Resea ch pape
Kinema ic and dynamic analysis o spa ial mul ibody sys ems
based on a o mula ion wi h ully Ca esian coo dina es and a
gene ic igid body
S´
e gio B. Gonçal es
a
, I o Roupa
b
, Paulo Flo es
c
, Miguel Ta a es da Sil a
a,*
a
IDMEC, Ins i u o Supe io T´
ecnico, Uni e sidade de Lisboa, Lisboa, Po ugal
b
ITI/LARSyS, Ins i u o Supe io T´
ecnico, Uni e sidade de Lisboa, Lisboa, Po ugal
c
CMEMS‑UMinho, Depa men o Mechanical Enginee ing, Uni e si y o Minho, Guima ˜
aes, Po ugal
ARTICLE INFO
Keywo ds:
Mul ibody dynamics o mula ions
Fully Ca esian coo dina es
Gene ic igid body
Kinema ic analysis
Dynamic analysis
ABSTRACT
This wo k in oduces he Fully Ca esian Coo dina es Fo mula ion wi h a Gene ic Rigid Body
(FCC-GRB), a no el global mul ibody o mula ion o h ee-dimensional mechanical sys em
analysis. The o mula ion’s in insic cha ac e is ics a e ho oughly de ailed and compa ed wi h
o he widely-used global o mula ions, enabling i s applica ion in bo h kinema ic and dynamic
analysis o complex mechanical sys ems and as a eaching ool in ad anced mul ibody dynamics
cou ses.
FCC-GRB o mula ion is ounded on wo main p emises: mul ibody sys ems a e desc ibed using
only Ca esian coo dina es, and he igid bodies a e modeled wi h a ixed and p ede e mined
s uc u e. Consequen ly, he kinema ic cons ain s a e desc ibed by lowe -deg ee equa ions and
he sys em mass ma ix is highly spa se. Addi ionally, he in oduc ion o he gene ic igid body
simpli ies he modeling p ocess by making he de ini ion o he bodies independen o sys em
opology. To educe he numbe o gene alized coo dina es, a educed modeling app oach using
less coo dina es o desc ibing he gene ic igid body is also in oduced and compa ed wi h he
ully-de ined al e na i e.
The o mula ion’s accu acy was alida ed h ough o wa d dynamic analysis o benchma k
p oblems. Simula ions demons a ed excellen ag eemen wi h e e ence da a, wi h bo h
modeling app oaches yielding compa able kinema ic esul s. The educed app oach o e ed as e
compu a ional pe o mance, pa icula ly in mo e complex models.
Abb e ia ions Desc ip ion
2D Two-dimensional
3D Th ee-dimensional
(con inued on nex page)
* Co esponding au ho .
E-mail add esses: [email p o ec ed] (S.B. Gonçal es), [email p o ec ed] (I. Roupa), [email p o ec ed]
(P. Flo es), [email p o ec ed] (M.T. Sil a).
Con en s lis s a ailable a ScienceDi ec
Mechanism and Machine Theo y
jou nal homepage: www.else ie .com/loca e/mechm
h ps://doi.o g/10.1016/j.mechmach heo y.2025.105955
Recei ed 23 Decembe 2024; Recei ed in e ised o m 3 Feb ua y 2025; Accep ed 10 Feb ua y 2025
Mechanism and Machine Theo y 209 (2025) 105955
A ailable online 8 Ma ch 2025
0094-114X/© 2025 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY license
( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
(con inued)
AR Angula Rela ion
CAR Cons an Angula Rela ion
CP Coinciden Poin
CV Coinciden Vec o
DAE Di e en ial Algeb aic Equa ions
DC D i e Cons ain
DoF Deg ee-o -F eedom
EoM Equa ions-o -Mo ion
FCC Fully Ca esian Coo dina es
FCC-GRB Fully Ca esian Coo dina es wi h a Gene ic Rigid Body
FD Fo wa d Dynamics
GRB Gene ic Rigid Body
GSJ G ounded Sphe ical Join
ICC In aclass Co ela ion
KC Kinema ic Cons ain
KJ Kinema ic Join
ODE O dina y Di e en ial Equa ions
OV O ien a ion o a Gene ic Vec o
PJ P isma ic Join
PRJ Pinned Re olu e Join
RB Rigid Body
RJ Re olu e Join
RoM Range-o -Mo ion
RGRB Reduced Gene ic Rigid Body
SJ Sphe ical Join
TP T ansla ion o a Gene ic Poin
UJ Uni e sal Join
Symbol (La in) Desc ip ion
03, I3Null and iden i y ma ices
b
τ
, b−
τ
Momen a ms o he equi alen o ce pai
τ
and −
τ
cP
1, cP
2, cP
3Scaling ac o s ep esen ing he local coo dina es o poin P in he local e e ence ame o he igid body
CP
iCons an ans o ma ion ma ix ha desc ibes he kinema ics o a gene ic poin P wi h espec o body i
Cs
iCons an ans o ma ion ma ix ha desc ibes he kinema ics o a gene ic ec o s wi h espec o body i
Cs1s2Cons an ans o ma ion ma ix ha ela es he gene ic ec o s s1 and s2
e
x
, e
y
, e
z
Basis ec o s ha de ine he global e e ence ame o he mul ibody sys em
Gene ic ex e nal o ce
k
RIn e nal eac ion o ces associa ed o he kinema ic cons ain o ype k
τ
, −
τ
Equi alen o ce pai o he ex e nal momen o o ce
τ
gGene alized o ces o he sys em
g2 iVeloci y-dependen ine ial o ce o he igid body i de ined in i s educed o m
g
iGene alized equi alen o ce o a concen a ed ex e nal o ce applied in body i
g
τ
iGene alized equi alen o ce o an ex e nal momen o o ce
τ
applied in body i
gΦGene alized in e nal o ces
gΦk
iGene alized in e nal o ces o he kinema ic cons ain o ype k wi h espec o body i
Iii Componen s o he ine ia enso o he igid body i
miMass o he igid body i
MMass ma ix o he sys em
MiMass ma ix o he igid body i
M2 iEqui alen mass ma ix o he igid body i de ined in i s educed o m
nb, nc, nq, n Numbe o bodies, kinema ic cons ain s, gene alized coo dina es and ec o s o he sys em
PGene ic poin P
P*Re e ence poin P*
OiO igin o he local e e ence ame o body i
q, ˙
q, ¨
qGene alized posi ions, eloci ies and accele a ions o he sys em
q3 i, ˙
q3 i, ¨
q3 iPosi ion, eloci y and accele a ion o he ully-de ined equi alen ec o o he educed igid body i
˙
q*
iGene alized i ual eloci ies o he igid body i
˙
q*
3 iGene alized i ual eloci ies o he igid body i desc ibed in e ms o he ully-de ined equi alen o m
P,
˙
P,
¨
PPosi ion, eloci y and accele a ion ec o s o he gene ic poin P in he global e e ence ame
P*, ˙
P*, ¨
P*Posi ion, eloci y and accele a ion ec o s o a e e ence poin P* in he global e e ence ame
˙
*
PVi ual eloci ies o he gene ic poin P
s, ˙
s, ¨
sPosi ion, eloci y and accele a ion ec o s o a gene ic uni ec o s
s*, ˙
s*, ¨
s*Posi ion, eloci y and accele a ion ec o s o a e e ence uni ec o s*
SCons an ans o ma ion ma ix ha con e s he educed igid body i in i s ully-de ined equi alen o m
Time
TKine ic ene gy
ui, ˙
ui, ¨
ui i, ˙
i, ¨
i wi, ˙
wi, ¨
wiPosi ion, eloci y and accele a ion o he gene ic igid body ec o s u, and w o body i in he global e e ence ame
uʹ
i, ʹ
i, wʹ
iPosi ion o he gene ic igid body ec o s u, and w in he local e e ence ame o body i
uSkew-symme ic ma ix o ec o u used o compu e he c oss p oduc
(con inued on nex page)
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
2
(con inued)
VPo en ial ene gy
Vi,Vi, ˙
ViT ans o ma ion ma ices ha con e he educed igid body i in i s ully-de ined equi alen o m
W*
iVi ual Powe gene a ed by he in e nal o ces o body i
˙
yS a e ec o including he gene alized eloci ies and accele a ions u ilized in he di ec in eg a ion me hod
Symbol (G eek) Desc ip ion
α
, βBaumga e s abiliza ion coe icien s
γRigh -hand-side ec o o he accele a ion equa ions o he sys em
γkCon ibu ions o he igh -hand-side ec o o he accele a ion equa ions o he kinema ic cons ain o ype k
θs1s2Angle be ween wo gene ic uni ec o s s1 and s2
θ*
s1s2,
˙
θ*
s1s2, ¨
θ*
s1s2Angula posi ion, eloci y and accele a ion o a e e ence angle θ* be ween wo gene ic uni ec o s s1 and s2
λLag ange mul iplie s o he sys em
λkLag ange mul iplie s o he kinema ic cons ain o ype k
ν
Righ -hand-side ec o o he eloci y equa ions o he sys em
ν
Vec o o he pa ial de i a i es o
ν
wi h espec o
ν
kCon ibu ions o he igh -hand-side ec o o he eloci y equa ions o he kinema ic cons ain o ype k
ξ,
η
, ζBasis ec o s ha de ine he local e e ence ame o a gene ic igid body
ξP
i,
η
P
i, ζP
iCoo dina e o he poin P in he ξ,
η
and ζ axes o he local e e ence ame o body i
ρ
iMass densi y o igid body i
τ
Gene ic ex e nal momen o o ce
τ
τ
in
kIn e nal join momen o o ce associa ed o join k
τ
in
kmIn e nal momen o o ce o join k associa ed o he m- h angula d i ing cons ain
Φ, ˙
Φ, ¨
ΦVec o o he kinema ic cons ain s and espec i e i s de i a i e and second de i a i e wi h espec o ime
ΦqJacobian ma ix o he sys em
Φ Vec o o he pa ial de i a i es o Φ wi h espec o ime
ΦkKinema ic cons ain equa ion o ype k
Φk
qCon ibu ions o he Jacobian ma ix om he kinema ic cons ain o ype k
ΩiGeome ic domain o he igid body i
1. In oduc ion
Mul ibody dynamics me hodologies ha e been success ully applied in he kinema ic and dynamic analysis o complex mechanical
sys ems, as hey allow o e icien modeling and simula ion o nume ous p oblems, p o iding esul s wi h a high physical meaning [1].
In ac , mul ibody me hodologies a e equen ly applied in many di e en a eas, namely ehicle dynamics [2–4], mecha onics and
obo ics [5–7], mechanisms and machines [8–12], biomechanics and medical de ices [13–18], co-simula ion wi h ini e elemen s
me hod [19–21], media and ideogames [22], and in eal- ime p oblems [23–27].
O e he las decades, se e al mul ibody sys ems o mula ions ha e been p oposed, a ying in he way he bodies a e de ined, he
na u e o he coo dina es u ilized, and he ype o cons ain and go e ning equa ions needed o desc ibe he sys em kinema ics and
dynamics [28–31]. In gene al, mul ibody sys ems o mula ions can be ca ego ized in o wo main ypes: ecu si e and global. Recu si e
o mula ions, also e e ed o as opological since he model de ini ion depends on he sys em opology, can be u he spli in o ully-
and semi- ecu si e app oaches. I should be no iced ha each ype o o mula ion comes wi h i s own se o ad an ages and d awbacks,
which signi ican ly impac he simplici y o he modeling p ocedu e, he sys ema iza ion o he equa ions o mo ion (EoM), as well as
compu a ional implemen a ion and e iciency [32]. Fo a mo e in-dep h discussion on he di e ences, bene i s, and p ac ical appli-
ca ions o each ype o mul ibody sys em o mula ion, he in e es ed eade is e e ed o he wo ks o Jal´
on e al. [33–35], Cuad ado
e al. [36], Bae e al. [37], Roupa e al. [30] o Yu e al. [38].
In global o mula ions, he posi ion and o ien a ion o each body is usually desc ibed in ela ion o he global e e ence ame
[28–30]. This ype o o mula ion ends o gene a e dependen gene alized coo dina es, equi ing he inco po a ion o cons ain
equa ions o desc ibe such dependencies and o model he mechanical sys em. Despi e inc easing he size o he p oblem o sol e, his
app oach g ea ly educes he dependency o he EoM om he sys em opology, allowing o an easy sys ema iza ion o he modeling
p ocedu e and de ini ion o he go e ning equa ions. By and la ge, wo main amilies o global o mula ions a e a ailable in li e a u e
o model mul ibody sys ems, chie ly he e e ence poin coo dina es [28,39,40], also e e ed o as Ca esian coo dina es, and he
na u al coo dina es [29,41]. Essen ially, hese wo o mula ions a y on he ype o coo dina es adop ed o model he bodies, which
he e o e a ec he s uc u e and complexi y o he kinema ic equa ions hese gene a e.
In Ca esian coo dina es o mula ion, igid bodies a e ypically desc ibed by a ixed kinema ic s uc u e composed o one poin and
a se o angula coo dina es o desc ibe espec i ely hei posi ion and o ien a ion [28,42]. This modeling app oach gene a es less
gene alized coo dina es pe igid body han he o he global o mula ions and makes he modeling o he igid bodies independen o
he sys em opology. Mo eo e , his o mula ion p oduces diagonal and cons an mass ma ices, and he join eac ion o ces and
momen s o o ce can be ob ained di ec ly om he EoM. Howe e , he use o angula a iables implies ha he kinema ic cons ain
equa ions a e desc ibed by nonlinea e ms, which e en ually penalize he compu a ional e iciency [29].
In na u al coo dina es, he igid bodies a e de ined eso ing only on he use o ec angula coo dina es o poin s and ec o s, and,
he e o e, he cons ain s equa ions o he mos common kinema ic pai s p esen a linea o quad a ic dependency on he gene alized
coo dina es [29,43]. Due o he ype o he elemen s ha cons i u e he igid body, he o mula ion wi h na u al coo dina es u ilizes
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
3
mo e coo dina es pe body han he Ca esian coo dina es. Howe e , a bene i associa ed wi h he na u al coo dina es app oach is he
possibili y o sha ing poin s and ec o s be ween bodies, de ining di ec ly kinema ic pai s. This modeling app oach allows o he
educ ion o he numbe o gene alized coo dina es and kinema ic cons ain equa ions needed o comple ely desc ibe he con igu-
a ion o mul ibody mechanical sys ems. The e o e, and depending on he opology o he sys em, he na u al coo dina es o mula ion
wi h an implici de ini ion o he join s ends o p oduce a smalle numbe o coo dina es han he Ca esian coo dina es [29,43]. The
desc ip ion o he mechanical sys em by means o poin s loca ed in ele an posi ions o each model segmen implies ha he
ma hema ical de ini ion o he igid bodies and hei mass ma ices is dependen on he sys em opology, making he modeling
p ocedu e less sys ema ic. Mo eo e , he use o sha ed elemen s also implies ha he bodies a e no comple ely independen , meaning
ha he sys em mass ma ix is coupled, and he join eac ion o ces canno be di ec ly ob ained when he EoM a e sol ed. To ob ain
he eac ion o ces di ec ly, an explici de ini ion o he join s can be adop ed, inc easing he numbe o gene alized coo dina es and
he compu a ional bu den [43].
Based on he na u al coo dina es, Gamei o e al. [44] p oposed a no el global o mula ion ha p ese es he main cha ac e is ics o
he na u al and Ca esian o mula ions, ha is, he bodies a e modeled wi h a p ede e mined kinema ic s uc u e and a e algeb aically
de ined using only Ca esian coo dina es. Simila app oaches ha e been p oposed by Uhla e al. [45], and Pappala do e al. [46–48].
Mo e ecen ly, Roupa e al. [30] p esen ed a de ailed desc ip ion o he heo e ical basis o he o mula ion o a plana case,
o mula ing he cons ain equa ions and con ibu ions o he Jacobian ma ix and igh -hand side ec o s o eloci y and accele a ion
o he mos common kinema ic cons ain s. The o mula ion was subsequen ly applied o he in e se and o wa d dynamic analysis o
di e en classical and biomechanical models, showing an excellen ag eemen wi h he benchma k da a a ailable in he li e a u e [30,
49–51]. Since he o mula ion was buil upon he wo a o emen ioned p emises, i was named as ully Ca esian coo dina es wi h a
gene ic igid body (FCC-GRB). I is impo an o no e ha he e m ully Ca esian coo dina es we e ini ially p oposed by Jal´
on and his
co-au ho s o desc ibe he na u al coo dina es o mula ion, as his o mula ion conside s only he use o Ca esian coo dina es o
desc ibe he kinema ics o he sys em. Howe e , as he kinema ic pai s a ise na u ally om he sha ing o poin s and ec o s, his
o mula ion la e became known as na u al coo dina es o mula ion [52].
Acco ding o Roupa e al. [30], he use o a ixed s uc u e o desc ibe he igid bodies allows o an easie sys ema iza ion o he
modeling p ocedu e, as hei de ini ion becomes independen o he opology o he sys em. I mus be highligh ed ha his app oach
di e s om he adi ional modeling wi h na u al coo dina es, whe e he de ini ion o he sys em a ies wi h he opology and
s uc u e o he sys em, app oxima ing he o mula ion o he one wi h Ca esian coo dina es. In u n, he exclusi e use o ec angula
coo dina es implies ha , as in he na u al coo dina es app oach, he cons ain equa ions p esen , a mos , a quad a ic dependency on
he gene alized coo dina es o he mos common kinema ic cons ain s [30]. Mo eo e , due o he ype o he elemen s ha compose
he igid body, i is possible o de e mine he kinema ics o a gene ic poin o ec o di ec ly om he gene alized coo dina es o he
sys em using a se o cons an ans o ma ion ma ices. This p ocedu e, simila o he one u ilized in he na u al coo dina es o mu-
la ion [29,53], sys ema izes he e alua ion o he coo dina es, eloci ies and accele a ion o any poin o ec o o he sys em,
simpli ying he kinema ic de ini ion o he cons ain equa ions, he applica ion o ex e nal o ces and momen s o he ma hema ical
de i a ion o he sys em mass ma ix. Since hese ans o ma ion ma ices a e cons an in ime, he con ibu ions o he Jacobian
ma ix a e mos ly cons an o linea on he gene alized coo dina es, meaning ha he con ibu ions o he igh -hand side ec o s o
eloci y and accele a ion can be null, linea , o quad a ic [30].
Thus, he p esen wo k ex ends au ho s’ p e ious de elopmen s [30] o o mula e spa ial mul ibody mechanical sys ems using ully
Ca esian coo dina es (FCC) oge he wi h a gene ic igid body (GRB) o mula ion. In he sequel o his p ocess, he main ing edien s
ela ed o kinema ics and dynamics o spa ial mul ibody sys ems a e p esen ed in a de ailed manne . Mo eo e , he main ea u es o
he p oposed o mula ion a e compa ed wi h na u al and Ca esian coo dina es, which allows o a discussion abou he main bene i s
and limi a ions o he p oposed app oach. Finally, he kinema ic and dynamic di e ences be ween modeling he gene ic igid body
wi h a ully-de ined and educed app oach a e also analyzed and hei compu a ional pe o mance examined.
The alidi y and accu acy o he o mula ion using bo h he ully-de ined and educed modeling app oaches a e assessed by
pe o ming a o wa d dynamic (FD) analysis o i e classical benchma k p oblems. The ajec o y o ele an poin s o he model, he
iola ion o he kinema ic cons ain s, he a ia ion o he mechanical ene gy and he compu a ional pe o mance a e compa ed wi h
he equi alen models a ailable in he Lib a y o Compu a ional Benchma k P oblems a ailable a In e na ional Fede a ion o he
P omo ion o Mechanism and Machine Science websi e (IFToMM) [54,55]. In o de o e alua e he di e ences in he compu a ional
ime be ween a ully-de ined o a educed modeling app oach, he FCC-GRB o mula ion is applied in he simula ion o a n- ou -ba
mechanism wi h a a iable numbe o segmen s.
One o he applica ions p e iously iden i ied o he plana FCC-GRB o mula ion is i s use in eaching mul ibody dynamics in
highe educa ion and ad anced cou ses [30]. Since he heo e ical p inciples a e main ained in he spa ial e sion, his o mula ion
could ha e an e en g ea e impac in he educa ional ield. The e o e, he main s eps equi ed o he comple e implemen a ion o he
o mula ion a e p esen ed in de ail h oughou his wo k, so ha , along wi h he plana app oach [30], hey can be di ec ly used as a
lea ning ool.
2. Fully Ca esian coo dina es o mula ion in spa ial mul ibody sys ems
2.1. Fundamen al equa ions
One o he main di e ences among global, semi- ecu si e, and ecu si e o mula ions lies in he ype o go e ning equa ions ha
desc ibe he sys em kinema ics and dynamics. This dis inc ion a ises om he ype o coo dina es used o desc ibe he sys em
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
4
con igu a ion and he s a egy employed o cha ac e ize he deg ees-o - eedom (DoF) o he model unde analysis. The o mula ion o
he kinema ic cons ain s and EoM o he FCC-GRB o mula ion is simila o o he global app oaches. The e o e, only he undamen al
equa ions needed o desc ibe he sys em kinema ics and dynamics a e p esen ed in his sec ion. The in e es ed eade is e e ed o he
plana e sion o he o mula ion [30] o he wo ks o Nik a esh [28], Jal´
on and Bayo [29], Haug [40], Shabana [39] and Ami ouche
[56] o a de ailed deduc ion o he equa ions.
2.1.1. Kinema ic analysis
The con igu a ion o a mul ibody model is ully de ined by a se o independen o dependen coo dina es, usually e e ed o as
gene alized coo dina es (q). Conside ing a sys em composed o nb bodies, he co esponding se o coo dina es can ma hema ically be
exp essed as
q={qT
1⋯qT
i…qT
nb }T(1)
whe e qi ep esen s he ec o o he gene alized coo dina es o he i- h body. Depending on he ype o coo dina es u ilized, he ec o
q can desc ibe he posi ion o a se o poin s and ec o s o he sys em, he global o ien a ion o he bodies, o he ela i e angula
displacemen s be ween adjacen elemen s.
In he p esence o dependen coo dina es, such as in he global o mula ions, he numbe o gene alized coo dina es (nq) su passes
he numbe o DoFs o he sys em, equi ing he de ini ion o a se o algeb aic equa ions o desc ibe hei dependencies. These a e
usually in oduced in he o m o kinema ic cons ain s (KC), and ga he ed in he ec o o he kinema ic cons ain s o he sys em as
Φ(q, ) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Φ1
⋮
Φi
⋮
Φnc
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
=0 (2)
in which Φi is he i- h se o kinema ic cons ain equa ions, and nc he numbe o kinema ic cons ain s. Besides desc ibing he de-
pendencies be ween he gene alized coo dina es, he ec o o kinema ic cons ain s also de ines he s uc u e and opology o he
sys em, and he kinema ics o he mo ion unde analysis. This deside a um is achie ed by ma hema ically desc ibing he geome ic
ela ions exis ing be ween he gene alized coo dina es, hus de ining he s uc u e o he bodies (e.g., KC o igid body ype), he
kinema ic pai s and hei co esponding DoFs (e.g., KC o join ype), among o he opological ea u es. Vec o Φ also includes
equa ions o exp ess he geome ic ela ions be ween he gene alized coo dina es and o he ex e nal a iables, allowing o he
de ini ion o opologic ela ions wi h ex e nal elemen s (e.g., KC wi h espec o global e e ence ame), he implemen a ion o passi e
and ac i e ac ua o s (e.g., KC o ac ua o ype), o he p esc ip ion o he DoFs o he sys em (e.g., KC o d i e ype).
Ma hema ically, ec o Φ is composed by a se o algeb aic equa ions in hei holonomic o m ha , when p ope ly sol ed, allows
o he compu a ion o he kinema ic consis en posi ions o he sys em. As some kinema ic cons ain s exhibi a nonlinea beha io , i
should be no ed ha Eq. (2) ep esen s a sys em o nonlinea equa ions. Di e en me hods ha e been p oposed o sol e he kinema ic
cons ain equa ions, a ying in hei complexi y and a e o con e gence [57,58]. A common app oach o compu e Eq. (2) deals wi h
he use o he i e a i e New on-Raphson me hod o , when in he p esence o edundan cons ain s, he i e a i e New on-Raphson
me hod wi h he leas squa e app oach [29,30].
The ull cha ac e iza ion o he sys em kinema ics equi es he calcula ion o he consis en eloci ies and accele a ions o he
mo ion unde analysis. Fo ha pu pose, he app oach a ailable in [29] can be applied, meaning ha he gene alized eloci ies o a
mul ibody sys em (˙
q) de ined wi h an FCC-GRB o mula ion can be di ec ly ob ained by sol ing he eloci y cons ain equa ions ( ˙
Φ)
in o de o ˙
q as
˙
Φ(q,
˙
q, ) = 0⇔Φq
˙
q=
ν
(3)
wi h
ν
( ) = − Φ (4)
whe e Φq ep esen s he Jacobian ma ix o he sys em, con aining he de i a i es o he kinema ic cons ain s wi h espec o he
gene alized coo dina es,
ν
is he igh -hand side ec o o eloci ies, and Φ deno es he ec o con aining he pa ial de i a i es o he
kinema ic cons ain equa ions wi h espec o ime. Simila ly, he gene alized accele a ions o he sys em ( ¨
q) can be ob ained sol ing
he accele a ion cons ain equa ions as
¨
Φ(q,
˙
q,
¨
q, ) = 0⇔Φq
¨
q=γ(5)
wi h
γ(q,
˙
q, ) =
ν
−(Φq
˙
q)q
˙
q(6)
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
5
in which γ and
ν
ep esen espec i ely he ec o o he igh -hand side ec o o accele a ions and he ec o con aining he pa ial
de i a i es o ec o
ν
wi h espec o ime.
2.1.2. Dynamic analysis
The dynamic analysis o a mul ibody sys em equi es he p ope es ablishmen o he EoM, as hese equa ions exp ess he ela ions
be ween he in e nal, ex e nal, ine ial, and eloci y-dependen o ces and momen s ha ac on he sys em and i s kinema ics [32]. In
he case o global o mula ions, he EoM a e gene ic and hei de i a ion independen o he sys em opology. In ac , his app oach
simpli ies he dynamic analysis o complex mul ibody sys ems, since i does no equi e he deduc ion o he EoM o each sys em unde
analysis, as in he case o ecu si e o mula ions [30]. Thus, he EoM o he FCC-GRB o mula ion wi h a gene ic igid body ollows he
same app oach applied in he na u al o Ca esian coo dina es and can be exp essed as [29,40,56]
M¨
q+ΦT
qλ=g(7)
whe e M ep esen s he mass ma ix o he sys em, λ is he ec o o Lag ange mul iplie s, and g deno es he ec o o he gene alized
o ces. Equa ion (7) desc ibes he ela ion be ween he ine ial (M¨
q), in e nal (ΦT
qλ) and ex e nal o ces (g) ha ac on he sys em.
When he objec i e o he analysis is o de e mine he dynamic esponse o he sys em o a se o know ex e nal o ces ( o wa d dy-
namics), Eq. (7) ep esen s a second o de o dina y di e en ial equa ion (ODE) ha needs o be sol ed o accele a ions, and sub-
sequen ly in eg a ed o e ime o ob ain he gene alized eloci ies and posi ions. Howe e , since bo h he gene alized accele a ions
and Lag ange mul iplie s a e unknown a iables, he sys em becomes unde de e mined, and a new se o equa ions needs o be
in oduced o allow o i s solu ion. Fo his pu pose, he posi ion cons ain equa ions, gi en by Eq. (2), can be conside ed, yielding he
ollow sys em
⎧
⎨
⎩
M¨
q+ΦT
qλ=g
Φ(q, ) = 0(8)
Equa ion (8) ep esen s a se o index-3 di e en ial algeb aic equa ions (DAEs), whe e he i s exp ession desc ibes he dynamics o
he sys em, and he second one ensu es he kinema ic consis ency o he mul ibody sys em. To o e come he nume ical p oblems o
sol ing highe index DAEs, Eq. (8) is usually eplaced by a DAE o lowe o de [59,60]. A common me hod o educe he o de o he
equa ions is o o ce he kinema ic consis ency o he sys em using he accele a ion cons ain equa ions, exp essed by Eq. (5), ins ead
o he posi ion cons ain equa ions. Thus, he EoM o a cons ained mul ibody mechanical sys em can be exp essed as a se o linea
algeb aic equa ions, which, when sol ed om a o wa d dynamics pe spec i e, ep esen a se o index-1 DAEs as
⎡
⎣MΦT
q
Φq0⎤
⎦{¨
q
λ}={g
γ}(9)
I mus be no iced ha he educ ion o he index o he DAEs can esul in he iola ion o he kinema ic cons ains du ing he
in eg a ion p ocedu e, leading o he appea ance o nume ical ins abili ies and d i p oblems [35,60,61]. O e he decades, di e en
me hods ha e been p oposed o deal wi h hese nume ical di icul ies, such as s abiliza ion p ocedu es [61–64], augmen ed
Lag angian o mula ions [59,65,66], penal y o mula ions [25,67–69], eloci y p ojec ions me hods [70–73] o o he pa i ion
me hodologies [74–81].
2.2. Fully Ca esian coo dina es wi h a gene ic igid body
One o he key cha ac e is ics ha dis inguishes he di e en global mul ibody o mula ions is he app oach u ilized o ep esen
he bodies ha compose he sys em unde s udy. The di e en me hodologies a ailable in luence he numbe and ype o he co-
o dina es and kinema ic cons ain s needed o p ope ly de ine he sys em, he me hods equi ed o apply he ex e nal o ces and
momen s, he de ini ion o he sys em mass ma ix, among o he ea u es.
When using Ca esian coo dina es o mula ions, he posi ion and o ien a ion o a gi en igid body a e desc ibed by he Ca esian
coo dina es o one poin and a se o angula - ela ed a iables. I applied in he s udy o spa ial sys ems, six coo dina es a e equi ed o
p ope ly de ine he six DoFs associa ed wi h a ee igid body, namely h ee Ca esian coo dina es o one e e ence poin , and h ee
Eule angles o Rod igues pa ame e s o desc ibe espec i ely he ansla ion o he body and i s o ien a ion. These ep esen a ions a e
cha ac e ized by he exis ence o singula con igu a ions, which can lead o he appea ance o nume ical issues du ing he dynamic
analysis o he sys em [39]. A common app oach o handle his issue is o eplace his ype o ep esen a ion wi h ano he (e.g. Eule
pa ame e s o he Rod igues o mula), inc easing he o al numbe o gene alized coo dina es pe body o se en. Howe e , he
inco po a ion o an ex a coo dina e leads o he appea ance o dependen coo dina es, equi ing he in oduc ion o one igid body
kinema ic cons ain [28,39,82].
An al e na i e solu ion o desc ibe he kinema ics o a mul ibody sys em is o u ilize he na u al coo dina es o mula ion, o iginally
p oposed by Jal´
on and his co-au ho s [29,41,83]. Wi h his o mula ion, he posi ion and o ien a ion o each igid body is de ined
eso ing solely o he use o Ca esian coo dina es o poin s o in e es and uni ec o s, eason why hey a e e e ed o as ully
ca esian coo dina es [29]. An impo an ea u e ela ed o he na u al coo dina es o mula ion is he possibili y o sha ing poin s and
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
6
ec o s be ween di e en bodies, esul ing in an implici de ini ion o se e al kinema ic join s (e.g., sphe ical join and e olu e join ).
As a esul , he numbe o gene alized coo dina es and join kinema ic cons ain s needed o desc ibe he con igu a ion o a sys em is
signi ican ly educed when compa ed wi h a p ocedu e based on explici de ini ion o he join s. This pa icula cha ac e is ic is, in
ac , he eason o he name “na u al” associa ed wi h his o mula ion, as he kinema ic join s a ise na u ally om he sha ing o
poin s and ec o s. This modeling p ocedu e is achie ed by adop ing a a iable con igu a ion o he de ini ion o a gi en igid body.
Depending on he opology o he sys em o be disc e ized, di e en ypes o bodies wi h a a iable numbe o poin s and ec o s can be
employed, such as wo poin s and wo ec o s, h ee non-collinea poin s and one ec o , ou o mo e non-collinea poin s. As an
example, a ypical modeling app oach o desc ibe a link as a igid body in a spa ial mechanism in ol es using wo poin s, usually
loca ed in i s ex emi ies, and wo uni ec o s, gene a ing a o al o 12 gene alized coo dina es. Consequen ly, a se o kinema ic
cons ain s equa ions needs o be in oduced, a leas six, o exp ess he ela ions be ween he dependen gene alized coo dina es.
These kinema ic es ic ions a e ypically in oduced in he o m o igid body kinema ic cons ain s, which include algeb aic ela ions
o ensu e: (i) he cons an dis ance be ween he poin s ha compose he body, ha is cons an dis ance condi ion; (ii) he uni a y na u e
o he uni ec o s, ha is uni a y module condi ion; (iii) he cons an angle be ween uni ec o s, ha is cons an angle condi ion; (i )
he geome ic ela ion be ween mul iple poin s and ec o s wi h espec o a e e ence ame composed by h ee selec ed ec o s, ha
is linea combina ion condi ions [29,43]. An al e na i e modeling app oach, which uses a smalle numbe o ec o s pe igid body,
can be u ilized o educe he dimension o he sys em o be sol ed. Taking as example he case o he link conside ed abo e, his can be
desc ibed by he Ca esian coo dina es o wo poin s and one uni ec o , esul ing in a o al o nine gene alized coo dina es. Despi e
equi ing less gene alized coo dina es and igid body cons ain s, his p ocedu e leads o he exis ence o a iable mass ma ices,
which depend on he gene alized eloci ies o he sys em. A common me hodology o handle his pa icula aspec consis s o
decomposing he igid body mass ma ix in wo e ms, one dependen o he gene alized coo dina es, which is assembled in sys em
mass ma ix, and a second a iable e m ha is ea ed as a eloci y-dependen ine ial o ce [29,53].
2.2.1. Fully-de ined igid body
In his wo k, a new app oach o model spa ial mul ibody sys ems wi h ully Ca esian coo dina es is p oposed. Con a ily o he
na u al coo dina es o mula ion, in which he de ini ion o each igid body is dependen on he opology o he sys em, he p esen ed
o mula ion conside s he concep o a gene ic igid body. Thus, each body can be desc ibed wi h a ixed and p ede e mined kinema ic
s uc u e composed o one ec o ha de ines he e e ence poin ( Oi) and h ee uni ec o s (u
i
,
i
, w
i
) o ep esen , espec i ely, i s
posi ion and o ien a ion in space (see Fig. 1a). Thus, he gene alized coo dina es ec o o a gi en igid body i (q
i
) de ined acco ding
o he p oposed FCC-GRB o mula ion can be w i en as
qi={ T
OiuT
i T
iwT
i}T(10)
Wi h he pu pose o keep he analysis simple, in wha ollows, he igid body e e ence poin Oi is loca ed a he cen e o mass
(CoM) o each body, and he igid body ec o s a e aligned wi h he p incipal axes o ine ia o he co esponding igid bodies. An
impo an ea u e associa ed wi h he FCC-GRB o mula ion is he de ini ion o he igid bodies eso ing solely o Ca esian co-
o dina es o poin s and ec o s. This modeling app oach has he ad an age o no equi ing he use o explici angula - ela ed a i-
ables, which e en ually p oduce kinema ic cons ain s ha ha e a linea o quad a ic dependency on he sys em gene alized
coo dina es.
The igid body de ini ion desc ibed in Eq. (10) gene a es 12 dependen gene alized coo dina es pe body, equi ing, a leas , six
cons ain equa ions o desc ibe he opologic ela ions be ween hem. As in he case o o mula ion wi h na u al coo dina es, he
opologic ela ions a e in oduced in he o m o igid body cons ain equa ions, in pa icula he uni a y module condi ion and he
Fig. 1. Rep esen a ion o he kinema ic s uc u e o a gene ic igid body de ined wi h FCC: a) Fully-de ined a ia ion (12 gene alized coo dina es);
b) Reduced a ia ion (9 gene alized coo dina es).
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
7
cons an angle condi ion. Howe e , in he FCC-GRB o mula ion, he kinema ic cons ain s equi ed o p ope ly de ine he bodies a e
independen o he opology o he segmen s being modeled. Fo he case o a igid body de ined wi h one poin and h ee ec o s, h ee
uni module condi ions need o be added o he ec o o he kinema ic cons ain s (Ф) o gua an ee he uni a y na u e o he igid body
ec o s and h ee cons an angle condi ions o ensu e ha hei ela i e o ien a ion emains cons an along he pe iod o analysis, as i
will be discussed in Sec ions 2.3.1 and 2.3.2.
When compa ed o o he common global o mula ions, de ining a mul ibody sys em wi h FCC-GRB ends o gene a e a la ge
numbe o gene alized coo dina es. In ac , he FCC o mula ion gene a es he same numbe o coo dina es pe igid body as he
na u al coo dina es o mula ion. Howe e , because he na u al coo dina es o mula ion allows o he sha ing o poin s and ec o s
be ween bodies, i ul ima ely p oduces ewe o al coo dina es. To add ess his, he nex sec ion explo es an al e na i e modeling
app oach whe e he base gene ic igid body is subs i u ed wi h a educed gene ic igid body, which comp ises ewe gene alized
coo dina es.
2.2.2. Reduced igid body
Knowing ha om wo non-collinea ec o s i is possible o ob ain a ec o basis in 3D using o hogonaliza ion me hods [84], he
kinema ic s uc u e o he gene ic igid body can be u he simpli ied o educe he o al numbe o gene alized coo dina es ha each
body gene a es. This p ocedu e is pe o med by conside ing one ec o desc ibing he posi ion o he e e ence poin ( Oi) and only wo
di ec ional uni ec o s (u
i
,
i
) o desc ibe he educed gene ic igid body (RGRB) (see Fig. 1b), esul ing in a o al o nine gene alized
coo dina es as
qi={ T
OiuT
i T
i}T(11)
I is impo an o no ice ha al hough he educed modeling app oach dec eases he o al numbe o gene alized coo dina es and
kinema ic cons ain s o igid body ype (elimina ing wo cons an angle condi ions and one uni module condi ion), i leads o he
appea ance o a iable mass ma ices. Simila o he na u al coo dina es o mula ion, his issue can be add essed by di iding he mass
ma ix in o wo componen s: one ha is inco po a ed in o he sys em mass ma ix and ano he ha is ea ed as an ex e nal o ce (see
Sec ion 4.2).
2.2.3. Rela ion be ween a educed and ully-de ined igid body
I should be no ed ha , depending on he modeling s a egy adop ed o desc ibe a gene ic igid body ( ully-de ined o educed), he
ec o con aining i s gene alized coo dina es (qi) di e s. Consequen ly, he kinema ic cons ain equa ions ha de ine bo h he o-
pology and he guiding o he sys em a e no iden ical. To add ess his issue, his sec ion in oduces he kinema ic ela ions equi ed o
con e he educed sys em in o i s ully-de ined equi alen . These ela ions p o ide a consis en amewo k o de ining he mul ibody
sys em and he necessa y ma hema ical ools o de i e exp essions speci ic o he educed app oach.
Despi e allowing o he dec ease o he numbe o gene alized coo dina es, he use o a educed igid body implies ha he ec o
basis, composed o ec o s ui, i and wi, and which de ines he ine ial ame o he body (ξ,
η
, ζ), is no ully de e mined. The e o e, a
p ocedu e o ans o m he se o gene alized coo dina es o he educed igid body i (qi) in o he equi alen ully-de ined ec o (q3 i)
is i s equi ed. This deside a um can be achie ed by de e mining a ec o basis using he igid body ec o s (ui and i) and
o hogonaliza ion me hods. Acco dingly, le one conside an auxilia y ec o wi pe pendicula o he plane de ined by he wo igid
body ec o s o body i, such ha
wi=ui× i(12)
Thus, i is possible o de ine a se o algeb aic equa ions ha ela e he se o educed gene alized coo dina es o he igid body i (qi)
and he ully-de ined equi alen ec o (q3 i) conside ing a ans o ma ion ma ix Vi, such ha
q3 i={ T
OiuT
i T
iwT
i}T=Viqi(13)
wi h
Vi=⎡
⎢
⎢
⎣
I30303
03I303
0303I3
0303
ui
⎤
⎥
⎥
⎦(12×9)
(14)
whe e I3 and 03 a e a 3 ×3 iden i y and null ma ices, espec i ely, and
ui ep esen s a skew-symme ic ma ix ha allows o he
compu a ion o he c oss p oduc p esen ed on Eq. (12), as
wi=
ui i=⎡
⎣
0−uizuiy
uiz0−uix
−uiyuix0⎤
⎦⎡
⎣ ix
iy
iz
⎤
⎦(15)
By algeb aically manipula ing Eq. (14), i is possible o ew i e i in e ms o a cons an ma ix S and he ans o ma ion ma ix Vi,
such ha
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
8
q3 i=SViqi(16)
wi h
S=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
I3030303
03I30303
0303I303
030303
1
2I3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦(12×12)
(17)
and
Vi=⎡
⎢
⎢
⎣
I30303
03I303
0303I3
03−
i
ui
⎤
⎥
⎥
⎦(12×9)
(18)
The exp ession o he eloci y ( ˙
q3 i) o he ully-de ined equi alen ec o can be ob ained by di e en ia ing Eq. (16) wi h espec
o ime, as
˙
q3 i={˙
T
Oi
˙
uT
i
˙
T
i
˙
wT
i}=S˙
Viqi+SVi
˙
qi(19)
By employing he ollowing c oss p oduc ela ion
iui=
uT
i i= −
ui i(20)
he exp ession p esen ed in Eq. (19) can be educed o
˙
q3 i=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
030303
030303
030303
03−1
2˙
i
1
2˙
ui
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Oi
ui
i
⎫
⎪
⎪
⎬
⎪
⎪
⎭
+
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
I30303
03I303
0303I3
03−1
2
i
1
2
ui
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙
Oi
˙
ui
˙
i
⎫
⎪
⎪
⎬
⎪
⎪
⎭
=⎡
⎢
⎢
⎢
⎢
⎢
⎣
˙
Oi
˙
ui
˙
i
ui
˙
i−
i
˙
ui
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=Vi
˙
qi(21)
In u n, he accele a ion (¨
q3 i) o he ully-de ined equi alen ec o can be ob ained by di e en ia ing he p e ious exp ession
wi h espec o ime, yielding
¨
q3 i={¨
T
Oi
¨
uT
i
¨
T
i
¨
wT
i}T=˙
Vi
˙
qi+Vi
¨
qi(22)
wi h
˙
Vi=⎡
⎢
⎢
⎣
030303
030303
030303
03−˙
i˙
ui
⎤
⎥
⎥
⎦(12×9)
(23)
I should be no ed ha by u ilizing he wo ans o ma ion ma ices V and ˙
V along wi h Eqs. (16) o (23), he kinema ics o he
educed igid bodies can be ully de e mined, p o iding he necessa y ma hema ical ela ions o conduc ing he kinema ic and dy-
namic analysis o he mul ibody sys em. Fo e alua ing he kinema ic cons ain s, Eq. (13) and ma ix V can be used di ec ly ins ead o
Eq. (16), he eby educing he compu a ional e o o his s ep.
2.3. Kinema ic cons ain s o igid body
The es ablishmen o he kinema ic cons ain s o igid body in he ec o o kinema ic cons ain s is equi ed o ensu e he igid
na u e o he body. These cons ain s a e de ined in he o m o algeb aic equa ions ha ela e he dependen gene alized coo dina es
ha desc ibe he igid body. I is wo h no ing ha aligning he igid body ec o s wi h he p incipal axes o ine ia o he segmen
implies he exis ence o a di ec ela ion be ween he gene alized coo dina es o he igid body and i s o a ion ma ix. Howe e , since
he o mula ion does no include angula a iables, he geome ic p ope ies in insic o he de ini ion o he body opology, namely
he o hogonali y o he basis ec o s de ining he local e e ence ame and hei uni a y na u e, mus be explici ly de ined as ki-
nema ic cons ain equa ions.
As desc ibed p e iously, wi h he de ini ion o a gene ic igid body, he modeling p ocedu e is signi ican ly simpli ied, equi ing he
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
9
3.2.5. Cons an angula ela ion be ween wo ec o s belonging o ully-de ined igid bodies
The cons an angula ela ion (CAR) condi ion ensu es ha wo di e en gene ic ec o s main ain hei ela i e di ec ion along he
ime. Besides ensu ing he geome ic ela ion be ween he wo uni ec o s, he CAR condi ion is also use ul o de ine join s cha -
ac e ized by wo o a ional DoFs. By cons aining he ela i e angle be ween he wo ec o s, he CAR condi ion allows o he ee
o a ion o he bodies in he planes o hogonal o he axes de ined by each ec o . This condi ion, oge he wi h he CP condi ion, can
be used o model uni e sal- ype join s. CAR condi ion can also be applied o de ine geome ic ela ions ha allow o he de ini ion o
p isma ic join s, slide s, among o he mechanical elemen s.
As in he case o he cons an angle condi ion, he CAR ela ion can be exp essed by one algeb aic equa ion ha o ces he do
p oduc o he wo ec o s o be equal h oughou he pe iod o analysis. Thus, le one conside wo gene ic uni ec o s s1 and s2
belonging espec i ely o bodies i and j (see Fig. 5), i is possible o de ine he CAR cons ain (ΦCAR) in i s homogenous o m as
ΦCAR(qi,qj)=sT
1s2−cos(θs1s2) = (Cs1
iqi)T(Cs2
jqj)−cos(θs1s2) = 0
ΦCAR =qT
iCs1s2qj−cos(θs1s2)
(64)
wi h
Cs1s2=[Cs1
i
TCs2
j](12×12)(65)
Equa ion (64) ep esen s one cons ain equa ion ha p esen s a quad a ic ela ion be ween he gene alized coo dina es o bodies i
and j, and, consequen ly, he con ibu ions o he Jacobian ma ix ha e a linea s uc u e in he o m, such ha
ΦCAR
q={qT
jCs1s2T
⏞⏟⏟⏞
ΦCAR
qi
qT
iCs1s2
⏞⏟⏟⏞
ΦCAR
qj}(1×24)
(66)
As he ma ices Cs1
i and Cs2
j a e cons an and no ime-dependen , ma ix Cs1s2 has also he same p ope ies. Hence, Eq. (64)
ep esen s a scle onomic ela ion, meaning ha i s con ibu ion o he igh -hand side ec o o eloci y (
ν
CAR) is null. In u n, he
quad a ic na u e o he CAR condi ion implies ha he con ibu ions o he igh -hand side ec o o accele a ion (γCAR) p esen a
quad a ic ela ion on he gene alized eloci ies o he sys em, as p esen ed below
ν
CAR =0 (67)
γCAR = − 2˙
qT
iCs1s2˙
qj= − 2˙
sT
1
˙
s2(68)
Al hough Eq. (64) p esen s a non-linea e m, his only needs o be compu ed once du ing he analysis. Since ma ix Cs1s2 is
cons an , i mus be e alua ed one ime a he beginning o he analysis. Thus, du ing he analysis o he mechanism, he CAR cons ain
condi ion is desc ibed only by algeb aic equa ions ha a e null, linea , o quad a ic in ype, ha ing a s ong in luence on he
compu a ional pe o mance o he o mula ion. Mo eo e , Eqs. (64) o (68) can be u he simpli ied i he ec o s being cons ained
a e igid body ec o s. Acco dingly, le one conside wo igid body ec o s (e.g., ui and j) belonging espec i ely o bodies i and j, he
CAR cons ain (ΦCAR) in i s homogenous o m can be gi en by
ΦCAR(qi,qj)=uiT j−cos(θui j)=0 (69)
ΦCAR
q={ T
j
⏞⏟⏟⏞
ΦCAR
ui
uT
i
⏞⏟⏟⏞
ΦCAR
j}(1×6)
(70)
γCAR = − 2˙
uT
i
˙
j(71)
In he case o igid body ec o s, he kinema ic cons ain equa ion, and he con ibu ions o Φq,
ν
and γ o he CAR condi ion
main ain he same deg ee o linea i y as in he case wi h he gene ic ec o , and hence i o e s he same ad an ages. Howe e ,
ob aining all he equi ed alues en ails ewe ma hema ical ope a ions, imp o ing he compu a ional e iciency o he o mula ion.
3.2.6. Cons an angula ela ion be ween wo ec o s belonging o educed igid bodies
The de ini ion o he CAR condi ion o he educed igid bodies can also use he do p oduc cons ain explo ed in he p e ious
sec ion and he ans o ma ion in he ully-de ined equi alen ec o p esen ed in Sec ion 2.2.3. Acco dingly, le one conside wo
gene ic uni ec o s s1 and s2 belonging espec i ely o bodies i and j de ined in i s educed o m, hen, he CAR cons ain (ΦCAR) in i s
homogenous o m can be exp essed as:
ΦCAR(qi,qj)=sT
1s2−cos(θs1s2) = (Cs1
iSViqi)T(Cs2
jSVjqj)−cos(θs1s2) = 0
ΦCAR =qT
iSViCs1s2SVjqj−cos(θs1s2)
(72)
Due o he dependency o he ma ices V on he gene alized coo dina es o he sys em, Eq. (58) ep esen s one cons ain equa ion
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
16
o highe deg ee. I is impo an o no e ha i s e alua ion eso s only on algeb aic mul iplica ions in ol ing linea ma ices and
ec o s dependen on he gene alized coo dina es o he sys em. This highe complexi y also esul s in mo e complex exp essions o
he con ibu ions o he Jacobian ma ix and ec o γ, in he o m
ΦCAR
q={qT
jVT
jSTCs1s2TVi
⏞⏟⏟⏞
ΦCAR
qi
qT
iVT
iSTCs1s2Vj
⏞⏟⏟⏞
ΦCAR
qj}(1×18)
(73)
ν
CAR =0 (74)
γCAR = − (qT
iVT
iSTCs1s2˙
Vj
˙
qj+qT
jVT
jSTCs1s2T˙
Vi
˙
qi+2˙
qT
iVT
iCs1s2Vj
˙
qj)
γCAR = − (sT
1Cs2
j
˙
Vj
˙
qj+sT
2Cs1
iV
⋅
iq
⋅
i+2˙
sT
1
˙
s2)(75)
3.3. Kinema ic cons ain s o d i ing ype
O en, he kinema ic analysis o a mul ibody sys em equi es he p esc ip ion o all he DoFs associa ed wi h he mechanical model.
This in o ma ion is included in he o m o kinema ic cons ain equa ions o d i ing ype, being usually di ided in o wo majo g oups:
(i) Linea d i ing cons ain s – comp ise he exp essions needed o p esc ibe he ansla ions o speci ic poin s o he o ien a ion o
ec o s; (ii) Angula d i ing cons ain s – include he equa ions equi ed o depic he angula o a ions be ween ec o s/bodies.
The d i ing cons ain s a e usually o mula ed using algeb aic equa ions ha ela e he gene alized coo dina es and he kinema ics
o he p esc ibed mo emen . Consequen ly, his ype o cons ain s p esen s a heonomic na u e, meaning ha hei exp essions ha e
an explici dependency on he ime ec o . As o he opological cons ain s, se e al condi ions can be applied o ully desc ibe he
kinema ics o he mechanical elemen s o d i e. In his wo k, pa icula ocus is gi en o he o mula ion o he linea ansla ion o one
poin , he o ien a ion o one ec o and he angula ela ion be ween wo ec o s.
I is no ewo hy ha , simila o he opological cons ain s, kinema ic cons ain s o d i ing ype can in ol e algeb aic exp essions
o p esc ibe he ela i e mo emen be ween wo elemen s wi hin he sys em. Al e na i ely, hey can encompass speci ic equa ions o
desc ibe he mo ion o one elemen o he sys em in ela ion o an ex e io elemen . In he i s g oup, one can conside he angula
ela ion cons ain o guide he angle be ween wo sys em ec o s, and in he second g oup he ansla ion o one poin , he o ien a ion
o one ec o in space, o he ela i e angle be ween a sys em ec o and an ex e nal ec o .
3.3.1. T ansla ion o one gene ic poin belonging o a ully-de ined body
The ansla ion o one gene ic poin (TP) condi ion ensu es ha one poin o in e es belonging o a gi en igid body o he model
ollows he ajec o y o one e e ence poin along he pe iod o analysis. Ma hema ically, his condi ion can be de ined by s a ing ha
he poin o he model sha es he same posi ion as he e e ence poin in he global e e ence ame. Hence, he TP condi ion can be
de ined in a simila way o he app oach used o modeling he CP condi ion. Acco dingly, le one conside a poin P belonging o body i
and he espec i e e e ence poin P*, in which he posi ion ( P*), eloci y (˙
P*) and accele a ion (¨
P*) a e known (see Fig. 6a), he TP
cons ain (ΦTP) in i s homogenous o m can be s a ed as
ΦTP(qi, ) = Pi− P*( ) = CP
iqi− P*( ) = 0(76)
Fig. 6. Schema ic ep esen a ion o he linea kinema ic cons ain s o d i e ype: a) TP condi ion; b) OV condi ion.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
17
Equa ion (76) ep esen s h ee linea equa ions ha en o ces he coo dina es o he poin P o be equal o hose o he e e ence
poin P*. Hence, his cons ain d i es h ee DoFs, which a e associa ed wi h he ansla ion o he body in space. Fo ha eason, he
TP condi ion is usually applied o ei he p esc ibe he posi ion o he en i e model in space by guiding he posi ion o one poin in he
model pa en body, o o guide bo h he posi ion and o ien a ion o he igid bodies o he sys em by minimizing he dis ance be ween
he p esc ibed posi ion o se e al e e ence poin s and hei co esponding in e es poin s [16].
As he second e m o he Eq. (76) does no depend on he gene alized coo dina es o he sys em, he con ibu ions o he TP
cons ain o he Jacobian ma ix a e cons an and equal o he ma ix C
ΦTP
q=[CP
i
⏞⏟⏟⏞
ΦTP
qi](3×12)
(77)
Since he TP condi ion p esen s a heonomic na u e, he con ibu ions o he igh -hand side ec o s o eloci y (
ν
TP) and accel-
e a ion (γTP) a e no null, being hei alues equal o he eloci y (˙
P*) and accele a ion (¨
P*) o he e e ence poin P*, espec i ely
ν
TP =˙
P*( )(78)
γTP =¨
P*( )(79)
3.3.2. T ansla ion o one gene ic poin belonging o a educed igid body
The o mula ion o he TP condi ion o a educed igid body is simila o ha p esen ed o a ully-de ined body. Since his
app oach equi es calcula ing he ully-de ined equi alen ec o by means o Eq. (16), bo h he kinema ic cons ain equa ions and he
con ibu ions o he Jacobian ma ix and γTP e lec he quad a ic na u e o he cons ain as a esul o he mul iplica ion o he
ans o ma ion ma ix V by he gene alized coo dina es o he body. Acco dingly, he exp ession o he TP condi ion in i s homog-
enous o m and he con ibu ions o he Φq,
ν
and γ a e de ined as
ΦTP(qi, ) = CP
iq3 i− *
P( ) = CP
iSViqi− *
P( ) = 0 (80)
ΦTP
q=[CP
iVi
⏞⏟⏟⏞
ΦTP
qi](3×9)
(81)
ν
TP =˙
*
P( )(82)
γTP =¨
*
P( ) − (CP
i
˙
Vi
˙
qi)(83)
3.3.3. O ien a ion o one gene ic ec o belonging o a ully-de ined body
The o ien a ion o a gene ic ec o (OV) cons ain is a kinema ic d i e condi ion ha guides he Ca esian componen s o one
ec o in space. This cons ain can be u ilized o p esc ibe he di ec ion o any gene ic ec o , enabling he de ini ion o he global
o ien a ion o he model segmen s o he o a ion axes o he kinema ic join s in space.
F om a ma hema ical poin o iew, he OV cons ain is simila o he TP condi ion. Howe e , he exp ession uses he ma ix C o a
gene ic ec o s ins ead o he one used o a gene ic poin P. Hence, he OV kinema ic cons ain (ΦOV) in i s homogenous o m can be
s a ed as
ΦOV(qi, ) = si−s*( ) = Cs
iqi−s*( ) = 0(84)
whe e si is a gene ic ec o belonging o body i, and s* ep esen s he p esc ibed Ca esian componen s o he e e ence ec o o ollow
along he pe iod o analysis (see Fig. 6b). Consequen ly, he espec i e con ibu ions o he Jacobian ma ix (ΦOV
q) and igh -hand side
ec o s o eloci y (
ν
OV) and accele a ion (γOV) a e simila o he ones p esen ed in Eqs. (77) o (79), bu conside ing ins ead he ma ix
subs i u ion e e ed be o e and he eloci y (˙
s*) and accele a ion (¨
s*) o he ec o s*.
I he ec o o d i e is a igid body ec o , hen he OV kinema ic cons ain can be simpli ied. Acco dingly, le one conside he
ec o u om he igid body i, he OV algeb aic condi ion and he espec i e con ibu ions o he Jacobian ma ix a e gi en by
ΦOV(qi, ) = ui−s*( ) = 0(85)
ΦOV
q=[I3
⏞⏟⏟⏞
ΦOV
ui](3×3)
(86)
In u n, he con ibu ions o ec o s
ν
and γ a e equal o ones p esen ed o a gene ic ec o s.
3.3.4. O ien a ion o one gene ic ec o belonging o a educed igid body
As in he case o he ully-de ined body, he OV cons ain equa ion o a educed igid body is simila o he one p esen ed in he TP
condi ion, conside ing he subs i u ion o he cons an ma ix C o a gene ic poin P by he equi alen ma ix o a gene ic ec o s. As
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
18
so, he OV kinema ic cons ain (ΦOV) o a educed igid body in i s homogenous o m can be s a ed as
ΦOV(qi, ) = Cs
iq3 i−s*( ) = Cs
iSViqi−s*( ) = 0(87)
Simila ly, he con ibu ions o he Jacobian ma ix and ec o s
ν
and γ will be analogous o he ones p esen ed in Eqs. (81) o (83),
using he equi alen ma ix Cs
i and he eloci y and accele a ion ec o s o he e e ence ec o s*.
3.3.5. Angula ela ion be ween wo ec o s belonging o ully-de ined igid bodies
The angula ela ion (AR) kinema ic d i e en o ces ha he angula displacemen be ween wo ec o s is equal o a p esc ibed
angle θ*. Consequen ly, his ype o cons ain can be used o guide he o a ional DoFs associa ed wi h he kinema ic join s, as i is
schema ically ep esen ed in Fig. 7.
Ma hema ically, he angula d i e cons ain can be de ined using he do p oduc be ween wo ec o s as i was desc ibed in
Sec ion 3.2.5. Howe e , since he angula displacemen be ween he wo ec o s a ies acco ding o he p esc ibed angle, he AR
condi ion becomes a heonomic cons ain , meaning ha addi ional e ms need o be included o he con ibu ions o he igh -hand
side ec o s o eloci y and accele a ion. Hence, le one conside he DoF associa ed wi h he ela i e angle be ween wo gene ic uni
ec o s s1 and s2 belonging espec i ely o bodies i and j, he AR kinema ic d i e cons ain can be de ined as
ΦAR(qi,qj, )=sT
1s2−cos(θ*
s1s2( ))=(Cs1
iqi)T(Cs2
jqj)−cos(θ*
s1s2( ))=0
ΦAR =qT
iCs1s2qj−cos(θ*
s1s2( ))(88)
whe e θ*
s1s2 is he p esc ibed angle be ween ec o s s1 and s2 along he ime. As in he CAR case, Eq. (88) exhibi a quad a ic ela ion
be ween he gene alized coo dina es o he sys em, meaning ha he compu a ional ad an ages iden i ied o he CAR condi ion a e
main ained. Mo eo e , as he second e m o Eq. (88) is independen o he gene alized coo dina es, he con ibu ions o he Jacobian
ma ix a e, in ac , equal o he ones p esen ed in Eq. (66). In con as , he con ibu ions o he ec o s
ν
and γ become dependen on he
angula eloci y (˙
θ*
s1s2) and angula accele a ion (¨
θ*
s1s2) o he p esc ibed angle θ*
s1s2, as p esen ed he ea e
ν
AR = − sin(θ*
s1s2( ))˙
θ*
s1s2( )(89)
γAR = − 2˙
qT
iCs1s2˙
qj−(cos(θ*
s1s2( ))(˙
θ*
s1s2( ))2+sin(θ*
s1s2( ))¨
θ*
s1s2( ))(90)
Equa ion (88) desc ibes one cons ain equa ion, which implies ha each AR kinema ic d i e guides one DoF o he mul ibody
sys em. Hence, one AR cons ain equa ion needs o be added o each o a ional DoF o he join being guided o allow o i s ull
kinema ic desc ip ion. I is impo an o no e ha , due o he use o he do p oduc , he AR condi ion p esen s limi a ions when d i ing
angles nea 0 and
π
ad. Fo hese pa icula cases, he ec o s become aligned, esul ing in he appea ance o linea dependen lines in
he Jacobian ma ix. Mo eo e , since he codomain o he cosine unc ion is posi i e o alues in he 1
s
and 4
h
quad an s and nega i e
in he 2
nd
and 3
d
quad an s, he AR cons ain can gene a e kinema ically consis en posi ions, which do no ma ch he p esc ibed
mo emen . In o de o o e come he add essed issues, he AR kinema ic d i e should be de ined in such a way ha i only d i es
angles be ween 0 and
π
ad o be ween
π
and 2
π
. An al e na i e app oach o a oid he issues ela ed o he domain o he do p oduc
equa ion is o in oduce addi ional d i ing cons ain s. Ins ead o elying on a single cons ain equa ion, his me hod employs a se o
AR equa ions wi h di e en ec o combina ions o desc ibe he same DoF. This app oach allows o expanding he ange o appli-
cabili y o he AR kinema ic d i e cons ain o co e he en i e RoM o he join . Howe e , his modeling s a egy in oduces
edundan cons ain s, equi ing nume ical me hods speci ically designed o handle such sys ems. Fo in e se dynamics analysis, he
kinema ic p oblem can be e icien ly sol ed using he New on-Raphson me hod wi h a leas squa es app oach, wi hou signi ican ly
Fig. 7. Schema ic ep esen a ion o he AR kinema ic d i e de ined be ween igid body i and j.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
19
impac ing compu a ional pe o mance [35,51]. In con as , o wa d dynamic analyses o guided sys ems may expe ience educed
compu a ional e iciency due o he inc eased p oblem dimensions. Addi ionally, he p esence o edundan cons ain s equi es
specialized me hodologies o ind a solu ion, such as hose discussed in Sec ion 2.1.2. A po en ial al e na i e o implemen ing hese
me hodologies is o di ide he simula ion in o mul iple subp oblems wi h sho e ime in e als, ensu ing ha each subp oblem
employs a se o ec o s ha sa is ies he AR condi ion domain.
Equa ions (88) o (90) can be u he simpli ied, i he AR d i e cons ain is de ined be ween wo igid body ec o s. Acco dingly,
le one conside wo igid body ec o s u and belonging o bodies i and j, he AR d i ing cons ain can be desc ibed using he do
p oduc as ollows
ΦAR(qi,qj)=uiT j−cos(θ*
ui j( ))=0 (91)
As in he gene ic case, he second e m o he Eq. (91) does no ha e an explici dependency on he gene alized coo dina es o he
sys em. Hence, he con ibu ions o he Jacobian ma ix a e equal o he ones p esen ed in Eq. (70). In u n, he con ibu ions o he
igh -hand side ec o o eloci y a e he same as he ones p esen ed in he gene ic case (see Eq. (89)), while he con ibu ions o he
ec o γ include he do p oduc o he eloci y o ec o s ui and j, as exp essed by Eq. (71), and he heonomic e ms p esen ed on he
gene ic case (see Eq. (90)).
3.3.6. Angula ela ion be ween wo ec o s belonging o educed igid bodies
The AR kinema ic d i e o a educed body uses he do p oduc de ini ion o desc ibe he angula ela ion be ween wo gene ic
ec o s. Howe e , he me hod o compu e he ully-de ined equi alen ec o s needs o be applied i s o allow o he calcula ion o
he componen s o he gene ic ec o s. Consequen ly, he AR kinema ic d i e in i s homogenous o m can be s a ed as
ΦAR(qi,qj)=(Cs1
iSViqi)T(Cs2
jSVjqj)−cos(θ*
ui j( ))=0
ΦAR =qT
iSViCs1s2SVjqj−cos(θ*
ui j( ))(92)
As he heonomic e m is independen o he gene alized coo dina es, he con ibu ions o he Jacobian ma ix a e equal o hose
ob ained o he CAR condi ion and gi en by Eq. (73). Since his e m is equal o he ully-de ined case, he con ibu ions o ec o
ν
a e
he same as he ones p esen ed in Eq. (89). Finally, he con ibu ions o he igh -hand side ec o o accele a ion include he quad a ic
e ms dependen on he eloci y and posi ion o he sys em as p esen ed in Eq. (75) and he heonomic e ms gi en by Eq. (90).
3.3.7. Angula ela ion be ween one ec o belonging o a ully-de ined igid body and one ex e nal ec o
An addi ional angula ela ion d i e o p esc ibe he ela i e mo ion o one ec o belonging o a igid body and an ex e nal ec o
is o mula ed. This condi ion can be used o guide he angula DoFs associa ed o he kinema ic join s de ined be ween a gi en body
and an ex e nal elemen , such as he pinned join s in oduced in Sec ion 3.4. In e ms o ma hema ical de ini ion, his ype o cons ain
can be o mula ed conside ing he do p oduc exp ession as p esen ed o he gene ic case wi h wo ec o s (see Sec ions 3.3.5 and
3.3.6). Hence, le one conside a gene ic uni ec o s1 belonging o a ully-de ined igid body i and an ex e nal uni ec o s*
2, he AR*
cons ain d i e in i s homogeneous o m can be es ablished as
ΦAR*(qi, ) = sT
1s*
2−cos(θ*
s1s*
2( ))=(Cs1
iqi)Ts*
2−cos(θ*
s1s*
2( ))=0 (93)
whe e θ*
s1s*
2 is he p esc ibed angle be ween ec o s1 and s*
2 along he pe iod o analysis. Since only one o he ec o s p esen s a
dependency on he gene alized coo dina es o he sys em, he con ibu ions o he Jacobian ma ix include only he e ms ela i e o i s
de i a i es wi h espec o he q, yielding
ΦAR*
q(qi, ) = {s*
2
TCs1
i
⏞⏟⏟⏞
ΦAR*
qi}(1×12)
(94)
When he ec o o guide is a igid body ec o , Eqs. (93) and (94) simpli ies, esul ing in
ΦAR*(qi, ) = uT
is*
2−cos(θ*
uis*
2( ))=0 (95)
ΦAR*
q(qi, ) = {s*
2
T
⏞⏟⏟⏞
ΦAR*
qi}(1×3)
(96)
whe e ui ep esen s he ec o u o igid body i. The Jacobian con ibu ions associa ed wi h Eqs. (94) and (96) do no p esen an
explici dependency on he gene alized coo dina es, which means ha i s alue can be de e mined a p io i i he ime ec o o he
analysis is known. This independency o he q ec o implies ha he con ibu ions o he igh -hand side ec o o eloci y (
ν
AR*) and
accele a ion (γAR*) ha e only he e ms ela i e o he heonomic e m, as gi en by Eqs. (89) and (90).
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
20
3.3.8. Angula ela ion be ween one ec o belonging o a educed igid body and one ex e nal ec o
In he p esence o a educed igid body de ini ion, he AR* cons ain d i e can be es ablished in a simila manne desc ibed abo e
using he ully-de ined equi alen ec o , ha is
ΦAR*(qi, ) = (Cs1
iSViqi)Ts*
2−cos(θ*
s1s*
2( ))=0 (97)
wi h
ΦAR*
q(qi, ) = ⎧
⎪
⎨
⎪
⎩s*T
2Cs1
iVi
⏞⏟⏟⏞
ΦAR*
qi⎫
⎪
⎬
⎪
⎭(1×9)
(98)
As ma ix Vi depends on he gene alized coo dina es, he con ibu ion o he Jacobian ma ix canno be de e mined a p io i as in he
ully-de ined case. This linea ela ion on he gene alized coo dina es implies ha addi ional e ms dependen on he gene alized
eloci ies o body i need o be included in he igh -hand side ec o o accele a ions besides he heonomic e ms p esen ed abo e, ha
is
γAR*= − (s*
2
TCs1
i
˙
Vi
˙
qi)−(cos(θ*
s1s*
2( ))(˙
θ*
s1s*
2( ))2+sin(θ*
s1s*
2( ))¨
θ*
s1s*
2( ))(99)
3.4. Kinema ic cons ain s wi h espec o he global e e ence ame
The ull de ini ion o he mul ibody mechanical sys em may equi e addi ional cons ain s o de ine he opologic ela ions be ween
he bodies and he global e e ence ame. These cons ain s allow o he es ablishmen o speci ic join s, such as ixed o pinned join s,
ha ela e he gene alized coo dina e o he sys em o ex e nal elemen s. Two main modeling app oaches can be adop ed o de ine
hese opologic ela ions.
The i s one u ilizes he de ini ion o g ound bodies, ha is, i ual massless bodies ha a e used o cons ain he sys em bodies.
This app oach has he main ad an age o using he same kinema ic cons ain equa ions ha a e u ilized o ela e wo igid bodies (see
Sec ion 3.2), simpli ying he modeling p ocedu e. Howe e , he de ini ion o he g ound bodies equi es he inco po a ion o ex a
coo dina es o he ec o o gene alized coo dina es (q), inc easing he complexi y o he analysis.
The second app oach employs speci ic kinema ic cons ain s o desc ibe he geome ic ela ions be ween poin s and ec o s o he
model and ex e nal elemen s. Consequen ly, no addi ional coo dina es need o be added o he ec o o gene alized coo dina es, being
hese ela ions ully-desc ibed by he use o algeb aic equa ions. This app oach equi es de ining speci ic equa ions based on he
mechanical elemen s being modeled. Ma hema ically, hese cons ain s a e simila o hose used o cons ain o d i e wo igid bodies
(see Sec ions 3.2 and 3.3). As he cons ain equa ions depend only on he gene alized coo dina es o one body, he con ibu ions o he
Jacobian ma ix and igh -hand side ec o o eloci y and accele a ion include only he e ms ela i e o ha body. The e o e, he
opological cons ain s wi h espec o global e e ence ame a e no de ailed in his wo k, howe e , he in e es ed eade is e e ed o
[30] o a de i a ion o his ype o kinema ic cons ain s.
4. Dynamics o 3D mul ibody sys ems wi h FCC-GRB o mula ion
The in e se and o wa d dynamic analysis o mul ibody sys ems equi es he assembly o he EoM as desc ibed by Eq. (7) o Eq. (9),
espec i ely. In bo h cases, beyond desc ibing he sys em opology and p esc ibing he guided DoFs ia he kinema ic cons ain
equa ions p esen ed in Sec ion 3, he implemen a ion o a mul ibody o mula ion also equi es he de ini ion o he sys em mass ma ix
and he de elopmen o me hods o applying ex e nal o ces and momen s o o ce. Hence, his sec ion explo es he ma hema ical
di e ences be ween using a ully-de ined e sus a educed de ini ion o a igid body in he dynamic analysis o mechanical sys ems
wi h FCC, p esen ing he main equa ions needed o e alua e ine ial and ex e nal o ces.
4.1. Mass ma ix o a ully-de ined gene ic igid body
One o he dis inc i e ea u es, which di e en ia es he FCC-GRB o mula ion om he na u al coo dina es o mula ion, is he
simplici y in he de ini ion o he sys em mass ma ix. This s aigh o wa dness is a di ec esul o he kinema ic s uc u e adop ed o
he gene ic igid body, which gene a es spa se and uncoupled mass ma ices [30].
The igid body mass ma ix can be de ined using he p inciple o he i ual powe , as desc ibed in Jal´
on and Bayo [29]. In ac , he
de i a ion o he mass ma ix o a spa ial igid body ollows he same app oach applied in he 2D o mula ion [30] and, o ha
eason, his wo k ocuses on p esen ing only he main di e ences. The in e es ed eade is e e ed o hese wo wo ks o a de ailed
de i a ion o he igid body mass ma ices.
By applying he p inciple o he i ual powe and he me hod o desc ibing he kinema ics o a gene ic poin P (see Sec ion 3.1.1),
i is possible o ob ain an exp ession ha ela es he mass ma ix o a gi en igid body i (Mi) and he cons an ma ix CP
i o a gene ic
poin P [29,30,53]
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
21
Mi=
ρ
i∫
Ωi
CP
i
TCP
idΩ(100)
whe e
ρ
i and Ωi a e espec i ely he mass densi y and he geome ic domain o he igid body i. By sol ing he in eg al p esen ed in Eq.
(100), and conside ing ha he e e ence poin is s a egically loca ed a he CoM and he igid bodies ec o s a e aligned wi h he
ine ia p incipal axes o he igid body being modeled, he inal o m o he mass ma ix o a gene ic igid body is ob ained as [29,53]
Mi=⎡
⎢
⎢
⎣
miI3030303
03IuiI30303
0303I iI303
030303IwiI3
⎤
⎥
⎥
⎦(12×12)
(101)
wi h
Iui=1
2(I
ηη
i+Iζζi−Iξξi)
I i=1
2(Iξξi+Iζζi−I
ηη
i)
Iwi=1
2(Iξξi+I
ηη
i−Iζζi)
(102)
whe e mi is he mass o igid body i and Iii he componen s o i s ine ia enso . I is impo an o no e ha by loca ing he e e ence
poin a he CoM and aligning he ec o s wi h he p incipal axes o ine ia, he mass ma ix o a gene ic igid body becomes diagonal,
wi h i s en ies equal o he mass and he ine ia ela ions along he igid body ec o s (Iui, I i, Iwi). I he e e ence poin is loca ed a
ano he poin o he ec o s a e no aligned, he o -diagonal en ies o he mass ma ix will p esen a dependency on he p oduc s o
ine ia.
F om a modeling app oach, he use o a gene ic igid body app oxima es he de ini ion o he mass ma ix om he one u ilized in
he Ca esian coo dina es, i.e., he ma ix becomes cons an and independen o he opology o he sys em. Mo eo e , i s en ies
p esen a highe physical meaning han he ones ob ained in he na u al coo dina es o mula ion, as hey di ec ly ep esen he ine ial
p ope ies o he body.
As men ioned abo e, he possibili y o sha ing ec o s be ween bodies implies ha he FCC o mula ion wi h a gene ic igid body
allows o bo h an explici and implici de ini ion o some kinema ic join s. Conside ing a ull explici model, he assembly o he
sys em mass ma ix is a s aigh o wa d p ocess, and i s en ies a e di ec ly he igid body mass ma ices. Conside ing he ec o o he
gene alized coo dina es p esen ed in Eq. (1), he global mass ma ix o he sys em will be cons an and diagonal, as shown below
M=⎡
⎢
⎢
⎢
⎢
⎣
M1
⋱
Mi
⋱
Mnb
⎤
⎥
⎥
⎥
⎥
⎦(nq×nq)
(103)
Despi e gene a ing mo e coo dina es, his modeling app oach has he main ad an age o gene a ing spa se and uncoupled mass
ma ices, which is a compu a ional ad an age i he p ope nume ical me hods a e used. In u n, he use o an implici modeling
app oach esul s in coupled mass ma ices, and consequen ly, less spa se ma ices. Thus, le one conside a ec o u sha ed by he
gene ic igid bodies i and j, he con ibu ions o each body o he global mass ma ix a e gi en by
M=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
M1
⋱
M OiMui+MujM iMwiM OjM jMwj
⋱
Mnb
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦(nq×nq)
(104)
I is impo an o no e ha in bo h cases, as he igid body mass ma ices a e independen o he gene alized coo dina es, he mass
ma ix o he sys em is cons an . This ac implies ha ma ix M only needs o be e alua ed once a he beginning o he analysis,
educing he numbe o calcula ions equi ed o each e alua ion o he EoM.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
22
4.2. Mass ma ix o a educed gene ic igid body
The mass ma ix gi en by Eq. (101) desc ibes ma hema ically he ine ial p ope ies o he gene ic igid body i, namely he mass
and he mass momen s o ine ia a ound he h ee axes ha cons i u e he local e e ence ame, which, acco dingly wi h he kinema ic
s uc u e adop ed o he gene ic igid body, a e de ined by he h ee igid body ec o s. The e o e, when in he p esence o a educed
igid body, he mass ma ix needs o also include he ine ia momen a ound a hi d i ual ec o w o ully desc ibe he ine ial
p ope ies o he body.
Once mo e, he p ocedu e p esen ed in Sec ion 2.2.3 can be u ilized o de e mine he espec i e ully-de ined equi alen ec o ,
and hen de ine he mass ma ix o he educed igid body i. As o he ully-de ined case, he p inciple o i ual powe can be used o
de e mine he equi alen mass ma ix. Thus, le one conside he ully-de ined equi alen ec o o he educed igid body i, he i ual
powe gene a ed wi hin his body by he ine ial o ces (W*
i) is gi en by
W*
i= −
ρ
∫˙
q*T
3 iCP
i
TCP
i
¨
q3 idΩ= −
ρ
∫˙
q*T
iVT
iCP
i
TCP
i(˙
Vi
˙
qi+Vi
¨
qi)dΩ(105)
whe e ˙
q*
i ep esen s he gene alized i ual eloci ies o he igid body i and ˙
q*
3 i deno es he i ual eloci ies desc ibed in e ms o he
ully-de ined equi alen o m. As he ans o ma ions ma ices Vi and ˙
Vi a e independen o he olume o body i, hese can be mo ed
ou side he in eg al. In his way, he in eg al in Eq. (105) is equal o he one in Eq. (100), which in u n is equal o he mass ma ix o a
ully-de ined igid body. Hence, Eq. (105) can be u he simpli ied as
W*
i= − ˙
q*T
iVT
i
ρ
⎛
⎝∫Ω
CP
i
TCP
idΩ⎞
⎠(˙
Vi
˙
qi+Vi
¨
qi)
W*
i= − ˙
q*T
i(VT
iMiVi
¨
qi+VT
iMi
˙
Vi
˙
qi)(106)
Equa ion (106) can be di ided in o wo main e ms. The i s one, dependen on he gene alized accele a ions o he sys em,
co esponds o he equi alen mass ma ix o he educed igid body (M2 i) as
M2 i=[VT
iMiVi](9×9)(107)
Ma ix M2 i can be assembled in he global mass ma ix o he sys em, ollowing he same app oach p esen ed in Eqs. (103) o (104).
Howe e , since ma ix Vi is dependen on he gene alized coo dina es, he global mass ma ix becomes dependen on he s a e o he
sys em. This ac implies ha o each e alua ion o he EoM, he global mass ma ix needs o be upda ed wi h he con ibu ions
ela i e o he educed igid bodies.
The second e m o Eq. (106) can be ea ed as a eloci y-dependen gene alized ine ial o ce (g2 i), which needs o be added o he
ec o o he gene alized o ces, such ha
g2 i=VT
iMi
˙
Vi
˙
qi(108)
The eloci y-dependen ine ial o ce is dependen on he s a e o he sys em, meaning ha he ec o o he gene alized o ce
ec o s needs o be also upda ed o each ime s ep.
4.3. Applica ion o ex e nal o ces and momen s o o ce
Since he ex e nal o ces and ex e nal momen s o o ce a e no necessa ily applied in he gene alized coo dina es o he sys em,
hose mus be ans o med in o equi alen gene alized o ces. I is impo an o no e ha he me hodology p oposed he ea e is
gene ic and can be applied o any ex e nal o ce o momen applied o he sys em. This me hod is also alid o s a e-dependen o ces
o momen s ha can be ep esen ed by an equi alen ex e nal o ce o momen o o ce. Howe e , in his pa icula case, he s a e o
he sys em needs o be i s ly de e mined o allow o he calcula ion o he magni ude, o ien a ion and applica ion poin o hese
o ces.
4.3.1. Ex e nal o ces o a ully-de ined igid body
The p inciple o i ual powe can be u ilized o de i e he equi alen gene alized o ce o any ex e nal o ce applied in he sys em.
This p inciple s a es ha he i ual powe p oduced by an ex e nal o ce applied in poin P belonging o body i is equal o he i ual
powe p oduced by he gene alized equi alen o ce, i.e., he p oduc o he gene alized equi alen o ce g
i by he ec o o i ual
eloci ies o igid body i [29,53], as p esen ed below
˙
*
P
T =˙
q*
i
Tg
i(109)
whe e ˙
*
P ep esen s he ec o o he i ual eloci ies o poin P. By applying he me hodology p esen ed in Sec ion 3.1.1, i is possible
o exp ess he algeb aic ela ion p esen ed in Eq. (109) as a unc ion o he gene alized coo dina es o he sys em, such ha
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
23
˙
q*
i
TCP
i
T =˙
q*
i
Tg
i⇒g
i=CP
i
T (110)
Equa ion (110) shows ha he equi alen gene alized o ce can be exp essed as he p oduc o he ec o o he ex e nal o ce and
he cons an ans o ma ion ma ix C o he poin P desc ibed in ela ion o body i (see Fig. 8a). This ela ion implies ha i bo h he
ex e nal o ce and i s applica ion poin a e cons an , as in he case o he conse a i e g a i a ional o ce, he equi alen gene alized
o ce is also cons an , meaning ha i s con ibu ion o he gene alized o ce ec o only needs o be compu ed once a he beginning o
he analysis. Simila ly, i only he applica ion poin o o ce is cons an in he local e e ence o he body (e.g., o ces gene a ed by
linea sp ings, dampe s, o ac ua o o ces wi h ixed a achmen poin s), ma ix C becomes cons an along he pe iod o analysis,
implying ha i can also be de ined a p io i.
4.3.2. Ex e nal momen o o ce o a ully-de ined igid body
As he FCC-GRB o mula ion does no u ilize angula coo dina es o desc ibe he igid bodies o ien a ions, he ex e nal momen s o
o ce canno be di ec ly applied o he sys em as a gene alized momen . Thus, each ex e nal momen o o ce needs o be i s con e ed
in o he equi alen o ces couple, which is hen applied o he sys em using he same app oach p esen ed in he p e ious sec ion.
Acco dingly, le one conside he ex e nal momen o o ce
τ
applied o body i, and he equi alen o ce couple
τ
and −
τ
, such ha
{
τ
=
τ
×b
τ
+ −
τ
×b−
τ
τ
+ −
τ
=0(111)
whe e b
τ
and b−
τ
a e espec i ely he momen a ms o o ce ec o s
τ
and −
τ
.
As he i ual powe p oduced by he gene alized equi alen momen o o ce (g
τ
i) is equal o he sum o he i ual powe p oduced
by he equi alen o ce couple [29,53], he ollowing exp ession yields
˙
q*
i
Tg
τ
i=˙
*
P
τ
T
τ
−˙
*
P −
τ
T −
τ
(112)
De ining he posi ion o poin P −
τ
a he o igin o he local e e ence ame o body i and o poin P
τ
a he ip o ec o s (see Fig. 8b)
and ecalling Eqs. (36) and (40), Eq. (112) can be w i en in e ms o he gene alized coo dina es o he sys em as
˙
q*
i
Tg
τ
i=˙
q*
i
T(CP
i
T−COi
i
T)
τ
⇒g
τ
i=Cs
i
T
τ
(113)
By loca ing he poin P −
τ
a he o igin o he e e ence, he con ibu ion o he o ce −
τ
o he momen is null, being only esponsible
by coun e ac ing he ansla ion p oduced by he o ce
τ
. The e o e, he ec o s should be de ined in such a way ha i is con ained in
he plane no mal o ec o
τ
and he no m o he p oduc Cs
iT is equal o he magni ude o he momen o o ce
τ
.
I mus be no ed ha his linea dependency on he ma ix C implies ha i he di ec ion o he ex e nal momen o o ce
τ
does no
change along he ime, he ma ix Cs
i is cons an and consequen ly only he p oduc p esen ed in Eq. (113) needs o be compu ed each
ime he EoM a e e alua ed.
4.3.3. Ex e nal o ces o a educed igid body
The applica ion o he ex e nal o ces o a igid body de ined in he educed o m ollows he same app oach p esen ed o he
ully-de ined igid body. Howe e , he exp essions p esen ed in Sec ion 4.3.1 need o be e ised conside ing he ully-de ined
equi alen ec o (q3 ) in oduced in Sec ion 2.2.3. Hence, and ecalling he p inciple o i ual powe o an ex e nal o ce (see
Fig. 8. a) Applica ion o an ex e nal o ce a poin P belonging o he gene ic igid body i; b) Applica ion o an ex e nal momen o o ce
τ
o he
gene ic igid body i and espec i e ans o ma ion in o he equi alen o ce couple
τ
and −
τ
.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
24
Eq. (109)), he gene alized equi alen o ce o a educed igid body can be desc ibed as
˙
q*
i
TVT
iCP
i
T =˙
q*
i
Tg
i⇒g
i=VT
iCP
i
T (114)
In con as o he ully-de ined case, he gene alized equi alen o ce o he educed igid body depends on he gene alized co-
o dina es o he sys em, meaning ha each ime he EoM a e e alua ed, he gene alized o ce ec o needs o be upda ed, e en when
cons an o ces a e applied.
4.3.4. Ex e nal momen o o ce o a educed igid body
The same p inciples p esen ed in Sec ion 4.3.2 can be applied o de e mine he gene alized momen o o ce o an ex e nal momen
τ
. Howe e , he ully-de ined equi alen ec o needs o be used o allow o he sys ema ic calcula ion o he con ibu ions o he
gene alized o ce ec o (g
τ
i).
˙
q*
i
Tg
τ
i=˙
q*
i
TVT
i(CP
i
T−COi
i
T)
τ
⇒g
τ
i=VT
iCs
i
T
τ
(115)
I should be no iced ha Eq. (115) p esen s an explici dependency on he gene alized coo dina es o he sys em, which implies
ha , e en o cons an momen s o o ce, he con ibu ions o ec o g need o be calcula ed o each ins an o ime. Ne e heless, as in
his pa icula case, he ec o s is cons an in he local e e ence ame o he body i, ma ix Cs
i is also cons an and can be de ined a
p io i du ing he p e-analysis p ocedu e.
4.4. In e nal eac ion o ces and momen s o o ce o a ully-de ined igid body
One o he majo objec i es, when pe o ming a dynamic analysis o mechanical sys ems, is he de e mina ion o he in e nal o ces
and momen s o o ce gene a ed du ing he mo ion o he sys em. F om a mechanical s andpoin , hese ep esen he o ces and
momen s ha need o be gene a ed wi hin he sys em o comply wi h he imposed kinema ic cons ain s. Thei physical meaning is
ela ed o he ype o kinema ic cons ain hey a e ob ained. Algeb aically, he in e nal o ces o he sys em (gΦ) can be calcula ed by
sol ing Eqs. (7) o (9) in o de o he in e nal o ces, such ha
gΦ={gΦ
1
T⋯gΦ
i
T⋯gΦ
nb
T}T=ΦT
qλ(116)
wi h
gΦ
i=gΦRB
i+gΦKJ
i+gΦDC
i(117)
whe e gΦ
i ep esen s he ec o o he in e nal o ces ac ing on body i, which, in u n, is equal o he sum o all con ibu ions om
kinema ic cons ain s and d i e s ela ed o his body, including igid body opological cons ain s (RB), kinema ic join opological
cons ain s (KJ), and d i ing cons ain s (DC).
The physical meaning o Eqs. (116) and (117) can be be e in e p e ed i he con ibu ions o each kinema ic d i e a e e alua ed
sepa a ely. In ac , he Lag angian mul iplie s associa ed wi h each kinema ic cons ain ep esen he magni ude o he in e nal o ces,
while he espec i e lines o he Jacobian ma ix hei di ec ion. This ele an cha ac e is ic o he Lag ange mul iplie s is o pa icula
ele ance, as i allows o he de e mina ion o he join eac ion o ces and momen s di ec ly om he EoM.
Taking as an example he case o a sphe ical join , de ined by he CP condi ion (see Sec ions 3.2.1 and 3.2.2), he gene alized
in e nal o ces gene a ed by his join a e gi en by
gΦCP =ΦCPT
qλCP =⎡
⎣CP
i
T
−CP
j
T⎤
⎦λCP =⎧
⎨
⎩
CP
i
TλCP
CP
j
T(−λCP)⎫
⎬
⎭(24×1)
(118)
By compa ing Eq. (118) wi h Eq. (110) and ha ing in mind ha λCP is a 3D ec o , i can be obse ed ha i exp esses he
applica ion o wo concen a ed o ces o equal magni ude and opposi e di ec ion (λCP
,−λCP) a poin P. These wo o ces ep esen he
ac ion- eac ion o ce couple gene a ed a he join . Hence, he join eac ion o ces ( CP
R) a e ob ained di ec ly om he EoM, as hey a e
equal o he Lag angian mul iplie s associa ed wi h he CP condi ion as
CP
R= CP
Ri= − CP
Rj={λCP}(3×1)(119)
In u n, he in e nal o ces gene a ed by he AR kinema ic d i e s desc ibe he equi alen o ces ha p oduce he in e nal d i ing
momen s o o ce. The e o e, he de e mina ion o he momen s o o ce a he join s in an FCC o mula ion is no a di ec p ocedu e,
equi ing addi ional calcula ions. Acco dingly, le one conside an angula d i ing cons ain be ween ec o s s1 and s2 belonging o
bodies i and j, and ecalling he con ibu ions o he Jacobian ma ix p esen ed in Sec ions 3.3.5 and 3.3.6, he gene alized in e nal
o ces o an AR d i e cons ain (gΦAR ) can be gi en by
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
25
igid body ec o s wi h join axes, simpli ying he de ini ion o sys em opology and enabling he gene a ion o mo e e icien
ma hema ical exp essions [33]. Addi ionally, he cons an na u e o ma ix C o poin s and ec o s ixed in he local e e ence ame
allows i o be compu ed in a p e-kinema ic s ep and assembled in o he Jacobian ma ix wi hou equi ing con inuous upda es du ing
EoM e alua ions.
Ne e heless, he FCC-GRB o mula ion is no wi hou limi a ions. Compa ed o bo h he Ca esian coo dina es and na u al co-
o dina es o mula ions, he FCC-GRB ypically gene a es a la ge numbe o gene alized coo dina es and kinema ic cons ain equa-
ions. Speci ically, e en when using he educed app oach, he numbe o gene alized coo dina es is la ge han in he classical
angula -based global o mula ions. Fo ins ance, when compa ed o a Ca esian coo dina es o mula ion wi h Eule pa ame e s, he
FCC-GRB o mula ion equi es app oxima ely wo addi ional gene alized coo dina es, and, consequen ly, wo addi ional kinema ic
cons ain s, pe igid body [28]. On he o he hand, while he FCC-GRB gene a es he same numbe o gene alized coo dina es pe igid
body as he na u al coo dina es, he abili y o his o mula ion o sha e poin s and ec o s be ween adjacen bodies can educe he o al
numbe o coo dina es and cons ain s in he sys em [29,53]. Howe e , he FCC-GRB also suppo s ec o sha ing, which implici ly
de ines sha ed ec o SV condi ions. Al hough no explo ed in his s udy, his capabili y could u he educe he o al numbe o
gene alized coo dina es and kinema ic cons ain s. Fo highly cons ained sys ems wi h a la ge numbe o e olu e join s, he FCC-GRB
may esul in a lowe o al coo dina e coun compa ed o he Ca esian coo dina es o mula ion.
Finally, one po en ial applica ion iden i ied o he plana FCC-GRB o mula ion is i s use in eaching mul ibody dynamics opics in
ad anced cou ses [30]. The main ea u es suppo ing his idea we e i s g ea e in ui i eness and simplici y in implemen ing and
modeling mul ibody sys ems compa ed o o he common global o mula ions. These p emises a e s ill alid o he spa ial o mula ion
wi h he ully-de ined app oach, namely: i) he in oduc ion o he gene ic igid body simpli ies he modeling o he mul ibody sys em,
allowing o easie sys ema iza ion o his p ocedu e. This is pa icula ly mo e no iceable when compa ed wi h he na u al coo dina es
o mula ion, whe e he model is s ongly dependen on he opology o he sys em; ii) he igid body mass ma ix emains diagonal and
cons an , wi h i s s uc u e independen o he sys em opology. Mo eo e , i s en ies a e di ec ly he ine ial p ope ies o he body
being modeled, esul ing in a mass ma ix wi h a highe physical meaning; iii) he igid bodies a e de ined eso ing only o Ca esian
coo dina es, meaning ha no backg ound knowledge on he pa ame iza ion o o a ions in spa ial models is equi ed, simpli ying he
implemen a ion o he o mula ion. This las poin was p ecisely one o he majo ad an ages a ibu ed o he na u al coo dina es
o mula ion, suppo ing i s use in educa ional applica ions [33]. Conside ing ha he FCC-GRB o mula ion is e en mo e in ui i e, i
can be mo e easily applied in he eaching o mul ibody opics in ad anced cou ses. Fo ha eason, he au ho s op ed o p esen ing in
de ail he speci ici ies o he o mula ion in 3D, so ha , oge he wi h he plana wo k [30], he unde g adua e and g adua e s uden s
can easily implemen he o mula ion and unde s and he esul s i p o ides. Hence, hese wo wo ks can be di ec ly used as an ed-
uca ion ool o suppo cou ses in he STEM a eas, p o iding insigh s o how o model and analyze simple and complex mechanical
sys ems in 3D.
CRediT au ho ship con ibu ion s a emen
S´
e gio B. Gonçal es: W i ing – e iew & edi ing, W i ing – o iginal d a , Visualiza ion, Valida ion, So wa e, Me hodology,
In es iga ion, Fo mal analysis, Concep ualiza ion. I o Roupa: W i ing – e iew & edi ing, Valida ion, Concep ualiza ion. Paulo
Flo es: W i ing – e iew & edi ing, Supe ision. Miguel Ta a es da Sil a: W i ing – e iew & edi ing, Valida ion, Supe ision, P ojec
adminis a ion, Concep ualiza ion.
Decla a ion o compe ing in e es
The au ho s decla e he ollowing inancial in e es s/pe sonal ela ionships which may be conside ed as po en ial compe ing
in e es s:
S´
e gio B. Goncal es, I o Roupa, Paulo Flo es, and Miguel Ta a es da Sil a epo s inancial suppo was p o ided by Fundaç˜
ao pa a
a Ciencia e a Tecnologia (FCT). I he e a e o he au ho s, hey decla e ha hey ha e no known compe ing inancial in e es s o
pe sonal ela ionships ha could ha e appea ed o in luence he wo k epo ed in his pape .
Acknowledgemen s
The au ho s acknowledge Fundaç˜
ao pa a a Ciˆ
encia e a Tecnologia (FCT) o i s inancial suppo ia he p ojec s LAETA Base
Funding (DOI: 10.54499/UIDB/50022/2020), LAETA P og amma ic Funding (DOI: 10.54499/UIDP/50022/2020), UIDB/04436/
2020 and UIDP/04436/2020, and Po uguese Reco e y and Resilience P og am (PRR) o i s inancial suppo ia IAPMEI/ANI/FCT
unde Agenda C645022399-00000057 (eGamesLab).
Supplemen a y ma e ials
Supplemen a y ma e ial associa ed wi h his a icle can be ound, in he online e sion, a doi:10.1016/j.mechmach heo y.2025.
105955.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 209 (2025) 105955
32
Da a a ailabili y
No da a was used o he esea ch desc ibed in he a icle.
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