{"id":98825,"date":"2020-07-17T08:39:08","date_gmt":"2020-07-17T06:39:08","guid":{"rendered":"https:\/\/botland.com.pl\/blog\/filtr-kalmana-podstawy-teoretyczne\/"},"modified":"2025-02-03T14:25:54","modified_gmt":"2025-02-03T13:25:54","slug":"kalmanuv-filtr-teoreticke-zaklady","status":"publish","type":"post","link":"https:\/\/botland.cz\/blog\/kalmanuv-filtr-teoreticke-zaklady\/","title":{"rendered":"Kalman\u016fv filtr &#8211; teoretick\u00e9 z\u00e1klady"},"content":{"rendered":"<span class=\"span-reading-time rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">\u010cas \u010dten\u00ed:<\/span> <span class=\"rt-time\"> 8<\/span> <span class=\"rt-label rt-postfix\">min.<\/span><\/span>\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"98825\" class=\"elementor elementor-98825 elementor-2821\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-ec03b9c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"ec03b9c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6c7e8469\" data-id=\"6c7e8469\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-430f3cb4 elementor-widget elementor-widget-text-editor\" data-id=\"430f3cb4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\u0160irok\u00e1 \u0161k\u00e1la technick\u00fdch v\u011bd pot\u0159ebuje k popisu nejr\u016fzn\u011bj\u0161\u00edch jev\u016f vhodn\u00e9 matematick\u00e9 n\u00e1stroje. Jak automatizace, tak robotika, teorie \u0159\u00edzen\u00ed a \u0159\u00eddic\u00edch syst\u00e9m\u016f, elektronika, elektroakustika, \u010d\u00edslicov\u00e9 zpracov\u00e1n\u00ed sign\u00e1l\u016f a p\u0159\u00edbuzn\u00e9 obory jsou zalo\u017eeny na line\u00e1rn\u00ed algeb\u0159e, oper\u00e1torov\u00e9m kalkulu, genetick\u00fdch algoritmech, numerick\u00fdch metod\u00e1ch, ale i fuzzy logice a dal\u0161\u00edch odv\u011btv\u00edch matematiky. Vedle matematick\u00fdch n\u00e1stroj\u016f obl\u00edben\u00fdch v technick\u00fdch v\u011bd\u00e1ch, jako je Laplaceova transformace, Z-transformace nebo metoda kone\u010dn\u00fdch prvk\u016f, algoritmus, kter\u00fd zp\u016fsobil revoluci mj. Jedn\u00e1 se o jeden z klasick\u00fdch n\u00e1stroj\u016f odhadu. Je to jeden z klasick\u00fdch n\u00e1stroj\u016f odhadu, kter\u00fd se pou\u017e\u00edv\u00e1 v aplikac\u00edch, jako je zpracov\u00e1n\u00ed sign\u00e1l\u016f, \u0159\u00edzen\u00ed elektrick\u00fdch pohon\u016f, autonomn\u00ed syst\u00e9my \u0159\u00edzen\u00ed vozidel a \u0159\u00edzen\u00ed procesor\u016f v po\u010d\u00edta\u010d\u00edch. V odborn\u00e9 literatu\u0159e je <strong>Kalman\u016fv filtr<\/strong> v\u011bt\u0161inou popisov\u00e1n v konkr\u00e9tn\u00edch aplika\u010dn\u00edch p\u0159\u00edpadech, jako jsou autonomn\u00ed roboty nebo line\u00e1rn\u00ed syst\u00e9my automatick\u00e9ho \u0159\u00edzen\u00ed s Gaussovsky rozlo\u017een\u00fdm \u0161umem, co\u017e velmi znesnad\u0148uje p\u0159ehledn\u00e9 p\u0159edstaven\u00ed problematiky po z\u00e1kladn\u00ed str\u00e1nce. N\u00e1\u0161 \u010dl\u00e1nek popisuje diskr\u00e9tn\u00ed Kalman\u016fv filtr pomoc\u00ed vektorov\u00e9ho a maticov\u00e9ho po\u010dtu a ilustruje praktick\u00fd p\u0159\u00edklad jeho pou\u017eit\u00ed.      <\/p>\n<h3>Co je Kalman\u016fv filtr?<\/h3>\n<p><strong>Kalman\u016fv filtr je speci\u00e1ln\u00ed algoritmus, kter\u00fd byl vyvinut a demonstrov\u00e1n na po\u010d\u00e1tku 60. let XX. stolet\u00ed. Rudolf Emil Kalman<\/strong> (1930-2016) &#8211; ma\u010farsko-americk\u00fd elektroin\u017een\u00fdr. Kalman\u016fv filtr lze pou\u017e\u00edt v jak\u00e9koli aplikaci, kde je sn\u00ed\u017een\u00e1 spolehlivost informac\u00ed o parametrech dynamick\u00fdch syst\u00e9m\u016f. Parametry dynamick\u00fdch syst\u00e9m\u016f. Pak m\u016f\u017eeme tak\u00e9 p\u0159edv\u00eddat, jak\u00e9 ud\u00e1losti mohou nastat p\u0159\u00ed\u0161t\u011b. To se prov\u00e1d\u00ed na z\u00e1klad\u011b dosud z\u00edskan\u00fdch informac\u00ed. V \u00favahu se pak bere mnoho podm\u00edn\u011bn\u00fdch faktor\u016f, kter\u00e9 se mohou m\u011bnit a m\u00edt v\u00fdznamn\u00fd vliv na v\u00fdsledek automatiza\u010dn\u00edho syst\u00e9mu, v n\u011bm\u017e je Kalman\u016fv filtr implementov\u00e1n. Tento algoritmus je vynikaj\u00edc\u00ed pro dynamick\u00e9 syst\u00e9my, jejich\u017e parametry se pr\u016fb\u011b\u017en\u011b m\u011bn\u00ed a ke spr\u00e1vn\u00e9 funkci nepot\u0159ebuj\u00ed velk\u00e9 mno\u017estv\u00ed pam\u011bti, pouze stav syst\u00e9mu p\u0159edch\u00e1zej\u00edc\u00ed dal\u0161\u00edmu. D\u016fle\u017eit\u00fdm faktorem ve prosp\u011bch tohoto algoritmu je jeho vysok\u00e1 rychlost, d\u00edky n\u00ed\u017e je \u00fasp\u011b\u0161n\u011b implementov\u00e1n v aplikac\u00edch re\u00e1ln\u00e9ho \u010dasu a vestavn\u00fdch syst\u00e9mech.       <\/p>\n<h3>Kalman\u016fv filtr &#8211; v jak\u00fdch aplikac\u00edch se vyplat\u00ed jej pou\u017e\u00edt?<\/h3>\n<p>Ne\u017e si uk\u00e1\u017eeme algoritmus, kter\u00fd ukazuje princip Kalmanova filtru, pou\u017eijeme jeden hardwarov\u00fd p\u0159\u00edklad jeho pou\u017eit\u00ed, kter\u00fdm je robot, kter\u00fd se dok\u00e1\u017ee pohybovat v hust\u011b zalesn\u011bn\u00fdch oblastech. Aby vestav\u011bn\u00fd naviga\u010dn\u00ed syst\u00e9m spr\u00e1vn\u011b fungoval, pot\u0159ebuje zn\u00e1t svou p\u0159esnou polohu. <\/p>\n<p>M\u016f\u017eeme tedy \u0159\u00edci, \u017ee n\u00e1\u0161 robot je definov\u00e1n stavovou prom\u011bnnou <span style=\"font-weight: 400;\">x<\/span><span style=\"font-weight: 400;\">k<\/span><span style=\"font-weight: 400;\">, <\/span>, kter\u00e1 je definov\u00e1na vektorem o dvou slo\u017ek\u00e1ch &#8211; poloze <span style=\"font-weight: 400;\">p <\/span> a rychlosti <span style=\"font-weight: 400;\">v<\/span><span style=\"font-weight: 400;\">:<\/span><\/p>\n<p id=\"gdcalert5\"><img decoding=\"async\" class=\"aligncenter size-full wp-image-2822\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/3.1.png\" alt=\"\" width=\"77\" height=\"29\">(3.1)<\/p>\n<p>Stav syst\u00e9mu je ur\u010den \u010d\u00edseln\u00fdmi \u00fadaji, kter\u00e9 definuj\u00ed jeho z\u00e1kladn\u00ed konfiguraci, a mohou to b\u00fdt \u010d\u00edseln\u00e9 \u00fadaje pro libovoln\u00e9 veli\u010diny. V na\u0161em p\u0159\u00edpad\u011b jsou t\u011bmito veli\u010dinami poloha a rychlost, ale u jin\u00fdch syst\u00e9m\u016f to m\u016f\u017ee b\u00fdt tak\u00e9 zb\u00fdvaj\u00edc\u00ed mno\u017estv\u00ed paliva v n\u00e1dr\u017ei v automobilu, teplota hrotu v ba\u0148ce p\u00e1jec\u00ed stanice, proud v budic\u00edm obvodu synchronn\u00edho gener\u00e1toru a mnoho dal\u0161\u00edch parametr\u016f, kter\u00e9 v z\u00e1vislosti na aplikaci vy\u017eaduj\u00ed sledov\u00e1n\u00ed hodnot. Robot v prezentovan\u00e9m p\u0159\u00edkladu je vybaven naviga\u010dn\u00edm syst\u00e9mem GPS, kter\u00fd je schopen ur\u010dit jeho sou\u0159adnice s p\u0159esnost\u00ed na deset metr\u016f. To je uspokojiv\u00fd v\u00fdsledek, ale vzhledem k prom\u011bnlivosti r\u016fzn\u00fdch parametr\u016f ter\u00e9nu, po n\u011bm\u017e se robot pohybuje, je nutn\u00e9, aby vestav\u011bn\u00fd syst\u00e9m GPS byl je\u0161t\u011b p\u0159esn\u011bj\u0161\u00ed. V nejhor\u0161\u00edm p\u0159\u00edpad\u011b by robot s p\u0159esnost\u00ed sou\u0159adnic na deset metr\u016f mohl dokonce spadnout do propasti.     <\/p>\n<p>Pro p\u0159esn\u00e9 ur\u010den\u00ed dr\u00e1hy robota je nutn\u00e9 zn\u00e1t parametry jednotliv\u00fdch sign\u00e1l\u016f ovl\u00e1daj\u00edc\u00edch jeho pohonnou jednotku &#8211; syst\u00e9m GPS pak &#8220;v\u00ed&#8221;, \u017ee kdy\u017e se robot pohybuje tak, \u017ee vykon\u00e1 t\u00e9m\u011b\u0159 p\u0159\u00edmo\u010dar\u00fd pohyb, je pravd\u011bpodobn\u00e9, \u017ee se bude v dal\u0161\u00edm kroku pohybovat stejn\u00fdm zp\u016fsobem a stejn\u00fdm sm\u011brem. Robot v\u0161ak st\u00e1le nebude dost\u00e1vat informace o okoln\u00edch faktorech, kter\u00e9 mohou ovlivnit jeho pohyb. Robot v\u0161ak st\u00e1le nep\u0159ij\u00edm\u00e1 informace o faktorech prost\u0159ed\u00ed, kter\u00e9 mohou ovlivnit jeho pohyb, nap\u0159. poryvy v\u011btru, kluzk\u00e1 p\u016fda, a tak\u00e9 nerovnosti ter\u00e9nu a p\u0159ek\u00e1\u017eky na silnici, kter\u00e9 d\u0159\u00edve nep\u0159edv\u00eddal (nap\u0159. le\u017e\u00edc\u00ed v\u011btve strom\u016f). <span style=\"font-size: 1rem;\">Pak m\u016f\u017ee po\u010det ot\u00e1\u010dek proveden\u00fdch silni\u010dn\u00edmi koly robota nespr\u00e1vn\u011b reprezentovat jeho pohyb, co\u017e vede k nespr\u00e1vn\u00e9 p\u0159edpov\u011bdi dal\u0161\u00edho kroku popisuj\u00edc\u00edho pohyb robota. <\/span>   <\/p>\n<p><span style=\"font-size: 1rem;\">Senzory GPS poskytuj\u00ed nep\u0159\u00edm\u00e9 informace o stavu syst\u00e9mu (robota), ale s omezenou p\u0159esnost\u00ed. Senzory GPS poskytuj\u00ed nep\u0159\u00edm\u00e9 informace o stavu syst\u00e9mu (robota), ale s omezenou p\u0159esnost\u00ed. Nicm\u00e9n\u011b se ukazuje, \u017ee \u00fadaje z\u00edskan\u00e9 ze senzor\u016f GPS lze vyu\u017e\u00edt v procesu odhadu k z\u00edsk\u00e1n\u00ed sou\u0159adnicov\u00fdch m\u011b\u0159en\u00ed se zv\u00fd\u0161enou p\u0159esnost\u00ed. Pr\u00e1v\u011b to umo\u017e\u0148uje algoritmus tzv. Zde p\u0159ich\u00e1z\u00ed ke slovu algoritmus zn\u00e1m\u00fd jako Kalman\u016fv filtr.    <\/span><\/p>\n<h3>Kalman\u016fv filtr &#8211; za\u010d\u00e1tek algoritmu<\/h3>\n<p>Pro zah\u00e1jen\u00ed algoritmu Kalmanova filtru zapi\u0161me rovnici stavov\u00e9ho vektoru (3.1) v maticov\u00e9m tvaru:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-2823\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/3.2.png\" alt=\"\" width=\"105\" height=\"87\">(3.2)<\/p>\n<p>Aktu\u00e1ln\u00ed polohu a rychlost objektu nezn\u00e1me, proto\u017ee existuje kone\u010dn\u00e1 mno\u017eina zahrnuj\u00edc\u00ed celou \u0161k\u00e1lu mo\u017en\u00fdch kombinac\u00ed polohy a rychlosti, kter\u00e9 se mohou uk\u00e1zat jako odpov\u00eddaj\u00edc\u00ed skute\u010dnosti, ale mezi nimi jsou i takov\u00e9, kter\u00e9 v\u00edce \u010di m\u00e9n\u011b odpov\u00eddaj\u00ed skute\u010dn\u00fdm hodnot\u00e1m. Algoritmus Kalmanova filtru p\u0159edpokl\u00e1d\u00e1, \u017ee ob\u011b stavov\u00e9 prom\u011bnn\u00e9 (ve zkouman\u00e9m p\u0159\u00edpad\u011b jsou to poloha a rychlost objektu) jsou rozd\u011bleny podle Gaussova rozd\u011blen\u00ed (norm\u00e1ln\u00ed rozd\u011blen\u00ed). Ka\u017ed\u00e1 prom\u011bnn\u00e1 m\u00e1 st\u0159edn\u00ed hodnotu, kter\u00e1 se nach\u00e1z\u00ed ve st\u0159edn\u00ed poloze n\u00e1hodn\u00e9ho rozd\u011blen\u00ed a je z\u00e1rove\u0148 o\u010dek\u00e1vanou hodnotou, a rozptyl, kter\u00fd definuje nejistotu m\u011b\u0159en\u00ed. Na prvn\u00ed pohled poloha a rychlost objektu spolu nesouvis\u00ed, co\u017e znamen\u00e1, \u017ee informace o stavu jedn\u00e9 prom\u011bnn\u00e9 (slo\u017eky stavov\u00e9ho vektoru) nebude dostate\u010dn\u00e1 k ur\u010den\u00ed druh\u00e9 prom\u011bnn\u00e9. Ve skute\u010dnosti poloha a rychlost objektu spolu souvisej\u00ed, tj. pravd\u011bpodobnost pozorov\u00e1n\u00ed ur\u010dit\u00e9 hodnoty prom\u011bnn\u00e9, kter\u00e1 ur\u010duje polohu objektu, z\u00e1vis\u00ed na jeho rychlosti.     <\/p>\n<p>Tato situace m\u016f\u017ee nastat nap\u0159\u00edklad tehdy, kdy\u017e zn\u00e1me p\u0159edchoz\u00ed polohu objektu a sna\u017e\u00edme se ur\u010dit jeho aktu\u00e1ln\u00ed polohu. Pokud je rychlost objektu p\u0159\u00edli\u0161 vysok\u00e1, pak dal\u0161\u00ed m\u011b\u0159en\u00ed polohy objektu bude vzd\u00e1len\u011bj\u0161\u00ed. Pokud se naopak objekt pohybuje ni\u017e\u0161\u00ed rychlost\u00ed, pak m\u011b\u0159en\u00ed polohy ve srovn\u00e1n\u00ed s p\u0159edchoz\u00edm m\u011b\u0159en\u00edm uk\u00e1\u017ee posunut\u00ed objektu o men\u0161\u00ed vzd\u00e1lenost ne\u017e p\u0159i vy\u0161\u0161\u00ed rychlosti. Vztah mezi polohou a rychlost\u00ed objektu je p\u0159i sledov\u00e1n\u00ed objektu velmi d\u016fle\u017eit\u00fd, proto\u017ee poskytuje cenn\u00e9 informace &#8211; m\u011b\u0159en\u00ed jednoho parametru usm\u011br\u0148uje potenci\u00e1ln\u00ed v\u00fdsledky n\u00e1sledn\u00fdch m\u011b\u0159en\u00ed stejn\u00fdch parametr\u016f. To je hlavn\u00ed \u00fa\u010del algoritmu Kalmanova filtru &#8211; shrom\u00e1\u017edit co nejv\u00edce \u00fadaj\u016f zat\u00ed\u017een\u00fdch nep\u0159esnost\u00ed m\u011b\u0159en\u00ed, aby bylo mo\u017en\u00e9 z\u00edskat co nejp\u0159esn\u011bj\u0161\u00ed v\u00fdsledek m\u011b\u0159en\u00ed.    <b> <\/b>Vztah mezi stavov\u00fdmi prom\u011bnn\u00fdmi je definov\u00e1n kovarian\u010dn\u00ed matic\u00ed a ka\u017ed\u00fdm prvkem v n\u00ed obsa\u017een\u00fdm. Pak prvek  <span style=\"font-size: 1rem;\">ur\u010duje stupe\u0148 korelace mezi i-tou stavovou prom\u011bnnou a j-tou stavovou prom\u011bnnou, p\u0159i\u010dem\u017e kovarian\u010dn\u00ed matice je \u010dtvercov\u00e1 matice, tj. jej\u00ed po\u010det \u0159\u00e1dk\u016f je stejn\u00fd jako po\u010det sloupc\u016f. <\/span><\/p>\n<h3>Maticov\u00fd z\u00e1pis<\/h3>\n<p>Pro modelov\u00e1n\u00ed stavu objektu je u\u017eite\u010dn\u00e9 pou\u017e\u00edt maticov\u00fd z\u00e1pis s vyu\u017eit\u00edm Gaussovy fuzzyfikace &#8211; pro ur\u010den\u00ed stavu objektu pro k-t\u00e9 m\u011b\u0159en\u00ed pak pot\u0159ebujeme dv\u011b rovnice. xk je vektor st\u0159edn\u00edch (skal\u00e1rn\u00edch) hodnot, co\u017e jsou o\u010dek\u00e1van\u00e9 hodnoty, a Pro modelov\u00e1n\u00ed stavu objektu je u\u017eite\u010dn\u00e9 pou\u017e\u00edt maticov\u00fd z\u00e1pis s vyu\u017eit\u00edm Gaussovy fuzzyfikace &#8211; pro ur\u010den\u00ed stavu objektu pro k-t\u00e9 m\u011b\u0159en\u00ed pak pot\u0159ebujeme dv\u011b rovnice. <sub><span style=\"font-weight: 400;\">.<\/span><span style=\"font-weight: 400;\">x<\/span>k<\/sub> je vektor st\u0159edn\u00edch (skal\u00e1rn\u00edch) hodnot, co\u017e jsou o\u010dek\u00e1van\u00e9 hodnoty, a P<sub>k<\/sub> je kovarian\u010dn\u00ed matice.<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter size-full wp-image-2824\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/4.1.4.2.png\" alt=\"\" width=\"562\" height=\"140\"><\/p>\n<p>V tomto p\u0159\u00edpad\u011b jsou stavov\u00fdmi prom\u011bnn\u00fdmi poloha a rychlost objektu, ale je t\u0159eba poznamenat, \u017ee stav m\u016f\u017ee b\u00fdt pops\u00e1n libovoln\u00fdm po\u010dtem prom\u011bnn\u00fdch reprezentuj\u00edc\u00edch libovoln\u00e9 parametry. Pak je nutn\u00e9 zaznamenat hodnoty prom\u011bnn\u00fdch pro sou\u010dasn\u00fd stav (v \u010dase k-1), aby bylo mo\u017en\u00e9 p\u0159istoupit k p\u0159edpov\u011bdi hodnot pro stav v \u010dase k. Nen\u00ed zn\u00e1mo, kter\u00fd stav odr\u00e1\u017e\u00ed skute\u010dn\u00e9 parametry objektu, ale to je pro proces predikce irelevantn\u00ed. F\u00e1zi predikce lze vizualizovat pomoc\u00ed vektoru F<sub>k<\/sub>. Ka\u017ed\u00fd bod, kter\u00fd se nach\u00e1z\u00ed ve stavov\u00e9m prostoru v aktu\u00e1ln\u00edm kroku, se p\u0159esune do nov\u00e9 oblasti &#8211; oblasti p\u0159edpov\u00eddan\u00fdch hodnot, do kter\u00e9 se syst\u00e9m p\u0159esune, pokud byl p\u0159edchoz\u00ed stavov\u00fd prostor spr\u00e1vn\u011b lokalizov\u00e1n. K nalezen\u00ed p\u0159edpov\u00eddan\u00e9ho stavov\u00e9ho prostoru p\u0159i znalosti aktu\u00e1ln\u00edch hodnot stavov\u00fdch veli\u010din ur\u010duj\u00edc\u00edch polohu a rychlost pou\u017eijeme soustavu rovnic:     <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2825\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/4.3.4.4.png\" alt=\"\" width=\"562\" height=\"203\"><\/p>\n<p>T\u00edmto zp\u016fsobem z\u00edsk\u00e1me predik\u010dn\u00ed matici, kter\u00e1 obsahuje informace o dal\u0161\u00edm (po k-1) stavu (k). Nicm\u00e9n\u011b st\u00e1le nen\u00ed jasn\u00e9, jak spr\u00e1vn\u011b aktualizovat kovarian\u010dn\u00ed matici. Zde je t\u0159eba ka\u017ed\u00fd bod obsa\u017een\u00fd v rozd\u011blen\u00ed vyn\u00e1sobit matic\u00ed A:  <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2826\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/4.5.4.6.4.7.4.8.png\" alt=\"\" width=\"574\" height=\"285\"><\/p>\n<h3>Vliv vn\u011bj\u0161\u00edch faktor\u016f<\/h3>\n<p>V t\u00e9to f\u00e1zi je\u0161t\u011b nebyly zohledn\u011bny v\u0161echny faktory, a to faktory, kter\u00e9 p\u0159\u00edmo nesouvisej\u00ed s vlastnostmi zkouman\u00e9ho syst\u00e9mu, ale vyskytuj\u00ed se nez\u00e1visle, ale mohou stav syst\u00e9mu tak\u00e9 ovlivnit. Nap\u0159\u00edklad je-li objektem popsan\u00fdm stavov\u00fdmi prom\u011bnn\u00fdmi elektrick\u00e1 lokomotiva (elektrick\u00e1 lokomotiva) &#8211; kdy\u017e strojvedouc\u00ed zv\u00fd\u0161\u00ed nap\u00e1jec\u00ed nap\u011bt\u00ed motor\u016f, nap\u0159. sn\u00ed\u017een\u00edm magnetick\u00e9ho toku v budic\u00edm obvodu, lokomotiva zrychl\u00ed. Podobn\u011b v popsan\u00e9m p\u0159\u00edpad\u011b robota m\u016f\u017ee palubn\u00ed naviga\u010dn\u00ed syst\u00e9m narazit na probl\u00e9my p\u0159i p\u0159enosu sign\u00e1lu do \u0159\u00eddic\u00edho syst\u00e9mu robota, aby oto\u010dil kola nebo zastavil, nap\u0159. kdy\u017e hroz\u00ed, \u017ee robota za j\u00edzdy p\u0159evr\u00e1t\u00ed v\u00edtr. Znalost t\u011bchto dodate\u010dn\u00fdch informac\u00ed o vn\u011bj\u0161\u00edch podm\u00ednk\u00e1ch lze zohlednit pomoc\u00ed vektoru   <\/p>\n<p><sub>k<\/sub>, co\u017e umo\u017e\u0148uje zahrnout korekci do predik\u010dn\u00ed matice:<\/p>\n<p>Za p\u0159edpokladu, \u017ee na z\u00e1klad\u011b \u0159\u00eddic\u00edch sign\u00e1l\u016f (nap\u0159. pln\u00edc\u00edho faktoru \u0159\u00eddic\u00edho sign\u00e1lu ot\u00e1\u010dek motoru) zn\u00e1me o\u010dek\u00e1vanou hodnotu zrychlen\u00ed a, z\u00edsk\u00e1me soustavu rovnic:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2827\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/5.1.5.2.png\" alt=\"\" width=\"557\" height=\"198\"><\/p>\n<p><sub>k<\/sub> je \u0159\u00eddic\u00ed vektor &#8211; v jednoduch\u00fdch syst\u00e9mech bez vn\u011bj\u0161\u00edch vliv\u016f lze tyto slo\u017eky zanedbat. Co kdy\u017e ale p\u0159edpov\u011bzen\u00e9 \u0159e\u0161en\u00ed p\u0159esto nen\u00ed dostate\u010dn\u011b p\u0159esn\u00e9? <\/p>\n<h3>Vn\u011bj\u0161\u00ed nejistota<\/h3>\n<p>Cel\u00fd proces predikce plynule pokra\u010duje, pokud je stav objektu zalo\u017een na v\u0161ech zohled\u0148ovan\u00fdch faktorech, v\u010detn\u011b vn\u011bj\u0161\u00edch faktor\u016f, jejich\u017e v\u00fdskyt nesouvis\u00ed s vlastnostmi samotn\u00e9ho syst\u00e9mu, kter\u00e9 ovliv\u0148uj\u00ed. To plat\u00ed tak dlouho, dokud zn\u00e1me maxim\u00e1ln\u00ed parametry vn\u011bj\u0161\u00edch faktor\u016f, v\u016f\u010di nim\u017e bude syst\u00e9m odoln\u00fd. Ale co kdy\u017e se objev\u00ed faktory, o kter\u00fdch nev\u00edme? Nap\u0159\u00edklad drony (stejn\u011b jako jin\u00e1 letadla) podl\u00e9haj\u00ed poryv\u016fm v\u011btru, kter\u00e9 mohou naru\u0161it dr\u00e1hu letu, a dokonce zp\u016fsobit sr\u00e1\u017eku s bl\u00edzk\u00fdm objektem (nap\u0159\u00edklad stromem). Robot na kole\u010dk\u00e1ch m\u016f\u017ee p\u0159i j\u00edzd\u011b uklouznout, co\u017e m\u00e1 za n\u00e1sledek, \u017ee se nem\u016f\u017ee pohybovat d\u00e1l, p\u0159esto\u017ee se kola ot\u00e1\u010dej\u00ed. Pak, proto\u017ee tyto faktory nejsou br\u00e1ny v \u00favahu, m\u016f\u017ee algoritmus ve f\u00e1zi p\u0159edpov\u011bdi selhat, co\u017e m\u016f\u017ee m\u00edt za n\u00e1sledek i nenapraviteln\u00e9 po\u0161kozen\u00ed syst\u00e9mu.      <\/p>\n<p>Aby se zv\u00fd\u0161ila pravd\u011bpodobnost t\u011bchto vysoce ne\u017e\u00e1douc\u00edch \u00fa\u010dink\u016f, mus\u00ed b\u00fdt do v\u00fdpo\u010dtu pro ka\u017ed\u00fd krok p\u0159edpov\u011bdi zahrnuta dal\u0161\u00ed nejistota. V p\u016fvodn\u00edm odhadu je pak p\u016fvodn\u00ed odhad roz\u0161\u00ed\u0159en o \u0159adu stav\u016f. P\u0159edpov\u00eddan\u00e9 \u0159e\u0161en\u00ed x<sub>k-1<\/sub>, se p\u0159esune do kovarian\u010dn\u00ed oblasti Q<sub>k<\/sub>. Jin\u00fdmi slovy lze tak\u00e9 \u0159\u00edci, \u017ee vn\u011bj\u0161\u00ed faktory, kter\u00e9 jsou zdrojem dodate\u010dn\u00e9 nejistoty, tvo\u0159\u00ed \u0161um kovarian\u010dn\u00ed matice Q<sub>k<\/sub>. V\u00fdsledkem takov\u00e9ho p\u016fsoben\u00ed je nov\u00e1 Gaussova \u0161mouha, kter\u00e1 m\u00e1 jinou kovarianci vzhledem k p\u016fvodn\u00ed Gaussov\u011b \u0161mouze, ale stejnou o\u010dek\u00e1vanou hodnotu stavov\u00e9 prom\u011bnn\u00e9. Pak se z\u00edsk\u00e1 roz\u0161\u00ed\u0159en\u00e1 kovarian\u010dn\u00ed matice zahrnut\u00edm faktoru Q<sub>k<\/sub>:     <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2828\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/6.1.6.2.png\" alt=\"\" width=\"554\" height=\"84\"><\/p>\n<p>Jin\u00fdmi slovy, aktualizovan\u00e1 &#8220;nejlep\u0161\u00ed&#8221; o\u010dek\u00e1van\u00e1 hodnota se p\u0159edpov\u00edd\u00e1 z p\u0159edchoz\u00ed o\u010dek\u00e1van\u00e9 hodnoty s p\u0159ihl\u00e9dnut\u00edm k vlivu vn\u011bj\u0161\u00edch podm\u00ednek. Tak\u00e9 nov\u011b odvozen\u00e1 vn\u011bj\u0161\u00ed nejistota, je v\u00fdsledkem kombinace d\u0159\u00edve vypo\u010dten\u00e9 vn\u011bj\u0161\u00ed nejistoty s roz\u0161\u00ed\u0159enou nejistotou v d\u016fsledku vlivu vn\u011bj\u0161\u00edch podm\u00ednek. V t\u00e9to f\u00e1zi m\u016f\u017eeme d\u00e1le zpracov\u00e1vat data z m\u011b\u0159ic\u00edch senzor\u016f.  <\/p>\n<h3>Aproximace odhadu na z\u00e1klad\u011b m\u011b\u0159en\u00ed<\/h3>\n<p>Testovan\u00fd syst\u00e9m m\u016f\u017ee b\u00fdt vybaven n\u011bkolika m\u011b\u0159ic\u00edmi senzory, kter\u00e9 poskytuj\u00ed informace pot\u0159ebn\u00e9 k ur\u010den\u00ed jeho stavu. A\u0165 u\u017e tyto senzory reaguj\u00ed na jak\u00e9koliv podn\u011bty, poskytuj\u00ed informace pot\u0159ebn\u00e9 k ur\u010den\u00ed polohy a rychlosti robota, tak\u017ee se pomoc\u00ed nich prov\u00e1d\u00ed nep\u0159\u00edm\u00e1 m\u011b\u0159en\u00ed. Je d\u016fle\u017eit\u00e9 si uv\u011bdomit, \u017ee m\u016f\u017ee doj\u00edt k nesouladu mezi jednotkami a m\u011b\u0159\u00edtkem ode\u010dtu a jednotkami a m\u011b\u0159\u00edtkem sledovan\u00e9ho stavu. Proto je popis sn\u00edma\u010d\u016f modelov\u00e1n pomoc\u00ed matice H<sub>k<\/sub>. Rozlo\u017een\u00ed ode\u010dt\u016f ze sn\u00edma\u010d\u016f lze p\u0159edpov\u011bd\u011bt z rovnic:    <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2829\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/7.1.7.2.png\" alt=\"\" width=\"546\" height=\"105\"><\/p>\n<p>Jednou z nejv\u011bt\u0161\u00edch v\u00fdhod algoritmu Kalmanova filtru je, \u017ee je imunn\u00ed v\u016f\u010di nep\u0159esnosti m\u011b\u0159en\u00ed prov\u00e1d\u011bn\u00fdch senzory (v d\u016fsledku \u0161umu). Senzory mohou b\u00fdt nespolehliv\u00e9, co\u017e m\u00e1 za n\u00e1sledek, \u017ee jak\u00fdkoli odhadovan\u00fd stav syst\u00e9mu m\u016f\u017ee b\u00fdt v\u00fdsledkem \u0159ady parametr\u016f sn\u00edman\u00fdch senzory. Kalman\u016fv filtr tento probl\u00e9m kompenzuje na minimum. Z ka\u017ed\u00e9ho pozorovan\u00e9ho \u00fadaje ze sn\u00edma\u010de m\u016f\u017eeme usuzovat, \u017ee testovan\u00fd syst\u00e9m byl v p\u0159esn\u011b definovan\u00e9m stavu. Vzhledem k p\u0159\u00edtomnosti nejistoty m\u011b\u0159en\u00ed v\u0161ak m\u016f\u017ee b\u00fdt n\u011bkolik p\u0159edpov\u00eddan\u00fdch stav\u016f v r\u00e1mci rozsahu ode\u010dt\u016f senzor\u016f pravd\u011bpodobn\u011bj\u0161\u00edch ne\u017e jin\u00e9. Za t\u00edmto \u00fa\u010delem zavedeme kovarian\u010dn\u00ed matici neur\u010ditosti R<sub>k<\/sub>, kter\u00e1 je v\u00fdsledkem \u0161umu vnesen\u00e9ho senzory, a vektor pr\u016fm\u011brn\u00fdch pozorovan\u00fdch \u00fadaj\u016f <sub>k<\/sub>     <span style=\"font-size: 1rem;\">.<br \/>\nW ten spos\u00f3b uzyskujemy dwa obszary rozmycia Gaussa &#8211; pierwszy wynikaj\u0105cy z przewidywanego odczytu, a drugi z rzeczywistego odczytu z czujnik\u00f3w.<br \/>\nDzi\u0119ki temu obszar poszukiwa\u0144 optymalnego rozwi\u0105zania znacz\u0105co zaw\u0119\u017ca si\u0119 i ca\u0142y algorytm mo\u017cna wykona\u0107 od nowa w ten sam spos\u00f3b, uzyskuj\u0105c jeszcze bardziej dok\u0142adne, tzn, jeszcze bardziej zgodne z rzeczywisto\u015bci\u0105 wyniki oblicze\u0144 na podstawie uzyskanych wcze\u015bniej danych pomiarowych.  <\/span><\/p>\n<h3>Kalmanovo vylep\u0161en\u00ed<\/h3>\n<p>Pro ur\u010den\u00ed matice definuj\u00edc\u00ed Kalmanovo zes\u00edlen\u00ed je nutn\u00e9 pou\u017e\u00edt jednorozm\u011brnou Gaussovu k\u0159ivku s ohledem na <span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> a st\u0159edn\u00ed hodnotu<\/span><span style=\"font-weight: 400;\">:<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2830\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/8.1.png\" alt=\"\" width=\"546\" height=\"84\"><\/p>\n<p>Vz\u00e1jemn\u00fdm vyn\u00e1soben\u00edm rovnic k\u0159ivek p\u0159edstavuj\u00edc\u00edch norm\u00e1ln\u00ed rozd\u011blen\u00ed pro p\u0159edpov\u00eddan\u00fd stav a skute\u010dn\u00fd stav zji\u0161t\u011bn\u00fd z \u00fadaj\u016f ode\u010dten\u00fdch z m\u011b\u0159ic\u00edch senzor\u016f z\u00edsk\u00e1me novou rovnici k\u0159ivky, jej\u00ed\u017e rozd\u011blen\u00ed zu\u017euje rozsah hodnot pro vektor po\u017eadovan\u00e9ho stavu:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2831\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/8.2-8.7.png\" alt=\"\" width=\"551\" height=\"485\"><\/p>\n<p><span style=\"font-weight: 400;\">Pomoc\u00ed maticov\u00e9ho z\u00e1pisu z\u00edsk\u00e1me:<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2832\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/8.8.8.9.8.10.png\" alt=\"\" width=\"564\" height=\"132\"><\/p>\n<p>kde K je matice definuj\u00edc\u00ed Kalman\u016fv zes\u00edlen\u00ed.<\/p>\n<h3>Kone\u010dn\u00e9 v\u00fdpo\u010dty<\/h3>\n<p>Na z\u00e1klad\u011b Gaussova rozd\u011blen\u00ed z\u00edsk\u00e1me dv\u011b rozd\u011blen\u00ed. Prvn\u00ed z\u00edsk\u00e1me z p\u0159edpov\u011bzen\u00e9ho m\u011b\u0159en\u00ed (9.1) a druh\u00e9 z pozorovan\u00e9ho m\u011b\u0159en\u00ed. <\/p>\n<p>Ze soustavy rovnic (9.3) ur\u010d\u00edme podm\u00ednky definuj\u00edc\u00ed spole\u010dnou plochu rozd\u011blen\u00ed (9.1) a (9.2).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2833\" src=\"https:\/\/botland.com.pl\/blog\/wp-content\/uploads\/2020\/01\/9.1-9.6.png\" alt=\"\" width=\"562\" height=\"470\"><\/p>\n<p>V z\u00e1vislosti na pot\u0159eb\u00e1ch aplikace lze cel\u00fd algoritmus n\u011bkolikr\u00e1t opakovat, aby se dos\u00e1hlo je\u0161t\u011b p\u0159esn\u011bj\u0161\u00edch v\u00fdsledk\u016f.<\/p>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-605df98 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"605df98\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-3ad5069\" data-id=\"3ad5069\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-cde2c9f elementor-widget elementor-widget-text-editor\" data-id=\"cde2c9f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div id=\"k-faq-accord\"><h2>Kalman\u016fv filtr &#8211; \u010casto kladen\u00e9 ot\u00e1zky<\/h2>\n    <div class=\"tabs\" itemscope=\"\" itemtype=\"https:\/\/schema.org\/FAQPage\">\n        <div class=\"tab\" itemscope=\"\" itemprop=\"mainEntity\" itemtype=\"https:\/\/schema.org\/Question\">\n            <input type=\"checkbox\" id=\"chck1\"><h3>\n                <label class=\"tab-label\" for=\"chck1\" itemprop=\"name\">Kalman\u016fv filtr &#8211; co to je?\n                <\/label><\/h3>\n            <div class=\"tab-content\" itemscope=\"\" itemprop=\"acceptedAnswer\" itemtype=\"https:\/\/schema.org\/Answer\">\n                <div itemprop=\"text\">\n<p>\nCo je Kalman\u016fv filtr? Kalman\u016fv filtr je algoritmus vytvo\u0159en\u00fd Rudolfem Emilem Kalmanem, ma\u010farsko-americk\u00fdm in\u017een\u00fdrem, kter\u00fd \u017eil v letech 1930-2016. Nejd\u016fle\u017eit\u011bj\u0161\u00ed vlastnost\u00ed filtru je jeho univerz\u00e1ln\u00ed pou\u017eitelnost. To znamen\u00e1, \u017ee jej lze \u00fasp\u011b\u0161n\u011b pou\u017e\u00edt ve v\u0161ech aplikac\u00edch, kde existuje riziko n\u00edzk\u00e9 spolehlivosti informac\u00ed. Sn\u00ed\u017een\u00e1 spolehlivost \u00fadaj\u016f se m\u016f\u017ee t\u00fdkat nap\u0159\u00edklad parametr\u016f definuj\u00edc\u00edch dynamick\u00e9 syst\u00e9my. Kalman\u016fv filtr se vyzna\u010duje vysokou rychlost\u00ed cel\u00e9ho algoritmu, co\u017e d\u00e1le roz\u0161i\u0159uje jeho pou\u017eit\u00ed.      \n<\/p>\n                <\/div>\n            <\/div>\n        <\/div>\n        <div class=\"tab\" itemscope=\"\" itemprop=\"mainEntity\" itemtype=\"https:\/\/schema.org\/Question\">\n            <input type=\"checkbox\" id=\"chck2\"><h3>\n                <label class=\"tab-label\" for=\"chck2\" itemprop=\"name\">Kalman\u016fv filtr &#8211; aplikace\n                <\/label><\/h3>\n            <div class=\"tab-content\" itemscope=\"\" itemprop=\"acceptedAnswer\" itemtype=\"https:\/\/schema.org\/Answer\">\n                <div itemprop=\"text\">\n<p>\nKalman\u016fv filtr se pou\u017e\u00edv\u00e1 v r\u016fzn\u00fdch typech dynamick\u00fdch syst\u00e9m\u016f. Zvl\u00e1\u0161tn\u00edm p\u0159\u00edpadem pou\u017eit\u00ed Kalmanova filtru jsou syst\u00e9my, kter\u00e9 pracuj\u00ed s parametry, kter\u00e9 se m\u011bn\u00ed tzv. spojit\u011b. To znamen\u00e1, \u017ee takov\u00fd syst\u00e9m nepot\u0159ebuje ke spr\u00e1vn\u00e9 funkci velk\u00e9 mno\u017estv\u00ed pam\u011bti. Posta\u010d\u00ed stav obvodu, kter\u00fd p\u0159edch\u00e1z\u00ed dal\u0161\u00edmu obvodu (jeho stav). Kalman\u016fv filtr lze \u00fasp\u011b\u0161n\u011b implementovat v <a href=\"https:\/\/botland.cz\/345-roboti\" data-wpel-link=\"internal\">robotech<\/a> vybaven\u00fdch mimo jin\u00e9 naviga\u010dn\u00edm syst\u00e9mem GPS. Dal\u0161\u00ed aplikace zahrnuj\u00ed aplikace re\u00e1ln\u00e9ho \u010dasu i vestav\u011bn\u00e9 syst\u00e9my.       \n<\/p>\n                <\/div>\n            <\/div>\n        <\/div>    \n    <\/div>\n<\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Rozlu\u0161t\u011bn\u00ed h\u00e1danky. Co jsou Kalmanovy filtry? Pozor &#8211; v\u00fdpo\u010dty! Ale bu\u010fte si jisti, \u017ee nejsou tak d\u011bsiv\u00e9, jak jsou vykreslov\u00e1ny.   <\/p>\n","protected":false},"author":7,"featured_media":60282,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[549,563],"tags":[699,653],"class_list":["post-98825","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-kompendium-cs","category-robotika","tag-robotyka-cs","tag-veda-cs"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Kalman\u016fv filtr - teoretick\u00e9 z\u00e1klady - Botland<\/title>\n<meta name=\"description\" content=\"Kalman\u016fv filtr - revolu\u010dn\u00ed metoda ur\u010dov\u00e1n\u00ed stavu automatick\u00fdch \u0159\u00eddic\u00edch syst\u00e9m\u016f. 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