Synthese des Steuersignals für einen hydraulischen Hexapod

63 ACC JOURNAL 2015, Volume 21, Issue 1 DOI: 10.15240/ ul/004/2015-1-007 SYNTHESIS OF CONTROL SIGNAL FOR HYDRAULIC HEXAPOD Voj ěch Klouček Cen e o Enginee ing Resea ch De elopmen , VÚTS, a.s., Design o Machine y, S á o ská 619, Libe ec XI - Růžodol I, 460 01 Libe ec, Czech Republic e-mail: oj ech.klouce[email p o ec ed] Abs ac The a icle desc ibes he hexapod wi h six linea hyd aulic mo o s and six deg ees o eedom. The hexapod consis s o a igid base pla e, mo able pla o m and six linea hyd aulic mo o s, which a e joined o he base pla e and he mo able pla o m by ball join s. In he i s pa o he a icle hexapod kinema ics is esol ed by using ma ix me hods o in es iga ion o spa ial mul ibody sys ems. The hexapod is used o he labo a o y ib a ion exci a ion equi alen o he ib a ion measu ed wi h accele ome e s in he eal pa ope a ion. The second pa o he a icle desc ibes he syn hesis o he con ol signal om he alues measu ed by accele ome e s. The compu a ion ou ines o he syn hesis a e p og ammed in Maple in e ace. In oduc ion The objec o s udy is a labo a o y hyd aulic hexapod used o dynamic es ing o mechanical componen s [1], [2] and subassemblies, e.g. ca sea s (Fig. 1). Dimensions o hexapod desc ibed he ein a e known om d awing documen a ion. Sou ce: a) Own using he CAD so wa e, b) h p://cxi. ul.cz/ak uali y/ak ualni-in o-od-nas/B_Hexapod.JPG Fig. 1: a) schema ic CAD model o he hexapod, b) he hexapod wi h a ca sea The essence o he expe imen s pe o med on he hexapod is moun ing an in es iga ed objec o he mo able pla o m and exci e he desi ed mo emen o ib a ion by hyd aulic mo o s o he hexapod. Each expe imen should simula e he eal ope a ion condi ions o componen s as p ecisely as possible. The e o e he desi ed mo emen is measu ed in he eal ope a ion by accele ome e s. 64 The e o e he aim o he wo k is accele ome e s’ signals con e sion o con ol signals o hyd aulic mo o s o he hexapod. Since wha is done is a p ocessing o la ge amoun s o measu ed da a, i equi es he con e sion o be simple and as . 1 A desc ip ion o he hexapod The hexapod consis s o he base pla e, mo able pla o m and six hyd aulic linea mo o s, which a e joined o he base pla e and pla o m by ball join s. The body o each hyd aulic mo o is ixed agains a o a y mo ion a ound i s longi udinal axis. The mo able pla o m has six deg ees o eedom owa ds he base pla e [3]. The e a e wo coo dina e sys ems on he hexapod: Fixed ones :a a a a Ax y z ( igid ixed wi h he base pla e) and mo able ones :b b b b Bx y z ( igidly ixed wi h he mo able pla o m). A a de aul posi ion o he pla o m bo h o he coo dina e sys ems coincide and a e a he middle poin o he pla o m’s uppe su ace (Fig. 2). Sou ce: Own using he CAD so wa e Fig. 2: a) hexapod a de aul posi ion, b) hexapod a gene al posi ion 2 Coo dina e sys ems ans o ma ion The hyd o mo o s a e numbe ed 1 o 6, ball join s on he base pla e 1 A o 6 A , ball join s on he mo able pla o m 1 B o 6 B (in Fig.1 only join s 1 B and 6 B ) a e iden i ied. Augmen ed adius ec o s o he a bi a y poin Q in he coo dina e sys ems a and b du ing mo ion :ba a e cons ained by he ans o ma ion equa ion aQ ab bQ  T , (1) whe e aQ is he augmen ed posi ion ec o o poin Q in he coo dina e sys em a , bQ is he augmen ed posi ion ec o o poin Q in he coo dina e sys em b and ab T is a ans o ma ion ma ix o mo ion :ba . 65 Augmen ed adius ec o s o join s , 1..6 i Ai in he coo dina e sys em a and join s , 1..6 i Bi in he coo dina e sys em b a e known om he dimensions o he hexapod and hey a e     , , , , , , , 11 TT aAi bBi aAi bBi aAi aAi aAi aAi bBi bBi bBi bBi x y z x y z                 uu u u . (2) T ans o ma ion ma ix du ing gene al mo ion :ba is   11 12 13 1 21 22 23 2 31 32 33 3 , , , 0, 0, 0 1 ab ab ab ab ab a a a a a a a a a a a a                         Su T S u O O , (3) whe e ab S is a di ec ional cosines ma ix and ab u adius ec o o poin B in he coo dina e sys em a . 3 Kinema ic solu ion Each hyd aulic mo o has a de aul leng h 0 L . The pis on s oke o i - h hyd aulic mo o om he de aul posi ion is i z . The cu en leng h o he i - h hyd aulic mo o (dis ance o join s i A and i B ) is 0, 1..6 i L z i . The e a e six condi ions o he cu en leng hs o hyd aulic mo o s in he a bi a y posi ion [3] o he mo able pla o m 0, 1..6 aBi aAi i L z i   uu . (4) Using a scala p oduc o de e mina ion o he ec o leng h, i is possible o con e (4) o he equa ion      2 0, 1..6 aBi aAi aBi aAi i L z i    u u u u . (5) I applies o he augmen ed adius ec o s i B in he coo dina e sys em a he ans o ma ion equa ion (1) 1 1 1 aBi ab ab bBi aBi ab bBi                   u S u u T O . (6) The physical meaning o a iable   0i Lz (cu en leng h o i - h hyd aulic mo o ) en o ces a condi ion 00 i Lz . Since he le hand side o (5) is a sum o second powe s, i holds ha     0 aBi aAi aBi aAi    u u u u . The pis on s oke o he i - h hyd aulic mo o he e o e is    0i aBi aAi aBi aAi zL   u u u u . (7) 66 I he ans o ma ion ma ix componen s a e known, sol ing o hyd aulic mo o s’ pis on s okes is easy. 4 Sol ing o ans o ma ion ma ix componen s The ask is based in such a way ha i is necessa y o compu e he componen s o he ans o ma ion ma ix om he measu ed signals o accele ome e s (known adius ec o s o n poin s, whe e n is he numbe o iaxial accele ome e s used). Since accele ome e s’ signals a e composed o he accele a ion alues in h ee mu ually pe pendicula di ec ions, i is necessa y i s o con e hese alues o he posi ion alues. Le he e be a coo dina e sys em in he space o he s udied body so ha a e moun ing he body o he hexapod mo able pla o m he coo dina e sys em coincides wi h he coo dina e sys em b . Le he e be n iaxial accele ome e s on he s udied body so ha he accele ome e s' axes a e pa allel wi h he axes o he coo dina e sys em b and hey do no lie on he s aigh line. The augmen ed adius ec o o j - h accele ome e in he coo dina e sys em b is known and i is , , , , 1.. 1 T bMj bMj bMj bMj bMj bMj x y z j n         u u . (8) The coo dina e sys ems a and b a e chosen so ha hey coincide in he de aul posi ion o he mo able pla o m. Le he componen s o he con e ed signal o j - h accele ome e be ,, j j j x y z . The augmen ed adius ec o s o he accele ome e s in he coo dina e sys em a a e , , , , 1.. 1 T aMj aMj aMj bMj j bMj j bMj j x x y y z z j n            u u . (9) Fo he adius ec o s o accele ome e s he ans o ma ion equa ion (1) also applies, he e o e , 1.. aMj ab bMj jn T . (10) 4.1 Sol ing ans o ma ion ma ix componen s using h ee accele ome e s The posi ion o he body in he 3D space is de ini ely de e mined by he coo dina es o i s h ee poin s which do no lie in a s aigh line. O hese nine coo dina es only six a e independen because he assump ion o a igid body implies h ee condi ions o cons an mu ual dis ances o hese poin s. When using h ee accele ome e s, i is 3n . We ha e nine a ailable signal componen s , , , 1.. j j j x y z j n . O hese only six componen s a e independen , e.g. 1 1 1 2 2 3 , , , , ,x y z y z z . By using (10) we ge a sys em o six linea equa ions o wel e unknown ans o ma ion ma ix componen s , , , 1..3 ij i a a i j  o each eco ded ime poin . Di ec ional cosines ma ix ab S con ains in columns he coo dina es o uni di ec ional ec o s o he coo dina e sys em’s b axes in he coo dina e sys em a . Le we deno e hese ec o s x b , y b , z b , whe e       11 21 31 12 22 32 13 23 33 , , , , , , , , T T T x y z a a a a a a a a a  b b b . (11) 67 Any wo o hese ec o s ha e o be pe pendicula and he leng h o each o hem is 1, which can be exp essed by using scala p oduc s 0, 0, 0, 1, 1, 1 x y y z z x x x y y z z      b b b b b b b b b b b b . (12) Fo calcula ions o ans o ma ion ma ix componen s i is necessa y o sol e he sys em o wel e equa ions o wel e unknowns. The equa ion sys em consis s o six linea equa ions ob ained om (10) and six non-linea equa ions (12). The solu ion o his equa ion sys em is necessa y o be done o each ime poin o he measu ed accele ome e s’ signals. As i is a sys em o non-linea equa ions, i is possible o sol e i by using he app op ia e i e a i e me hod. Howe e , wi h he high numbe o he measu ed alues, his ask is ime consuming and demanding on compu e capaci y. 4.2 Sol ing ans o ma ion ma ix componen s using ou accele ome e s When using ou accele ome e s, i is 4n . We ha e wel e a ailable signal componen s , , , 1.. j j j x y z j n . Six o hem a e independen again. I we assume ha he s udied body is ideally igid and accele ome e s a e ideally ixed o he body, he componen s o hese signals comply wi h six condi ions o he cons an dis ances be ween he accele ome e s. These six condi ions a e equi alen wi h he equa ions (12), because hey ep esen he ac ha he s udied body is igid and hus he mo able coo dina e sys em emains o hono mal. I we subs i u e he measu ed alues o equa ion (10) and i we use all wel e componen s o he accele ome e signals, we ob ain a sys em o wel e linea equa ions o wel e unknown componen s o he ans o ma ion ma ix (3). We will w i e his sys em in a ma ix o m SM x p , (13) whe e S M is a ma ix o he equa ion sys em (squa e, 12 h o de ), x ec o o unknowns, and p ec o o igh hand sides. The ma ix S M con ains componen s o ec o s bMj u , ec o x con ains wel e unknown ans o ma ion ma ix componen s, and ec o p con ains componen s o ec o s aMj u . The sys em (13) has jus one solu ion i and only i he de e minan o he ma ix o equa ion sys em 0 SS DM . I his condi ion is accomplished, hen he equa ion sys em is easy o sol e, e.g. by using Gauss elimina ion o ma ix in e se, because he ma ix S M con ains many ze o componen s. I applies he heo em 0 S D i and only i he accele ome e s 1 o 4 do no lie in a plane. We will p o e his heo em. Le he e be h ee ec o s wi h he s a poin in he loca ion o he accele ome e 1 and end poin s in he loca ions o he accele ome e s 2, 3, 4. The componen s o hese ec o s will be pu in o columns o ma ix V M , so ha   2 1 3 1 4 1 ,, V bM bM bM bM bM bM    M u u u u u u . (14) 68 De e minan o ma ix V M we deno e VV DM . By di ec calcula ion o he de e minan s S D and V D i is possible o e i y he equali y 3 SV DD . (15) The condi ion 0 S D is ue i and only i he de e minan 0 V D , which esul s om he equali y (15). A de e minan is nonze o i and only i all i s columns a e linea ly independen , which occu s only i he accele ome e s 1 o 4 do no lie on a plane. This condi ion is necessa y and su icien . The p oo is inished. Conclusion In his pape we explo ed kinema ics o he hyd aulic hexapod wi h six deg ees o eedom. Fu he mo e, he me hod o con e sion o he ans o ma ion ma ix componen s o s okes o hexapod hyd aulic mo o s was p oposed. Fo a simula ion o mo ion, measu ed in a eal ope a ion using accele ome e s, i is necessa y o con e he measu ed da a o he ans o ma ion ma ix componen s a each ime. The e a e wo me hods deduced: When using h ee accele ome e s and when using ou accele ome e s. I was ma hema ically deduced ha while using ou accele ome e s, he con e sion is signi ican ly as e and easie . Acknowledgemen s The esea ch was made possible by VÚTS, a.s., esea ch p ojec NPU - L012, 99253/5 and Technical uni e si y o Libe ec, Facul y o Mechanical Enginee ing, Depa men o he Design o Machine Elemen s and Mechanism. Li e a u e [1] PETŘÍK, J.; MARTONKA, R.: Measu ing pla o m o sea es ing. P oceedings o he 52 h In e na ional Con e ence o Machine Design Depa men s. pp. 184-187, ISBN 978- 80-248-2450-5. Š Technical Uni e si y o Os a a, 2011. [2] FLIEGEL, V.; MARTONKA, R.: Elec o-dynamic measu ing equipmen . P oceedings o he Con e ence MMa MS. pp. 101-103, ISBN 978-80-553-0731-2. Technical Uni e si y o Košice, 2011. [3] FLIEGEL, V.; MARTONKA, R.: Hexapod- he Pla o m wi h 6DOF. P oceedings o he 54 h In e na ional Con e ence o Machine Design Depa men s. pp. 111-116, ISBN 978-80-7372-986-8. Technical Uni e si y o Libe ec, 2013. Ing. Voj ěch Klouček, Ph.D. 69 SYNTÉZA ŘÍDICÍHO SIGNÁLU PRO HYDRAULICKÝ HEXAPOD Článek popisuje hexapod se šes i lineá ními hyd omo o y a šes i s upni olnos i. Ten o hexapod se skládá z uhé základní desky, pohybli é plošiny a šes i lineá ních hyd omo o ů, k e é jsou připojeny k základní desce a pohybli é plošině pomocí kulo ých čepů. V p ní čás i článku je řešena kinema ika hexapodu za yuži í ma ico ých me od zkoumání p os o o ých ázaných mechanických sys émů. Hexapod je použí án p o labo a o ní ybuzení ib ací, k e é jsou ek i alen ní ib acím naměřeným pomocí akcele ome ů eálném p o ozu. D uhá čás článku popisuje syn ézu řídicího signálu z hodno naměřených akcele ome y. Výpoč o é algo i my syn ézy jsou nap og amo ány p os ředí so wa e Maple. SYNTHESE DES STEUERSIGNALS FÜR EINEN HYDRAULISCHEN HEXAPOD Diese A ikel besch eib eine Hexapod-Ein ich ung mi sechs Linea mo o en und sechs F eihei sg aden. De Hexapod bes eh aus eine s a en G undpla e, eine beweglichen Pla o m und sechs linea en o o en, die mi de G undpla e und de mobilen Pla o m mi els Kugelbolzen e bunden sind. Im e s en Teil des A ikels wi d die Kinema ik des Hexapods mi Hil e on Ma ix-Me hoden zu Un e suchung on Meh kö pe sys emen un e such . De Hexapod wi d im Labo zu An egung on Schwingungen genu z , die äqui alen zu den im ealen Be ieb mi Hil e des Beschleunigungsmesse s gemessenen Vib a ionen sind. De zwei e Teil des Tex es besch eib die Syn hese des S eue signals aus den on Beschleunigungsmesse n gemessenen We en. Die Syn hese-Algo i hmen sind in So wa e Maple p og ammie . SYNTEZA SYGNAŁU STEROWANIA DLA HYDRAULICZNEGO SZEŚCIONOGA A ykuł opisuje sześcionóg z sześcioma liniowymi silnikami hyd aulicznymi i sześcioma s opniami swobody. Sześcionóg składa się ze sz ywnej pły y głównej, uchomej pla o my i sześciu liniowych silników hyd aulicznych, k ó e są połączone z pły ą główną i uchomą pla o mą za pomocą swo zni kulis ych. W pie wszej części a ykułu p zeds awiono kinema ykę sześcionoga s osując macie zowe me ody badania p zes zennych mechanicznych sys emów wieloobiek owych. Sześcionóg wyko zys ywany jes do labo a o yjnego wywoływania d gań, k ó e odpowiadają wib acjom namie zonym p zy pomocy akcele ome ów w zeczywis ym świecie. D uga część a ykułu opisuje syn ezę sygnału s e ującego z wa ości namie zonych p zez akcele ome y. Algo y my obliczeniowe syn ezy zap og amowano w op og amowaniu Maple.