Kinematic and inverse dynamic analysis using mixed and fully Cartesian coordinates with a generic rigid body

Resea ch pape
Kinema ic and in e se dynamic analysis using mixed and ully
Ca esian coo dina es wi h 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:
Fully Ca esian coo dina es
Gene ic igid body
Mixed coo dina es
Kinema ic analysis
In e se dynamics
Ad anced educa ion
ABSTRACT
The p opaga ion o e o s along a kinema ic chain, caused by using d i e s compu ed om noisy
da a, can a ec he accu acy o he kinema ic and dynamic ou comes. Minimizing he e ec o
such e o s is c ucial, pa icula ly when adi ional smoo hing echniques p o e o be ine ec i e.
This wo k expands he mul ibody o mula ion wi h Fully Ca esian Coo dina es and a Gene ic
Rigid Body (FCC-GRB) o he in e se dynamic analysis o spa ial mechanical sys ems using mixed
coo dina es (MC). This me hod conside s he inco po a ion o angula a iables, enabling he
de e mina ion o he kinema ic consis en posi ions ha bes i he e e ence da a, while
simul aneously compu ing he join angula d i e s. The accu acy and compu a ional pe o -
mance o he o mula ion a e e alua ed using bo h nume ical- and op imiza ion-based me hods in
he s udy o wo mechanisms guided wi h pe u bed da a.
The esul s show ha implemen ing an MC me hodology wi h FCC-GRB can be easily pe -
o med wi hou comp omising he heo e ical ounda ions o he classical o mula ion. This
app oach e icien ly compu es bo h he posi ions and d i e s o he model simul aneously,
a oiding he p opaga ion o e o s along he kinema ic chain. Nume ical me hods based on he
New on-Raphson algo i hm gene a ed posi ions closely ma ching he e e ence da a, while
op imiza ion-based me hods ensu ed a s ic e ul illmen o he opological cons ain s.
1. In oduc ion
The in e se dynamic analysis o mul ibody mechanical sys ems ypically equi es p io compu a ion o kinema ic posi ions om
noisy o missing expe imen al da a [1,2]. A classic example is he biomechanical analysis o human mo ion. Typically, when acqui ing
expe imen al da a, a se o e lec i e ma ke s is placed on he subjec ’s skin a speci ic loca ions, o en co esponding o bony emi-
nences [3–5]. Due o he ela i e mo ion o so issues, such as he skin, muscles, endons, and ligamen s, in ela ion o he bones, a
phenomenon usually e e ed o as so issue a i ac (STA), he acqui ed da a is p one o expe imen al e o s. These can a ec he
quali y o he measu ed ou comes, making i challenging o use he da a a a clinical le el [6–10]. This a i ac is pa icula ly di icul
o emo e om expe imen al da a wi hou elying on specialized eal- ime medical imaging echniques (e.g., [11–13]), as i o en has
he same equency as he mo emen i sel and is ask-dependen . Fo ins ance, i s magni ude, equency, and iming may a y
* 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.2025.106080
Recei ed 24 Ma ch 2025; Recei ed in e ised o m 8 May 2025; Accep ed 15 May 2025
Mechanism and Machine Theo y 214 (2025) 106080
A ailable online 5 June 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/ ).
Nomencla u e Head
Abb e ia ions
3D Th ee-Dimensional
AR Angula Rela ion
CNO Cons ained Nonlinea Op imiza ion
CoM Cen e o Mass
DoF Deg ee o F eedom
ED Euclidean Dis ance
EoM Equa ions o Mo ion
FCC-GRB Fully Ca esian Coo dina es wi h Gene ic Rigid Body
KA Kinema ic Analysis
LS Leas Squa es
MCR Mixed Coo dina es Rela ion
MO Mul i-Objec i e
NRM New on-Raphson Me hod
STA So Tissue A i ac
TC T acking Cons ain
TOC T acking O ien a ion Cons ain
WLS Weigh ed Leas Squa es
Symbol (La in)
03, I3Null and iden i y ma ices
CP
iCons an ans o ma ion ma ix o a gene ic poin P wi h espec o body i
Cs
iCons an ans o ma ion ma ix 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
F, FObjec i e unc ion o he op imiza ion p oblem
FqJacobian ma ix o he objec i e unc ion F
F∗Op imiza ion goals o he mul i-objec i e goal a ainmen p oblem
gGene alized o ces o he sys em
gΦGene alized in e nal o ces
MMass ma ix o he sys em
nb, nc, nd, nTC Numbe o bodies, kinema ic cons ain s, gene alized angula a iables and acking cons ain 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
qlb, qub Lowe and uppe bounds o he gene alized coo dina es o he op imiza ion
qnSolu ion a o he NRM o i e a ion n
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
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
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
Vi,Vi,˙
ViT ans o ma ion ma ices ha con e he educed igid body i in o i s ully-de ined equi alen o m
wnWeigh coe icien o he n- h acking cons ain
wMO Weigh coe icien s ec o o he MO op imiza ion
WNRM-WLS and op imiza ion weigh ma ix
Symbol (G eek)
γRigh -hand-side ec o o he accele a ion equa ions o he sys em
γkRigh -hand-side ec o o he accele a ion equa ions o he kinema ic cons ain o ype k
Y Slack a iable o he mul i-objec i e goal a ainmen p oblem
ε
New on-Raphson i e a ion ole ance
θVec o o he gene alized angula a iables
θkk- h angula gene alized coo dina e o he model
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
2
depending on he mo emen being analyzed, he indi idual’s an h opome ic cha ac e is ics, he body segmen and ana omical join
in ol ed, and he muscle ac i a ion pa e n. As a esul , au oma ic de ec ion becomes di icul , and s anda d il e ing echniques a e
o en ine ec i e o i s a enua ion [6,14].
Ob aining high-quali y kinema ic and kine ic inpu s is c ucial, as hey ha e a signi ican in luence on he quali y o he dynamic
ou comes [15]. A common p ocedu e o ensu ing he quali y o kinema ic inpu s in ol es compu ing he kinema ic consis en po-
si ions o a gi en model. Essen ially, his p ocedu e de e mines he posi ions, eloci ies, and accele a ions o he sys em ha comply
wi h a se o opological cons ain s, which de ine ma hema ically he model, and d i ing cons ain s ha guide i s mo emen [1].
Me hodologies based on mul ibody sys em o mula ions p esen hemsel es as an e icien and accu a e app oach o pe o ming he
kinema ic and dynamic analysis o complex mechanical models [16]. Two classical app oaches a e commonly employed o compu e
he kinema ic consis en posi ions [17]. The i s add esses he p oblem h ough a o wa d kinema ics app oach. The join angula
d i e s, which usually desc ibe he angula deg ees-o - eedom (DoFs) o he model, a e compu ed om he expe imen al da a in a
p e-analysis phase. These d i e s a e hen applied o he model du ing he kinema ic analysis (KA) o de e mine he posi ion and
o ien a ion o all he segmen s [1,17]. The second me hod u ilizes e e ence poin s and ec o s o de e mine he kinema ic consis en
posi ions ha minimize he di e ences om hese e e ences, ollowing an in e se kinema ics app oach [17,18]. In o de o add ess
he inaccu acies in he expe imen al da a, such as STA, se e al me hods ha e been p oposed. Some o hem use speci ic algo i hms
designed o mi iga e hese e ec s (e.g., [19–21]), while o he s ely on nume ical o op imiza ion me hods oge he wi h
mul ibody-based models (e.g., [7,9,10,22,23]). The choice o he mos app op ia e me hod depends on he desi ed le el o accu acy,
mo emen in analysis, o compu a ional pe o mance equi emen s.
Se e al mul ibody o mula ions a e a ailable in he li e a u e, p esen ing di e en ad an ages and a ge applica ions [17,24–31].
Recen ly,based on he wo k o Gamei o e al. [32], Roupa e al. [28] and Gonçal es e al. [29] p esen ed he heo e ical basis o he
Fully Ca esian Coo dina es wi h a Gene ic Rigid Body (FCC-GRB) o mula ion o he plana and spa ial analysis o mul ibody sys ems.
Due o he ype o coo dina es used o desc ibe he sys em and he kinema ic s uc u e adop ed o model he segmen s, his o mula ion
p ese es some o he main cha ac e is ics epo ed o he mos common global o mula ions, namely he na u al coo dina es [17,27,
33] and Ca esian coo dina es [24–26].
Simila ly o he na u al coo dina es o mula ion, in he FCC-GRB o mula ion, he mul ibody sys em is desc ibed using only
ec angula coo dina es o poin s and uni ec o s, implying ha no angula - ela ed a iables a e equi ed o desc ibe he o ien a ion
o he bodies. Fu he mo e, he kinema ics o any poin o ec o belonging o he model can be exp essed as a linea ans o ma ion o
a se o cons an ans o ma ion ma ices (ma ices C and V) and he gene alized coo dina es o he sys em. These ea u es imply ha
he cons ain equa ions o he mos common kinema ic cons ain s p esen , a he mos , a quad a ic dependency on he gene alized
coo dina es o he sys em, and, consequen ly, he espec i e con ibu ions o he Jacobian ma ix and igh -hand side ec o s o e-
loci y and accele a ion a e quad a ic, linea , cons an , o null [28,29].
Con e sely o he na u al coo dina es o mula ion, in which he de ini ion o he bodies depends on he opology o he sys em, he
igid bodies a e de ined wi h a p ede e mined kinema ic s uc u e composed o one e e ence poin and wo o h ee uni ec o s.
Mo eo e , i he e e ence poin is loca ed a he cen e o mass (CoM) o he body, he igid body mass ma ices become diagonal wi h
hei en ies equal o he ine ial p ope ies o he segmen s being modeled [28,29]. The use o a gene ic igid body app oxima es he
p oposed o mula ion o he Ca esian coo dina es app oach. Hence, some o he ad an ages o en associa ed wi h his o mula ion,
namely he easie sys ema iza ion o he modeling p ocedu e, and he s aigh o wa dness and high physical meaning o he mass
ma ices, also apply o he FCC-GRB o mula ion, simpli ying i s compu a ional implemen a ion [28,29].
Some au ho s explo ed a hyb id mul ibody o mula ion based on na u al coo dina es, which also uses angula a iables [34,35].
This app oach, o iginally p oposed by Jal´
on and Bayo [17], conside s ha he se o gene alized coo dina es includes no only he
coo dina es ha de ine he model opology bu also he angula d i e s ha guide he sys em, allowing o he simul aneous de e -
mina ion o he consis en posi ions and o ien a ions o he model and hei espec i e angula DoFs. Due o he hyb id na u e o he
coo dina es ha compose he sys em, his o mula ion is commonly e e ed o as mixed coo dina es (MC) [17,34,35].
The modeling app oach wi h MC is pa icula ly use ul when pe o ming he kinema ic analysis o sys ems ha equi e he p io
calcula ion o inpu d i e s om expe imen al da a, which may be subjec o measu emen e o s. By minimizing he dis ance be ween
he expe imen al da a and he co esponding elemen s o he mechanical model (e.g., poin s, ec o s, o ien a ions), his me hodology
θ∗
kRe e ence angle o he k- h angula gene alized coo dina e o he model
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
kRigh -hand-side ec o o he eloci y equa ions o he kinema ic cons ain o ype k
τ
in
kIn e nal join momen o o ce associa ed o join k
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
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
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
3
enables o ind he kinema ic consis en posi ions ha bes i he inpu da a o e ime. Fu he mo e, as he en i e posi ion is minimized
a he same ime, small e o s due o da a inconsis ency a e no p opaga ed along he kinema ic chain, as in he case o he kinema ic
analysis wi h d i e s. These ad an ages a e pa icula ly sui able o he biomechanics ield, whe e expe imen al da a is o en
con amina ed wi h STA, which can esul in ma ke displacemen s o a ew cen ime e s om hei o iginal posi ion [8,9,36]. The use o
an MC app oach allows o a enua ion o he e ec o hese a i ac s, while simul aneously compu ing he join angula d i e s di ec ly
om he expe imen al ma ke s. In addi ion, his s ep elimina es he need o p e-p ocessing p ocedu es ha in ol e he calcula ion o
he h ee-dimensional (3D) angles om expe imen al da a, which is o en a complex issue, especially o use s wi h less expe ise in he
opic.
Thus, he p esen wo k examines he use o an MC app oach in andem wi h an FCC-GRB o mula ion o pe o m he kinema ic and
in e se dynamic analysis o spa ial mul ibody sys ems. Th ee di e en app oaches o sol ing he kinema ic equa ions a e conside ed,
namely he New on-Raphson i e a i e me hod (NRM) oge he wi h a weigh ed leas squa es (WLS) minimiza ion, a cons ained
nonlinea op imiza ion (CNO) and a mul i-objec i e (MO) goal a ainmen op imiza ion, o e alua e hei in luence on he accu acy o
he econs uc ion o he kinema ic consis en posi ions om he expe imen al da a, he iola ion o he kinema ic cons ain s and
compu a ional ime. Wi h he pu pose o assessing i s applicabili y, he FCC-GRB o mula ion wi h an MC app oach is applied in he
analysis o wo mechanisms, conside ing, as inpu s, da a wi h and wi hou pe u ba ions.
Acco ding o Roupa e al. [28] and Gonçal es e al. [29], one o he majo ad an ages o modeling mul ibody sys ems wi h an
FCC-GRB o mula ion is i s ease o implemen a ion and modeling o complex mechanical sys ems. This ac implies ha , in addi ion o
he classical applica ions usually a ibu ed o he global mul ibody o mula ions, he FCC-GRB o mula ion is sui able o ad anced
educa ion pu poses. As he inco po a ion o he angula a iables does no signi ican ly change he modeling p ocedu e, he
s aigh o wa dness epo ed o he plana and spa ial o mula ion is s ill applicable in he mixed coo dina es a ia ion. Consequen ly,
he s eps equi ed o inco po a e angula coo dina es in o he o mula ion a e p esen ed in de ail, so ha , oge he wi h he plana and
spa ial FCC-GRB s udies [28,29], his wo k can se e as an educa ional ool o suppo ing he eaching o mul ibody- ela ed opics.
2. Fo mula ion
2.1. Fundamen al aspec s o he FCC-GRB o mula ion
In he FCC-GRB o mula ion, mul ibody sys ems a e modeled using a se o igid bodies de ined wi h a p e-de e mined kinema ic
s uc u e. This app oach ollows he me hodology used in he Ca esian coo dina es o mula ion, p esen ing ad an ages in he sys-
ema iza ion o he modeling p ocedu e and de ini ion o he sys em mass ma ices. Howe e , con a y o Ca esian coo dina es
o mula ion case, which uses angula - ela ed coo dina es, he ma hema ical s uc u e adop ed o de ining each body uses only
ec angula coo dina es o poin s and ec o s. Speci ically, he s uc u e adop ed o he igid body conside s he use o one ec o ha
de ines he e e ence poin ( Oi), usually loca ed a i s cen e o mass, and h ee non-coplana uni ec o s (ui, i, wi), as schema ically
ep esen ed in Fig. 1a. The e o e, he ec o o he gene alized coo dina es o he gene ic igid body i (qi) is de ined as [29]
qi={ T
OiuT
i T
iwT
i}T(1)
Vec o Oi allows o desc ibe he ansla ion mo emen s o he igid body i, while he ec o s ui, i, wi p o ide i s o ien a ion. Since he
h ee uni ec o s de ine a ec o basis in space, i is possible o compu e he kinema ics o any gene ic poin (P) o ec o (s) belonging
o he body di ec ly om i s gene alized coo dina es using a se o ans o ma ion ma ices, as p esen ed below [29]
Fig. 1. Rep esen a ion o he kinema ic s uc u e o a gene ic igid body i wi h a gene ic poin P and ec o s belonging o i , modeled in i s: a) ully-
de ined o m; b) educed o m.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
4
P=CP
iqi(2)
and
s=Cs
iqi(3)
whe e P and s ep esen , espec i ely, he posi ion ec o o he gene ic poin P and ec o s in he global e e ence ame, and CP
i and
Cs
i he espec i e ans o ma ion ma ices.
As i is possible o es ablish a ec o basis using o hogonaliza ion me hods wi h wo non-collinea ec o s, he de ini ion o he
gene ic igid body can be educed o wo uni ec o s (see Fig. 1b)
qi={ T
OiuT
i T
i}T(4)
The educed de ini ion, gi en by Eq. (4), has he ad an age o equi ing less gene alized coo dina es and kinema ic cons ain s han
he ully-de ined o m, educing he dimensions o he p oblem o sol e. Howe e , he de ini ion o he kinema ic cons ain s ha
desc ibe he opology o he sys em equi es he compu a ion o he ully-de ined equi alen ec o (q3 i) om he educed o m. This
p ocedu e is pe o med using a se o ans o ma ion ma ices S and V, which p esen s an explici dependency on he gene alized
coo dina es o he sys em, such as [29]
q3 i=SViqi(5)
wi h
S=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
I3030303
03I30303
0303I303
030303
1
2I3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦(12×12)
(6)
and
Vi=⎡
⎢
⎢
⎣
I30303
03I303
0303I3
03−
i
ui
⎤
⎥
⎥
⎦(12×9)
(7)
This dependency implies ha he deg ee o he kinema ic cons ain s inc eases, and, consequen ly, he espec i e con ibu ions o
he Jacobian ma ix (Φq). Mo eo e , he igid body mass ma ix becomes dependen on he gene alized coo dina es, also gene a ing
eloci y-dependen ine ial o ces [29].
Thus, conside ing a mechanical model composed o nb bodies, he posi ion and o ien a ion o he mul ibody sys em is algeb aically
de ined as
Fig. 2. Rep esen a ion o he kinema ic s uc u e and he espec i e DoFs o a e olu e join de ined be ween ec o s1 o igid body i and ec o s2
o igid body j and o a sphe ical join de ined be ween ec o s s3 and s4 o igid bodies j and ec o s s5 and s6 o igid body j +1 modeled using an
FCC-GRB o mula ion wi h MC.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
5

q={qT
1⋯qT
i⋯qT
nb }T(8)
whe e q is he ec o o gene alized coo dina es o he en i e sys em, including all he bodies de ined in he ully-de ined and educed
o ms.
2.2. Mixed coo dina es in an FCC-GRB o mula ion
When including an MC app oach in o he FCC-GRB o mula ion, he angula d i e s (θk) ha guide he model a e also ea ed as
gene alized coo dina es o he sys em o be sol ed (see Fig. 2). This app oach has he ad an age o allowing o he simul aneous
compu a ion o he coo dina es and join angles o he model ha bes i he expe imen al da a, wi hou equi ing a p e-p ocessing
s ep o calcula e he d i e s o he model. By minimizing he dis ance be ween he elemen s o he model and he equi alen elemen s
o he expe imen al da a, he me hod mi iga es he e ec o he expe imen al e o s in he kinema ic econs uc ion o he mo emen ,
enabling o ind he op imal model con igu a ion ha , on a gi en imes ep, bes - i s he expe imen al da a.
Ma hema ically, he MC app oach in ol es he addi ion o a se o angula a iables (θ) o he ec o o he gene alized coo dina es
q={qT
1⋯qT
i⋯qT
nb θ1⋯θk⋯θnd }T(9)
whe e θk is he alue o he angula displacemen o he k- h d i e o he model. Conside ing his app oach, he numbe o angula
a iables o add should be equal o he numbe o DoFs o model using his app oach (nd). I is wo h men ioning ha he o al numbe
o angula DoFs o he model can be highe , being he o he DoFs guided using he adi ional app oach wi h kinema ic cons ain
d i e s as desc ibed in sec ion 2.4.1.
The inclusion o he angula coo dina es inc eases he o al numbe o gene alized coo dina es o he sys em. The e o e, a se o
angula -based cons ain equa ions needs o be added o he ec o o he kinema ic cons ain s o he sys em (Φ) o exp ess he o-
pological dependencies be ween he FCC-GRB gene alized coo dina es and he angula coo dina es (see Sec ion 2.3). Mo eo e ,
addi ional linea acking cons ain s need o be inco po a ed in o he ec o Φ o i he model o he expe imen al da a, e ec i ely
guiding he model h oughou his p ocess. I should be no ed ha , when analyzing (bio)mechanical sys ems, he e e ence da a
ypically ep esen he posi ion o o ien a ion o ele an poin s o ec o s in he model. Hence, wo main ypes o acking cons ain s
a e commonly used, namely he acking cons ain (TC) o guiding poin s and he acking o ien a ion cons ain (TOC) o guiding
ec o s (see Sec ion 2.4.2).
2.3. Kinema ic cons ain s – Topologic de ini ion
As i happens in o he mul ibody o mula ions, he modeling o mechanical sys ems wi h FCC-GRB equi es he ma hema ical
de ini ion o hei opology. Gonçal es e al. p esen ed a de ailed desc ip ion o he kinema ic cons ain equa ions needed o a
comp ehensi e ep esen a ion o a mechanical sys em in 3D, using he mos common kinema ic join s [29]. Consequen ly, he p esen
wo k ocuses only on he kinema ic equa ions equi ed o ully in eg a e he MC in o he FCC-GRB o mula ion.
2.3.1. Mixed coo dina es ela ion o a ully-de ined body ep esen a ion
F om he opological poin o iew, he use o an MC app oach equi es he explici de ini ion o he geome ic ela ions be ween
he gene alized coo dina es o he FCC-GRB o mula ion and he newly in oduced angula coo dina es. This p ocedu e can be
desc ibed by one algeb aic cons ain equa ion ha imposes a gi en angle be ween wo uni ec o s belonging o each o he bodies
ela ed by he DoF. Le one conside wo gene ic uni ec o s s1 and s2 belonging espec i ely o igid bodies i and j (see Fig. 2), he
mixed coo dina es ela ion (MCR) is desc ibed in i s homogenous o m as
ΦMCR(qi,qj,θk)=sT
1s2−cosθk=0 (10)
whe e θk is he angula a iable ha desc ibes he angula displacemen be ween ec o s s1 and s2 (θs1s2). By applying he linea
ans o ma ion exp essed in Eq. (3), he p e ious equa ion can be desc ibed in e ms o he gene alized coo dina es o he sys em as
ΦMCR(qi,qj,θk)=(Cs1
iqi)T(Cs2
jqj)−cosθk=0
ΦMCR =qT
iCs1s2qj−cosθk
(11)
wi h
Cs1s2=[Cs1
i
TCs2
j](12 ×12)(12)
Equa ion (11) exp esses a quad a ic ela ion be ween he gene alized coo dina es o bodies i and j. Howe e , con a y o he
p emises o he FCC-GRB o mula ion, i also includes nonlinea e ms ha a e dependen on he gene alized angula a iables.
The e o e, he con ibu ions o he Jacobian ma ix (Φq), comp ising he pa ial de i a i es o he kinema ic cons ain equa ions wi h
espec o he gene alized coo dina es o he sys em, include bo h he linea e ms associa ed wi h he gene alized coo dina es o he
wo cons ained bodies and an addi ional con ibu ion associa ed wi h he angula a iable θk, such ha
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
6
ΦMCR
q={qT
jCs1s2T
⏞⏟⏟⏞
ΦMCR
qi
qT
iCs1s2
⏞⏟⏟⏞
ΦMCR
qj
sinθk
⏞⏟⏟⏞
ΦMCR
θk}(1×25)
(13)
Due o he cons an na u e o ma ices C, ma ix Cs1s2 is also cons an and ime independen [29]. Mo eo e , he gene alized
angula a iables do no p esen an explici dependency on he ime, indica ing ha Eq. (11) ep esen s a scle onomic cons ain . Thus,
he con ibu ions o he igh -hand side ec o o he eloci ies o he sys em (
ν
) a e null
ν
MCR =0 (14)
In u n, he quad a ic and anscenden al na u e o he MCR condi ion implies ha he con ibu ions o he igh -hand side ec o o
he accele a ions (γ) include quad a ic e ms ha depend on he gene alized eloci ies o bodies i (˙
qi) and j (˙
qj), as well he gene alized
angula eloci ies (˙
θk), such ha
γMCR = − 2˙
qT
iCs1s2˙
qj−˙
θ2
kcosθk= − 2˙
sT
1˙
s2−˙
θ2
kcosθk(15)
I should be no ed ha Eq. (11) ep esen s one kinema ic cons ain equa ion ha desc ibes one o a ional DoF associa ed wi h he
kinema ic join de ined by bodies i and j. Consequen ly, o ully desc ibe he o a ional DoFs associa ed wi h a speci ic kinema ic join ,
i is necessa y o include, a leas , one MCR cons ain equa ion pe DoF in he ec o o he kinema ic cons ain s, and he espec i e
con ibu ions o he Jacobian ma ix and igh -hand side ec o s o eloci y and accele a ion.
Equa ions (13) o (15) can be simpli ied when he ec o s ha desc ibe he MCR condi ion a e di ec ly igid body ec o s (e.g., ui
and j). In his case, he de ini ion o he MCR condi ion in i s homogenous o m is gi en by
ΦMCR(qi,qj,θk)=uiT j−cosθk(16)
This simplici y in he e alua ion o he MCR kinema ic cons ain is also ansla ed in o he calcula ion o he espec i e con i-
bu ions o Φq,
ν
and γ
ΦMCR
q={ T
j
⏞⏟⏟⏞
ΦMCR
ui
uT
i
⏞⏟⏟⏞
ΦMCR
j
sinθk
⏞⏟⏟⏞
ΦMCR
θk}(1×25)
(17)
ν
MCR =0 (18)
γMCR = − 2˙
uT
i˙
j−˙
θ2
kcosθk(19)
2.3.2. Mixed coo dina es ela ion o a educed body ep esen a ion
As p e iously men ioned, a educed app oach can be adop ed o model he igid bodies. In his case, he kinema ic cons ain
equa ions needed o include he MC a e simila o hose used in he ully-de ined case. Howe e , he linea ans o ma ion p esen ed in
Eq. (5) should be applied o calcula e he ully-de ined equi alen ec o , esul ing in he ollowing exp ession
ΦMCR(qi,qj,θk)=(Cs1
iSViqi)T(Cs2
jSVjqj)−cosθk=0
ΦMCR =qT
iSViCs1s2SVjqj−cosθk
(20)
The dependence o he ma ices V on he gene alized coo dina es o he sys em implies an inc ease in he deg ee o he cons ain
equa ions. Ne e heless, as discussed in [29], he e alua ion o he MCR cons ain is highly e icien , as i only conside s he algeb aic
mul iplica ion o linea ma ices and ec o s dependen on he gene alized coo dina es o he sys em. This aspec is also alid o he
con ibu ions o he Jacobian ma ix and igh -hand side ec o o accele a ions
ΦMCR
q={qT
jVT
jSTCs1s2TVi
⏞⏟⏟⏞
ΦMCR
qi
qT
iVT
iSTCs1s2Vj
⏞⏟⏟⏞
ΦMCR
qj
sinθk
⏞⏟⏟⏞
ΦMCR
θk}(1×25)
(21)
γMCR = − (qT
iVT
iSTCs1s2˙
Vj˙
qj+qT
jVT
jSTCs1s2T˙
Vi˙
qi+2˙
qT
iVT
iCs1s2Vj˙
qj)−˙
θ2
kcosθk
γMCR = − (sT
1Cs2
j˙
Vj˙
qj+sT
2Cs1
i˙
Vi˙
qi+2˙
sT
1˙
s2)−˙
θ2
kcosθk
(22)
whe e ˙
Vi is he ime de i a i e o he ans o ma ion ma ix V o he body i de ined ollowing he me hodology p esen ed in [29].
Since Eq. (20) main ains i s scle onomic na u e, he con ibu ions o he igh -hand side ec o o eloci ies o he educed case emain
null.
2.4. Kinema ic cons ain s – D i e s de ini ion
In addi ion o de ining he opology o he sys em, he kinema ic analysis o a mul ibody sys em equi es he desc ip ion o all he
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
7
DoFs associa ed wi h he mechanical model unde analysis. Two main app oaches can be conside ed o d i e hese DoFs, namely a
me hodology based on ansla ional and angula d i e s, o a me hod based on he minimiza ion o a se o acking cons ain s.
In he i s app oach, he angula DoFs need o be calcula ed om he expe imen al da a, being subsequen ly ed in he sys em in he
o m o kinema ic cons ain s o angula d i e ype ha need o be sa is ied o each ime ame. The p esence o a i ac s in he
expe imen al da a will be e lec ed in he co esponding d i e s, and, consequen ly, in he kinema ic consis en posi ions compu ed
du ing he analysis. A de ailed desc ip ion o he mos common d i e ypes used in FCC-GRB o mula ion can be ound in [29].
The second me hod uses a se o acking cons ain s and acking o ien a ion cons ain s, which exp ess geome ic ela ions be-
ween ele an poin s and ec o s o he model and he espec i e expe imen al da a, being minimized along he analysis. This me hod
allows o an adap a ion o he model o he p esc ibed da a, mi iga ing he e ec o possible expe imen al e o s in he compu a ion o
he kinema ic consis en posi ions. I is wo h no ing ha o he ypes o acking cons ain s can also be inco po a ed. Fo example, he
angula ela ion used o guide he model in he i s me hod could be e o mula ed as a acking cons ain o be minimized. Howe e ,
when applied o expe imen al da a acqui ed using ma ke -based op oelec onic MOCAP sys ems, his s a egy con as s wi h he
p ima y objec i e o au oma ically de e mining join angula d i e s ha bes ep oduce he obse ed poin posi ions, equi ing also a
p e-kinema ic s ep o calcula e he e e ence d i e alue. As his app oach alls ou side he scope o his s udy, i will no be explo ed.
Ne e heless, in sys ems whe e only he posi ions o ce ain poin s and he join angles a e a ailable, he me hod could be adap ed o
include such da a, making i possible o compu e a solu ion ha i s bo h he poin posi ions and minimizes he di e ences in angula
d i e s.
2.4.1. Angula d i e s
Wi hin he FCC-GRB o mula ion, he angula DoFs can be desc ibed using an AR condi ion. This kinema ic cons ain imposes ha
he ela i e angle be ween wo ec o s o he sys em is equal o a p esc ibed angle. Ma hema ically, he AR condi ion is simila o he
o mula ion used o desc ibe he MCR condi ion. Howe e , i conside s ha he angula ela ion is made using he p esc ibed angle (θ∗
k)
ins ead o he gene alized angula a iables (θk). Thus, le one conside wo gene ic uni ec o s s1 and s2 belonging espec i ely o
igid bodies i and j de ined in hei ully-de ined o m, he AR condi ion can be desc ibed in he homogenous o m as
ΦAR(qi,qj, )=sT
1s2−cosθ∗
s1s2( ) = (Cs1
iqi)T(Cs2
jqj)−cosθ∗
s1s2( ) = 0
ΦAR =qT
iCs1s2qj−cosθ∗
s1s2( )
(23)
whe e θ∗
s1s2 is he p esc ibed angle ha desc ibes he angula DoF be ween ec o s s1 and s2 along he ime. The AR condi ion exp esses
a quad a ic ela ion be ween he gene alized coo dina es o he sys em, implying ha hei con ibu ions o he Jacobian ma ix a e
linea and equal o
ΦAR
q={qT
jCs1s2T
⏞⏟⏟⏞
ΦAR
qi
qT
iCs1s2
⏞⏟⏟⏞
ΦAR
qj}(1×24)
(24)
Con a y o he MCR condi ion, he second e m o Eq. (23) p esen s an explici dependency on he ime ec o . Hence, he AR
condi ion p esen s a heonomic na u e, meaning ha he con ibu ions o he igh -hand-side ec o o eloci ies and accele a ions also
include e ms dependen on he ime
ν
AR = − ˙
θ∗
s1s2( )sinθ∗
s1s2( )(25)
γAR = − 2˙
qT
iCs1s2˙
qj−((˙
θ∗
s1s2( ))2cosθ∗
s1s2( ) + ¨
θ∗
s1s2( )sinθ∗
s1s2( ))(26)
Simila ly o he MCR condi ion, Eq. (23) ep esen s a single kinema ic cons ain ha go e ns one DoF. The e o e, o each DoF o
he model ha is guided using an angula d i e cons ain , i is necessa y o inco po a e, a leas , one AR condi ion in o he ec o o
kinema ic cons ain s.
When he ec o s o cons ain a e igid body ec o s, Eqs. (23) o (26) can be simpli ied, ob aining
ΦAR(qi,qj, )=uiT j−cosθ∗
ui j( ) = 0 (27)
Fo his case, he con ibu ions o he Jacobian ma ix and igh -hand side ec o s o eloci ies and accele a ions a e iden ical o
hose exp essed by Eqs. (17) o (19), conside ing only he e ms co esponding o he FCC-GRB gene alized coo dina es plus he
heonomic e ms p esen ed in Eqs. (25) and (26).
I a educed igid body de ini ion is used o he de ini ion o he model segmen s, he AR condi ion can be de i ed om Eq. (23),
conside ing he compu a ion o he ully-de ined equi alen ec o as p esen ed in Eq. (5), yielding
ΦAR(qi,qj, )=qT
iSViCs1s2SVjqj−cosθ∗
ui j( ) = 0 (28)
As o he ully-de ined case, he de ini ion o he AR condi ion in he educed o m is simila o he MCR condi ion (see Eq. (20)).
Howe e , he angula gene alized coo dina es a e subs i u ed by he p esc ibed angle. The e o e, he con ibu ions o Φ,
ν
and γ a e
once again iden ical o hose p esen ed in Eqs. (21) o (22), conside ing only he e ms co esponding o he gene alized coo dina es o
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
8
he sys em and he heonomic e ms exp essed by Eqs. (25) and (26).
2.4.2. T acking cons ain s
The o mula ion o he TC condi ion is simila o he linea ansla ion d i e desc ibed in [29]. Howe e , ins ead o en o cing he
dis ance be ween a model poin and a e e ence poin o be ze o along he h ee axes o he global e e ence ame, i minimizes he
dis ance be ween hem. Le one conside a gene ic poin P belonging o he igid body i de ined in i s ully-de ined o m, he TC
condi ion can be desc ibed in i s homogenous o m as
ΦTC(qi, ) = Pi− P∗( ) = ⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Pxi− P∗
x
Pyi− P∗
y
Pzi− P∗
z
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
=CP
iqi− P∗( ) = 0(29)
whe e P∗ ep esen s he p esc ibed coo dina es o he e e ence poin P* in he global e e ence ame. Eq. (29) ans o ms in o h ee
linea cons ain equa ions ha guide h ee DoFs associa ed wi h poin P, implying ha he con ibu ions o he Jacobian ma ix a e
cons an and equal o he co esponding ma ix C
ΦTC
q=⎡
⎢
⎣CP
i
⏞⏟⏟⏞
ΦTC
qi⎤
⎥
⎦(3×12)(30)
I should be no ed ha he con ibu ions can be assembled in o he Jacobian ma ix o he sys em du ing he kinema ic p e-
p ocessing s eps, elimina ing he need o i s upda e du ing he analysis and, consequen ly, educing i s compu a ional e o . In
u n, he heonomic na u e o he TC condi ion means ha he con ibu ions o he igh -hand side ec o s o he eloci ies and ac-
cele a ions a e no null, p esen ing explici e ms depending on he eloci y (˙
P∗) and accele a ion (¨
P∗) o he e e ence poin P*, as
p esen ed below
ν
TC =˙
P∗( )(31)
γTC =¨
P∗( )(32)
In a simila way o he kinema ic cons ain s, Eq. (29) can be adap ed o he educed de ini ion o a igid body using he linea
ans o ma ion p esen ed in Eq. (5). Thus, le one conside a e e ence poin P* belonging o a igid body i de ined in i s educed o m,
he TC condi ion and he co esponding con ibu ions o Φ,
ν
and γ a e gi en by
ΦTC(qi, ) = CP
iq3 i− P∗( ) = CP
iSViqi− P∗( ) = 0(33)
ΦTC
q=⎡
⎢
⎣CP
iVi
⏞⏟⏟⏞
ΦTC
qi⎤
⎥
⎦(3×9)
(34)
TC =˙
P∗( )(35)
γTC =¨
P∗( ) − (CP
i˙
Vi˙
qi)(36)
The use o he ma ix Vi inc eases he deg ee o he kinema ic cons ain , implying ha i s con ibu ions o he Jacobian ma ix
become dependen on he sys em gene alized coo dina es, and, consequen ly, i equi es i s upda e in each ime s ep.
The acking cons ain s can also be applied o map he o ien a ion o a se o guiding ec o s. In his case, he kinema ic cons ain
equa ions in hei homogeneous o m a e simila o hose p esen ed o a e e ence poin P*, conside ing he ma ix Cs o a gene ic
ec o s (see Eq. (3)) ins ead o ma ix CP. The e o e, le one conside a gene ic uni ec o s belonging o igid body i and he espec i e
guiding ec o s∗, he acking o ien a ion cons ain can be gi en by
ΦTOC(qi, ) = si−s∗( ) = Cs
iqi−s∗( ) = 0(37)
o in he educed o m
ΦTOC(qi, ) = Cs
iq3 i−s∗( ) = Cs
iSViqi−s∗( ) = 0(38)
Ma hema ically, he TOC o mula ion is simila o he TC condi ion, which means ha he con ibu ions o he Jacobian ma ix
(ΦTOC
q) a e simila o hose gi en by Eqs. (30) o (34), howe e , using he ma ix Cs
i. In u n, he con ibu ions o he igh -hand side
ec o o eloci ies (
ν
TOC) and accele a ions (γTOC) a e equal o hose exp essed by Eqs. (31) and (32) o (35) and (36), conside ing he
subs i u ion o he ans o ma ion ma ix CP
i and he eloci y and accele a ion ec o s o he e e ence poin by he equi alen en i ies
o ec o s∗, espec i ely ˙
s∗and ¨
s∗.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
9
kinema ic analysis wi h he NRM (see Figs. C.1 e- in SM C), wi h he MO wi h TC p esen ing he highe alues (~10
–5
m) and he MO
wi h ED wi h he lowe ones (10
–14
o 10
–9
m). Rega ding he CNO, no di e ences we e obse ed be ween he TC o he ED cons ain in
he objec i e unc ion, showing di e ences o he e e ence da a in he o de o 10
–8
o 10
–6
m.
4.1.2. Compu a ional e iciency
To begin wi h, i mus be said ha all ou me hods con e ged quickly o he op imal solu ion, conside ing as an ini ial guess he
posi ion o he p e ious ime ame. Howe e , he numbe o i e a ions pe ime s ep and he compu a ional ime depended on he
magni ude o he pe u ba ion imposed on he e e ence da a and he me hod used o pe o m he kinema ic analysis (see Table 1).
As expec ed, he NRM me hod exhibi ed a lowe numbe o i e a ions and as e con e gence. In he absence o pe u ba ions and
conside ing a ole ance e o o 10
–10
, his me hod con e ged in app oxima ely 2.6 s (8.6 ms/ ame) o he simple model, aking
a ound 3 % mo e ime han he kinema ic analysis wi h AD. The inc ease o he i e a ion ole ance in ou o de s o magni ude esul ed
in a dec ease o app oxima ely 1 i e a ion pe ime ame and 21.8 % o he compu a ional ime. The inclusion o he pe u ba ions in
he TC d i e s esul ed in an inc ease o he compu a ional ime o 8.2 % (9.6 ms/ ame) o he no mal STA case and 51.7 % (13.5 ms/
ame) o he ex eme case.
The op imiza ion-based me hods o he unpe u bed case equi ed mo e ime o ind a solu ion han he NRM wi h WLS, anging
om 350 % (40.0 ms/ ame) o he MO wi h ED objec i e unc ion o 1928 % (170.0 ms/ ame) in he case o he CNO wi h in e io -
poin algo i hm. The addi ion o he pe u ba ions o he e e ence da a esul ed in a simila numbe o i e a ions pe ime ame and
compu a ional imes o he unpe u bed case. A signi ican di e ence in he compu a ional pe o mance was obse ed be ween he
CNO wi h he de aul algo i hm (in e io -poin me hod) and he MO a ainmen p oblem, wi h he la e aking app oxima ely 78 %
less ime o he unpe u bed case. The use o he SQP algo i hm allowed o educe signi ican ly he op imiza ion ime o he CNO,
eaching an a e age alue o 42 ms/ ame. An in luence o he objec i e unc ion can also be ound in he MO op imiza ion, wi h he
ED-based unc ion, which gene a es a lowe numbe o objec i e unc ions, aking less ime han he one based on he TC condi ion.
4.2. Fi e-ba linkage wi h pe u ba ions in all e e ence poin s
4.2.1. Compu a ional accu acy
The esul s ob ained o he i e-ba linkage wi h pe u ba ions in all poin s we e simila o hose obse ed in he p e ious model
(see Fig. 5 and Fig. C.2 in SM C). The NRM wi h LS and WLS allowed o achie e a solu ion ha ul ils wi h he i e a ion ole ance
de ined o he analysis. Fo he unpe u bed case, bo h me hods achie ed di e ences o he e e ence da a in he o de o 10
–16
o
10
–12
, while he op imiza ion me hods yielded alues anging om 10
–8
and 10
–4
. Howe e , an adap a ion o he gene alized posi ions
o he model poin s o he noisy e e ence poin s was obse ed when only he LS app oach was u ilized, leading o iola ions o he
opological cons ain s (maximum alues o ΦTop
io 3.9 ×10
–3
o a 7.5 mm pe u ba ion and 1.4 ×10
–2
o a 30.0 mm pe u ba ion).
The elaxa ion o he TC weigh s allowed o he minimiza ion o his issue, ob aining alues in he o de o 10
–5
and 10
–4
o he
analyses wi h an i e a ion ole ance o 10
–6
and 10
–10
(see Table 2).
The op imiza ion me hods enabled he compu a ion o a se o kinema ic consis en posi ions ha i he e e ence da a, while
main aining he maximum cons ain iola ion unde 10
–10
. Al hough he di e en algo i hms gene a ed sligh ly di e en solu ions,
none di ec ly ollowed he noisy da a used as inpu . A signi ican di e ence in he kinema ic pa e ns was obse ed be ween he esul s
gene a ed using pa allel and sequen ial op imiza ion, pa icula ly o he pe u bed cases. The inabili y o p o ide an ini ial guess nea
he solu ion o he pa allel me hod esul ed in a noisy pa e n, as he me hod con e ged o simila posi ions ha also ul illed he
opological cons ain s bu did no necessa ily ep esen he global minimum (see Fig. 5b and d). Mo eo e , some con e gence
p oblems we e ound in some ime ames, e en o he unpe u bed case, leading o he appea ance o some noisy a i ac s in he
esul s (see Table 2 and Fig. C.2 in SM C).
Table 1
No malized compu a ional ime, a e age numbe o i e a ions pe ame, mean cons ain iola ions pe ame and maximum alue o he cons ain
iola ion o he opological kinema ic cons ain s.
No malized Time [ms/ ame] N. I e a ions ‖Φ‖Max ΦTop
i
eP0 P7.5 P30 P0 P7.5 P30 P0 P7.5 P30 P0 P7.5 P30
KA AD 10
–6
6.6 - - 3.3 - - 2.0 ×10
–7
- - 7.9 ×10
–7
- -
  10
–10
8.6 - - 4.2 - - 2.8 ×10
–12
- - 2.7 ×10
–11
- -
NRM LS 10
–6
7.0 7.4 9.1 3.0 3.3 4.1 1.1 ×10
–7
3.1 ×10
–7
2.3 ×10
–7
7.4 ×10
–7
2.4 ×10
–3
8.9 ×10
–3
  10
–10
8.9 11.1 13.5 4.0 5.0 6.3 9.9 ×10
–13
1.9 ×10
–11
2.5 ×10
–11
3.5 ×10
–12
2.4 ×10
–3
8.9 ×10
–3
NRM WLS 10
–6
6.9 7.5 9.0 3.0 3.3 4.0 1.1 ×10
–7
2.4 ×10
–7
2.4 ×10
–7
7.4 ×10
–7
1.2 ×10
–5
5.7 ×10
–5
  10
–10
8.8 9.6 13.5 4.0 4.9 6.5 1.0 ×10
–12
1.8 ×10
–11
2.4 ×10
–11
2.4 ×10
–12
1.2 ×10
–5
5.7 ×10
–5
CNO IP 10
–10
171.4 173.1 179.2 24.0 24.1 24.3 5.0 ×10
–11
4.9 ×10
–11
5.2 ×10
–11
9.8 ×10
–11
9.9 ×10
–11
9.9 ×10
–11
SQP 10
–10
42.3 42.1 43.0 3.3 3.3 3.4 1.1 ×10
–12
1.8 ×10
–12
4.3 ×10
–12
7.7 ×10
–12
3.6 ×10
–11
9.6 ×10
–11
MO TC 10
–10
90.8 51.6 51.8 12.3 6.6 5.8 5.4 ×10
–11
9.1 ×10
–12
7.1 ×10
–12
9.9 ×10
–11
9.2 ×10
–11
9.9 ×10
–11
ED 10
–10
40.0 41.0 43.3 4.9 5.0 5.1 5.4 ×10
–12
6.6 ×10
–12
9.6 ×10
–12
9.9 ×10
–11
9.6 ×10
–11
9.7 ×10
–11
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
16

Fig. 5. Compa ison be ween he KA esul s and e e ence da a o P4 o he i e-ba linkage model wi h a maximum pe u ba ion o 52 mm o all
poin s conside ing he NRM (le ) and op imiza ion algo i hms ( igh ): Top (a and b) – x coo dina e; Middle (c and d) – y coo dina e; Bo om (e and
) – z coo dina e.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
17
4.2.2. Compu a ional e iciency
The esul s o he mo e complex model wi h pe u ba ions in all e e ence poin s show ha he NRM is signi ican ly as e han he
bes op imiza ion case. Con a y o he simple model, whe e he compu a ional imes we e simila , he inc ease in model complexi y
esul ed in a 57.7 % inc ease in compu a ional ime o he NRM wi h WLS wi h an i e a ion ole ance o 10
–6
compa ed o he ki-
nema ic analysis wi h AD. This di e ence was signi ican ly highe when he i e a ion ole ance was educed o 10
–10
, leading o an
inc ease o 79.4 % in he a e age ime pe ame, e en hough an equi alen numbe o i e a ions pe ame was obse ed.
Fo he unpe u bed case, he CNO wi h SQP was as e han he MO wi h bo h TC and ED objec i e unc ions. This end e e ed
when he pe u ba ions we e added o he inpu da a, wi h he MO wi h TC unc ion con e ging as e . This in e sion can be un-
de s ood by he lowe numbe o i e a ions and unc ion e alua ions ha his me hod equi es o con e ge. Mo eo e , wi h he in-
c ease o he pe u ba ion, he no malized ime pe ame educed o he MO cases.
The use o a pa alleliza ion s a egy wi h six pools esul ed in a educ ion o he compu a ional cos , anging om i e imes as e
in he unpe u bed case o wo imes in he maximum pe u ba ion case. The lowe e iciency o pa alleliza ion o he mo e pe u bed
cases can be explained by he educ ion o he ela i e di e ence in he numbe o i e a ions equi ed o ind a solu ion be ween he
sequen ial and pa alleliza ion cases. In con as , he educ ion o he CNO was app oxima ely wo imes o all cases.
In gene al, using one addi ional acking poin led o an inc ease in compu a ional cos . This di e ence was mo e isible in he
op imiza ion analyses, wi h inc eases anging om 3.0 ×10
–2
% o he MO wi h TC and lowe pe u ba ion o 43.4 % o he MO wi h
ED and highe pe u ba ion. The esul s o he NRM wi h WLS and an i e a ion ole ance o 10
–10
showed inc eases anging om 2.5 %
o 6.8 %.
4.3. Fi e-Ba linkage wi h pe u ba ion on P
4
and P
9
To simula e he e ec o STA in ma ke s mo e p one o his a i ac and e alua e i s in luence on he p opaga ion o e o s along he
kinema ic chain, a hi d condi ion, whe e only P
4
and P
9
we e pe u bed, was es ed, e ealing signi ican di e ences be ween me hods
(see Fig. 6 and Fig. C.3 in SM C). The esul s sugges ha bo h he NRM-WLS and he op imiza ion me hods can co ec he pe u -
ba ions in oduced ea lie in he kinema ic chain. The posi ions achie ed o he las poin o he open chain closely ma ched he
e e ence posi ion. In con as , he kinema ic analysis wi h AD and pe u bed d i e s exhibi ed subs an ial disc epancies in e ms o he
kinema ic pa e ns, e lec ing he noise in oduced in he da a. This di e ence is clea ly e iden in he Euclidean dis ance plo , which
showed a maximum dis ance o 0.05 m o he smalle pe u bed case and 0.21 m o he maximum pe u bed case. The esul s indica e
ha he NRM wi h WLS ends o gene a e smalle di e ences compa ed o he op imiza ion me hods; howe e , i comes a he cos o
no ensu ing such s ic compliance o he kinema ic cons ain s.
Fig. 7 p esen s one o he join angula d i e s compu ed o P4 using he MC app oach. The esul s show ha e en when in o-
ducing pe u ba ions in he poin ha de ines his join and he nea es one P
9
, which assis s in acking he segmen longi udinal
o a ions, he me hod can compu e a consis en angle ha ma ches he e e ence d i e , while educing he noise. Fo he no mal STA
alues, he angle pa e n ollows he e e ence wi hou signi ican oscilla ions (see Fig. 7a). On he o he hand, o he maximum
pe u bed case, al hough he compu ed angle ollows he expec ed end, i s ill exhibi s oscilla ions h oughou he analysis, which a e
mo e no iceable in he MO-ED case (see Fig. 7b). I is wo h men ioning ha he magni ude o hese oscilla ions is signi ican ly lowe
han wha is obse ed when he d i e s a e compu ed om he pe u bed da a.
The compa ison be ween he MC app oach and he use o smoo hed noisy angula d i e s is p esen ed in Fig. 8. The esul s show
ha he MC me hod using NRM-WLS leads o posi ion and eloci y p o iles close o he e e ence da a han any o he smoo hed d i e
Table 2
No malized compu a ional ime, a e age numbe o i e a ions pe ame, mean cons ain iola ions pe ame and maximum alue o he cons ain
iola ion o he opological kinema ic cons ain s.
No malized Time [ms/ ame] N. I e a ions ‖Φ‖Max ΦTop
i
eP0 P7.5 P30 P0 P7.5 P30 P0 P7.5 P30 P0 P7.5 P30
KA AD 10
–6
16.2 - - 3.6 - - 8.2 ×10
–8
- - 1.1 ×10
–6
- -
  10
–10
18.4 - - 4.0 - - 6.6 ×10
–12
- - 7.8 ×10
–11
- -
NRM LS 10
–6
24.8 36.2 36.2 3.7 5.5 5.6 5.7 ×10
–8
3.2 ×10
–7
3.3 ×10
–7
1.1 ×10
–6
3.9 ×10
–3
1.4 ×10
–2
  10
–10
26.10 56.7 57.6 4.0 8.6 8.8 1.2 ×10
–11
3.4 ×10
–11
3.4 ×10
–11
4.3 ×10
–11
3.9 ×10
–3
1.4 ×10
–2
NRM WLS (8 e p s) 10
–6
25.30 38.8 50.8 3.8 5.8 7.8 3.1 ×10
–8
3.4 ×10
–7
4.2 ×10
–7
9.8 ×10
–7
7.0 ×10
–5
2.5 ×10
–4
  10
–10
30.10 60.0 83.5 4.6 9.2 12.9 7.1 ×10
–12
3.1 ×10
–11
4.3 ×10
–11
1.1 ×10
–10
7.0 ×10
–5
2.5 ×10
–4
NRM WLS (9 e p s) 10
–6
26.6 39.7 41.7 3.8 5.8 8.1 3.2 ×10
–8
3.4 ×10
–7
4.2 ×10
–7
9.0 ×10
–7
7.9 ×10
–5
2.8 ×10
–4
  10
–10
31.20 61.5 89.2 4.6 9.2 13.4 5.3 ×10
–12
3.5 ×10
–11
4.2 ×10
–11
1.6 ×10
–10
7.9 ×10
–5
2.8 ×10
–4
CNO (8 e p s) SQP 10
–10
1199.2 1605.2 1384.7 34.1 46.7 48.1 1.2 ×10
–15
1.3 ×10
–14
2.4 ×10
–13
1.6 ×10
–14
3.8 ×10
–13
3.3 ×10
–11
SQP Pa 10
–10
548.2 703.1 554.9 77.0 82.1 83.9 6.3 ×10
–7
4.7 ×10
–6
2.3 ×10
–7
1.9 ×10
–4
1.4 ×10
–3
8.8 ×10
–11
CNO (9 e p s) SQP 10
–10
1420.3 1786.0 1848.6 38.1 45.9 46.9 1.6 ×10
–15
8.3 ×10
–15
1.1 ×10
–14
2.8 ×10
–14
3.5 ×10
–13
1.0 ×10
–12
MO (8 e p s) TC 10
–10
1668.5 847.1 504.9 57.7 30.6 22.6 4.4 ×10
–11
1.4 ×10
–11
7.1 ×10
–12
9.9 ×10
–11
2.2 ×10
–10
9.6 ×10
–11
ED 10
–10
2120.7 1016.1 616.4 78.3 40.1 30.7 4.4 ×10
–11
2.6 ×10
–11
3.3 ×10
–11
1.0 ×10
–10
1.0 ×10
–10
1.0 ×10
–10
TC Pa 10
–10
343.9 278.8 255.8 78.8 66.1 60.2 3.7 ×10
–11
1.3 ×10
–11
1.3 ×10
–11
1.0 ×10
–10
1.0 ×10
–10
9.9 ×10
–11
ED Pa 10
–10
424.7 327.5 278.9 117.3 91.5 75.0 4.8 ×10
–11
2.3 ×10
–11
3.0 ×10
–11
9.9 ×10
–11
1.0 ×10
–10
1.0 ×10
–10
MO (9 e p s) TC 10
–10
1748.5 847.4 683.1 56.7 28.4 23.0 4.5 ×10
–11
1.5 ×10
–11
9.9 ×10
–12
1.0 ×10
–10
1.0 ×10
–10
9.4 ×10
–11
ED 10
–10
2243.3 1150.1 884.1 76.5 40.6 32.1 4.7 ×10
–11
2.7 ×10
–11
3.1 ×10
–11
1.0 ×10
–10
1.0 ×10
–10
1.0 ×10
–10
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
18
cases (see Fig. 8a-b), as well as o lowe Euclidean dis ances (see Fig. 8c). Al hough smoo hing he expe imen al da a educes he
in luence o he pe u bed da a on he kinema ic esul s, he Euclidean dis ances emain app oxima ely one o de o magni ude highe
han hose ob ained wi h he MC app oach. Fo he e e ence cu o equency o 6 Hz, commonly adop ed in gai analysis, he a e age
educ ion in dis ance was app oxima ely 21.7 % (see Fig. 8c). Applying a s ic e cu o equency u he educed bo h he Euclidean
dis ance and he magni ude o he high- equency oscilla ions obse ed in he join d i e signals (−34.9 %), app oxima ing also he
eloci y cu es o he e e ence da a. Howe e , i should be no ed ha cu o alues a ound 3 Hz may be o e ly es ic i e, po en ially
a enua ing meaning ul mo ion componen s o he signal and he eby comp omising he accu acy o bo h kinema ic and dynamic
ou comes [46].
The esul s ob ained wi h he 3 Hz il e u he emphasize he ad an ages o he MC app oach o e angula -based d i ing me hods,
pa icula ly in minimizing he p opaga ion o a i ac s along he kinema ic chain. Despi e educing noise le els in he join angles o
alues simila o hose achie ed by he MC me hod (see Fig. 8d), he magni ude o he di e ences in he poin posi ions emained
highe (see Fig. 8c). This ou come s ems om he absence o a global i ing p ocess in he angula -based d i ing me hods, implying
ha pe u ba ions in oduced in ea lie segmen s o he chain a e di ec ly p opaga ed h oughou he sys em. Fu he mo e, i should
be highligh ed ha i ing he model o he expe imen al da a enhances no only posi ional accu acy bu also he accu acy o eloci y
and accele a ion pa e ns, ul ima ely imp o ing he quali y o he dynamic ou comes.
Fig. 6. Compa ison be ween he KA esul s and e e ence da a o P6 o he i e-ba linkage model wi h a maximum pe u ba ion o 52 mm in P4 and
P9 conside ing he AD wi h pe u bed d i e s, NRM-WLS and MO wi h ED: a) x coo dina e; b) y coo dina e; c) z coo dina e; d) Euclidean dis ance
be ween he KA ou come and e e ence da a.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
19
Fig. 7. Compa ison be ween he join angula d i e compu ed using MC and e e ence da a o poin P4 in he i e-ba linkage model wi h NRM-
WLS and MO-ED: a) maximum pe u ba ion o 13 mm in P4 and P9; b) maximum pe u ba ion o 52 mm in P4 and P9.
Fig. 8. Compa ison be ween he KA esul s and e e ence da a o he i e-ba linkage model wi h a maximum pe u ba ion o 52 mm in P4 and P9
conside ing he AD wi h pe u bed d i e s wi h and wi hou il e ing, and NRM-WLS: a) x coo dina e o poin P6; b) x eloci y o poin P6; c)
Euclidean dis ance be ween he KA ou come and e e ence da a o poin P6; d) Join angula d i e o poin P4.
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
20
5. Discussion
The p esen s udy expands he ully Ca esian o mula ion wi h a gene ic igid body o encompass he in e se kinema ic and
dynamic analysis o mul ibody sys ems wi h Mixed Coo dina es and e alua es he accu acy and e iciency o di e en me hods. The
adop ion o an MC o mula ion enables he compu a ion o he kinema ically consis en posi ions o he model ha be e i he
e e ence da a, while simul aneously calcula es he angula d i e s o he model, which desc ibe he DoFs associa ed wi h he kine-
ma ic join s. This modeling app oach enables he e alua ion o he majo kinema ic ou comes in a single analysis, elimina ing he need
o addi ional s eps o compu e he d i e s o he model.
Besides simpli ying he modeling p ocedu e and p e- and pos -kinema ic s eps, he p esen ed app oach also o e s ad an ages
ela ed o he accu acy o he solu ion. By minimizing he dis ance be ween he acking ma ke s and he co esponding poin s o he
model, he me hod de e mines a kinema ically consis en posi ion ha accu a ely ep esen s he o e all posi ion o he model in
ela ion o he expe imen al da a. This me hodology add esses he issues commonly associa ed wi h he use o angula d i e s, namely
he p opaga ion o e o s along he kinema ic chain due o he compu a ion o d i e s om noisy o inconsis en da a. Mo eo e , he
use o di e en weigh s can o ce he model o ollow poin s ha a e less p one o expe imen al e o s o a e mo e ele an o he
analysis. I should be no ed ha his s udy speci ically ocused on open-chain models, whe e e o p opaga ion is ypically mo e
p onounced due o he absence o opological cons ain s o en o ce posi ional consis ency along he chain. None heless, since he MC
me hod wi h FCC-GRB main ains he same p inciples o he global FCC-GRB o mula ion, i does no ace he limi a ions commonly
associa ed wi h ecu si e o mula ions in he analysis o closed-loop sys ems [29,47], and, consequen ly, i can be applied o analyze
such sys ems wi hou equi ing signi ican modi ica ions.
F om he implemen a ion poin o iew, inco po a ing he angula coo dina es in o he o mula ion is a s aigh o wa d p ocess.
The angula DoFs o he sys em, which a e adi ionally modeled using angula d i e s, a e he e ea ed as angula a iables ha a e
e alua ed du ing he p ocess o compu ing he kinema ically consis en posi ions. The AR kinema ic cons ain s, which desc ibe he
angula ela ion be ween wo ec o s o he model, a e eplaced by an MCR condi ion. In ac , he wo condi ions a e simila , being he
heonomic e m o he AR cons ain equa ions subs i u ed by a simila e m dependen on he gene alized coo dina es o he sys em.
This ela ion implies ha he in e se dynamic analysis o a sys em de ined using MC is also a s aigh o wa d p ocess. The simplici y o
his p ocedu e is ela ed o he ac ha one sys em modeled wi h MC and FCC-GRB can be easily con e ed o a classical FCC-GRB
model, by ans o ming he angula gene alized a iables in o angula d i e s and by emo ing he acking cons ain s. The e o e,
he physical meaning o he in e nal o ces compu ed using he Lag ange me hod is he same, no equi ing addi ional s eps besides he
ones al eady pe o med in he classical o mula ion. I is wo h men ioning ha al hough his wo k de ails he implemen a ion o a
spa ial MC o mula ion based on FCC-GRB, his app oach can be applied in o he global o mula ions, conside ing he speci ic
cha ac e is ics o each o mula ion [17,34,35]. Mo eo e , ansi ioning o a 2D FCC-GRB o mula ion can be easily pe o med by
applying he same concep s as hose employed in he spa ial o mula ion, bu using he equi alen plana exp essions [28]. In his
pa icula case, he key di e ence lies in he lowe minimum numbe o acking cons ain equa ions needed o ully desc ibe he
sys em kinema ics, as he igid bodies and kinema ic join s ha e ewe DoFs.
The esul s ob ained o he wo analyzed models suppo he p e iously men ioned ad an ages. Bo h he NRM and op imiza ion
me hods we e able o ind a kinema ically consis en solu ion, while simul aneously de e mining he angula d i e s o he model. The
NRM me hod was signi ican ly as e han he di e en op imiza ion condi ions es ed in his wo k. This esul is a di ec consequence
o he na u e o he NRM algo i hm, which exhibi s a quad a ic con e gence nea he solu ion [39,40]. This ac implies ha i an ini ial
guess nea he solu ion is gi en o he i e a ion algo i hm, i will con e ge quickly o he solu ion. The use o he NRM me hod o sol e
kinema ic equa ions is no wi hou limi a ions. The classical p oblems ypically associa ed wi h he NRM algo i hm can occu ,
including con e gence issues when ini ia ing wi h a poo ini ial guess o one a om he solu ion, o e shoo e ec s due o he bad
beha io o he unc ion nea he solu ion, o null-de i a i e unc ions [39,40,48].
Rega ding he op imiza ion me hods, he MO a ainmen goal p o ed o be he as e solu ion o he mo e complex model. The
esul s indica e an in luence o he ype o objec i e unc ion u ilized. Despi e gene a ing mo e equa ions han he ED condi ion, he use
o he TC-based objec i e unc ion esul ed in lowe compu a ional imes o he mo e complex model, a di e ence in pa explained by
he lowe numbe o i e a ions equi ed o ind a solu ion. This end was no obse ed in he simple model, indica ing ha he ela i e
di e ences in he compu a ional imes may be s ongly a ec ed by he complexi y o he model unde analysis and he op imiza ion
inpu s. In ac , he simula ions o he cases wi h highe le els o pe u ba ion con e ged as e han he ones wi h lowe le els,
equi ing also less i e a ions o ind an op imal solu ion. In con as , he CNO wi h SQP con e ged signi ican ly as e han wi h he IP
algo i hm. Con e sely o he MO case, he inc ease in he pe u ba ion le els esul ed in highe compu a ional imes and a highe
numbe o i e a ions o ind a solu ion. I should be no ed ha he g adien s o he equali y cons ain equa ions and objec i e unc ions
we e p o ided o he op imize . The use o ini e di e ences signi ican ly inc eased he simula ion ime; howe e , his di e ence was
no quan i ied in his analysis.
In gene al, he inco po a ion o he angula coo dina es led o an inc ease in he compu a ional e o o all condi ions when
compa ed o he classical FCC-GRB o mula ion wi h angula d i e s. Ne e heless, i is wo h no ing ha he p e- and pos -p ocessing
s eps equi ed o compu ing he d i e s a e no included in he analysis, and no i ing o he e e ence da a is pe o med. In con as ,
in he MC analyses, bo h he compu a ion o he gene alized coo dina es and he angula d i e s a e ca ied ou simul aneously and a e
accoun ed in he compu a ional cos s.
In e ms o accu acy, he kinema ic analyses wi h he NRM me hod did no yield esul s as exac as hose ob ained wi h he
op imiza ion me hods. Since he TC and TOC kinema ic equa ions a e ea ed as equali y cons ain s ha mus be ul illed, he me hod
gene a es kinema ic posi ions ha , in o de o adap he model o he noisy e e ence poin s, may iola e some o he cons ain s. The
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
21

use o di e en weigh s o he opological and acking cons ain s enables he mi iga ion o his di icul y. By elaxing he weigh s
associa ed wi h he acking cons ain s, he me hod assigns less signi icance o hese equa ions, p io i izing he ul illmen o he
opological cons ain s. This idea is suppo ed by he esul s ob ained o he analyses wi h pe u ba ions. The maximum alue o he
opological cons ain s iola ion du ing he analysis dec eased by wo o de s o magni ude when a weigh o 10
–3
was used. This
di e ence can be e en inc eased i lowe weigh s a e assigned o he TC equa ions, o highe weigh s a e gi en o he opological
cons ain s. This be e i ing o he expe imen al poin s is also suppo ed by he lowe ED o he e e ence poin s. O e all, he NRM
me hod p esen ed lowe alues han he op imiza ion-based me hods. This di e ence is pa icula ly no iceable in he cases wi hou
pe u ba ions, meaning ha he NRM can be he p e e able choice in e ms o accu acy when he inpu da a does no p esen noise o
a i ac s.
One o he ad an ages o using op imiza ion me hods is he ea men o he acking cons ain equa ions as objec i e unc ions o
be minimized du ing he dynamic analysis. This app oach ensu es ha only he opological cons ain s a e ea ed as equali y con-
s ain s, hus p e en ing iola ions o hese cons ain s. All he sequen ial op imiza ions esul ed in maximum cons ain iola ions o
he opological cons ain s on he o de o 10
–10
and 10
–11
, wi hin he numbe o i e a ions and unc ion e alua ion de ined o he
analysis and conside ing a cons ain ole ance o 10
–10
. I should be no ed ha o some posi ions o he i e-ba linkage model, he
CNO wi h he in e io -poin algo i hm p esen ed con e gence p oblems due o he sys em ma ix becoming close o singula o badly
scaled, eaching he maximum numbe o i e a ions.
Despi e educing he a e age ime o he analysis, he use o a pa allel app oach o sol e he op imiza ion p oblem esul ed in a
noisy pa e n. This ou come can be explained by using he ini ial posi ion as he op imiza ion ini ial guess o all ime ames. The
me hod con e ged o simila posi ions ha comply wi h he op imiza ion equali y cons ain s, bu do no ep esen he global minimal
poin . Hence, he kinema ic posi ions should be pos -p ocessed o a oid un ealis ic accele a ions, as hese could impac he dynamic
ou comes. An al e na i e solu ion o mi iga e his issue is o i s pe o m an analysis using he NRM and WLS and hen use he esul s
om his analysis as ini ial guesses o he op imiza ion p oblem. Ano he app oach is o subdi ide he en i e p oblem in o as many
sub-p oblems as he a ailable h eads, sol ing he kinema ic analysis sequen ially wi hin each sub-p oblem. Howe e , while his
s a egy add esses he issue o noisy posi ions wi hin each sub-p oblem, i may in oduce he challenge o ensu ing da a con inui y a
he bounda ies be ween sub-p oblems.
Hence, conside ing he ad an ages and limi a ions o he es ed me hods, op imiza ion algo i hms should be he p e e ed choice
when analysis accu acy is c i ical, while he NRM wi h WLS should be employed when compu a ion e o is an impo an ac o , o he
inpu da a does no p esen noise o pe u ba ions.
I is wo h men ioning ha a 3-second analysis pe iod sampled a 100 Hz was used in his s udy, as his du a ion exceeds he ypical
cycle imes obse ed in human mo ion. Fo example, a ull gai cycle a no mal cadence las s app oxima ely 1.1 s [49–51]. While his
du a ion migh be conside ed sho o simula ion-based analyses [52], he p esen s udy ocuses exclusi ely on he in e se dynamic
analysis o (bio)mechanical sys ems, which does no equi e in eg a ion s eps. The e o e, as long as he ini ial guess o each ime
ame is close o he ac ual solu ion, bo h he nume ical and op imiza ion-based me hods a e expec ed o con e ge wi hou nume ical
p oblems. In he app oach adop ed he e, he ou pu om one ime ame se es as he ini ial guess o he subsequen ame, u he
ensu ing p oximi y o he solu ion and p omo ing as con e gence. This assump ion may no hold o e y as mo ions; howe e , in
such cases, an inc eased acquisi ion equency is ypically equi ed o accu a ely cap u e he dynamics o he mo emen [53], he eby
mi iga ing his issue.
A ele an ea u e o he MC o mula ion wi h FCC-GRB is ha i la gely p ese es he p inciples o he classical FCC-GRB app oach.
The e o e, he ad an ages epo ed o he classical o mula ion a e s ill main ained. Howe e , he inclusion o angula coo dina es in
he ec o o he gene alized coo dina es implies ha he sys em is no longe exclusi ely de ined using Ca esian coo dina es.
Consequen ly, kinema ic cons ain equa ions inco po a ing nonlinea e ms dependen on angula a iables a e gene a ed. Since
hese a iables do no di ec ly de ine he igid bodies, hey do no impac he de ini ion o he igid body opological cons ain s o he
compu a ion o o he poin s o ec o s o he model, as he de ini ion o he ans o ma ion ma ices C and V emains equal. The e o e,
mos o he kinema ic cons ain s main ain hei linea o quad a ic na u e, meaning ha hei con ibu ions o he Jacobian ma ix
and igh -hand side ec o s o eloci ies and accele a ions a e s ill null, linea o quad a ic wi h espec o he gene alized coo dina es.
This independence o he igid bodies’ de ini ion om he angula gene alized a iables implies ha he de ini ion o he sys em mass
ma ices is he same as in he classical o mula ion, conside ing he elimina ion me hod desc ibed in sec ion 2.6.
Al hough o wa d dynamics is no he p ima y ocus o his hyb id o mula ion o o he analyses p esen ed in his s udy, he MC
o mula ion wi h he FCC-GRB amewo k has he po en ial o be adap ed o he o wa d dynamics analysis o mechanical sys ems.
While he inc eased dimensionali y o he sys em is expec ed o aise compu a ional demands, no addi ional nume ical challenges a e
an icipa ed beyond hose commonly encoun e ed in adi ional FCC-GRB simula ions, such as singula con igu a ions, edundan
cons ain s, ill-condi ioned Jacobian and mass ma ices, o in eg a ion s abili y p oblems [29,41,54]. Fu u e wo k also aims o explo e
ex ending he MC wi h FCC-GRB o inco po a e lexible bodies o adop ing al e na i e sol ing s a egies based on di e en op imi-
za ion algo i hms o a i icial in elligence app oaches [37,38,55–58].
The simila i ies be ween he classical FCC-GRB o mula ion and he FCC-GRB wi h MC in e ms o implemen a ion and modeling
p ocedu e imply ha he s aigh o wa dness associa ed wi h he classical app oach emains applicable in he second case. The po-
en ial applica ions iden i ied o he classical o mula ion, such as eaching mul ibody dynamic opics in highe educa ion, a e s ill
alid [28,29]. Al hough he MC a ia ion includes angula a iables, i does no equi e a high le el o expe ise in 3D o a ion
pa ame iza ion. This cha ac e is ic aligns wi h he ad an ages o en highligh ed when u ilizing he na u al coo dina es o mula ion in
ad anced educa ional con ex s [35].
The in insic p ope ies o he FCC-GRB o mula ion also make i pa icula ly sui able o he biomechanics o mo ion ield. Since
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
22
he poin s and ec o s ha compose he gene ic igid body ha e a di ec ela ion wi h he ine ial in o ma ion ound in an h opo-
me ical da abases, he modeling p ocess becomes s aigh o wa d. The inco po a ion o he MC simpli ies he in e se dynamic
analysis o biomechanical models, as he e e ence poin s and ec o s can be de ined in he measu e ha hey di ec ly ep esen he
ma ke s used in he expe imen al acquisi ion o mo emen . Mo eo e , he igid body ec o s can be easily aligned wi h he ana omical
join axes, di ec ly de ining he o a ion di ec ions. I he gene alized angula d i e s a e de ined o ma ch he con en ion es ablished
o he join angles ([3,4]), he me hodology can di ec ly p o ide he join angles.
Finally, as highligh ed in he in oduc ion, expe imen al da a acqui ed using adi ional ma ke -based op oelec onic mo ion
cap u e sys ems a e p one o STA, which, due o hei equency cha ac e is ics and unp edic able na u e, a e di icul o au oma ically
de ec and a enua e using con en ional smoo hing echniques. By i ing he biomechanical mul ibody model, ypically de ined using
measu emen s aken du ing s a ic acquisi ions wi h he ma ke s placed a he co ec ana omical landma ks, he MC helps mi iga e
STA e ec s, he eby imp o ing bo h kinema ic and dynamic analyses. This ad an age is pa icula ly signi ican , as bo h dynamic and
muscle analysis ou comes a e highly sensi i e o pe u ba ions in kinema ic da a [15,59,60]. Hence, in addi ion o simpli ying he p e-
and pos -p ocessing s eps, he MC me hod enhances he accu acy o kinema ic da a, ul ima ely imp o ing he eliabili y o biome-
chanical analysis.
I is also impo an o men ion ha he use o he MC app oach inc eases he complexi y o he sys em, pa icula ly when sol ed
om a o wa d dynamics pe spec i e, due o he inc ease in he numbe o coo dina es and kinema ic cons ain equa ions. When
applied in an in e se kinema ic con ex , he inc ease o he sys em complexi y has less impac on he compu a ional pe o mance o he
analysis, making i mo e sui able o his ype o s udy. In ac , he kinema ic analysis wi h NRM and WLS o he i e-ba linkage ook
app oxima ely 30 ms pe ame o con e ge, indica ing i s po en ial o eal- ime applica ions. This ime could be imp o ed by using a
educed de ini ion o he model o by using a as e compiled p og amming language ins ead o an in e p e ed one. In ce ain sce-
na ios, employing an MC o mula ion can o e addi ional modeling ad an ages when applied om a o wa d dynamics o p edic i e
s andpoin [35]. The explici use o angula a iables can simpli y he con ol o speci ic sys ems ha ely on angula inpu s and
acili a e he modeling o he ac ion o ex e nal o que ac ua o s, such as passi e and ac i e exoskele ons and wheelchai s [61–65].
6. Conclusions
This wo k explo es he use o an MC o mula ion wi h FCC-GRB in he in e se kinema ic and dynamic analysis o spa ial mul ibody
sys ems. The majo s eps equi ed o ully implemen he o mula ion, bo h wi h ully-de ined and educed igid bodies, a e desc ibed
in de ail. The o mula ion is alida ed in he analysis o wo models wi h di e en le els o complexi y. The accu acy and compu-
a ional di e ences be ween using a nume ical me hod based on he NRM and op imiza ion algo i hms a e analyzed o e alua e he
applicabili y o he o mula ion wi hin di e en con ex s. The esul s indica e ha in e ms o accu acy, op imiza ion-based me hods
ensu e he kinema ic consis ency o he sys em, albei he inc ease in compu a ional e o . The use o an NRM wi h WLS allows o he
educ ion o he compu a ional cos , as i enables o con ol he iola ion o cons ain s a a opological le el by de ining app op ia e
weigh s.
F om an implemen a ion poin o iew, he use o an MC o mula ion wi h FCC-GRB o e s he main ad an age o simul aneously
compu ing he kinema ically consis en posi ion and d i e s associa ed wi h he join angula DoFs. This p ocedu e is achie ed by
ea ing he join displacemen angles as gene alized coo dina es o he sys em and by minimizing he dis ance be ween a se o poin s
and ec o s o he model and hei espec i e e e ence da a. As a esul , he e is no need o p e- o pos -p ocessing s eps o ob ain all
kinema ic ou comes, a oiding also he e o s associa ed wi h he use o angula d i e s, such as he p opaga ion o expe imen al e o s
along he kinema ic chain.
Despi e equi ing he explici use o angula a iables, he adop ion o MC does no equi e complex pa ame iza ions o 3D angle
o a ions, main aining he simplici y o implemen a ion ypically a ibu ed o he na u al coo dina es and classical FCC-GRB
o mula ion. The e o e, i is he au ho s’ belie ha his o mula ion emains sui able no only o eaching mul ibody- ela ed
opics bu also o o he a eas ha equi e kinema ic analyses o complex sys ems wi h noisy expe imen al da a, such as biome-
chanics o mo emen .
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 , 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, 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 .
S.B. Gonçal es e al.
Mechanism and Machine Theo y 214 (2025) 106080
23
Acknowledgmen 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.
106080.
Da a a ailabili y
No da a was used o he esea ch desc ibed in he a icle.
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