >   = #   +C+C    &        !     3    *++      !  ;<= (,     1&           Filter estimates (posterior means) vs. True state 9 8 7 E[x(t)] 6 5 4 3 True x PF estimate PF−EKF estimate PF−UKF estimate 2 1 0 10 20 30 40 50 60 Time  ;&.=   * 5        #  <             !               +       .  :  :        .           ,    +  @ - @  .          .  . &              -    4                      ,.  ! 6                3         "       # #     3                 ! 5      #   3  &        )F0"        )G0  $   )  ;&<= <) )    01  -   ..;G 3       ..(; #      # ..(; >   =    ..(; >   =   ...-       ...-  ..(, 3       ..&( #      # ..&( >   =    ..&( >   =   ...;       ...G '    $      *++  !       #   #          3          !  ;&=  #                           4                       (;      −3 Interest rate 14 x 10 12 10 8 6 4 0 50 100 150 200 250 0 50 100 150 200 250 0.21 Volatility 0.2 0.19 0.18 0.17 0.16 0.15 0.14 Time (days)  ;&&= "   #            !   3                                3       4 "   #                                       #                           "                                # *      #        3                                             8 Æ     4                  8 3 E              3     4           #                   $  7                          N 3N                                         4                       S      #                                                      #                   8        0           4   S                  (G               7         4 Q     0    Æ   #          "      3              4 3  "      "   9  ;D                                                           *                    "                    4        /                4   "     S                          S           5  #       3    #                  C  &  ,      #      3 4+35   C  D C                    "     # $ %& '         69    C94..G(<.D  1 4-;<&('D  K81>8    ((D<,4-G4C4(, + KQX  8     E   "         )""" 0    &    "     (( <=((GW('( <--; %()  E  + 9          "   1      G<=D(;WD,- <-;( %') 1 * E 8 8 +    8 88 C K  C   >1C   C8 %,) $ 1 C  C 6 KN95  C > +  6       74"     4    1    = ><          )   %    (       &   &   K  E  ? + <--D %D) $  0    2  #    (   ( + $ >K   C #    K   <--- %;) $  0    + 6Q 8 7 0  8 K  9   + C           (   %   <& '=-,,W--( &... %G) 8 K  S    4     E  "    1  C# KR4 6 60R1 (<. 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Æ    4  "      0    "     &   (   ( + $ ,"&((-  E  E  8 &..< %,<) 1    +    8 *   4         4     "     0    )   %    & /  /  9 ? C # + &..< %,&)  8 *  8  6  6   3    " =                = $     (   ( + $     <--; %,()  8 *  1    +    #       6          >???  &        / %    %   ,&%%- ? ? 8  C S  &...  %,')  8 * 1    +   8  6  K    #   4   98 9  ?   41 +P         3> &   (   )     DDDWD;& +  >  &... %,,) 2 9 Y  2 E 9   ?      9   + 4C        &    '  %   '- (D (=&.<,W&.&& <-;,