Mecha onic design o a 3 deg ees o eedom pa allel kinema ics
manipula o wi h in eg a ed o ce pla e o human balance e alua ion
and ehabili a ion
✰
F ancisco J. Campa
*
, Mikel Diez , Ja ie Co al, E ik Macho, Saioa He e o, Cha les Pin o
Depa men o Mechanical Enginee ing, Uni e si y o he Basque Coun y UPV/EHU, Escuela de Ingenie ía de Bilbao, Plaza Ingenie o To es Que edo 1, Bilbao 48013,
Spain
ARTICLE INFO
Associa e Edi o : Michael G. Ruppe
Keywo ds:
Human balance
Dynamic pos u og aphy, Rehabili a ion
Pa allel kinema ics
Fo ce pla e
ABSTRACT
One o he main sequelae o s oke is hemipa esis o pa ial hemiplegia. This condi ion causes pa ien s o unde go
leng hy ehabili a ion p ocesses o eco e balance and gai . Thus, i is equi ed o ha e ools o sys ema ize
ehabili a ion asks while moni o ing he e olu ion o he pa ien o e ime. This pape p esen s a new p o o ype
based on a 3PRS pa allel kinema ic manipula o wi h a o ce pla e in he end e ec o o ehabili a e and e alua e
balance by measu ing he Cen e o P essu e o he pa ien , a widely s udied pa ame e in he ield o pos u -
og aphy. The p o o ype has been designed o be able o measu e in a eliable and epea able way and, as i is a
medical de ice, i includes design conside a ions ela ed o ease o use, isual eedback and sa e y. The pape
desc ibes he kinema ics, dynamics, mecha onic modelling and mechanical design o he pa allel kinema ics
machine, as well as he design o he o ce pla e in cha ge o de e mining he loca ion o he Cen e o P essu e.
Also, a o mula ion is p oposed o co ec he in luence o he ine ial and g a i a o y o ces on he op pla e
du ing he mo ion o he pla o m on he measu emen . Finally, he p o o ype has been e alua ed expe imen ally
o compa e he mo ion wi h he de eloped models and o de e mine he p ecision and obus ness agains noise
and mo ion ela ed e ec s.
Abb e ia ions
TUG Timed Up and Go es
SPPB Sho Physical Pe o mance Ba e y
DGI Dynamic Gai Index
BBS Be g Balance Scale
COP Cen e o P essu e
AP An e opos e io
ML Mediola e al
DOF Deg ees o F eedom
PRS P isma ic-Re olu e-Sphe ical join s
PID P opo ional-In eg al-De i a i e Con olle
FEM Fini e Elemen s Me hod
K
zz
ZZ e m o he s i ness ma ix o he FEM model
V-M Von Mises equi alen s ess
SF Sa e y Fac o
OWR Ope a ional Wo kspace o Ro a ion
H
G
Angula momen um o he op pla e
J
i
Ac ua o ine ia momen in mo o axis
c
i
Ac ua o iscous ic ion in mo o axis
τ
Ci
Ac ua o Coulomb ic ion in mo o axis
m
i
Mass o pla e ha links e olu e join and ac ua o
m
BC
Manipula o ba s mass
I
BC
Manipula o ba s ine ia in hei mass cen e
m
MP
Mobile pla o m o al mass
I
xG,yG,zG
Mobile pla o m ine ia momen s in he mass cen e
p Ac ua o s pi ch
K
P opo ional gain o he posi ion con olle
K
p
P opo ional gain o he eloci y con olle
K
i
In eg al gain o he eloci y con olle
1. In oduc ion
Globally, s oke is he second leading cause o dea h and he hi d
✰This pape was ecommended o publica ion by Associa e Edi o : Michael G. Ruppe
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (F.J. Campa).
Con en s lis s a ailable a ScienceDi ec
Mecha onics
jou nal homepage: www.else ie .com/loca e/mecha onics
h ps://doi.o g/10.1016/j.mecha onics.2024.103278
Recei ed 8 Ma ch 2024; Recei ed in e ised o m 24 Oc obe 2024; Accep ed 20 No embe 2024
Mecha onics 105 (2025) 103278
A ailable online 25 No embe 2024
0957-4158/© 2024 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY-NC license (
h p://c ea i ecommons.o g/licenses/by-
nc/4.0/ ).
leading cause o disabili y [1]. S oke is caused by he dea h o some
b ain cells due o lack o oxygen, usually caused by he loss o blood low
om ei he a blockage o up u e o an a e y. The incidence o s oke
has a ied in ecen decades, wi h he incidence doubling in de eloping
coun ies and almos hal ing in de eloped coun ies [2]. On a e age,
s oke is su e ed 15 yea s ea lie , mainly due o unheal hy habi s in
de eloped coun ies. Depending on he se e i y, s oke can esul in
e e y hing om impai ed language and/o mo o skills, senso y al e -
a ions, o dea h [3].
Speci ically, he loss o mobili y due o o al o pa ial hemipa esis,
loss o s eng h o mo o esponse o he uppe o lowe limbs o hal o
he body, is one o he mos se ious long- e m e ec s. In [4] i is shown
ha 32 % o s oke pa ien s ha e se e e p oblems o a e simply unable
o walk, compa ed o abou 9 % o all o he condi ions. This means ha
he mos basic daily asks a e hampe ed by hese mobili y- ela ed
sequelae. Mobili y is conside ed a c ucial ac o in heal hy ageing as i
gene ally de e mines a pe son’s abili y o change posi ion o o mo e
a ound.
When i comes o ehabili a ing pa ien s a ec ed by neu ological
damage, i is essen ial o add ess ini ially he unc ion o balance, hen
he gai and, inally o in pa allel wi h he second, he ans e o abili y
o mo e an objec om one posi ion o ano he . I is he e o e necessa y
o ha e ools ha allow an accu a e measu emen o he deg ee o loss o
balance and gai unc ion, bo h o he ini ial diagnosis and o moni o
he pa ien ’s e olu ion du ing ehabili a ion. A majo obs acle in he
diagnosis and ea men o balance is ha i depends on he in eg a ion
in he cen al ne ous sys em o in o ma ion ecei ed om h ee main
senso y sys ems: es ibula , isual and soma osenso y [5].
Nowadays, unc ional measu emen es s a e used o diagnose bal-
ance unc ion, among which he e a e mo e han hi y p o ocols,
including he TUG, he SPPB, he DGI o he BBS [6–9]. These es s di e
in he ype o exe cises and he in e p e a ion o he esul s, since, while
some me hods in ol e quan i a i e measu emen s o ime o numbe o
epe i ions, o he s ocus on quali a i e aspec s ha a e no always
assessed by he same pe son. Also, none o hem is sui able o all pa-
ien s, and in mos cases, hey a e no able o disc imina e be ween
pa ien s om di e en clinical g oups: age, s a e o he ehabili a ion
p ocess o senso y sys em a ec ed [10–12]. Hence, he e is a need o
de ices capable o aking objec i e measu emen s o assess he pa ien
and ha a e able o combine a a ie y o exe cises, om s anding up-
igh on an inclined loo , o walking, o pe o ming unc ional exe cises
ha simula e e e yday ac ions such as picking up a glass om a
cupboa d.
On he o he hand, i is widely accep ed in he li e a u e ha he
Cen e o P essu e (COP) is a pa ame e ela ed o he s a e o he bal-
ance unc ion [13,14].The COP is he poin o applica ion o he esul-
an o ce applied o he su ace whe e he pa ien is s anding, and in
s a ic condi ions i should be aligned wi h he Cen e o Mass (COM) o
he pe son, see Fig. 1, whe e he eac ion on he ee is depic ed and he
o al eac ion ha balances he weigh is applied in he COP. Howe e , i
is s ill no clea which indica o s deduc ed om he COP mo ion a e he
mos ele an o diagnose balance. In his sense, di e en me ics ob-
ained om he COP mo ion ha e been p oposed in he li e a u e
[15–18]: maximum, minimum and mean COP displacemen in he
an e opos e io (AP) and mediola e al (ML) di ec ions, shape o he
con idence ellipse in a s a okinesig am, maximum, minimum and mean
COP eloci ies in AP and ML di ec ions, and Fou ie analysis o he COP
mo ion. In [19] i is shown ha some o hese indica o s show a good
co ela ion wi h BBS and TUGT es s.
In an a emp o ind a solu ion o he p oblem o balance e alua ion
and ehabili a ion, di e en equipmen can al eady be ound on he
ma ke . The Biodex Balance Sys em® is a il ing pla o m sys em o ain
he balance unc ion [20]. BalanceTu o om MediTouch is a mo ing
eadmill pla o m wi h a mo ing bel ha allows o wa d, backwa d
and la e al mo emen o s a ic and dynamic balance ehabili a ion
[21]. The Hube 360 is a mul i-axial mo ing pla o m ha inco po a es
o ce senso s in bo h he pla o m and he handg ips [22]. In [23,24] he
design o a 2-deg ee-o - eedom pla o m o ankle and lowe unk
ehabili a ion is shown. The pla o m has a gimball-based design
achie ing o a ion a ound ho izon al axes by means o wo mo o s, one
o hem no ixed o he ixed ame. This sys em measu es he o ces
exe ed by he pa ien on he pla o m and he mo emen o he pa ien
wi h an IMU.
O he comme cial sys ems in eg a e no only he mo emen o he
pla o m, bu also i ual en i onmen s o pa ien aining. Fo
example, [25,26] show a 6 deg ees o eedom (DOF) pla o m based on
he S ewa a chi ec u e wi h a eadmill ha allows he pa ien o walk
while simula ing di e en en i onmen s o pe u ba ions. In e ms o
balance o gai analysis, i is a e y comple e solu ion as i in eg a es
bo h he o ce senso s on he eadmill and he possibili y o ack he
body using mo ion cap u e.
Sys ems in a esea ch s age can also be ound in he li e a u e. Fo
example, [27] shows he design o a 3 DOF sphe ical pla o m ha al-
lows he pla o m on which he pa ien s ands o be o a ed a ound he
h ee axes, allowing il ing mo emen s. In he alida ion s udy, ma ke s
placed on he subjec s we e used o es ima e hei mo emen s. One
limi a ion o his sys em is ha he use o ma ke s would no be possible
o implemen in pa ien s due o he high p epa a ion ime in ol ed.
O he p o o ype can be ound in [28], whe e he design p ocedu e o a
3PRS pa allel mechanism is desc ibed o he e alua ion o e igo in
pa ien s.
In his pape , a new pa allel pla o m based on he 3PRS a chi ec u e
is p oposed o he assessmen o balance in pa ien s who ha e su e ed a
s oke. In compa ison wi h he S a e o he A , he pla o m, named
OREKA, p esen s a much wide mo ing o ce pla e and an easie and
wide access o he sys em. The pape is o ganized as ollows. In Sec ion
2 he design o he sys em is p esen ed, om i s equi emen s o he
kinema ics, dynamics and mechanical design. In Sec ion 3 he expe i-
men al alida ion o he mecha onic sys em is p esen ed, desc ibing he
o ce pla e pe o mance and i s accu acy in Sec ion 4. Finally, he main
conclusions a e p esen ed in Sec ion 5.
2. Mecha onic design
2.1. Design equi emen s
The equi emen s we e de ined oge he wi h he medical s a om
Hospi al Go liz o m he basque public heal hca e sys em o co e he
Fig. 1. Cen e o P essu e (COP) and Cen e o Mass (COM) alignmen in
s a ic condi ions.
F.J. Campa e al.
Mecha onics 105 (2025) 103278
2
unc ional needs: e gonomics, use expe ience, ange o mo ion and
eloci y, sa e y aspec s and measu emen s o be ca ied ou . Addi ion-
ally, he UNE-EN ISO 12100, UNE-EN ISO 14971 and UNE-EN 60204-1
s anda ds ela ed o medical and elec ical de ices we e consul ed.
Hence, he ollowing basic design equi emen s we e es ablished:
- Maximum il ange o he pla o m o ±15◦in any di ec ion wi h
espec o he ho izon al plane, wi h a maximum il ing eloci y o 20
deg/s.
- Pa ien weigh up o 150 kg.
- P ecision o 2 mm in he measu emen o he COP posi ion.
- Easy access o he de ice, la ge pla o m o do exe cises in di e en
pos u es: s anding, andem, semi andem.
- De o ma ions lowe han 1 mm in any mo ing pa .
- Sa e y measu es as an eme gency bu on, an en apmen de ec o , o
a all de ec o .
2.2. Kinema ics o he manipula o
The manipula o mus in oduce a dis u bance on he o ce pla e ha
s imula es he esponse o he pa ien . I was decided o in oduce a
e ical mo ion and a il ing a ound any ho izon al axis. Hence, a 3PRS
ype pa allel kinema ics mechanism has been p oposed [29]. In he
mo ing pla o m he e a e h ee S-join s a poin s B
i
in a iangula
a angemen o base 2R and heigh D, as shown in Fig. 2, wi h P being
he o igin o he mobile e e ence sys em UVW, wi h he U-axis aligned
wi h PB
3
and he W axis pe pendicula o he mo ing pla o m. The
pla o m is connec ed o he linea ac ua o s h ough h ee equal B
i
C
i
ba s o leng h L. These a e joined o he linea ac ua o s h ough he
R-join s a poin s C
i
, and ansla e e ically acco ding o he a ia ion o
s
i
, coo dina es o he join space. The o igin o s
i
coo dina es is in poin s
A
i
, which shape a iangle wi h base 2H and heigh N. The o igin o he
ixed e e ence ame XYZ is a poin O, he di ec ion o he X-axis
coinciding wi h OA
3
and being Z-axis in e ical di ec ion.
2.2.1. Pa asi ic mo ions
The selec ed a angemen wi h ba s B
1
C
1
and B
3
C
3
in he same plane
minimizes pa asi ic mo ions in he pla o m [30] and allows a wide
access a ending o he equi emen s o he design. Indeed, he numbe
o DOF o he pla o m a e 3: e ical ansla ion in Z-axis acco ding o
z
P
, o a ion a ound he X-axis acco ding o
ψ
, and o a ion a ound he
Y-axis acco ding o angle θ. Tha means ha he pa asi ic mo ions a e
he ansla ions in X-axis and Y-axis, x
P
and y
P
espec i ely, and he
o a ion φ a ound Z-axis. A e imposing he kinema ic es ic ions o he
p esen a angemen , whe e S-join s B
1
and B
3
emain in he y =0 plane
and S-join B
2
in he x =0 plane, i is demons a ed ha he displace-
men in he Y-axis in P and he o a ion a ound Z a e ze o [30], see Eq.1.
xP= − Dsin
ψ
sinθyP=0ϕ=0 (1)
Consequen ly, he o a ion ma ix R ha ela es he mo ing e e -
ence sys em wi h he ixed one is:
R=⎡
⎣
cosθsin
ψ
sinθcos
ψ
sinθ
0 cos
ψ
−sin
ψ
−sinθsin
ψ
cosθcos
ψ
cosθ⎤
⎦(2)
2.2.2. In e se and di ec kinema ic p oblem
Imposing he condi ion ha he ec o s C
i
B
i
ha e a cons an leng h o
L [31], he posi ion o he ac ua o s s
i
is ob ained as a unc ion o he
posi ion o he pla o m de ined by z
P
,
ψ
and θ.
s1=zP+Rsinθ+
L2− (Rcosθ−H+Dsin
ψ
sinθ)2
√
s2=zP+Dcosθsin
ψ
+
L2− (N−Dcos
ψ
)2
√
s3=zP−Rsinθ+
L2− (H−Rcosθ+Dsin
ψ
sinθ)2
√
(3)
On he o he hand, he posi ion o he pla o m is calcula ed as a
Fig. 2. Kinema ic model o he manipula o decoupling he mechanism om he ac ua o s.
F.J. Campa e al.
Mecha onics 105 (2025) 103278
3
unc ion o he ac ua o s posi ion s
i
, sol ing he ollowing sys em o
h ee nonlinea equa ions, one pe ba , using he New on–Raphson
me hod whe e he ini ial guess is he de aul posi ion a ze o il ing:
( − Dsin
ψ
sinθ−Rcosθ+H)2+ (zP+Rsinθ−s1)2=L2
(Dcos
ψ
−N)2+ (zP+Dcosθsin
ψ
−s2)2=L2
( − Dsin
ψ
sinθ+Rcosθ−H)2+ (zP−Rsinθ−s3)2=L2
(4)
2.2.3. Mechanism syn hesis
To achie e he desi ed wo kspace wi h o a ions up o ±15◦, a syn-
hesis p ocedu e has been ca ied ou ollowing he one desc ibed in
[31]. I has been conside ed ha he mo ing pla o m whe e he pa ien
is going o be posi ioned, mus ha e a minimum wid h o 500 mm and
leng h o 600 mm o allow se e al s anding poses as equi ed. A ending
o hese equi emen s, he ollowing dimensions a e achie ed: Table 1
2.3. Dynamics and con ol modelling
2.3.1. In e se dynamic p oblem
To be able o compu e he equi ed o ces o mo e he manipula o ,
he in e se dynamic p oblem o he manipula o has been sol ed,
ollowing he guidelines o [32], whe e he use o he P inciple o ene gy
equi alence is p oposed. Ins ead o s udying he dynamics o he com-
ple e pa allel mechanism, which is a complex ask due o he cons ain s
o closed kinema ic chains, he P inciple o ene gy equi alence s a es
ha he mechanical ene gy o each o he n subsys ems in which he
manipula o can be disassembled is equal o he ene gy o he comple e
manipula o , p o ided ha hey mo e as i hey we e assembled. The
subsys ems selec ed o he s udy o his mechanism ha e been he
mobile pla o m, he h ee ba s and he h ee ac ua o s. This me hod
allows analysing he dynamics o each subsys em in a simple way. Thus,
o a subsys em j whose gene alized coo dina es a e q
j
, he equa ions o
mo ion a e ob ained om he Lag ange’s equa ions:
j=d
d
∂
Lj
∂
˙
qj
−
∂
Lj
∂
qj
(5)
whe e
j
is he gene alized o ce on he elemen .
The condi ion ha all subsys ems mo e as he assembled mechanism
implies ha q
j
is a unc ion o he gene alized coo dina es o he
assembled mechanism q
s
. The e o e, he i ual displacemen s δq
j
a e
ela ed o he i ual displacemen δq
s
by he co esponding Jacobians:
δqj=
∂
qj
∂
qs
δqs=Jjδqs(6)
Also, he i ual wo k in he assembled mechanism is equal o he
o al o he i ual wo ks in all he subsys ems. The i ual wo k o he
join o ces is no compu ed as i is cancelled wi h he one in he adjacen
elemen .
δWs=δqT
s s=∑
n
j=1
δWj=∑
n
j=1
δqT
j j(7)
Subs i u ing Eq. (6) in Eq.7, he o ces
s
o apply on join s B
i
o mo e
he mechanism a e:
s=⎧
⎨
⎩
F1
F2
F3⎫
⎬
⎭
=∑
n
j=1
JT
j j(8)
2.3.2. Con ol
To model he global dynamic beha iou o he manipula o and he
con ol, he app oach p oposed by [33] has been ollowed. The dy-
namics o he manipula o a e decoupled om he dynamics o he ac-
ua o s, in such a way ha he o ces F
i
on he ac ua o able C
i
o mo e
he manipula o a e pe cei ed as a dis u bance F
id
by he ac ua o s, as i
is shown in Fig. 2.
The posi ion con ol o he manipula o is based on a join -space
con ol app oach, as i is shown in he Simulink model in Fig. 3. The
posi ion command in wo kspace coo dina es (z
c
,
ψ
c
, θ
c
) is ed h ough
he in e se kinema ic p oblem o calcula e he posi ion commands in he
join space (s
10
, s
20
, s
30
), which a e con olled using a cascaded PID
con ol o posi ion, eloci y and cu en . Then, he di ec kinema ic
p oblem is applied o calcula e he end e ec o posi ion (z
s
,
ψ
s
, θ
s
) om
he ac ua o s simula ed posi ion (s
1 s
, s
2 s
, s
3 s
).
Rega ding he modelling o he posi ion con ol o he ac ua o s in
Fig. 4, a P con olle is used in he posi ion loop, and a PI in he eloci y
and cu en loops. This con ol app oach was selec ed as a comp omise
be ween a low acking e o and a good esponse agains ex e nal dis-
u bances, which will come mainly om he weigh o he pa ien abo e
he o ce pla e apa om he ic ion in he mechanical componen s.
The linea posi ion command s
i0
is con e ed o angula posi ion
h ough he a iable s2 h =2
π
/p conside ing he lead p o he ball sc ew.
K
, K
p
and K
i
a e espec i ely, he p opo ional gain o he posi ion
con ol, and he p opo ional and in eg al gains o he mo o eloci y
con ol. As he cu en loop uns as e , he con e sion om cu en o
o que is assumed o be immedia e h ough he mo o o que cons an
K
, hus simpli ying he cu en loop.
A 1 DOF o a ional model has been assumed o model each i- h
ac ua o dynamics, conside ing only he o al ine ia, J
aci
in Fig. 4. The
dis u bances a e he o ce F
id
ha mechanism does on he ac ua o ,
which is con e ed o dis u bance o que conside ing he ball sc ew lead
p h ough he a iable h2s =p/(2
π
), he weigh o he ac ua o able
MG
aci
, and he Coulomb and iscous ic ion, which we e ini ially es i-
ma ed and hen iden i ied expe imen ally once he p o o ype was buil .
Finally, he simula ed angula posi ion o he mo o is con e ed again o
linea posi ion s
is
making use o he a iable h2s.
2.4. Mechanical design
A e conside ing o he equi emen s such as weigh , o al olume o
maximum ope a ing heigh , he design o he pa allel mechanism
con e ged o he model p esen ed in Fig. 5. The mobile pla o m consis s
o wo 7075-T6 aluminum pla es measu ing 800 ×800 mm, wi h a
minimum hickness o 12 mm and ein o cemen ibs wi h a maximum
hickness o 20 mm o inc ease i s bending s i ness, as shown in Fig. 6.
The uppe pla e is he one on which he pa ien s ands, and he lowe one
is a icula ed o he ba s. Fou uniaxial o ce senso s link he pla es, in
such a way ha he weigh o he op pla e and he pa ien is ansmi ed
o he bo om pla e h ough he senso s. The alida ion o he design in
e ms o s eng h and s i ness is shown in he ollowing subsec ions.
2.4.1. S eng h alida ions
To alida e sel -designed pa s, a FEM model o he whole mecha-
nism in Fig. 5 was de eloped. E e y link o he kinema ic chain was
assembled wi h he co esponding join o ensu e a ealis ic o ce
ansmission, see ed lines in Fig. 7. S a ing om he CAD geome y, a
simpli ied e sion wi hou cham e s o bol holes was de eloped, so a
simple mesh is ob ained, imp o ing he compu a ional cos . The wo
pla es o he mobile pla o m a e joined by he ou o ce senso s, which
a e subs i u ed by beams in he model.
The simpli ied FEM model consis s o a o al o 24 di e en pa s
wi hou conside ing he ixed elemen and ac ua o s, which a e eplaced
by bounda y condi ions. These pa s a e connec ed by 23 kinema ic
condi ions ha ep esen he join s. Fo he s a ic s udy, a e se e al
ini ial i e a ions i has been shown ha he wedges and he sphe ical
Table 1
Dimensions o he kinema ic model o he pla o m.
L (mm) R (mm) D (mm) H (mm) N (mm)
400 350 600 450 700
F.J. Campa e al.
Mecha onics 105 (2025) 103278
4
join ha dly unde go any de o ma ion, so as i can be seen in Table 2,
hey will be conside ed as igid solids.
A sensi i i y analysis o he mesh has been pe o med o selec he
elemen size in each pa o ob ain a good compu a ional accu acy. The
esul ing model consis s o 230,000 highe o de e ahed on elemen s.
The a e age compu a ion ime o each analysis posi ion is o he o de
o 2 min on a compu a ion s a ion wi h 4 co es wo king in pa allel and
32 Gb o RAM memo y.
Fig. 3. Mecha onic model o he manipula o .
Fig. 4. Dynamic model o he i =1 ac ua o .
Fig. 5. OREKA pla o m CAD model.
Fig. 6. Loca ion o he senso s in he mobile pla o m and de ails o he ibs.
F.J. Campa e al.
Mecha onics 105 (2025) 103278
5
S eng h simula ions ha e been conduc ed conside ing a 1.5 kN
e ical o ce in he mos c i ical o ien a ion o he mechanism. Fo a
sa e-side design, he applica ion poin o he o ce has been changed
along he op pla e un il he c i ical posi ion o each elemen was
achie ed.
As bo h he ma e ials used in he p o o ype a e conside ed duc ile,
Von Mises equi alen s ess (V-M s ess) has been calcula ed. Consid-
e ing he s eng h o he espec i e ma e ial, he sa e y ac o (SF) o
each pa was calcula ed o alida e he s eng h o he p o o ype, see
Table 2. All he sa e y ac o s a e bigge han uni y, alida ing hen he
p o o ype. In ac , he highes sa e y ac o s in some pa s a e due o he
de o ma ion equi emen in Sec ion 2.1.
2.4.2. S a ics: e ical s i ness
I is well-known ha he s i ness o pa allel manipula o s a ies o e
he wo kspace. To ensu e ha he s i ness design equi emen is me , an
analysis o he s i ness will be ca ied ou by means o a FEM model. Fo
he s udy, only he elemen s o he s i ness ma ix in ol ing he e ical
load componen F
z
will be conside ed since he o igin o he applied
o ces comes om he weigh o he pa ien s.
Fu he mo e, he s i ness analysis is going o be limi ed o K
zz
as
epo ed du ing he equi emen s s age. When pa ien s s ep on o he
pla o m o pe o m he balance es s, his s i ness componen has
special signi icance in e ms o con idence.
Bo h he base ame and he a el guides o he d i es ha e a e y
high s i ness, so ha he analysis o he s uc u al beha iou in he
wo king space can be ca ied ou independen ly o he Z coo dina e a
which he manipula o is posi ioned. The s udy shall conside only he
possible dependence o he s uc u al beha iou o he manipula o on
he o ien a ions.
Likewise, he ope a ional wo kspace o o a ion (OWR) is de ined by
he maximum and minimum o ien a ions ha he manipula o can each
o ul il i s unc ion wi hou in e e ence be ween i s pa s, (-15◦, +15◦)
in he case o he p esen 3PRS manipula o . Consequen ly, o he s udy
o he s uc u al beha iou p oposed he e, he ope a ing ange o o a-
ion in each o he o ien a ions has been disc e ised in in e als o 3◦,
esul ing in 11 alues o each o a ion. This makes a o al o 121
analysis poin s. Fig. 8 shows K
zz
a ia ion o e he OWR, whe e K
zz
alue is ep esen ed in he z axis as well as wi h he colo ba . Also, he
mos ep esen a i e alues o K
zz
a e p esen ed in Table 3:
Fo he K
zz
componen , he maximum alue is a loca ion (3◦,15◦),
while he minimum is a loca ion (-15◦,15◦). F om he alues in Table 3,
he highes alue is wice as high as he lowes alue and, in addi ion, he
mean and s anda d de ia ion indica e ha he s i ness alues ac oss he
map a e high. Howe e , as can be seen in Fig. 8, he s i ness has a
maximum o a pi ch o
ψ
=3◦and dec eases mo e sha ply when
dec easing
ψ
han when inc easing i . Ano he no ewo hy aspec is ha
he minimum o he unc ion occu s o a alue o
ψ
=-15◦whe e, in
addi ion, i p esen s a maximum sensi i i y o he a ia ion o he
pa ame e θ.
E alua ion exe cises o s oke pa ien s will comp ise independen
o ien a ion a ia ions. Mainly wo a ia ions, θ a ying be ween
(-13◦,6◦) while
ψ
emains ze o and, hen θ cons an and
ψ
a ying up o
maximums o (-8◦,8◦). Fo he mos common ajec o ies o eseen o he
use o he machine in equilib ium ea men and diagnosis, see second
ow in Table 3, he s i ness is high and uni o m du ing he o a ions
associa ed wi h he mos common mo emen s. Fo a pa ien o abou
150 kg, as equi ed, he pla o m would unde go displacemen s smalle
han 1 mm in mos o he exe cises. This esul e i ies he imposed
equisi e in he design s age.
2.5. Fo ce pla e design and COP measu emen
To measu e he posi ion o he pa ien ’s COP du ing he es s, he
o ce pla e is composed by he wo aluminium pla es measu ing 800 ×
800 mm and he ou uniaxial o ce senso s. The senso s measu e he
o ce in he di ec ion pe pendicula o he pla o m su ace, so ha by
combining hei alues i is possible o deduce he posi ion o he COP.
The senso s a angemen in he o ce pla e can be seen in Fig. 6.
To de e mine whe e he COP is loca ed on he op pla e su ace, a
mobile e e ence sys em X’Y’Z’ is loca ed on he op pla e su ace, in he
cen e O’ o he ec angle de ined by he senso s loca ed in S
i
. I s di-
mensions a e 2a x 2b, whe e a =250 mm and b =300 mm. I mus be
aken in o accoun ha he pla o m will be il ed, and he ine ial o ces
and weigh on he op pla e will be measu ed also, apa om he weigh
o he pa ien .
Assuming ha mg and m a e espec i ely he weigh and he mas o
Fig. 7. Symbolic ep esen a ion o he kinema ic cons ain s o he FEM model.
Table 2
Numbe o pa s and ma e ials o FEM model and hei pe o mance. (*) Fo
comme cial elemen s.
Pa Ma e ial Elas ic
beha iou
V-M
s ess
(MPa)
S eng h
(MPa)
SF
Uppe pla e 7075-T6
Aluminium
Elas ic 39.9 500 12.53
Senso (*) AISI 1045
S eel
Beam As pe da ashee ins uc ions
Lowe pla e 7075-T6
Aluminium
Elas ic 2.4 500 208.33
Wedge 7075-T6
Aluminium
Rigid 1.11 450 405.41
Sphe ical
join (*)
AISI 1045
S eel
Rigid As pe da ashee ins uc ions
T-link AISI 1045
S eel
Elas ic 201.75 450 2.23
Ball bea ing
uni (*)
AISI 1045
S eel
Elas ic As pe da ashee ins uc ions
Suppo
pla e
7075-T6
Aluminium
Elas ic 36.62 500 13.65
Fig. 8. Kzz s i ness map in he OWR.
F.J. Campa e al.
Mecha onics 105 (2025) 103278
6
he op pla e and w is he esul an e ical o ce ecei ed in he con ac
wi h he pa ien , hei p ojec ion o e he pla e su ace in a il ed po-
si ion is calcula ed using he ansposed o a ion ma ix R
T
in Eq. (2). In
s a ic condi ions and wi h he pla e ho izon al, w should be equal o he
weigh o he pa ien , bu in a gene al posi ion i is exp essed as ollows.
w=⎧
⎨
⎩
wxʹ
wyʹ
wzʹ⎫
⎬
⎭
=RT⎧
⎨
⎩
0
0
−w⎫
⎬
⎭
=⎧
⎨
⎩
wsinθ
−wsin
ψ
cosθ
−wcos
ψ
cosθ⎫
⎬
⎭
(9)
Hence, s udying he ansla ional dynamics o he op pla e in Z’
di ec ion, pe pendicula o he pla e, Eq. (10) is ob ained, whe e
iz
a e
he o ces measu ed by he senso s and a
Gz’
is he Z’ componen o he
accele a ion o he mass cen e o he op pla e.
∑
4
i=1
izʹ− (mg +w)cos
ψ
cosθ=maGzʹ(10)
The accele a ion o he mass cen e o he op pla e is calcula ed as
ollows as a unc ion o he accele a ion o P, he o igin o he mobile
e e ence sys em UVW in Fig. 2, he angula eloci y
ω
and angula
accele a ion
α
o he pla e and he ela i e posi ion ec o PG be ween
he mass cen e and P:
aG=aP+
α
×PG +
ω
× (
ω
×PG)(11)
To ob ain a
P
, i s , he posi ion o P in he global ame XYZ is:
OP =⎧
⎨
⎩
−Dsin
ψ
sinθ
0
z⎫
⎬
⎭
(12)
Di e en ia ing:
d2OP
d 2=⎡
⎢
⎢
⎣
cos
ψ
sinθsin
ψ
cosθ0
0 0 0
0 0 1
⎤
⎥
⎥
⎦⎧
⎪
⎪
⎨
⎪
⎪
⎩
¨
ψ
¨
θ
¨
z
⎫
⎪
⎪
⎬
⎪
⎪
⎭
+...
...⎡
⎢
⎢
⎣
−sin
ψ
sinθ2cos
ψ
cosθ
0 0
0 0
⎤
⎥
⎥
⎦{˙
ψ
2+
˙
θ2
˙
ψ
˙
θ}
(13)
Applying he ansposed o a ion ma ix, he accele a ion o P in he
mo ing ame is:
aP=RTd2OP
d 2(14)
Finally, he angula eloci y and accele a ion in he mo ing ame
a e:
ω
=⎧
⎨
⎩
˙
ψ
˙
θcos
ψ
−
˙
θsin
ψ
⎫
⎬
⎭
α
=⎧
⎨
⎩
¨
ψ
¨
θcos
ψ
−
˙
θ˙
ψ
sin
ψ
−(¨
θsin
ψ
+
˙
θ˙
ψ
cos
ψ
)⎫
⎬
⎭
(15)
F om Eq. (10), he o ce w ecei ed om he pa ien is:
w=∑
4
i=1 iz −mgcos
ψ
cosθ−maGzʹ
cos
ψ
cosθ(16)
Rega ding he o a ional dynamics o he op pla e, he heo em o
he angula momen um is applied in X’ and Y’. Taking in o accoun he
symme y in Y’Z’ o he pla e, and he ac ha he cen e o g a i y G o
he pla e is in ha plane, see Table 4, he angula momen um H
G
is:
HG= { e1e2e3}⎡
⎢
⎢
⎣
IxʹG0 0
0IyʹG−CxʹG
0−CxʹGIzʹG
⎤
⎥
⎥
⎦⎧
⎪
⎪
⎨
⎪
⎪
⎩
ω
xʹ
ω
yʹ
ω
zʹ
⎫
⎪
⎪
⎬
⎪
⎪
⎭
=...
... =⎧
⎪
⎪
⎨
⎪
⎪
⎩
IxʹG
ω
xʹ
IyʹG
ω
yʹ−CxʹG
ω
zʹ
−CxʹG
ω
yʹ+IzʹG
ω
zʹ
⎫
⎪
⎪
⎬
⎪
⎪
⎭
(17)
whe e: e
1
, e
2
, e
3
a e he uni ec o s o he mo ing ame; I
x’G
, I
y’G,
I
z’G
a e he momen s o ine ia wi h espec o he axis o a ame wi h o igin
in G and pa allel o he mo ing ame; C
x’G
is he p oduc o ine ia wi h
espec o he wo planes o ha ame ha in e sec in he x
G’
axis. Upon
di e en ia ion:
dHG
d =⎧
⎨
⎩
IxʹG
α
xʹ
IyʹG
α
yʹ−CxʹG
α
zʹ
−CxʹG
α
yʹ+IzʹG
α
zʹ⎫
⎬
⎭
+
ω
×HG(18)
The o ce w is applied on he COP, poin Q, and c ea es a momen
wi h espec o he mass cen e G ha is exp essed as:
NGW=GQ ×w=⎧
⎨
⎩
GQyʹwzʹ−GQzʹwyʹ
GQzʹwxʹ−GQxʹwzʹ
GQxʹwyʹ−GQyʹwxʹ⎫
⎬
⎭
(19)
Conside ing ha he COP will be on he op pla e su ace, z’
Q
=0, and
ha he dis ance O’G
x’
is ze o, as shown in Table 4, he ec o GQ is:
GQ = − OʹG+OʹQ=⎧
⎨
⎩
xʹQ
−OʹGyʹ+yʹQ
−OʹGzʹ⎫
⎬
⎭
(20)
The op pla e lies on he ou senso s, which impose a es ic ion o
he displacemen in X’ and Y’ and measu e he o ce in Z’. Hence, he
momen o hose o ces wi h espec o he mass cen e is:
NGF=∑
4
i=1
GSi× i=∑
4
i=1⎧
⎨
⎩
GSiyʹ izʹ−GSizʹ iyʹ
GSizʹ ixʹ−GSixʹ izʹ
GSixʹ iyʹ−GSiyʹ ixʹ⎫
⎬
⎭
(21)
Hence, he heo em o he angula momen um in X’ and Y’ esul s in
he ollowing equa ions:
xʹ)(−OʹGyʹ+yʹQ)wzʹ+OʹGzʹwyʹ+...
... ∑
4
i=1(GSiyʹ izʹ−GSizʹ iyʹ)=(dHG
d )xʹ
yʹ) − OʹGzʹwxʹ−xʹQwzʹ+...
... ∑
4
i=1(GSizʹ ixʹ−GSixʹ izʹ)=(dHG
d )yʹ
(22)
In Eq. (22), he e ms dependen on he o ces ix’ and iy’ a e
neglec ed because hey a e no measu ed by he uniaxial o ce senso s.
This is a simpli ica ion ha in oduces an e o in he measu emen ,
howe e , hey a e ela i ely small e ms due o he small il o he able
Table 3
Cha ac e is ic alues o K
zz
in he OWR.
K
zz
(N/mm) Min. Max. A . De .
-15◦<
ψ
<15◦, -15◦<θ <15◦1.59⋅10
3
3.12⋅10
3
2.80⋅10
3
283.96
-13◦<
ψ
<6◦,
-8◦<θ <8◦
2.93⋅10
3
3.04⋅10
3
2.96⋅10
3
30
Table 4
Ine ial and geome ical p ope ies o he o ce pla e.
i- h senso i =1i =2i =3i =4
GS
ix’
(mm) -250 -250 250 250
GS
iy’
(mm) -317.15 282.85 282.85 −317.15
GS
iz’
(mm) -1.69
O’G (mm) O’G
x’
O’G
y’
O’G
z’
017.15 -7.31
I
axisG
(kgm
2
)I
x’G
I
y’G
I
z’G
0.4758 0.4639 0.939
C
x’G
(kgm
2
) -0.0001943
F.J. Campa e al.
Mecha onics 105 (2025) 103278
7
and he ac ha hey a e mul iplied by GSiz’, wo o de s o magni ude
smalle han GSiz’ and GSiz’, as seen in Table 4.
As a esul , he posi ion o he COP depends on he pla o m mass
cen e posi ion, he ine ial e m due o he pla e mo ion, and he o ces
measu ed. Subs i u ing Eq. (9) in o Eq. (22), he coo dina es o he COP,
x’
Q
and y’
Q
, a e calcula ed as a unc ion o he pla o m posi ion as:
xʹQ=OʹGzʹ anθ
cos
ψ
+...
... 1
wcos
ψ
cosθ[(dHG
d )yʹ
+∑
4
i=1(GSixʹ izʹ)]
yʹQ=OʹGyʹ−OʹGzʹ an
ψ
−...
1
wcos
ψ
cosθ[(dHG
d )xʹ
−∑
4
i=1(GSiyʹ izʹ)]
(23)
2.6. Desc ip ion o he p o o ype
Once he mechanical design o all he elemen s has been comple ed,
he comme cial elemen s a e selec ed. Th ee 3.6 kW synch onous mo-
o s om he manu ac u e B&R model 8LSA55.DA030S000-3 a e
selec ed, which a e con olled by ACOPOS P3 con olle s, model
8EI013HWS10.0200-1. Each mo o has a sa e y b ake ha is ac i a ed in
he e en o a loss o powe o he machine. These mo o s a e connec ed
o ball bea ing linea guides model EGC-BS-KF-GK om FESTO wi h a
pi ch o 40 mm/ e . The mo o -guide assembly comple es he e ical
ac ua o equi ed o d i e he mechanism. Each e ical ac ua o has a
maximum linea speed o 2 m/s, and a maximum con inuous o ce o
1800 N, i.e. hey ha e been o e sized so ha each one sepa a ely could
suppo up o 180 kg o weigh . The ods connec ing he pla o m and
he linea guides a e connec ed o he pla o m by Hephais SRJ024C
sphe ical join s, and o he linea guides by SKF SY20TR ball bea ing
e olu ion join s. Mo ion con ol is p o ided by he con olle on he
B&R panel PC 3100. This con ol is closed, i.e. he use o he machine is
only able o p og am a se ies o p ede e mined mo emen s, ag eed as
sui able be ween he UPV/EHU eam and Hospi al Go liz. The panel PC,
mo o con olle s and main elec ical componen s a e ins alled in a
sepa a e cabine so he medical s a can use i mo e com o ably.
As explained in Sec ion 2.5, he mobile pla o m is a o ce pla e and is
esponsible o aking measu emen s o de e mine he posi ion o he
COP du ing he exe cises p og ammed in he machine. The ou senso s
ins alled in he pla o m a e In e ace GWMC based on s ain gauges
wi h a maximum ange o up o 2000 N. The senso s signals a e
ampli ied by In e ace SGA ampli ie s, in eg a ed in o he machine
s uc u e.
On he o he hand, h ee senso ized hand es s ha e been ins alled.
One is on al, and wo a e la e al a abou 45◦, so ha he pa ien can
hold on i necessa y o as indica ed in he exe cise. Each o he suppo s
has h ee uniaxial o ce senso s based on s ain gauges o 250 N ange
model TE ELAF-T1 M, ha e lec he ex en o which he pa ien leans
on he hands o main ain balance du ing he exe cise. The da a collec ed
by all he senso s is p o ided o he ehabili a ion doc o ia USB, who
has a so wa e o p ocessing hem, accessing quan i a i e pa ame e s
abou he e olu ion o he pa ien ’s pe o mance du ing he ehabili a-
ion phase.
Finally, as can be seen in Fig. 9, he mechanism is suppo ed by a
igid s uc u e o med by h ee le els o 4 and 8 mm ho izon al s eel
pla es joined by 4 mm hick s eel b aces in S . And ew’s C oss. Thus, he
machine dimensions a e 1300 ×970 ×1400 mm wi h a o al weigh o
300 kg, which gua an ee he s abili y o he assembly. The whole as-
sembly is p o ec ed by an aluminium ai ing which ensu es ha he
pa ien canno come in o con ac wi h any o he elec ical elemen s o
he sys em. To ensu e he sa e y o he pa ien , addi ional secu i y sen-
so s ha e been ins alled. In his way, he pe ime e o he mo ing
pla o m is su ounded by a physical ba ie on which pho oelec ic
ba ie senso s (in blue in Fig. 10) ha e been ins alled, so ha i he
pa ien , o any eason, inse s a limb be ween he pla o m and he
ai ing, he beam is in e up ed, and he sys em s ops wi hou any ha m
o he pa ien .
3. Expe imen al alida ion o he manipula o dynamics
The mecha onic model used in he design has been alida ed o
check i i can p edic he dynamic beha iou o he manipula o . The
pa ame e s used a e in Table 5. Fi s , he ine ial pa ame e s ha e been
ob ained om he CAD model and manu ac u e da ashee s: J
i
is he
mo o s ine ia, m
i
is he mass o he couplings be ween he ac ua o s and
he mechanism, m
BC
and I
BC
a e he mass o he ba s and hei ine ia in
hei cen e o g a i y, m
MP
is he mobile pla o m o al mass and I
x’G,
I
y’G,
I
z’G
he ine ia o he mobile pla o m in he cen e o g a i y in axis
X’, Y’ and Z’. Also, he iscous c
i
and Coulomb ic ion
τ
Ci
o he mo o s
ha e been iden i ied expe imen ally by unning a se ies o es s a
di e en cons an angula eloci ies wi h he mo o s in isola ion.
Rela ing he o que and speed, he end is i ed o a linea ela ion
whe e he slope is he iscous ic ion and he alue a ze o speed is he
Coulomb ic ion. Finally, he ball sc ew lead p and he con ol gains
used, which a e he same o he h ee ac ua o s, a e shown.
Se e al es s ha e been conduc ed ep oducing he exe cises p o-
posed by he medical s a wi h no pe son on he o ce pla e. They a e
pe iodic o a ions be ween wo angula posi ions o he able. He e, he
esul s o wo exe cises a e shown: il ing only in an e o-pos e io (AP,
a ound X axis) di ec ion om -12.5◦ o 6.4◦and il ing in medio-la e al
(ML, a ound Y axis) di ec ion om -8◦ o 8◦, bo h wi h a apezoidal
Fig. 9. Schema ic ep esen a ion o he p o o ype.
F.J. Campa e al.
Mecha onics 105 (2025) 103278
8
eloci y p o ile. All he expe imen al signals ha e been sampled a 50
Hz, and low pass il e ed a 10 Hz. Two compa isons be ween he model
and he expe imen s a e shown: in Fig. 11 he o que o he mo o s and
in Fig. 12 he acking e o in linea coo dina es s
i
, ha is, con e ing
he angula e o measu ed in he mo o encode o a linea e o
applying he lead p o he ball sc ew.
The esul s ob ained in bo h mo ions by he simula ion a e simila o
he measu emen s, conside ing all he simpli ica ions made and he ac
ha he dynamic pa ame e s used a e he heo e ical ones. In ac , only
ic ion in he mo o s has been expe imen ally iden i ied, neglec ing he
ic ion in he join s, o he a ia ions in mass o ine ia due o
geome ical de ia ions du ing he manu ac u ing o he componen s.
Rega ding he o que, he highes e o s appea clea ly in he second
ac ua o , whe e an a e age e o up o 0.5 Nm appea s. As o he
acking e o , he e a e peaks o e o up o 1 mm in he measu ed
posi ion ha a e ela ed o a small ib a ion ha appea s in he
manipula o in he in e sions o he mo emen . I is due o he ine ial
o ces exci ing he i s modal equency o he manipula o a ound 12
Hz, bu hey a e no dange ous o he pa ien om he clinical poin o
iew. In gene al, he beha iou o he manipula o in isola ion is in line
wi h he model p edic ion, bu he esul s could be imp o ed applying a
dynamic pa ame e iden i ica ion ins ead o using he heo e ical dy-
namic pa ame e s [34].
4. Expe imen al e alua ion o he o ce pla e pe o mance
4.1. In luence o noise and mo ion ela ed o ces
The o ce pla e was e alua ed i s in e ms o noise ejec ion in s a ic
and dynamic condi ions, ha is, du ing mo ion. Rega ding he noise, as
he o ce signal measu ed in he senso s a e subjec o elec omagne ic
in e e ence, he measu ed posi ion o he Cen e o P essu e p esen s
oscilla ions e en wi h no load abo e he pla e. Unde dynamic condi-
ions, he mo ion o he pla o m i sel also in oduces ine ial and
g a i a ional e ec s ha a e measu ed by he o ce senso s and mus be
compensa ed as shown in Sec ion 2.5. The expe imen al me hodology
has been he ollowing. Fi s , he COP coo dina es we e measu ed du ing
he execu ion o he exe cises shown in Table 6. These ep esen he
opposi e ex emes in e ms o ampli ude and eloci y o mo ion om a
o al o 25 p e-p og ammed exe cises a ailable.
Then, o e alua e he in luence o noise and mo ion ela ed e ec s,
he s anda d de ia ion o he COP coo dina es in each es was calcu-
la ed. I was expec ed ha he signal- o-noise a io would depend on he
load applied on he pla e, he highe he o ce measu ed, he lesse he
in luence o noise. To p o e ha , in s a ic condi ions, 6 combina ions o
calib a ed masses om 5.2 o 41.8 kg we e loca ed on he pla o m and
measu ed o 30 s. To loca e he masses a ix u e pla e wi h holes e e y
25 mm wi h a ole ance o 0.1 mm was used. The ix u e pla e was
cen ed in he ec angle o med by he senso s wi h he aid o a cus om
ix u e designed o place i using he con ou o he op pla e as a
e e ence, as i is shown in Fig. 13. All he masses we e machined wi h a
lug in he cen e o one side o ix hem in he holes o he ix u e pla e.
The esul is shown as Ex. 0 in Fig. 14 ollowing he nomencla u e o
Table 6. The igu e, in loga i hmic axis, ep esen s he s anda d de ia-
ion o he COP coo dina es as a unc ion o he mass loca ed on he
pla o m o he condi ions in Table 6. Wi h a 5.2 kg mass, he s anda d
de ia ion eaches 0.72 mm o he y coo dina e o he COP and alls o
0.08 mm o bo h coo dina es wi h 41.8 kg. Hence, he in luence o noise
in s a ic condi ions is below 0.1 mm o a weigh ha could be consid-
e ed as low o an adul pa ien and i will be e en lowe o weigh ie
pa ien s.
In dynamic condi ions, he calib a ed masses we e no used as he
heigh o hei mass cen e combined wi h he il ing o he able
in oduced a mo ion in he COP ha did no allow isola ing he in luence
o he noise and he mo ion. So, all he measu emen s we e done unde
he condi ions in Table 6 wi h no mass abo e he able and, ins ead, a
i ual mass loca ed in he cen e o he able was simula ed adding ¼ o
he i ual weigh o he o ce signal o each senso .
In Fig. 14 he s anda d de ia ion in he measu ed COP coo dina es is
Fig. 10. P o o ype buil and ins alled in Hospi al Go liz.
Table 5
Ine ial and geome ical p ope ies o he elemen s o he manipula o . PID
con olle gains.
i- h ac ua o i =1i =2i =3
J
i
(kg⋅m
2
)0.0025
m
i
(kg) 1.188
m
BC
(kg) 1.186
I
BC
(kg⋅m
2
)0.025
m
MP
(kg) 52.31
I
x’G,y’G,z’G
(kg⋅m
2
) I
x’G
=2.539 I
y’G
=2.51 I
z’G
=5.034
c
i
(Nm⋅s/ ad) 0.00325 0.00155 0.00125
τ
Ci
(Nm) 0.49 0.5 0.49
MG
aci
, (N) 62.8 73.82 87.9
p (m/ e ) 0.04
K
(1/s) 50
K
p
(A⋅s/ ad) 2
K
i
(A/ ad) 40
F.J. Campa e al.
Mecha onics 105 (2025) 103278
9
