เพนด, มผกผ, งก, ามภาษา, ในบทความน, ไว, ให, านและผ, วมแก, ไขบทความศ, กษาเพ, มเต, มโดยสะดวก, เน, องจากว, เด, ยภาษาไทยย, งไม, บทความด, งกล, าว, กระน, ควรร, บสร, างเป, นบทความโดยเร, วท, งกฤษ, inverted, pendulum, เป, นป, ญหาพ, นฐานท, ใช, ในการเร, ยนการสอนและในการสา. lingkkhamphasa inbthkhwamni miiwihphuxanaelaphurwmaekikhbthkhwamsuksaephimetimodysadwk enuxngcakwikiphiediyphasaithyyngimmibthkhwamdngklaw krann khwrribsrangepnbthkhwamodyerwthisudephndulmphkphn xngkvs Inverted pendulum epnpyhaphunthanthiichinkareriynkarsxnaelainkarsathitkarprayuktthvsdirabbkhwbkhum ephndulmphkphnepnrabbthimicudsmdulxyurxbaeknhmundwyknsxngcud idaekcudthiephndulmtngtrngxyuinaenwding aelacudthiephndulmxyuthingtwlnginding aetcudthimiesthiyrphaphemuximmitwkhwbkhumnncamicudediywkhux cudthiaeknthingtwlngethann imwaeracaplxyephndulmthicudidktam ephndulmcatklngsucudniesmx karthicathaihephndulmnisamarthtngtrnginaenwdingidnnkhunkbkaristwkhwbkhumthiehmaasmekhaipinrabbsungmiidhlakhlaywithi 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x t displaystyle x t epnrayakhxngrthinaekn x displaystyle x thiaeprphntamewla aelasmkarlakrangc Lagrangian khxngrabbepndngtxipni 4 L T V displaystyle L T V ody T displaystyle T khuxphlngnganclnkhxngrabb aela V displaystyle V khuxphlngnganskykhxngrabb rupphaphaesdngrupaebbkhxngephndulmphkphnaelatwaeprthiichinkarsrangaebbcalxngthangkhnitsastr L 1 2 M v 1 2 1 2 m v 2 2 m g ℓ cos 8 displaystyle L frac 1 2 Mv 1 2 frac 1 2 mv 2 2 mg ell cos theta ody v 1 displaystyle v 1 epnkhwamerwkhxngkhxngtwrth v 2 displaystyle v 2 epnkhwamerwkhxngcudsunyklangmwl m displaystyle m khxngmwlbnaethngephndulm thngni v 1 displaystyle v 1 aela v 2 displaystyle v 2 samarthekhiynihxyuinrupkhxng x displaystyle x aela 8 displaystyle theta dngtxipni v 1 2 x 2 displaystyle v 1 2 dot x 2 v 2 2 d d t x ℓ sin 8 2 d d t ℓ cos 8 2 displaystyle v 2 2 left frac d dt left x ell sin theta right right 2 left frac d dt left ell cos theta right right 2 thakarldrup v 2 displaystyle v 2 idphlepn v 2 2 x 2 2 ℓ x 8 cos 8 ℓ 2 8 2 displaystyle v 2 2 dot x 2 2 ell dot x dot theta cos theta ell 2 dot theta 2 aethnsmkarkhangtnlnginsmkarlakrangc idwa L 1 2 M m x 2 m ℓ x 8 cos 8 1 2 m ℓ 2 8 2 m g ℓ cos 8 displaystyle L frac 1 2 left M m right dot x 2 m ell dot x dot theta cos theta frac 1 2 m ell 2 dot theta 2 mg ell cos theta aelasmkarkarekhluxnthi d d t L x L x F displaystyle frac mathrm d mathrm d t partial L over partial dot x partial L over partial x F d d t L 8 L 8 0 displaystyle frac mathrm d mathrm d t partial L over partial dot theta partial L over partial theta 0 aethnthi L displaystyle L insmkarkhangtncaidsmkarthixthibaykareluxnthikhxngephndulmaebbphkphndngni M m x m ℓ 8 cos 8 m ℓ 8 2 sin 8 F displaystyle left M m right ddot x m ell ddot theta cos theta m ell dot theta 2 sin theta F ℓ 8 g sin 8 x cos 8 displaystyle ell ddot theta g sin theta ddot x cos theta caehnidwasmkarthiidepnsmkarimechingesnsungyakthicanaipxxkaebbtwkhwbkhum inthangptibtiphuxxkaebbcaniymaeprngsmkarimechingesnihepnsmkarechingesnkxn odysmmutiwaaethngephndulmaekwngxyuinchwngmumelksungpramanepn 0 displaystyle 0 id 8 0 displaystyle theta approx 0 thngniephuxkhwamngaytxkarxxkaebbtwkhwbkhum aelangaytxkarxthibayphvtikrrmkhxngrabbduephim aekikhthvsdirabbkhwbkhum Segway ephndulmphkphnaebbfuruta Furuta pendulum xangxing aekikh edwid brrecidphngschy phakhwichawiswkrrmiffa khnawiswkrrmsastr culalngkrnmhawithyaly rabbkhwbkhumphlwt karwiekhraah karxxkaebb aelakarprayukt Dynamical Control Systems Analysis Design and Applications sankphimphaehngculalngkrnmhawithyaly 2551 ISBN 978 974 03 2205 4 M W Spong and M Vidyasagar Dynamics and Control of Root Manipulators John Wiley 1989 Katsuhiko Ogata Modern control engineering Edition 5 Prentice Hall 2010 ISBN 0136156738 9780136156734 1 Simple Inverted Pendulum Cart Dynamics Lagrangian Development by Jaspen Patenaudeaehlngkhxmulxun aekikhsuxkarsxnekiywkbthvsdirabbkhwbkhum mitwxyangepnephndulmaebbphkphn khxngmhawithyalykharenkiemllxn Katsuhiko Ogata Modern control engineering Edition 5 Prentice Hall 2010 ISBN 0136156738 9780136156734 Franklin et al 2005 Feedback control of dynamic systems 5 Prentice Hall ISBN 0 13 149930 0 Norman S Nise Control Systems Engineering Edition 6 John Wiley amp Sons 2010 ISBN 0470547561 9780470547564 edwid brrecidphngschy phakhwichawiswkrrmiffa khnawiswkrrmsastr culalngkrnmhawithyaly rabbkhwbkhumphlwt karwiekhraah karxxkaebb aelakarprayukt Dynamical Control Systems Analysis Design and Applications sankphimphaehngculalngkrnmhawithyaly 2551 ISBN 978 974 03 2205 4 wibuly aesngwiraphnthusiri phakhwichawiswkrrmekhruxngkl khnawiswkrrmsastr culalngkrnmhawithyaly karkhwbkhumrabbphlsastr Control of Dynamic Systems sankphimphaehngculalngkrnmhawithyaly 2548 ISBN 974 13 3393 5 ekhathungcak https th wikipedia org w index php title ephndulmphkphn amp oldid 9269167, wikipedia, วิกิ หนังสือ, หนังสือ, ห้องสมุด,
