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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.
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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
.
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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)
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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)
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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)

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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.
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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.