From f545407e26b3860ebb19b39036efe9cf4859fcf9 Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Mon, 3 Oct 2022 20:12:51 +0200 Subject: [PATCH 01/19] probni commit --- content/2022/robot-valjak.md | 26 ++++++++++++++++++++++++++ 1 file changed, 26 insertions(+) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index b9daf65..ae7d161 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -2,3 +2,29 @@ title: Robot valjak summary: Robot u obliku valjka je projekat rađen na letnjem kampu za stare polaznike 2022. godine od Nikole Ristanovića i Filipa Bajraktarija. --- + + +### Uvod + +Uvod treba da sadrži sledeće stvari: + +- Opis i motivaciju projekta, odnosno kako ste došli do ideje i šta ste radili. +- Pregled literature. Ukratko opišite šta su drugi radili pre vas. + + +### Aparatura i metoda + +#### Dinamika sistema + +Kao što je objašnjeno u uvodu, postoje dve vrste sfernih robota koji se razlikuju po načinu konvertovanja električne energije koju motor dobija iz bateriju u mehaničku energiju koja dovodi do kretanja robota. U robotu valjku, 2D verziji sfernog robota, je implementirana druga metoda kojom se pomoću elektromotornog pogona klatno valjka izvodi iz ravnotežnog položaja što dovodi do momenta sile usled tangencijalne komponente gravitacione sile što dovodi do rotacije robota. + +##### Horizontalno kretanje + +Potencijalne, kinetičke i rotacione energije za telo i klatno valjka dati su sledećim izrazima: + +$$U_1 = 0$$ +$$U_2 = -M_2gd\cos(\theta _ 1 + \theta _ 2)$$ +$$K_1 = \frac{1}{2}M_1(r\omega _ 1)^2$$ +$$K_2 = \frac{1}{2}M_2((r\omega _ 1 - d\cos(\theta _ 1 + \theta _ 2)(\omega _ 1 + \omega _ 2))^2 + (d\sin(\theta _ 1 + \theta _ 2)(\omega _ 1 + \omega _ 2))^2)$$ +$$T_1 = \frac{1}{2}J_1(\omega _ 1)^2$$ +$$T_2 = \frac{1}{2}J_1(\omega _ 1 + \omega _ 2)^2$$ \ No newline at end of file From 965128c820a8e72f31b82bbecea73dbe3ce2186d Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Wed, 5 Oct 2022 21:26:26 +0200 Subject: [PATCH 02/19] dodate jednacine za kretanje valjka i zapoceto poglavlje transfer funkcije --- content/2022/robot-valjak.md | 34 ++++++++++++++++++++++++++++++++-- 1 file changed, 32 insertions(+), 2 deletions(-) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index ae7d161..b939498 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -20,11 +20,41 @@ Kao što je objašnjeno u uvodu, postoje dve vrste sfernih robota koji se razlik ##### Horizontalno kretanje -Potencijalne, kinetičke i rotacione energije za telo i klatno valjka dati su sledećim izrazima: +Pre nego što se krene na pravljenje bilo kog hardverskog projekta neophodno je temeljno razumeti dinamiku (fiziku) sistema kako bi mogao da se simulira realni sistem na računaru i na taj način olakša projektovanje i programiranje hardvera. Grana fizike koja se bavi proučavanjem kinematike složenih fizičkih sistema se naziva Ojler-Lagranžova mehanika. Ona nam služi da složene fizičke sisteme predstavimo matematički isključivo pomću energija za dati sistem. Potencijalne, kinetičke i rotacione energije za telo i klatno valjka dati su sledećim izrazima: $$U_1 = 0$$ $$U_2 = -M_2gd\cos(\theta _ 1 + \theta _ 2)$$ $$K_1 = \frac{1}{2}M_1(r\omega _ 1)^2$$ $$K_2 = \frac{1}{2}M_2((r\omega _ 1 - d\cos(\theta _ 1 + \theta _ 2)(\omega _ 1 + \omega _ 2))^2 + (d\sin(\theta _ 1 + \theta _ 2)(\omega _ 1 + \omega _ 2))^2)$$ $$T_1 = \frac{1}{2}J_1(\omega _ 1)^2$$ -$$T_2 = \frac{1}{2}J_1(\omega _ 1 + \omega _ 2)^2$$ \ No newline at end of file +$$T_2 = \frac{1}{2}J_1(\omega _ 1 + \omega _ 2)^2$$ + +Za razliku od klasične Njutnove mehanike za Lagranžovu mehaniku je potrebno definisati odgovarajuću Ojler-Lagranžove funkcije, koja se definiše na sledeći način: +$$L = K_1 + K_2 + T_1 + T_2 - U_1 - U_2$$ +Primenivši Ojler-Lagranžovu jednačinu na dva sistema, klatno i valjak, +$$\frac{d}{dt}\left(\frac{\partial L}{\partial \omega _ 1}\right)\ -\ \frac{\partial L}{\partial \omega _ 1}\ = \ -T + T_f$$ +$$\frac{d}{dt}\left(\frac{\partial L}{\partial \omega _ 2}\right)\ -\ \frac{\partial L}{\partial \omega _ 2}\ = \ T$$ +dobijaju se sledeće jednačine kojima je u potpunosti opisana dinamika sistema: +$$-T + T_f\ =\ \alpha_1(J_1 + J_2 + M_1r^2 + M_2r^2 + M_2d^2 - 2M_2rd)\ +\ \alpha_2(J_2 - M_2rd + M_2d^2)\ +\ M_2gd(\theta_1 + \theta_2)$$ +$$T\ =\ \alpha_1(J_1 - M_2rd + M_2d^) + \alpha_2(J_2 + M2d^2) + M_2gd(\theta_1 + \theta_2)$$ +gde su iskorišćene aproksimacije za male uglove i male ugaone brzine. + +#### Simulacije + +Simulacije služe da se na što realističniji način predstavi realni svet i dobije bolji uvid u to kako će se sistem ponašati. Pored osećaja za sistem, simulacije omogućavaju lakše testiranje softvera kao i testiranje sistema u različitim prirodnim uslovima. + +##### Transfer funkcije + +Transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaznog signala, za sve moguće ulazne vrednosti. Za bilo koji kontrolni sistem postoji referentni ulaz ili pobuda koji prolazi kroz transfer funkciju kako bi se proizveo odgovor sistema na odgovarajuću pobudu. + +Transfer funkcija (funkcija prenosa) je definisana kao Laplasova transformacija izlazne promenjljive i Laplasove transformacije ulazne promenjljive pod pretpostavkom da su svi početni uslovi jednaki nuli. +$$G(s)\ =\ \frac{C(s)}{R(s)}$$ +Način na koji se formira transfer funkcija kontrolnog sistema je sledeća: +1. Formiraju se jednačine sistema +2. Uzimaju se Laplasove transformacije jednačina, pretpostavljajući da su početni uslovi jednaki nuli +3. Određuju se ulazi i izlazi sistema +4. Na kraju se uzima odnos Laplasove transformacije izlaza i ulaza što je tražena transfer funkcija + +Takođe, važno je naglasiti da izlaz i ulaz nekog sistema ne mora nužno da bude istih fizičkih dimenzija. + +U radu su korišćene tri transfer funkcije koje se uz data pravila mogu izvesti iz fizičkih jednačina za horizontalno kretanje valjka. To su transfer funkcija ugla inklinacije klatna valjka i napona na motoru, transfer funkcija ugaone brzine klatna valjka i napona na motoru i, na kraju, ugaona brzina valjka i napona motora. \ No newline at end of file From 9ff405344c77a50ba351e82a3c2cc56dc8585342 Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Wed, 5 Oct 2022 21:57:24 +0200 Subject: [PATCH 03/19] zapocet PID --- content/2022/robot-valjak.md | 7 ++++++- 1 file changed, 6 insertions(+), 1 deletion(-) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index b939498..93109c3 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -50,6 +50,7 @@ Transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaz Transfer funkcija (funkcija prenosa) je definisana kao Laplasova transformacija izlazne promenjljive i Laplasove transformacije ulazne promenjljive pod pretpostavkom da su svi početni uslovi jednaki nuli. $$G(s)\ =\ \frac{C(s)}{R(s)}$$ Način na koji se formira transfer funkcija kontrolnog sistema je sledeća: + 1. Formiraju se jednačine sistema 2. Uzimaju se Laplasove transformacije jednačina, pretpostavljajući da su početni uslovi jednaki nuli 3. Određuju se ulazi i izlazi sistema @@ -57,4 +58,8 @@ Način na koji se formira transfer funkcija kontrolnog sistema je sledeća: Takođe, važno je naglasiti da izlaz i ulaz nekog sistema ne mora nužno da bude istih fizičkih dimenzija. -U radu su korišćene tri transfer funkcije koje se uz data pravila mogu izvesti iz fizičkih jednačina za horizontalno kretanje valjka. To su transfer funkcija ugla inklinacije klatna valjka i napona na motoru, transfer funkcija ugaone brzine klatna valjka i napona na motoru i, na kraju, ugaona brzina valjka i napona motora. \ No newline at end of file +U radu su korišćene tri transfer funkcije koje se uz data pravila mogu izvesti iz fizičkih jednačina za horizontalno kretanje valjka. To su transfer funkcija ugla inklinacije klatna valjka i napona na motoru, transfer funkcija ugaone brzine klatna valjka i napona na motoru i, na kraju, ugaona brzina valjka i napona motora. + +##### PID simulacije + +Kako bi se razumeo PID kontroler potrebno je definisati odredjene termine. *Željena vrednost* je obično vrednost koju unosi korisnik. Na primer, za sistem za grejanje to bi bila željena temperatura prostorije. *Izlaz* je kontrolisana vrednost PID kontrolera. U automobilu bi to bila količina goriva koja je potrebna da bi se održavala konstantna brzina. *Greška* je vrednost koju koristi PID kontroler da bi odredio kako da manipuliše izlazom da bi se dostigla željena vrednost. Greška se računa kao razlika između željene vrednosti i trenutne vrednosti sistema. \ No newline at end of file From 739866b919ec02a0b08b2c8d80f9fd2a14ccd50b Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Fri, 7 Oct 2022 16:36:18 +0200 Subject: [PATCH 04/19] little changes --- content/2022/robot-valjak.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index 93109c3..8a8b72e 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -62,4 +62,4 @@ U radu su korišćene tri transfer funkcije koje se uz data pravila mogu izvesti ##### PID simulacije -Kako bi se razumeo PID kontroler potrebno je definisati odredjene termine. *Željena vrednost* je obično vrednost koju unosi korisnik. Na primer, za sistem za grejanje to bi bila željena temperatura prostorije. *Izlaz* je kontrolisana vrednost PID kontrolera. U automobilu bi to bila količina goriva koja je potrebna da bi se održavala konstantna brzina. *Greška* je vrednost koju koristi PID kontroler da bi odredio kako da manipuliše izlazom da bi se dostigla željena vrednost. Greška se računa kao razlika između željene vrednosti i trenutne vrednosti sistema. \ No newline at end of file +Kako bi se razumeo PID kontroler potrebno je definisati odredjene termine. *Željena vrednost* je obično vrednost koju unosi korisnik. Na primer, za sistem za grejanje to bi bila željena temperatura prostorije. *Izlaz* je kontrolisana vrednost PID kontrolera. U automobilu bi to bila količina goriva koja je potrebna da bi se održavala konstantna brzina. *Greška* je vrednost koju koristi PID kontroler da bi odredio kako da manipuliše izlazom kako bi se dostigla željena vrednost. Greška se računa kao razlika željene vrednosti i trenutne vrednosti sistema. \ No newline at end of file From e09ff8658a8c61450be4308d2646274e7b85858d Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Sat, 8 Oct 2022 13:34:17 +0200 Subject: [PATCH 05/19] PID --- content/2022/robot-valjak.md | 10 +++++++++- 1 file changed, 9 insertions(+), 1 deletion(-) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index 8a8b72e..1aebf5a 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -60,6 +60,14 @@ Takođe, važno je naglasiti da izlaz i ulaz nekog sistema ne mora nužno da bud U radu su korišćene tri transfer funkcije koje se uz data pravila mogu izvesti iz fizičkih jednačina za horizontalno kretanje valjka. To su transfer funkcija ugla inklinacije klatna valjka i napona na motoru, transfer funkcija ugaone brzine klatna valjka i napona na motoru i, na kraju, ugaona brzina valjka i napona motora. +TODO: Analiza stabilnosti transfer funkcija + ##### PID simulacije -Kako bi se razumeo PID kontroler potrebno je definisati odredjene termine. *Željena vrednost* je obično vrednost koju unosi korisnik. Na primer, za sistem za grejanje to bi bila željena temperatura prostorije. *Izlaz* je kontrolisana vrednost PID kontrolera. U automobilu bi to bila količina goriva koja je potrebna da bi se održavala konstantna brzina. *Greška* je vrednost koju koristi PID kontroler da bi odredio kako da manipuliše izlazom kako bi se dostigla željena vrednost. Greška se računa kao razlika željene vrednosti i trenutne vrednosti sistema. \ No newline at end of file +Kako bi se razumeo PID kontroler potrebno je definisati odredjene termine. *Željena vrednost* je obično vrednost koju unosi korisnik. Na primer, za sistem za grejanje to bi bila željena temperatura prostorije. *Izlaz* je kontrolisana vrednost PID kontrolera. U automobilu bi to bila količina goriva koja je potrebna da bi se održavala konstantna brzina. *Greška* je vrednost koju koristi PID kontroler da bi odredio kako da manipuliše izlazom kako bi se dostigla željena vrednost. Greška se računa kao razlika željene vrednosti i trenutne vrednosti sistema. + +U radu su simulirana tri PID kontrolera. Prvi PID kontroler odžava konstantan ugao inklinacije klatna u odnosu na horizontalnu podlogu po kojoj se valjak kreće. Drugi PID kontroler održava konstantnu ugaonu brzinu valjka. Poslednji PID kontroler održava konstantan ugaoni pređeni put valjka. Blok dijagram kontrolnog sistema je predstavljen na slici: + +Kao što je u prethodnom poglavlju rečeno, transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaznog signala, za sve moguće ulazne vrednosti. Ukoliko je transfer funkcija predstavljena preko Laplasovih transformacija onda je takva funkcija prenosa definisana na kontinualnom domenu. Takav oblik transfer funkcija je od koristi prilikom teorijskih analiza kao što je analiza polova i nula. Nazalost, u praksi nam treba diskretna reprezentacija sistema što proizilazi iz toga da mikrokontroler rade isključivo nad diskretnim domenom vrednosti. Postoje razni algoritmi koji vrše željenu konverziju. Jedan od njih, koji je i koriščen u radu, *Zero-order Hold* metoda + +## Istraživanje i rezultati \ No newline at end of file From 826e613c6711ea3e6c32f857ac4a26f168622425 Mon Sep 17 00:00:00 2001 From: NikolaRistanovic Date: Wed, 12 Oct 2022 14:01:21 +0200 Subject: [PATCH 06/19] Dodat uvod. --- content/2022/robot-valjak.md | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index 1aebf5a..3434271 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -8,8 +8,12 @@ summary: Robot u obliku valjka je projekat rađen na letnjem kampu za stare pola Uvod treba da sadrži sledeće stvari: -- Opis i motivaciju projekta, odnosno kako ste došli do ideje i šta ste radili. -- Pregled literature. Ukratko opišite šta su drugi radili pre vas. +Motivacija iza projekta je bila suprotstavljane ideje popularnog robota - balansera. Ideja je bila da pomoću klatna koje uvek teži da se vrati u stabilan položaj +robot može biti konstruisan sa manjom preciznošću što bi ubrzalo proces pravljenja u slučaju masovne proizvodenje, kao i vreme dolaska do prvog funkcionalnog prototipa, budući da robot može da radi uz bilo kakav (ne)kontrolisan pogon motora. Takođe zbog oblika robota sve vitalne komponente se sadržane unutar konture, čineći robota izdržljivijim i jednostavnijim za popravku. + +Reference su skoro ne postojeće budući da se ideja o robotu dobila bez drugog uzora/slične ideje. + +???kako da formulišem da su refernece skoro ne postojeće i da sam sam došao na ideju robota??? ### Aparatura i metoda From 38becf3780b21e573f0e66046772b344ea2724a4 Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Sat, 15 Oct 2022 19:06:01 +0200 Subject: [PATCH 07/19] transfer funkcije v2 --- content/2022/robot-valjak.md | 34 ++++++++++++++++++++++++++++------ 1 file changed, 28 insertions(+), 6 deletions(-) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index 3434271..8eae4dd 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -43,13 +43,13 @@ $$-T + T_f\ =\ \alpha_1(J_1 + J_2 + M_1r^2 + M_2r^2 + M_2d^2 - 2M_2rd)\ +\ \alph $$T\ =\ \alpha_1(J_1 - M_2rd + M_2d^) + \alpha_2(J_2 + M2d^2) + M_2gd(\theta_1 + \theta_2)$$ gde su iskorišćene aproksimacije za male uglove i male ugaone brzine. -#### Simulacije +#### Modelovanje robota Simulacije služe da se na što realističniji način predstavi realni svet i dobije bolji uvid u to kako će se sistem ponašati. Pored osećaja za sistem, simulacije omogućavaju lakše testiranje softvera kao i testiranje sistema u različitim prirodnim uslovima. ##### Transfer funkcije -Transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaznog signala, za sve moguće ulazne vrednosti. Za bilo koji kontrolni sistem postoji referentni ulaz ili pobuda koji prolazi kroz transfer funkciju kako bi se proizveo odgovor sistema na odgovarajuću pobudu. +Transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaznog signala, za sve moguće ulazne vrednosti. Za bilo koji kontrolni sistem postoji referentni ulaz ili pobuda koji prolazi kroz transfer funkciju kako bi se proizveo (simulirao) odgovor sistema na odgovarajuću pobudu. Transfer funkcija (funkcija prenosa) je definisana kao Laplasova transformacija izlazne promenjljive i Laplasove transformacije ulazne promenjljive pod pretpostavkom da su svi početni uslovi jednaki nuli. $$G(s)\ =\ \frac{C(s)}{R(s)}$$ @@ -58,13 +58,35 @@ Način na koji se formira transfer funkcija kontrolnog sistema je sledeća: 1. Formiraju se jednačine sistema 2. Uzimaju se Laplasove transformacije jednačina, pretpostavljajući da su početni uslovi jednaki nuli 3. Određuju se ulazi i izlazi sistema -4. Na kraju se uzima odnos Laplasove transformacije izlaza i ulaza što je tražena transfer funkcija +4. Na kraju, odgovrajući odnos Laplasove transformacije izlaza i ulaza predstavlja željenu transfer funkciju -Takođe, važno je naglasiti da izlaz i ulaz nekog sistema ne mora nužno da bude istih fizičkih dimenzija. +Takođe, važno je naglasiti da izlaz i ulaz nekog sistema ne mora nužno da bude istih fizičkih dimenzija. Na primer, DC motori se pokreću kada se na njihov ulaz dovede konstantan napon, a kao izlaz sistema može da se prati broj obrtaja motora ili obrtni momenat motora. Merne jedinice za date primere bi redom bile (obrtaja po sekundi)/(napon motora) i (Njutn metar)/(napon motora). -U radu su korišćene tri transfer funkcije koje se uz data pravila mogu izvesti iz fizičkih jednačina za horizontalno kretanje valjka. To su transfer funkcija ugla inklinacije klatna valjka i napona na motoru, transfer funkcija ugaone brzine klatna valjka i napona na motoru i, na kraju, ugaona brzina valjka i napona motora. +TODO: Grafici, polovi i osnovne aritmetičke operacije nad tf -TODO: Analiza stabilnosti transfer funkcija +U radu se kao pobuda sistema koristio isključivo napon. To bi značilo da je smanjenje ili povećanje ugaone brzine robota prouzrokovano smanjenju ili povećanju referentnog napona na ulazu motora. Kako bi se znalo na koji način treba implementirati odogovarajuće kontrolere potrebno je analizirati stabilnost odgovarajućih transfer funkcija karakterističnih za datog robota, gde kontroler predstavlja sistem koji nezavisno može da održava unete referentne vrednosti zadate od strane korisnika. Na primer, ukoliko bi korisnik želeo da se robot kreće konstantnom ugaonom brzinom $\omega$, kontroler treba samostalno da podešava napon na motoru tkd se ugaona brzina $\omega$ održava konstantnom. + +Prva transfer funkcija koja je analizirana u radu je transfer funkcija ugaone brzine robota u zavisnosti od ulaznog napona. + +$$\dfrac{\Omega_1}{V} = \dfrac{\ \ \ \ \ \ \ \ \ \ A + Bs^2}{C + Ds + Es^2 + Fs^3}$$ +$A = -2dgK_tM_2$ +$B = K_t(-2J_2 + d(-2d+r)M_2)$ +$C = dgRT_vM_2 - 2dgK_eK_tM_2$ +$D = -T_vK_eK_t + dgRJ_1M_2 + dgr^2RM_1M_2 + dgr^2RM_2^2$ +$E = RT_vJ_2 - 2J_2K_eK_t + d^2RT_vM_2 - K_eK_t(J_1 + r^2M_1 + (2d^2 - 3dr + r^2M_2)$ +$F = d^2RJ_1M_2 + d^2r^2RM_1M_2 + RJ_2(J_1 + r^2(M_1 + M_2))$ + +Druga transfer funkcija koja je analizirana u radu je transfer funkcija ugaone brzine klatna u odnosu na osovinu valjka u zavisnosti od ulaznog napona. + +$$\dfrac{\Omega_2}{V} = \dfrac{\ \ \ \ \ \ G + Fs + Es^2}{A + Bs + Cs^2 + Ds^3}$$ +$A = dgRT_vM_2 - 2dgK_eK_tM_2$ +$B = -T_vK_eK_t + dgRJ_1M_2 + dgr^2RM_1M_2 + dgr^2RM_2^2$ +$C = RT_vJ_2 + d^2RT_vM_2 - K_eK_t(J_1 + 2J_2 + r^2M_1 + (2dd - 3dr + r^2)M_2)$ +$D = RJ_1J_2 + d^2RJ_1M_2 + d^2r^2RM_1M_2 + r^2RJ_2(M_1 + M_2)$ +$E = K_t(J_1 + 2J_2 + r^2M_1 + (2d^2 - 3dr + r^2)M_2)$ +$F = K_tT_v$ + +Koristeći prethodno dve definisane funkcije mogu se izvesti dve nove transfer funkcije za koje važi da se njihovim diferenciranjem dobijaju polazne transfer funkcije: transfer funkcija ugaonog pređenog puta robota u zavisnosti od napona na ulazu motora i transfer funkcija ugla klatna u odnosu na osovinu valjka u zavisnosti od ulaznog napona. Treća, i poslednja, transfer funkcija predstavlja zavisnost ugla inklinacije klatna u odnosu na podlogu i dobija se kao paralelna veza prethodne dve transfer funkcije. ##### PID simulacije From 2a0001626a12215da637cb7412090a44617a5635 Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Sun, 16 Oct 2022 00:11:48 +0200 Subject: [PATCH 08/19] istrazivanje i rezultati --- content/2022/robot-valjak.md | 10 +- .../predjeni_put_valjka_od_napona.svg | 791 ++++++++++++++++++ 2 files changed, 799 insertions(+), 2 deletions(-) create mode 100644 static/images/2022/robot-valjak/predjeni_put_valjka_od_napona.svg diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index 8eae4dd..3977d3a 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -86,7 +86,7 @@ $D = RJ_1J_2 + d^2RJ_1M_2 + d^2r^2RM_1M_2 + r^2RJ_2(M_1 + M_2)$ $E = K_t(J_1 + 2J_2 + r^2M_1 + (2d^2 - 3dr + r^2)M_2)$ $F = K_tT_v$ -Koristeći prethodno dve definisane funkcije mogu se izvesti dve nove transfer funkcije za koje važi da se njihovim diferenciranjem dobijaju polazne transfer funkcije: transfer funkcija ugaonog pređenog puta robota u zavisnosti od napona na ulazu motora i transfer funkcija ugla klatna u odnosu na osovinu valjka u zavisnosti od ulaznog napona. Treća, i poslednja, transfer funkcija predstavlja zavisnost ugla inklinacije klatna u odnosu na podlogu i dobija se kao paralelna veza prethodne dve transfer funkcije. +Koristeći prethodno dve definisane funkcije mogu se izvesti dve nove transfer funkcije za koje važi da se njihovim diferenciranjem dobijaju polazne transfer funkcije: transfer funkcija ugaonog pređenog puta robota u zavisnosti od napona na ulazu motora i transfer funkcija ugla klatna u odnosu na osovinu valjka u zavisnosti od ulaznog napona. Treća, i poslednja, transfer funkcija predstavlja zavisnost ugla inklinacije klatna u odnosu na podlogu u zavisnosti od napona motora i dobija se kao paralelna veza prethodne dve transfer funkcije. ##### PID simulacije @@ -96,4 +96,10 @@ U radu su simulirana tri PID kontrolera. Prvi PID kontroler odžava konstantan u Kao što je u prethodnom poglavlju rečeno, transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaznog signala, za sve moguće ulazne vrednosti. Ukoliko je transfer funkcija predstavljena preko Laplasovih transformacija onda je takva funkcija prenosa definisana na kontinualnom domenu. Takav oblik transfer funkcija je od koristi prilikom teorijskih analiza kao što je analiza polova i nula. Nazalost, u praksi nam treba diskretna reprezentacija sistema što proizilazi iz toga da mikrokontroler rade isključivo nad diskretnim domenom vrednosti. Postoje razni algoritmi koji vrše željenu konverziju. Jedan od njih, koji je i koriščen u radu, *Zero-order Hold* metoda -## Istraživanje i rezultati \ No newline at end of file +## Istraživanje i rezultati + +Tokom rada na projektu, uspešno je realizovana hardverska implementacija robota i prikupljeni su podaci o tome kako sistem reaguje na uzastopne odskočne odzive. Tokom istraživanja su izmerena tri odskočna odziva koja odgovaraju redom transfer funkcijama ugaone brzine robota u zavisnosti od ulaznog napona motora, ugaone brzine klatna u odnosu na osovinu valjka u zavisnosti od ulaznog napona motora i ugla inklinacije klatna u odnosu na podlogu u zavisnosti od napona motora. + +Prilikom dizajniranja klatna valjka centar mase korpe i komponenti (baterije, arduino i ostali moduli) se ne nalazi u osnosimetričnoj ravni korpe već je blago izbačen sa strane što dovodi do toga da nula u merenju inklinacije ugla nije prava nula. Takođe, ovome je potpomogla i činjenica da se na DC motoru nalazi reducer zbog kog ne postoji prava nula već postoji interval uglova koji mogu biti prava nula. Za male uglove moment zemljine teže na klatno se izjednačava sa momentom sile koju reducer proizvodi na klatno. Drugim rečima, javalja se mali interval oko prave nule za koje se klatno nalazi u indiferentnoj ravnoteži i u kojoj krajnji položaj zavisi isključivo od početnog impulsa. Ukoliko je veći početni impuls veća je i greška merenja ugla inklinacije. Ovaj fenomen se može predstaviti na grafiku ispod gde se vidi da robot konstantno *driftuje* u jednu stranu. Za dalji rad na ovom robotu treba da se poravna centar mase sistema kao i da se poveća masa klatna kako bi se smanji opseg uglova koji mogu biti prava nula i na taj način postići preciznija merenja. + +Druga dva merenja predstavljaju kako u realnom sistemu napon na motoru utiče na ugaonu brzinu klatna i ugao inklinacije klatna. Takođe, jedan od ciljeva ovog projekta je bila i numerička identifikacija sistema koja nažalost nije urađena zbog gore navedenih problema kao i nedostatka vremena, ali idalje ostaje kao mogućnost za neki budući rad na sličnu temu. \ No newline at end of file diff --git a/static/images/2022/robot-valjak/predjeni_put_valjka_od_napona.svg b/static/images/2022/robot-valjak/predjeni_put_valjka_od_napona.svg new file mode 100644 index 0000000..742dacf --- /dev/null +++ b/static/images/2022/robot-valjak/predjeni_put_valjka_od_napona.svg @@ -0,0 +1,791 @@ + + + + + + + + 2022-10-15T21:35:32.701745 + image/svg+xml + + + Matplotlib v3.5.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + From 31ec61bce877e3ac4b2868881856a139bd76a2d6 Mon Sep 17 00:00:00 2001 From: NikolaRistanovic Date: Sun, 16 Oct 2022 00:53:45 +0200 Subject: [PATCH 09/19] Dodavanje teksta za hardver Ispravke sitnih greski --- content/2022/robot-valjak.md | 21 +++++++++++++++++++-- 1 file changed, 19 insertions(+), 2 deletions(-) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index 8eae4dd..4d5adbb 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -6,8 +6,6 @@ summary: Robot u obliku valjka je projekat rađen na letnjem kampu za stare pola ### Uvod -Uvod treba da sadrži sledeće stvari: - Motivacija iza projekta je bila suprotstavljane ideje popularnog robota - balansera. Ideja je bila da pomoću klatna koje uvek teži da se vrati u stabilan položaj robot može biti konstruisan sa manjom preciznošću što bi ubrzalo proces pravljenja u slučaju masovne proizvodenje, kao i vreme dolaska do prvog funkcionalnog prototipa, budući da robot može da radi uz bilo kakav (ne)kontrolisan pogon motora. Takođe zbog oblika robota sve vitalne komponente se sadržane unutar konture, čineći robota izdržljivijim i jednostavnijim za popravku. @@ -96,4 +94,23 @@ U radu su simulirana tri PID kontrolera. Prvi PID kontroler odžava konstantan u Kao što je u prethodnom poglavlju rečeno, transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaznog signala, za sve moguće ulazne vrednosti. Ukoliko je transfer funkcija predstavljena preko Laplasovih transformacija onda je takva funkcija prenosa definisana na kontinualnom domenu. Takav oblik transfer funkcija je od koristi prilikom teorijskih analiza kao što je analiza polova i nula. Nazalost, u praksi nam treba diskretna reprezentacija sistema što proizilazi iz toga da mikrokontroler rade isključivo nad diskretnim domenom vrednosti. Postoje razni algoritmi koji vrše željenu konverziju. Jedan od njih, koji je i koriščen u radu, *Zero-order Hold* metoda +### Hardver + +Pod hardverom se podrazumevaju svi vidljivi delovi robota, poput spojeva, mikrokontrolera, senzora, elektromehanickih uredjaja(motora), napajanja i slicno. + +#### Lista uredjaja + +1. MPU6050 IMU(Inertial Mesurement Unit) +2. Arduino Nano +3. HC05 Bluetooth modul +4. L298N Dual H-Most za kontrolu DC Motora +5. LM2598 DC-DC Stepdown regulator napona +6. JGA25-370 DC motor +7. 4S1P Li-ion baterija +8. E38S6G5 Opticki inkrementalni enkoder + + + + + ## Istraživanje i rezultati \ No newline at end of file From dfecfb6c531e695bafadba2661afab64d5b175fd Mon Sep 17 00:00:00 2001 From: NikolaRistanovic Date: Sun, 16 Oct 2022 02:53:52 +0200 Subject: [PATCH 10/19] Dodat tekst vezan za harver i pojedinacne delove hardvera --- content/2022/robot-valjak.md | 29 +++++++++++++++++++++++++++++ 1 file changed, 29 insertions(+) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index 802ef80..c4be646 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -97,6 +97,7 @@ Kao što je u prethodnom poglavlju rečeno, transfer funkcija predstavlja odnos ### Hardver Pod hardverom se podrazumevaju svi vidljivi delovi robota, poput spojeva, mikrokontrolera, senzora, elektromehanickih uredjaja(motora), napajanja i slicno. +*Sema za povezivanje pojedincanih komponenti je prilozena na kraju dokumenta* #### Lista uredjaja @@ -109,6 +110,34 @@ Pod hardverom se podrazumevaju svi vidljivi delovi robota, poput spojeva, mikrok 7. 4S1P Li-ion baterija 8. E38S6G5 Opticki inkrementalni enkoder +#### Princip rada + +##### Arduino Nano i HC05 + +Uredjaj koji vrsi prikupljanje i obradu podataka i signala sa senzora, kao i generisanjem signala za upravljane motorom je Arduino Nano. Ovo je 16-bitni mikrokontroler koji na sebi poseduje Atmega328p cip. Ovaj mikrokontoler sa HC05 Bluetooth modulom komunicira pomocu UART-a na BAUD rate-u od 9600 bitova po sekundi, omogucavajuci bezicni prenos podataka. Napaja se pomocu napona od 5 volti sa mikrokontrolera. *Paznja! iako modul radi na 5V prihvata napon na signalnim pinovima (RX,TX) je ogranicen na 3.3V, tako da je obavezna upotreba naponskog razdelnika kao bi se napon signala sa mikrokontrolera(5V) spustio na odgovarajuci nivo* + +##### MPU6050 + +MPU6050 je akcelerometar i ziroskop koji se koristi za merenje ugla inklinacije tega robota. MPU6050 je sa mikrokontrolerom povezan pomocu I2C protokola i napaja se pomocu 5 volti sa mikrokontrolera. Ovaj senzor mikrokontroleru dostavlja podatke sa ziroskopa i akcelerometra pomocu kojih se u softveru izracunava ugao inklinacije koji je neophodan za odredjivanje ugaonog ubrzanja robota i PID kontrolu. + +##### Li-ion baterija i LM2598 Stepdown regulator + +Napajanje koje se u projektu koristi su 4 redno vezane Samsung 18650-35E Litijum-jonske baterije zbog svog kapaciteta od 5000mAh (miliamper casova) i maksimalne struje 2A (Ampera) koje mogu propustiti. Baterije se pune eksternim punjacem pomocu balansirajuceg konektora na njima. Napon sa baterije se sprovodi do LM2598 regulatora gde se napon spusta na 12V kako bi dalje mogao da se dovede do H-mosta, buduci da je u trenutnoj konfiguraciji njegov napon ogranicen na maksimum od 12V. Potom se struja iz H-mosta pomocu njegovog integrisanog regulatora spusta na 5V i dovodi do mikrokontrolera i ostalih senzora. Napon baterije se meri na mikrokontroleru kroz naponski razdelnik kako bi se napon od 16.8V spustio na 5V koji su bezbedni za mikrokontroler koji proverava trenutan napon/preostali kapacitet baterije. +*Procenjena maksimalna snaga potrosena u robotu je $12V * ~2A(Maksimalna utrosena struja celog robota) = 24W$ nakon regulatora, imajuci ovo u vidu struja koja protice kroz bateriju se moze izracunati kao $I = P/V$ gde je I - struja kroz bateriju, P - Utrosena snaga i V - napon baterije, gde je I = 24W / 16.8V(Nominalni napon baterije) = ~1.42A gde je onda minimalno vreme rada robota 5000mAh / 1.42A = ~3.5h +/- 10%(zbog gresaka u merenju, efikasnosti regulatora...)* + +##### DC motor i L298N H-most + +Buduci da pinovi mikrokontrlera nemaju snagu da pokrenu motor, kao pokretac motora se postavlja H-most koji pomocu signala koje dobija od strane mikrokontrolera pokrece motor. Dva signala koje H-most dobija su digitalni signali : + +1. Koji svojim naponom od 0V ili 5V odredjuje smer i +2. Koji koristeci PWM(Pulse Width Modulation) brzim paljenjem i gasenjem kontrolise napon koji je doveden do motora od 0-12V + +JGA25370 je DC motor koji radi na naponu od V=12V i Maksimalnoj struji od I=2A (U stanju zakocenja osovine). Motor u sebi poseduje reduktor koji umanjuje broj obrtaja motora u sekundi ali povecava obrtni momenat gde je broj obrtaja $RPM_max = 250$ obrtaja u minutu i obrtni momenat $T = 1.4kg/cm$. Ovaj motor je jedina komponenta koja pokrece robota. + +##### E38S6G5 Opticki inkrementalni enkoder + +E38S6G5 je opticki enkoder koji se koristi za merenje predjenog puta robota. E38S6G5 ima rezoluciju od 600 pulseva po obrtaju osovine sto nam daje preciznost od 0.6 stepeni ili ~0.01rad. Princip rada optickog enkodera je da se analizom izlaznih talasa moz odrediti da li se enkoder okrenuo u pozitivnom ili negativnom smeru. U konkretnom slucaju na pin 3 mikrokontrolera je povezan 1. od 2 izlaza enkodera, prilikom svake rastuce ivice ovog pina mikrokontroler belezi napon na 2. izlazu enkodera, u slucaju da je on pozitivan mikrokontroler to belezi kao obrt u pozitivnom smeru, a u slucaju da je napon negativam mikrokontroler to belezi kao obtr u negativnom smeru. Kada za svaki ovaj dogadjaj od nekog brojaca saberemo ili oduzmemo 1 mozemo dobiti predjeni put. *Ovo nije realni predjeni put buduci da se ne uracunava veoma potencijalni zanos robota u slucaju da stoji u mestu, u softveru se ovo resava oduzimanjem ugla inklinacije od predjenog puta.* + From e2c595bf58960590b2def948abcfad6f28f5d693 Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Sun, 16 Oct 2022 14:58:56 +0200 Subject: [PATCH 11/19] merging --- content/2022/robot-valjak.md | 20 ++++++++++++++------ 1 file changed, 14 insertions(+), 6 deletions(-) diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index c4be646..e4a2cf5 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -18,11 +18,13 @@ Reference su skoro ne postojeće budući da se ideja o robotu dobila bez drugog #### Dinamika sistema -Kao što je objašnjeno u uvodu, postoje dve vrste sfernih robota koji se razlikuju po načinu konvertovanja električne energije koju motor dobija iz bateriju u mehaničku energiju koja dovodi do kretanja robota. U robotu valjku, 2D verziji sfernog robota, je implementirana druga metoda kojom se pomoću elektromotornog pogona klatno valjka izvodi iz ravnotežnog položaja što dovodi do momenta sile usled tangencijalne komponente gravitacione sile što dovodi do rotacije robota. +Postoje dve vrste sfernih robota koji se razlikuju po načinu konvertovanja električne energije koju motor dobija iz baterije u mehaničku energiju koja dovodi do kretanja robota[^1]. Prva vrsta robota su sferni roboti sa Unutrašnjim Pogonskim Sistemom (*eng. Inside Driving Unit*). Takvi roboti u sebi sadže uglavnom pogonske točkove koji zahvaljujući sili trenja sa unutrašnje strane proizvode moment sile, ili spreg sila ukoliko postoje više takvih pogonskih točkova, koja dovodi do rotacije cele konture robota. S druge strane, postoji druga vrsta sfernih robota koji konvertuju potencijalnu energiju gravitacije u kinetičku energiju rotacije. Takvi roboti izvode njihov centar mase iz ravnotežnog položaja usled čega se javlja moment sile gravitacije na klatno koja teži da *klatno* vrati u prvobitan položaj. Zbog nedovoljno jake sile trenja između robota i podloge robot počinje da se kreće. U robotu valjku, 2D verziji sfernog robota, je implementirana druga metoda kojom se pomoću elektromotornog pogona klatno valjka, izvodi iz ravnotežnog položaja. U slučaju robota valjka klatno predstavlja skup komponenti (mikrokontroleri, baterije i ostala elektronika) koji se nalazi ispod ose simetrije valjka. + +[^1]: Control System for a Spherical Robot, Masaki Nagai ##### Horizontalno kretanje -Pre nego što se krene na pravljenje bilo kog hardverskog projekta neophodno je temeljno razumeti dinamiku (fiziku) sistema kako bi mogao da se simulira realni sistem na računaru i na taj način olakša projektovanje i programiranje hardvera. Grana fizike koja se bavi proučavanjem kinematike složenih fizičkih sistema se naziva Ojler-Lagranžova mehanika. Ona nam služi da složene fizičke sisteme predstavimo matematički isključivo pomću energija za dati sistem. Potencijalne, kinetičke i rotacione energije za telo i klatno valjka dati su sledećim izrazima: +Pre nego što se krene na pravljenje bilo kog hardverskog projekta neophodno je temeljno razumeti mehaniku sistema kako bi mogao da se simulira realni sistem na računaru i na taj način olakša projektovanje i programiranje hardvera. Grana fizike koja se bavi proučavanjem mehanike složenih fizičkih sistema se naziva Lagranžova mehanika. Ona nam služi da složene fizičke sisteme predstavimo matematički isključivo pomću energija za dati sistem. Potencijalne, kinetičke i rotacione kinetičke energije za telo i klatno valjka su date u sledećim izrazima: $$U_1 = 0$$ $$U_2 = -M_2gd\cos(\theta _ 1 + \theta _ 2)$$ @@ -31,15 +33,21 @@ $$K_2 = \frac{1}{2}M_2((r\omega _ 1 - d\cos(\theta _ 1 + \theta _ 2)(\omega _ 1 $$T_1 = \frac{1}{2}J_1(\omega _ 1)^2$$ $$T_2 = \frac{1}{2}J_1(\omega _ 1 + \omega _ 2)^2$$ -Za razliku od klasične Njutnove mehanike za Lagranžovu mehaniku je potrebno definisati odgovarajuću Ojler-Lagranžove funkcije, koja se definiše na sledeći način: +Za razliku od klasične Njutnove mehanike za Lagranžovu mehaniku je potrebno definisati odgovarajući Lagranžian, koji se definiše na sledeći način: $$L = K_1 + K_2 + T_1 + T_2 - U_1 - U_2$$ -Primenivši Ojler-Lagranžovu jednačinu na dva sistema, klatno i valjak, +Iskoristivši Ojler-Lagranžovu jednačinu na dva sistema, klatno i valjak, $$\frac{d}{dt}\left(\frac{\partial L}{\partial \omega _ 1}\right)\ -\ \frac{\partial L}{\partial \omega _ 1}\ = \ -T + T_f$$ $$\frac{d}{dt}\left(\frac{\partial L}{\partial \omega _ 2}\right)\ -\ \frac{\partial L}{\partial \omega _ 2}\ = \ T$$ dobijaju se sledeće jednačine kojima je u potpunosti opisana dinamika sistema: $$-T + T_f\ =\ \alpha_1(J_1 + J_2 + M_1r^2 + M_2r^2 + M_2d^2 - 2M_2rd)\ +\ \alpha_2(J_2 - M_2rd + M_2d^2)\ +\ M_2gd(\theta_1 + \theta_2)$$ $$T\ =\ \alpha_1(J_1 - M_2rd + M_2d^) + \alpha_2(J_2 + M2d^2) + M_2gd(\theta_1 + \theta_2)$$ -gde su iskorišćene aproksimacije za male uglove i male ugaone brzine. +gde su iskorišćene aproksimacije za male uglove i male ugaone brzine. Ukoliko važi +$$\theta_1 + \theta_2 \ll 1 \ \ ,\ \ \omega_1 + \omega_2 \ll 1$$ +mogu se primeniti sledeće aproksimacije: +- $\sin(\theta_1 + \theta_2) \approx \theta_1 + \theta_2$ +- $\cos(\theta_1 + \theta_2) \approx 1$ +- $\omega_1 + \omega_2 \approx 0$. + #### Modelovanje robota @@ -47,7 +55,7 @@ Simulacije služe da se na što realističniji način predstavi realni svet i do ##### Transfer funkcije -Transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaznog signala, za sve moguće ulazne vrednosti. Za bilo koji kontrolni sistem postoji referentni ulaz ili pobuda koji prolazi kroz transfer funkciju kako bi se proizveo (simulirao) odgovor sistema na odgovarajuću pobudu. +Transfer funkcija predstavlja odnos između ulaznog i ulaznog signala upravljačkog sistema, za sve moguće ulazne vrednosti. Za bilo koji kontrolni sistem postoji referentni ulaz ili pobuda koji prolazi kroz transfer funkciju kako bi se proizveo (simulirao) odziv sistema na odgovarajuću pobudu. Transfer funkcija (funkcija prenosa) je definisana kao Laplasova transformacija izlazne promenjljive i Laplasove transformacije ulazne promenjljive pod pretpostavkom da su svi početni uslovi jednaki nuli. $$G(s)\ =\ \frac{C(s)}{R(s)}$$ From 26a848b6afb0b82de79517762cf37be833bdfbfa Mon Sep 17 00:00:00 2001 From: FilipBajraktari Date: Sat, 5 Nov 2022 17:47:29 +0100 Subject: [PATCH 12/19] PID kontroler i jos po nesto --- content/2022/robot-valjak.md | 84 +- .../2022/robot-valjak/predjeni put robota.png | Bin 0 -> 40763 bytes .../2022/robot-valjak/predjeni_put_valjka.svg | 1 + .../predjeni_put_valjka_od_napona.svg | 791 ------------------ .../images/2022/robot-valjak/prva.drawio2.svg | 4 + .../2022/robot-valjak/smer_kretanja.svg | 263 ++++++ .../robot-valjak/ugao_inklinacije_klatna.png | Bin 0 -> 45484 bytes .../robot-valjak/ugao_inklinacije_klatna.svg | 1 + .../robot-valjak/ugaona_brzina_klatna.png | Bin 0 -> 52419 bytes .../robot-valjak/ugaona_brzina_klatna.svg | 1 + 10 files changed, 332 insertions(+), 813 deletions(-) create mode 100644 static/images/2022/robot-valjak/predjeni put robota.png create mode 100644 static/images/2022/robot-valjak/predjeni_put_valjka.svg delete mode 100644 static/images/2022/robot-valjak/predjeni_put_valjka_od_napona.svg create mode 100644 static/images/2022/robot-valjak/prva.drawio2.svg create mode 100644 static/images/2022/robot-valjak/smer_kretanja.svg create mode 100644 static/images/2022/robot-valjak/ugao_inklinacije_klatna.png create mode 100644 static/images/2022/robot-valjak/ugao_inklinacije_klatna.svg create mode 100644 static/images/2022/robot-valjak/ugaona_brzina_klatna.png create mode 100644 static/images/2022/robot-valjak/ugaona_brzina_klatna.svg diff --git a/content/2022/robot-valjak.md b/content/2022/robot-valjak.md index e4a2cf5..591720a 100644 --- a/content/2022/robot-valjak.md +++ b/content/2022/robot-valjak.md @@ -6,13 +6,7 @@ summary: Robot u obliku valjka je projekat rađen na letnjem kampu za stare pola ### Uvod -Motivacija iza projekta je bila suprotstavljane ideje popularnog robota - balansera. Ideja je bila da pomoću klatna koje uvek teži da se vrati u stabilan položaj -robot može biti konstruisan sa manjom preciznošću što bi ubrzalo proces pravljenja u slučaju masovne proizvodenje, kao i vreme dolaska do prvog funkcionalnog prototipa, budući da robot može da radi uz bilo kakav (ne)kontrolisan pogon motora. Takođe zbog oblika robota sve vitalne komponente se sadržane unutar konture, čineći robota izdržljivijim i jednostavnijim za popravku. - -Reference su skoro ne postojeće budući da se ideja o robotu dobila bez drugog uzora/slične ideje. - -???kako da formulišem da su refernece skoro ne postojeće i da sam sam došao na ideju robota??? - +Motivacija iza projekta je bila suprotstavljane ideje popularnog robota - balansera. Ideja je bila da pomoću klatna koje uvek teži da se vrati u stabilan položaj robot može biti konstruisan robot sa manjom preciznošću što bi ubrzalo proces pravljenja u slučaju masovne proizvodenje, kao i vreme dolaska do prvog funkcionalnog prototipa, budući da robot može da radi uz bilo kakav (ne)kontrolisan pogon motora. Takođe zbog oblika robota sve vitalne komponente se sadržane unutar konture, čineći robota izdržljivijim i jednostavnijim za popravku. ### Aparatura i metoda @@ -33,6 +27,8 @@ $$K_2 = \frac{1}{2}M_2((r\omega _ 1 - d\cos(\theta _ 1 + \theta _ 2)(\omega _ 1 $$T_1 = \frac{1}{2}J_1(\omega _ 1)^2$$ $$T_2 = \frac{1}{2}J_1(\omega _ 1 + \omega _ 2)^2$$ +![Poprečni presek robota valjka](/images/2022/robot-valjak/smer_kretanja.svg) + Za razliku od klasične Njutnove mehanike za Lagranžovu mehaniku je potrebno definisati odgovarajući Lagranžian, koji se definiše na sledeći način: $$L = K_1 + K_2 + T_1 + T_2 - U_1 - U_2$$ Iskoristivši Ojler-Lagranžovu jednačinu na dva sistema, klatno i valjak, @@ -51,7 +47,7 @@ mogu se primeniti sledeće aproksimacije: #### Modelovanje robota -Simulacije služe da se na što realističniji način predstavi realni svet i dobije bolji uvid u to kako će se sistem ponašati. Pored osećaja za sistem, simulacije omogućavaju lakše testiranje softvera kao i testiranje sistema u različitim prirodnim uslovima. +Kako bi se postigao bolji uvid u to kako će se robot ponašati, ključan korak ovog rada je bio kompjutersko modelovanje robota - simulacije. Pored samog 3D modela, potrebno je bilo napraviti i niz programa (kompjuterskih simulacija) koji imaju za ulogu da analitički kao i grafički predstave kako bi se robot valjak ponašao pod uticajem različith pobuda i na taj način izbegnu potencijalne greške koje možda ne bi mogle tako lako da se predvide. ##### Transfer funkcije @@ -96,11 +92,49 @@ Koristeći prethodno dve definisane funkcije mogu se izvesti dve nove transfer f ##### PID simulacije -Kako bi se razumeo PID kontroler potrebno je definisati odredjene termine. *Željena vrednost* je obično vrednost koju unosi korisnik. Na primer, za sistem za grejanje to bi bila željena temperatura prostorije. *Izlaz* je kontrolisana vrednost PID kontrolera. U automobilu bi to bila količina goriva koja je potrebna da bi se održavala konstantna brzina. *Greška* je vrednost koju koristi PID kontroler da bi odredio kako da manipuliše izlazom kako bi se dostigla željena vrednost. Greška se računa kao razlika željene vrednosti i trenutne vrednosti sistema. +U oblasti automatike postoji niz algoritama (kontrolera) koji imaju za cilj neki vid stabilizacije. To može da bude stabilizacije oko položaja labilne ravnoteže kao što je to slučaj sa problemom *Inverted Pendulum*[^2] gde treba da se stabilizuje matematičko klatno u položaju labilne ravnoteže, ili stabilizacija u nekom proizvoljnom položaju koji nalaže zadati problem. Jedan od algoritama koji se pokazao najboljim u praksi je PID kontroler čije su najveće prednosti brza i precizna stabilizacija. + +[^2]: A Comparative Study of Inverted Pendulum, S. Nemitha, B. Vijaya Bhaskar and S. Rakesh Kumar + +Kako bi se razumeo PID kontroler potrebno je definisati odredjene termine. *Željena vrednost* je obično vrednost koju unosi korisnik. Na primer, za sistem za grejanje to bi bila željena temperatura prostorije. *Izlaz* je kontrolisana vrednost PID kontrolera, tj. izlaz može da bude bilo šta na šta možemo direktno da delujemo a šta utiče na promenu vredsnoti koju želimo da kontrolišemo. U primeru sa grejačem, uticali bismo na snagu grejača što bi direktno uticalo na temperaturu prostorije. *Greška* je vrednost koju koristi PID kontroler da bi odredio na koji način je potrebno manipulisati izlazom kako bi se dostigla željena vrednost. Greška se računa kao razlika željene vrednosti i trenutne vrednosti sistema. + +###### Matematika PID kontrolera + +Teorijski posmatrano, transfer funkcije i PID kontroleri su definisani na kontinualnom domenu, tj. definisane su na način koji je nama blizak. Za svaki trenutak u vremenu možemo tačno odrediti vrednost izlaza transfer funkcije. Takva interpretacija nam odgovara ukoliko želimo da ispitujemo osobine same transfer funkcije, ali ukoliko želimo da u realnom svetu implementiramo takav model potrebna nam transfer funkcija definisana na diskretnom domenu. U tom slučaju znamo tačan izlaz sistema ali na svakih *dt* sekundi. Prvi razlog zbog kojeg se uvodi ovakav model je taj što bi bilo nemogući predstaviti beskonačno mnogo vredsnoti kompjuterski, a drugi razlog je taj što je odabir vremenskog intervala *dt* ograničen odozdo brzinom učitavanja vrednosti sa perifernih uređaja (senzora) i brzinom izvršavanja programa. Transfer funkcija PID kontrolera u kontinualnom domenu je predstavljena sledećim izrazom: + +$$C(s) = K_p + \dfrac{K_i}{\ s} + K_ds$$ + +$K_p$ - proporcionalni koeficijent +$K_i$ - integralni koeficijent +$K_d$ - diferencijalni koeficijent + +U tabeli ispod se može videti kako svaki koeficijent utiče na ishod PID kontrolera: + +| $\ \ \ $ PID | RISE TIME | OVERSHOOT | SETTLING TIME | STEADY-STATE ERROR | +| -------- | -------- | -------- | -------- | -------- | +| $\ \ \ \ \ K_p$ | Smanjuje | Povećava | Male promene | Smanjuje | +| $\ \ \ \ \ K_p$ | Smanjuje | Povećava | Povećava | Smanjuje | +| $\ \ \ \ \ K_p$ | Male promene| Smanjuje | Smanjuje | Nema promena | -U radu su simulirana tri PID kontrolera. Prvi PID kontroler odžava konstantan ugao inklinacije klatna u odnosu na horizontalnu podlogu po kojoj se valjak kreće. Drugi PID kontroler održava konstantnu ugaonu brzinu valjka. Poslednji PID kontroler održava konstantan ugaoni pređeni put valjka. Blok dijagram kontrolnog sistema je predstavljen na slici: +Kao i svaku drugu transfer funkciju, PID kontroler možemo diskretizovati koristeći razne metode. Metoda koja je korišćena u ovom radu je ZOH metoda, *Zero-order hold*, implementirana u biblioteci *Python Control Systems Library*[^3]. -Kao što je u prethodnom poglavlju rečeno, transfer funkcija predstavlja odnos između signala upravljačkog sistema i ulaznog signala, za sve moguće ulazne vrednosti. Ukoliko je transfer funkcija predstavljena preko Laplasovih transformacija onda je takva funkcija prenosa definisana na kontinualnom domenu. Takav oblik transfer funkcija je od koristi prilikom teorijskih analiza kao što je analiza polova i nula. Nazalost, u praksi nam treba diskretna reprezentacija sistema što proizilazi iz toga da mikrokontroler rade isključivo nad diskretnim domenom vrednosti. Postoje razni algoritmi koji vrše željenu konverziju. Jedan od njih, koji je i koriščen u radu, *Zero-order Hold* metoda +[^3]: Python Control Systems Library 0.9.2 + +U radu su simulirana tri PID kontrolera. Prvi PID kontroler odžava konstantan ugao inklinacije klatna u odnosu na horizontalnu podlogu po kojoj se valjak kreće. Drugi PID kontroler održava konstantnu ugaonu brzinu valjka. Poslednji PID kontroler održava konstantan ugaoni pređeni put valjka. Sva tri PID kontrolera mogu nezavisno da se implementiraju ali ukoliko izlaz jednog PID kontrolera prosleđujemo sledećem PID kontroleru možemo da postignemo isti efekat samo će sistem u ovom slučaju biti manje oscilatoran što doprinosti brzini stabilizacije, *rising time*, i grešci stabilizacije, *steady state error*. Blok dijagram kontrolnog sistema je predstavljen na slici: + +![Poprečni presek robota valjka](/images/2022/robot-valjak/prva.drawio2.svg) + +###### Diferencijator + +Diferencijator je specijalni oblik transfer funkcije koji ima za ulogu da prima ulazni signal a da kao izlaz prosleđuje funkciju koja odgovara prvom izvodu ulaza. Ovde treba naglasiti da je nemoguće napraviti idealni diferencijator već se na ovaj način postiže aproksimacija željenog efekta. Diferencijator je transfer funcija prvog reda sa nulom u nuli koja okarakterisana frekvencijom odsecanja, *cut-off frequency*. + +$$D(s) = \dfrac{\ \ \ \omega s \ }{s + \omega}$$ + +Mana ovakve implementacije diferencijatora je postojanje kašnjenja, *lag*, o čemu treba voditi računa prilikom implemntacije realnog sistema. Kašnjenje se može smanjiti povećanjem frekvencijom odsecanja ali će na taj način žrtvovati tačnost izlaznog signala. + +###### Komplementarni filter + +Ideja iza komplementarnog filtera je da se uzmu spori signali sa akcelerometra i brzi signali sa žiroskopa i na kraju ih spojimo u jedan signal. Akcelerometar je dobar pokazatelj orijentacije u statičkim uslovima, dok je žiroskop dobar pokazatelj nagiba u dinamičkim uslovima. Dakle, ideja je da se signali sa akcelerometra puste kroz *low-pass* filter a signali sa žiroskopa da se propuste kroz *high-pass* filter i onda novodobijeni signali iskombinuju. *Low-pass* filter nam omogućava da propustimo frekvencije ispod zadate odsečen frekvencije, dok nam *high-pass* filter omogučava da propustimo frekvencije iznad zadate odsečen frekvencije. Na taj način obezbeđujemo da u svakom trenutku ceo signal podleže oba filtera. ### Hardver @@ -122,38 +156,44 @@ Pod hardverom se podrazumevaju svi vidljivi delovi robota, poput spojeva, mikrok ##### Arduino Nano i HC05 -Uredjaj koji vrsi prikupljanje i obradu podataka i signala sa senzora, kao i generisanjem signala za upravljane motorom je Arduino Nano. Ovo je 16-bitni mikrokontroler koji na sebi poseduje Atmega328p cip. Ovaj mikrokontoler sa HC05 Bluetooth modulom komunicira pomocu UART-a na BAUD rate-u od 9600 bitova po sekundi, omogucavajuci bezicni prenos podataka. Napaja se pomocu napona od 5 volti sa mikrokontrolera. *Paznja! iako modul radi na 5V prihvata napon na signalnim pinovima (RX,TX) je ogranicen na 3.3V, tako da je obavezna upotreba naponskog razdelnika kao bi se napon signala sa mikrokontrolera(5V) spustio na odgovarajuci nivo* +Uređaj koji vrši prikupljanje i obradu podataka i signala sa senzora, kao i generisanjem signala za upravljane motorom je Arduino Nano. Ovo je 16-bitni mikrokontroler koji na sebi poseduje Atmega328p čip. Ovaj mikrokontoler sa HC05 Bluetooth modulom komunicira pomoću UART-a na BAUD rate-u od 9600 bitova po sekundi, omogućavajući bežični prenos podataka. Napaja se pomoću napona od 5 volti sa mikrokontrolera. *Paznja! Iako modul radi na 5V, napon na signalnim pinovima (RX, TX) je ograničen na 3.3V, tako da je obavezna upotreba naponskog razdelnika kao bi se napon signala sa mikrokontrolera (5V) spustio na odgovarajući nivo* ##### MPU6050 -MPU6050 je akcelerometar i ziroskop koji se koristi za merenje ugla inklinacije tega robota. MPU6050 je sa mikrokontrolerom povezan pomocu I2C protokola i napaja se pomocu 5 volti sa mikrokontrolera. Ovaj senzor mikrokontroleru dostavlja podatke sa ziroskopa i akcelerometra pomocu kojih se u softveru izracunava ugao inklinacije koji je neophodan za odredjivanje ugaonog ubrzanja robota i PID kontrolu. +MPU6050 je akcelerometar i žiroskop koji se koristi za merenje ugla inklinacije klatna robota. MPU6050 je sa mikrokontrolerom povezan pomocu I2C protokola i napaja se pomoću 5V sa mikrokontrolera. Ovaj senzor mikrokontroleru dostavlja podatke sa žiroskopa i akcelerometra pomoću kojih se u softveru izračunava ugao inklinacije koji je neophodan za određivanje ugaone brzine robota i PID kontrolu. ##### Li-ion baterija i LM2598 Stepdown regulator -Napajanje koje se u projektu koristi su 4 redno vezane Samsung 18650-35E Litijum-jonske baterije zbog svog kapaciteta od 5000mAh (miliamper casova) i maksimalne struje 2A (Ampera) koje mogu propustiti. Baterije se pune eksternim punjacem pomocu balansirajuceg konektora na njima. Napon sa baterije se sprovodi do LM2598 regulatora gde se napon spusta na 12V kako bi dalje mogao da se dovede do H-mosta, buduci da je u trenutnoj konfiguraciji njegov napon ogranicen na maksimum od 12V. Potom se struja iz H-mosta pomocu njegovog integrisanog regulatora spusta na 5V i dovodi do mikrokontrolera i ostalih senzora. Napon baterije se meri na mikrokontroleru kroz naponski razdelnik kako bi se napon od 16.8V spustio na 5V koji su bezbedni za mikrokontroler koji proverava trenutan napon/preostali kapacitet baterije. +Napajanje koje se u projektu koristi su 4 redno vezane Samsung 18650-35E Litijum-jonske baterije zbog svog kapaciteta od 5000mAh i maksimalne struje od 2A. Baterije se pune eksternim punjacem pomoću balansirajućeg konektora na njima. Napon sa baterije se sprovodi do LM2598 regulatora gde se napon spušta na 12V kako bi dalje mogao da se dovede do H-mosta, budući da je u trenutnoj konfiguraciji njegov napon ograničen na maksimalnih 12V. Potom se struja iz H-mosta pomocu njegovog integrisanog regulatora spusta na 5V i dovodi do mikrokontrolera i ostalih senzora. Napon baterije se meri na mikrokontroleru kroz naponski razdelnik kako bi se napon od 16.8V spustio na 5V koji su bezbedni za mikrokontroler. Baterija može da radi oko 3.5h u aktivnom režimu robota. + + *Procenjena maksimalna snaga potrosena u robotu je $12V * ~2A(Maksimalna utrosena struja celog robota) = 24W$ nakon regulatora, imajuci ovo u vidu struja koja protice kroz bateriju se moze izracunati kao $I = P/V$ gde je I - struja kroz bateriju, P - Utrosena snaga i V - napon baterije, gde je I = 24W / 16.8V(Nominalni napon baterije) = ~1.42A gde je onda minimalno vreme rada robota 5000mAh / 1.42A = ~3.5h +/- 10%(zbog gresaka u merenju, efikasnosti regulatora...)* ##### DC motor i L298N H-most -Buduci da pinovi mikrokontrlera nemaju snagu da pokrenu motor, kao pokretac motora se postavlja H-most koji pomocu signala koje dobija od strane mikrokontrolera pokrece motor. Dva signala koje H-most dobija su digitalni signali : +Budući da pinovi mikrokontrlera nemaju snagu da pokrenu motor, kao pokretač motora se postavlja H-most koji pomoću signala koji dobija od strane mikrokontrolera pokreće motor. Dva signala koje H-most dobija su digitalni signali: 1. Koji svojim naponom od 0V ili 5V odredjuje smer i -2. Koji koristeci PWM(Pulse Width Modulation) brzim paljenjem i gasenjem kontrolise napon koji je doveden do motora od 0-12V +2. Koji koristeci PWM (Pulse Width Modulation) brzim paljenjem i gašenjem kontroliše napon koji je doveden do motora u opsegu od 0-12V -JGA25370 je DC motor koji radi na naponu od V=12V i Maksimalnoj struji od I=2A (U stanju zakocenja osovine). Motor u sebi poseduje reduktor koji umanjuje broj obrtaja motora u sekundi ali povecava obrtni momenat gde je broj obrtaja $RPM_max = 250$ obrtaja u minutu i obrtni momenat $T = 1.4kg/cm$. Ovaj motor je jedina komponenta koja pokrece robota. +JGA25370 je DC motor koji radi na naponu od 12V i maksimalnoj struji od 2A. Motor u sebi poseduje reduktor koji umanjuje broj obrtaja motora u sekundi ali povecava obrtni momenat gde je broj obrtaja $RPM_{max} = 250$ obrtaja u minutu i obrtni momenat $T = 1.4kg/cm$. Ovaj motor je jedina komponenta koja pokreće robota. ##### E38S6G5 Opticki inkrementalni enkoder -E38S6G5 je opticki enkoder koji se koristi za merenje predjenog puta robota. E38S6G5 ima rezoluciju od 600 pulseva po obrtaju osovine sto nam daje preciznost od 0.6 stepeni ili ~0.01rad. Princip rada optickog enkodera je da se analizom izlaznih talasa moz odrediti da li se enkoder okrenuo u pozitivnom ili negativnom smeru. U konkretnom slucaju na pin 3 mikrokontrolera je povezan 1. od 2 izlaza enkodera, prilikom svake rastuce ivice ovog pina mikrokontroler belezi napon na 2. izlazu enkodera, u slucaju da je on pozitivan mikrokontroler to belezi kao obrt u pozitivnom smeru, a u slucaju da je napon negativam mikrokontroler to belezi kao obtr u negativnom smeru. Kada za svaki ovaj dogadjaj od nekog brojaca saberemo ili oduzmemo 1 mozemo dobiti predjeni put. *Ovo nije realni predjeni put buduci da se ne uracunava veoma potencijalni zanos robota u slucaju da stoji u mestu, u softveru se ovo resava oduzimanjem ugla inklinacije od predjenog puta.* +E38S6G5 je optički enkoder koji se koristi za merenje pređenog puta robota. E38S6G5 ima rezoluciju od 600 pulseva po obrtaju osovine što nam daje preciznost od 0.6 stepeni ili ~0.01rad. Princip rada optičkog enkodera je da se analizom izlaznih talasa može odrediti da li se enkoder okrenuo u pozitivnom ili negativnom smeru. U konkretnom slučaju na pin 3 mikrokontrolera je povezan prvi od dva izlaza enkodera, gde se prilikom svake rastuće ivice ovog pina mikrokontroler beleži napon na drugom izlazu enkodera. U slučaju da je napon pozitivan mikrokontroler to belezi kao obrt u pozitivnom smeru, a u slučaju da je napon negativam mikrokontroler to belezi kao obrt u negativnom smeru. Kada za svaki ovaj dogadjaj od nekog brojaša saberemo ili oduzmemo 1 mozemo dobiti ukupni ugao za koji se enkoder zarotirao. +## Istraživanje i rezultati +Tokom rada na projektu, uspešno je realizovana hardverska implementacija robota i prikupljeni su podaci o tome kako sistem reaguje na uzastopne odskočne odzive. Tokom istraživanja su izmerena tri odskočna odziva koja redom odgovaraju transfer funkcijama ugla inklinacije klatna u odnosu na podlogu u zavisnosti od napona motora, ugaone brzine klatna u odnosu na podlogu u zavisnosti od ulaznog napona motora i ugaonog pređenog puta robota u zavisnosti od ulaznog napona motora. +![Ugao inklinacije robota valjka](/images/2022/robot-valjak/ugao_inklinacije_klatna.png) +![Ugaona brzina klatna robota valjka](/images/2022/robot-valjak/ugaona_brzina_klatna.png) -## Istraživanje i rezultati +![Predjeni put robota valjka](/images/2022/robot-valjak/predjeni_put_valjka.svg) -Tokom rada na projektu, uspešno je realizovana hardverska implementacija robota i prikupljeni su podaci o tome kako sistem reaguje na uzastopne odskočne odzive. Tokom istraživanja su izmerena tri odskočna odziva koja odgovaraju redom transfer funkcijama ugaone brzine robota u zavisnosti od ulaznog napona motora, ugaone brzine klatna u odnosu na osovinu valjka u zavisnosti od ulaznog napona motora i ugla inklinacije klatna u odnosu na podlogu u zavisnosti od napona motora. +Prilikom dizajniranja klatna valjka centar mase korpe i komponenti (baterije, arduino i ostali moduli) se ne nalazi u osnosimetričnoj ravni korpe već je blago izbačen sa strane što dovodi do toga da nula u merenju inklinacije ugla nije prava nula. Takođe, ovome je potpomogla i činjenica da se na DC motoru nalazi reducer zbog kog ne postoji prava nula već postoji interval uglova koji mogu biti prava nula. Za male uglove moment zemljine teže na klatno se izjednačava sa momentom sile koju reducer proizvodi na klatno. Drugim rečima, javalja se mali interval oko prave nule za koje se klatno nalazi u indiferentnoj ravnoteži i u kojoj krajnji položaj zavisi isključivo od početnog impulsa. Ukoliko je veći početni impuls veća je i greška merenja ugla inklinacije. Ovaj fenomen se može predstaviti na grafiku ispod gde se vidi da robot konstantno *driftuje* u jednu stranu. Za dalji rad na ovom robotu treba da se poravna centar mase sistema kao i da se poveća masa klatna kako bi se smanjio opseg uglova koji mogu biti prava nula i na taj način postići preciznija merenja. -Prilikom dizajniranja klatna valjka centar mase korpe i komponenti (baterije, arduino i ostali moduli) se ne nalazi u osnosimetričnoj ravni korpe već je blago izbačen sa strane što dovodi do toga da nula u merenju inklinacije ugla nije prava nula. Takođe, ovome je potpomogla i činjenica da se na DC motoru nalazi reducer zbog kog ne postoji prava nula već postoji interval uglova koji mogu biti prava nula. Za male uglove moment zemljine teže na klatno se izjednačava sa momentom sile koju reducer proizvodi na klatno. Drugim rečima, javalja se mali interval oko prave nule za koje se klatno nalazi u indiferentnoj ravnoteži i u kojoj krajnji položaj zavisi isključivo od početnog impulsa. Ukoliko je veći početni impuls veća je i greška merenja ugla inklinacije. Ovaj fenomen se može predstaviti na grafiku ispod gde se vidi da robot konstantno *driftuje* u jednu stranu. Za dalji rad na ovom robotu treba da se poravna centar mase sistema kao i da se poveća masa klatna kako bi se smanji opseg uglova koji mogu biti prava nula i na taj način postići preciznija merenja. +Druga dva merenja predstavljaju kako u realnom sistemu napon na motoru utiče na ugaonu brzinu klatna i ugao inklinacije klatna. Na drugom grafiku se može jasno videti gore opisan problem. Referentna nula je pomerena za neku vrednost $d\theta$ i razlikuje se od stvarne nule. Takođe, jedan od ciljeva ovog projekta je bila i numerička identifikacija sistema koja nažalost nije urađena zbog nedostatka vremena kao i gore navedenih problema, ali idalje ostaje kao mogućnost za neki budući rad na ovu ili neku sličnu temu. -Druga dva merenja predstavljaju kako u realnom sistemu napon na motoru utiče na ugaonu brzinu klatna i ugao inklinacije klatna. Takođe, jedan od ciljeva ovog projekta je bila i numerička identifikacija sistema koja nažalost nije urađena zbog gore navedenih problema kao i nedostatka vremena, ali idalje ostaje kao mogućnost za neki budući rad na sličnu temu. +## Zaključak \ No newline at end of file diff --git a/static/images/2022/robot-valjak/predjeni put robota.png b/static/images/2022/robot-valjak/predjeni put robota.png new file mode 100644 index 0000000000000000000000000000000000000000..6ea2a7beca658221c9d31b98c3591b849b3777eb GIT binary patch literal 40763 zcmdSBbyQXDyEVKvow5Px6qHc9OF}78B&9n9X{39Dh&0kAElPJuryxiv-67r5{oea| z&i6a#jQ1Pk`}<=U3|#JYuRE@5&Uwvig($z2!^fe-fglL~nY^?r1fd`x2)2NQ0e-?i zJo5+qCG0Gt<*a7^+S$$ctvRGXClO9g z+y7j^VgJ^GbHey<6ZjBp2YGEL2qG{>{)XjAe0>8!>3PqjrPSR&?aaD)tG}AU+@Jr< z{1FZ1bL__EA3?2Y3hHFiFAD;7_pb1;RV_mFuBEm)@-P-<#u1cis=-6bJm)xV3a*`W z{E3}85t@{8J-n0u)vH%RUMFl~j~@?zenxzDb_V_ft3F!?5gKf~)KE@Eg;1x)hSgK{ z-Dc`~^vE97lVG1X*hpDL1&{ixS6OdsBR-+|MMZex;^GQq@8H>Q<6!qg4~~yz+}(vz1Z+d)Xg$!P#4lKUFE*It5)zia 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Kontroler ugaonog pređenog puta
Kontroler ugaonog pr...
Kontroler ugaone brzine robota
Kontroler ugaone brz...
Kontroler ugla inklinacije
Kontroler ugla inkli...
PLANT
PLANT
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KF
Diferencijator
Diferencijator
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KF
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Filtriran ugao inklinacije
Filtriran ugao inklinacije
Ugao inklinacije klatna
Ugao inklinacije klatna
Ugaona brzina klatna
Ugaona brzina klatna
Ugao rotacije osovine
Ugao rotacije osovine
PLANT
PLANT
KF - Komplementarni filter
KF - Komplementarni filter
Text
Text
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