Resea ch A icle
Selec ion o MEMS Accele ome e s o Til Measu emen s
Se giusz Auczak,1Robe G epl,2and Maciej Bodnicki1
1Ins i u e o Mic omechanics and Pho onics, Facul y o Mecha onics, Wa saw Uni e si y o Technology,
ul. Boboli 8, 02-525 Wa saw, Poland
2Ins i u e o Solid Mechanics, Mecha onics and Biomechanics, Facul y o Mechanical Enginee ing, B no Uni e si y o Technology,
Technick´
a2896/2,61669B no,CzechRepublic
Co espondence should be add essed o Se giusz Łuczak; s.luczak@mch .pw.edu.pl
Recei ed 23 Decembe 2016; Accep ed 5 Feb ua y 2017; Published 30 Ma ch 2017
Academic Edi o : Pie o Siciliano
Copy igh © 2017 Se giusz Łuczak e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License,
which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
In o de o build a il senso ha ing a desi ed sensi i i y and measu ing ange, one should selec an app op ia e ype, o ien a ion,
and ini ial posi ion o an accele ome e . Va ious cases o il measu emen s a e conside ed: de e mining exclusi ely pi ch, axial
il , o bo h pi ch and oll, whe e Ca esian componen s o he g a i y accele a ion a e measu ed by means o low-g uni-, bi-, i-,
o mul iaxial mic omachined accele ome e s. 15 di e en o ien a ions o such accele ome e s a e dis inguished (each illus a ed
wi h espec i e g aphics) and ela ed o he ele an ma hema ical o mulas. Resul s o he pe o med expe imen al s udy e ealed
inhe en misalignmen s o he sensi i e axes o mic omachined accele ome e s as la ge as 1∘. Some o he p oposed o ien a ions
make i possible o a oid a necessi y o using he mos misaligned pai s o he sensi i e axes; some inc ease he accu acy o
il measu emen s by ac i a ing all he sensi i e axes o educing he e ec s o aniso opic p ope ies o mic omachined iaxial
accele ome e s; o he o ien a ions make i possible o educe a necessa y numbe o he sensi i e axes a ull measu emen ange. An
inc ease o accu acy while using mul iaxial accele ome e s is discussed. P ac ical guidelines o an op imal selec ion o a pa icula
mic omachined accele ome e o a speci ic case o il measu emen a e p o ided.
1. In oduc ion
Mic omachined accele ome e s belonging o Mic oelec-
omechanical Sys ems (MEMS) a e employed in many kinds
o a ious de ices [1], owing o hei small dimensions (cu -
en ly e en less han 2 mm), easy in eg a ion wi h elec onics,
high shock- esis ance, high eliabili y, sa is ac o y accu acy,
low powe consump ion, and low cos . They ealize measu e-
men s o a ious ypes o accele a ion (whe he cons an o
a iable), i s de i a i es (je k, jounce) and in eg als ( eloci y,
posi ion). New unexpec ed applica ions eme ge cons an ly
in, o example, sma wa ches, unde wa e equipmen like
scuba di ing compu e s (e.g., Vype Ai and No o, Cob a
3, DX, D6i No o by Suun o Company), new gene a ion o
sa e mo o cycle ABS [2] o ac ion con ol sys ems (e.g.,
Mo o cycle S abili y Con ol by Bosch Co p. [3]), o no el
designs o commonly known objec s, like he elec onic
gaming die p esen ed in [4]. One o he mos s a egic ields o
applica ion is a ious ypes o na iga ion sys ems, p oposed,
o example, in [5, 6].
Besides ypical measu emen s o accele a ion ela ed o
ib a ion o mo ion, ano he basic applica ion o accele om-
e e s is il measu emen s [7]. Such measu emen s a e
employed in many de ices, usually o he pu poses o
posi ioning, aligning, le eling, o na iga ing. Howe e , hese
measu emen s a e co ec only unde s a ic o quasi-s a ic
condi ions. Mo eo e , in he case o quasi-s a ic condi ions,
especially a conside able equencies, addi ional e o s due
o ampli ude a enua ion [8] mus be aken in o accoun .
As a as speci ic applica ions o il measu emen a e
conce ned, accele ome e s a e embedded in such equipmen
as a ious po able digi al de ices (e.g., cellula phones,
pho o came as, compu e p ojec o s, GPS ecei e s, and
game consoles). They can be applied di ec ly in small de ices
like mobile mic o obo s [9] (especially when hey a e a pa
o he only na iga ion sys em a ailable, like in he case o
some in-pipe o biomime ic obo s, p oposed, e.g., in [10, 11]),
o indi ec ly, as embedded in sma phones, as epo ed, e.g.,
in [12]. One can also ind a lo o un ypical applica ions
o MEMS accele ome e s used o il measu emen s ela ed
Hindawi
Jou nal o Senso s
Volume 2017, A icle ID 9796146, 13 pages
h ps://doi.o g/10.1155/2017/9796146
2Jou nal o Senso s
o li e sciences, o example, in es iga ion o mo emen s o
animals [13] o humans [14–16] and ehabili a ion equipmen ,
o example, exoskele ons p esen ed in [17–19], o name only
a ew.
Due o he ac ha di e en ma hema ical o mulas can
be employed and di e en ypes o MEMS accele ome e s
a e a ailable, il measu emen s can be ealized in a ious
ways. Besides, o ien a ion o he accele ome e is ye ano he
impo an issue comp ehensi ely add essed in he pape .
In o de o pe o m a il measu emen , he mos ad an a-
geous solu ion is usually o apply a iaxial MEMS accele om-
e e and use a c angen unc ion [20]. Ne e heless, he e
a e cases, speci ied in he pape , when o he ypes o MEMS
accele ome e s a e mo e p e e able han iaxial senso s, o
example, due o hei inhe en impe ec ions. Reasons o
employing ma hema ical o mulas o he han a c angen
unc ion a e minu ely discussed in [20]. Fo ins ance, i he
accu acy o simple il measu emen s is no sa is ac o y, usu-
allycomplexsolu ionsa esea ched o ,based, o example,on
da a usion. Ye , some imes a sa is ac o y inc ease o accu acy
can be ob ained by a anging an app op ia e o ien a ion o
he MEMS accele ome e .
Despi e common applica ion o MEMS accele ome e s,
i is ha d o ind any sa is ac o y in o ma ion on he a o e-
men ioned p oblems in he ela ed publica ions. The e o e,
we decided o add ess he issue o wha ype o MEMS
accele ome e (uni-, bi-, i-, o mul iaxial) should be used
in o de o mee equi emen s ela ed o ce ain case o il
measu emen , de ining also i s o ien a ion and ini ial posi-
ion as well as analyzing he espec i e sensi i i y esul ing
om applica ion o a gi en ma hema ical o mula.
Owing o he guidelines p esen ed in he pape , on he one
hand,i ispossible obuild hesimples il measu emen uni
basedonMEMSaccele ome e s,beinga hesame ime he
mos cos -e ec i e ye ensu ing a sa is ac o y accu acy, and,
on he o he hand, o ob ain possibly high accu acy a he cos
o complica ing he measu emen .
2. Componen Angles o Til
The mos common way o exp essing il is o de e mine
i s wo componen angles pi ch 𝛼and oll 𝛾(de ined wi h
acco dance o ae onau ical nomencla u e, as, o example, in
[21]) illus a ed in Figu e 1 agains he componen s o he
g a i a ional accele a ion 𝑔.O en,i is heauxilia yangle𝛽
ha is de ined as he oll, o example, in [22–28], as in some
cases such app oach is mo e con enien , since accele ome e s
espond di ec ly o his angle and no o angle 𝛾.
A bi a y il angle 𝜑(axial il ) included be ween accel-
e a ion 𝑔and axis 𝑧has been esol ed in o wo componen
angles 𝛼(pi ch) and 𝛾( oll) o al e na i ely auxilia y angle 𝛽.
Angles 𝛼and 𝛽a e con ained wi hin e ical planes 𝑥0𝑥𝑧0and
𝑘𝑦𝑧0, ( he planes a e gene ally no o hogonal wi h espec o
one ano he ), ha is, angles be ween ho izon al plane 𝑥0𝑦0
and axis 𝑥o 𝑦, espec i ely. The coo dina e sys em 𝑥0𝑦0𝑧0
is immobile (i s axis 𝑧0is e ical), whe eas he coo dina e
sys em xyz is ixed o hemobile il senso loca eda i s
o igin. Angles 𝛼and 𝛽a e in e dependen and hei alues
esul om he alue o angle 𝜑. In o de o illus a e 𝛽angle,
Figu e 1: Componen s o he il and he g a i a ional accele a ion.
line khas been c ea ed as a esul o he in e sec ion o e ical
plane 𝑦𝑧0and ho izon al plane 𝑥0𝑦0.
Angle 𝛾be ween axis 𝑦and 𝑦0is con ained wi hin a il ed
plane 𝑦0𝑦𝑧;i is heanglebywhich he il senso o a es
a ound axis x, while il ed in wo phases ( he i s phase is
o a ion by pi ch 𝛼a ound axis ywhile i s ill o e laps axis 𝑦0).
I shouldbeno ed ha mos o he ela ed o mulasp esen ed
la e in he ex a e analogous o angle 𝛼and 𝛽.Rela ion
be ween he conside ed angles is as ollows [21, 29]:
𝛾21 =a csin (sin 𝛽
cos 𝛼), (1)
𝛼=0󳨐⇒𝛾1 =𝛽1.(2)
Howe e , de e mining oll 𝛾on he basis o (1) usually
esul s in a dec ease o he ela ed accu acy [29] (see (21)).
F om his poin on, angles 𝛼,𝛽,𝛾,and𝜑will be designa ed
wi h addi ional subsc ip s, depending on he ype o he con-
side ed il measu emen (single- o dual-axis, as illus a ed
in Figu e 2, o any o hese wo) and he o mula employed
o hei de e mina ion, as speci ied in Nomencla u e. The
subsc ip s ha e been in oduced in o de o allow us o
unequi ocally dis inguish be ween a ious cases in oduced
la e in he ex .
I was accep ed ha , in case o single-axis il measu e-
men s, he only componen il angle ha appea s is pi ch 𝛼.
3. Ma hema ical Fo mulas
The o mulas p esen ed in his sec ion can be ound in
nume ous ela ed publica ions, o example, [20, 22–35]. As
i esul s om Figu e 1, he componen il angles can be
calcula ed basically as ollows:
𝛼01 =a csin 𝑔
𝑔,(3)
𝛽01 =a csin 𝑔
𝑔.(4)
Jou nal o Senso s 3
360∘
g
(a)
360∘360∘
g
(b)
Figu e 2: (a) Single-axis il senso ; (b) dual-axis il senso .
Howe e , he e ical componen accele a ion 𝑔allows
us o use he ollowing o mulas:
𝛼22 =a ccos√𝑔2
+𝑔2

𝑔,(5)
𝛽22 =a ccos√𝑔2
+𝑔2

𝑔,(6)
𝛼23 =a c an 𝑔
√𝑔2
+𝑔2
,(7)
𝛽23 =a c an 𝑔
√𝑔2
+𝑔2
(8)
and addi ionally,
𝛾03 =a c an 𝑔
𝑔.(9)
I il is o be measu ed o e one axis only, (5)–(8) ge
simpli ied, as one o he componen s o accele a ion (𝑔o
𝑔) will be hen o ze o alue (e.g., (8) becomes iden ical o
(9)). So, he equa ions will ge ans o med as ollows:
𝛼12 =a ccos 𝑔
𝑔,(10)
𝛽12 =a ccos 𝑔
𝑔,(11)
𝛼13 =a c an 𝑔
𝑔,(12)
𝛽13 =a c an 𝑔
𝑔.(13)
While de e mining he a bi a y il angle 𝜑, which in ac
is an axial il de ined in [33] and may be ega ded as a pseudo
dual-axis il measu emen , he ollowing ela ions a e alid:
𝜑02 =a ccos 𝑔
𝑔,(14)
𝜑21 =a csin √𝑔2
+𝑔2

𝑔,(15)
𝜑23 =a c an √𝑔2
+𝑔2

𝑔.(16)
I 𝑔=0o 𝑔=0, hen he measu emen is ac ually a
single-axis il measu emen , whe e 𝜑is de e mined ins ead
o 𝛼.
Applica ion o (14)–(16) may be in e es ing in such
measu emen s o axial il as, o example, di ec ional d illing
[36] o while moni o ing an objec agains losing i s balance,
since pi ch and oll a e incon enien o be used in such cases.
Addi ionally, as discussed in Sec ion 8, unde ce ain
condi ions, measu emen o axial il may be mo e ad an a-
geous han measu emen o pu e pi ch, as a as he esul an
accu acy is conce ned.
Besides he p esen ed o mulas, he e a e s ill o he ways
o de e mining he il angles. As sugges ed in [24], i is
ad an ageous in some cases o combine an a c sine o mula
((3) o (4), (15)) wi h an a c cosine o mula ((5) o (6) and (10)
o (11), (14)) in a simple way o as a weigh ed a e age [30, 37].
4. O ien a ions o he Accele ome e
Possibili ies o applying ce ain accele ome e in pa icula
il measu emen a e illus a ed in Table 1.
Fo mulas o de e mining pi ch and oll gi en in Table 1
a e lis ed o each case in he o de o he highes measu e-
men sensi i i y (see Sec ion 5); colo s o he sensi i e axes
o he accele ome e s, as hey appea in he p esen ed igu es,
ag ee wi h he accep ed con en ion (see Nomencla u e).
I should be added ha ob aining a ull measu emen
ange o he componen il angles is possible by obse ing
he sign o he e ical componen accele a ion 𝑔,whichis
posi i eo e hedomaino ⟨−90∘;90∘⟩, while nega i e o e
he emaining angula ange ( hus, a leas wo sensi i e axes
o he accele ome e a e equi ed) [31].
Addi ional commen s o he cases speci ied in Table 1:
(i) Cases 1-A, 2-A, 2-B, 5-A, and 5-B a e commonly
applied and a e well known.
(ii) Case 1-B ep esen s a pseudo dual-axis il (axial il )
e e ed o in [33].
4Jou nal o Senso s
Table 1: Til senso s e sus ypes o accele ome e .
Numbe Type o accele ome e Type o il senso
AB
1
Uniaxial
x0
y0
z0

x0y0
z0

Measu emen ange 𝛼=⟨−90∘,90∘⟩ 𝜑=⟨0∘,180∘⟩
Ma hema ical o mula (3) (14)
2Biaxial
x0y0
z0

x0
y0
z0

Measu emen ange 𝛼=⟨0∘,360∘⟩𝛼=⟨−90∘,90∘⟩𝛾=⟨−90∘,90∘⟩
Ma hema ical o mula (12); (10); (3) 𝛼:(3) 𝛾:(4)
∗
3Biaxial

x0y0
z0

Measu emen ange 𝛼=⟨0∘,360∘⟩𝛾=⟨0∘,360∘⟩
Ma hema ical o mula 𝛼:(3) 𝛾:(6)
∗
Jou nal o Senso s 5
Table 1: Con inued.
Numbe Type o accele ome e Type o il senso
AB
4Biaxial

x0
y0
z0

Measu emen ange 𝛼=⟨0∘,360∘⟩𝛾=⟨0
∘,360∘⟩
Ma hema ical o mula 𝛼:(5) 𝛾:(9);(4)
∗
5T iaxial

x0y0
z0

x0
y0
z0

Measu emen ange 𝛼=⟨0∘,360∘⟩𝛼=⟨0∘,360∘⟩𝛾=⟨0∘,360∘⟩
Ma hema ical o mula (12); (10); (3) 𝛼:(7);(5);(3) 𝛾:(8)
∗;(9);(6)
∗;(4)
∗
6T iaxial

x0y0
z0

x0y0
z0

Measu emen ange 𝛼=⟨0∘,360∘⟩𝛼=⟨0∘,360∘⟩𝛾=⟨0∘,360∘⟩
Ma hema ical o mula (12); (10); (3) 𝛼:(7);(5);(3) 𝛾:(8)
∗;(9);(6)
∗;(4)
∗

6Jou nal o Senso s
Table 1: Con inued.
Numbe Type o accele ome e Type o il senso
AB
7T iaxial
x0y0
z0


x0y0
z0

Measu emen ange 𝜑=⟨0∘,360∘⟩𝛼=⟨0∘,360∘⟩𝛾=⟨0∘,360∘⟩
Ma hema ical o mula (16); (15); (14) 𝛼:(7);(5);(3) 𝛾:(8)
∗;(9);(6)
∗;(4)
∗
8Mul iaxial

x0y0
z0

x0y0
z0

Measu emen ange 𝛼=⟨0∘,360∘⟩𝛼=⟨0∘,360∘⟩𝛾=⟨0∘,360∘⟩
Ma hema ical o mula (12)†;(10)
†;(3)
†𝛼:(7)
†;(5)
†;(3)
†𝛾:(8)
‡;(9)
†;(6)
‡;(4)
‡
Jou nal o Senso s 7
Table 1: Con inued.
Numbe Type o accele ome e Type o il senso
AB
9Mul iaxial x0y0
z0

Measu emen ange 𝜑=⟨0∘,360∘⟩
Ma hema ical o mula (16)†; (15)†;(14)
†
A: single-axis il senso .
B: dual-axis il senso .
∗Roll mus be de e mined using (1).
†Accele ome e indica ions mus be i s con e ed in o Ca esian componen s (see (17)).
‡Accele ome e indica ions mus be i s con e ed in o Ca esian componen s (see (17)); hen oll mus be de e mined wi h (1).
8Jou nal o Senso s
Table 2: Ini ial posi ions o he accele ome e s.
Numbe # Case (Table 1) Ini ial posi ion o he accele ome e
11-A𝑠1‖𝑥0
21-A𝑠1‖𝑧0
31-B𝑠1‖𝑧0
41-B𝑠1⊥𝑧0
52-A𝑠1‖𝑥0𝑠2‖𝑧0
62-B𝑠1‖𝑥0𝑠2‖𝑦0
73-B𝑠1‖𝑥0𝑠2‖𝑧0
84-B𝑠1‖𝑦0𝑠2‖𝑧0
95-A𝑠1‖𝑥0𝑠2‖𝑦0𝑠3‖𝑧0
10 5-B 𝑠1‖𝑥0𝑠2‖𝑦0𝑠3‖𝑧0
11 6-A 𝑠1‖𝑥0𝑠2‖𝑧0𝑠3‖𝑦0
12 6-B 𝑠1‖𝑥0𝑠2‖𝑧0𝑠3‖𝑦0
13 7-A 𝑠1∦𝑥0𝑠2∦𝑦0𝑠3‖𝑧0
14 7-B 𝑠1‖𝑧0𝑠2‖𝑦0𝑠3‖𝑥0
15 8-A ∡𝑠1𝑥0=35.3∘∡𝑠2𝑦0=35.3∘
16 8-B ∡𝑠1𝑥0=35.3∘∡𝑠2𝑦0=35.3∘
17 9-A ∡𝑠1𝑧0=54.7∘∡𝑠2𝑧0=54.7∘𝑠1∉𝑥0𝑧0𝑠2∉𝑦0𝑧0
1,2,3,and4: sensi i e axes o he accele ome e .
#1–#4: a bi a y posi ion o he accele ome e chip, ypically like in 1-A, o 5-A.
#1, #3: when il angles <45∘a e o be de ec ed wi h highe sensi i i y.
#2, #4: when il angles >45∘a e o be de ec ed wi h highe sensi i i y.
#3-#4: addi ionally, exp essions 1∉
00,1∉
00a e ue while il ed.
#10, #16: i =0, hen(9)and(13)a e hesame.
#15, #16: a he same ime: 30= 144.7∘,40= 144.7∘.
#17:a hesame ime:30= 54.7∘,40= 54.7∘,3∉
00,and4∉
00.
(iii) Case 2-A, ypically used o single-axis il measu e-
men , can be accep ed o dual-axis il measu emen
using an un ypical o ien a ion illus a ed in cases 3-
B and 4-B, which a e an al e na i e o case 2-B; he
esul ing ad an age is ob aining he ull measu emen
ange o bo h pi ch and oll, using only 2 sensi i e
axes. Choice be ween 3-B and 4-B is ela ed o he
sensi i i y o he measu emen s: a 3-B he highes
sensi i i y is ob ained o 𝛼=0
∘and 𝛽=90
∘,whe eas
a 4-B o 𝛼=90
∘and 𝛽=0
∘; he sensi i i y i sel is
a iable, jus as in case 2-B.
(i ) Cases 5-A and 6-A a e no ecommended i i is
s i en o ob aining a highe sensi i i y o he mea-
su emen , since none o he accele ome e sensi i e
axes should o e lap axis 𝑦0while il ed, jus as in case
7-A. Howe e , addi ional calcula ions may be hen
necessa y o con e angle 𝜑in o 𝛼.Analogously, he
same applies acco dingly o cases 8-A and 9-A.
( ) Fo cases 6-A, 6-B, and 7-B, see Sec ion 7 o explana-
ion.
( i) Cases 8-A, 8-B, and 9-A e e o a senso p esen ed
in[38],whichhasbeenusedasanexampleo a
mul iaxial accele ome e . Applica ion o such senso
makesi possible oinc ease hesensi i i yo il
measu emen s e en by 13% [30].
While using a mul iaxial accele ome e , be o e calcula -
ing il angles, Ca esian componen s o he g a i a ional
accele a ion mus be de e mined i s . Wi h ega d o he
accele ome e p esen ed in cases 8-A, 8-B, and 9-A, he
ollowing o mulas a e ue [38]:
𝑔=1
2⋅sin 54.7∘(𝑔1+𝑔4),
𝑔=1
2⋅sin 54.7∘(𝑔2+𝑔3),
𝑔=1
4⋅cos 54.7∘(𝑔1+𝑔2+𝑔3+𝑔4),
(17)
whe e 𝑔1,𝑔2,𝑔3,and𝑔4a e p ojec ions o he ec o o
he g a i a ional accele a ion on o he sensi i e axes o he
accele ome e , which a e inclined by 54.7∘(colo s o he
sensi i e axes, ep esen ed by he bold a ows, deno e hei
numbe s wi h acco dance o he accep ed nomencla u e).
Excep o selec ing an app op ia e case o il measu e-
men (as speci ied in Table 1), i is also impo an o se he
ini ial posi ion o he accele ome e in an ad an ageous way.
The ele an da a ela ed o he cases illus a ed in Table 1 a e
included in Table 2.
5. Sensi i i y o Til Measu emen s
E en hough he nominal alues o he il angles calcula ed
acco ding o he p esen ed equa ions a e he same, ye
hey a e de e mined wi h di e en sensi i i y, hus di e en
accu acy. So, applica ion o pa icula o mula signi ican ly
a ec s he accu acy o he measu emen .
Jou nal o Senso s 9
5.1. Sensi i i y Rela ed o Angles 𝛼,𝛽,and𝜑.The app op ia e
o mulas can be de i ed om (3)–(8), as demons a ed in
[20]. They e e o pi ch 𝛼in hesamemeasu eas o oll(while
exp essed by he auxilia y angle 𝛽), so a il angle 𝜓has been
in oduced. A ela ion be ween he sensi i i y 𝑘01 (exp essed
in ad−1)and he il angle𝜓01 can be de ined as [24]
𝑘01 =𝑔cos 𝜓01.(18)
Likewise, o il angles 𝜓02 and 𝜓03 we can ob ain [20]
𝑘02 =𝑔sin 𝜓02,
𝑘03 =𝑔=cons .(19)
Sensi i i ies 𝑘01 and 𝑘02, ela ed o hea csineand hea c
cosine o mulas,dec easedown oze oa a il angleo 90
∘
o 0∘, espec i ely, wha is e y disad an ageous. Howe e ,
applica ion o he a c angen o mulas ((7)-(8), (12)-(13), and
(16))ensu esacons an anda hesame ime hehighes alue
o he sensi i i y.
As a as he axial il 𝜑is conce ned, sensi i i y 𝑘01 e e s
o 𝜑02,𝑘02 e e s o 𝜑01,and𝑘03 e e s o 𝜑03.
5.2. Sensi i i y Rela ed o Angle 𝛾.On he basis o (1), he
ela ed sensi i i y can be de e mined as
𝑘21 =𝑔cos 𝛼√(cos2𝛼−sin2𝛽)
sin2𝛼⋅sin2𝛽+cos2𝛼⋅cos2𝛽.(20)
The sensi i i y 𝑘21 is exp essed on he plo in g/ ad; hus,
i is a iable o e he ange o ca. ⟨0;10⟩m/s2 ad. I should
be no ed ha no e e y combina ion o angles 𝛼and 𝛽is
possible, since he angles a e in e dependen .
In he case o (9), he ela ed sensi i i y can be exp essed
by he ollowing o mula:
𝑘23 =𝑔cos 𝛼(21)
which is also a iable o e he ange o ⟨0;10⟩m/s2 ad, his
ime exac ly as 𝑘01. Bo h o mulas a e s ongly nonlinea .
Howe e , sensi i i y 𝑘23 is a leas equal o 𝑘21 (o en highe ).
Plo o (20) is illus a ed in Figu e 3.
In nume ous applica ions, accu acy o il measu emen s
(he e ep esen ed by he sensi i i y) is an impo an issue
(e en hough he e a e some i ial cases when his p oblem
is o no conce n). Then, he mos con enien ma hema ical
ela ionship should be used, aking in o accoun , i s o
all, he equi ed accu acy and hen simplici y o he ela ed
compu a ions ( hus, he esul an speed and cos o he
measu emen uni ). Mo eo e , i mus no be o e looked ha
he highe he accu acy o he measu emen s he highe he
numbe o he sensi i e axes o he accele ome e .
To conclude, i can be s a ed ha in mos o he cases he
bes solu ion is o use a c angen o mulas, ha is, (7)–(9),
(12)-(13), and (16), since hey ea u e he highes sensi i i y,
p o ided ha he numbe o he a ailable sensi i e axes
is su icien . Howe e , he e a e also cases when he o he
o mulas a e p e e ed, as minu ely discussed in [20].
0
50
Be a (deg)
0
50
Alpha (deg)
−50 0
−50
0.5
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k21
Figu e 3: Va iabili y o he sensi i i y 𝑘21 exp essed by (20).
I is wo h men ioning ha he sensi i e axes need no
o be o hogonal. O he spa ial con igu a ions a e accep -
able, and some imes e en mo e ad an ageous. Then, highe
sensi i i y can be ob ained o e some angula egions a
he cos o i s lowe alue o e he o he egions. Ano he
disad an age is he ac ha he ela ed compu a ionsa e hen
mo e complica ed, as b ie ly discussed in [30].
6. Accele ome e Selec ion
The equi ed numbe o he sensi i e axes esul s om ype o
il measu emen , he measu emen ange, and he employed
ma hema ical o mula. Type o he selec ed accele ome e
mus comply wi h his numbe ( he numbe o i s sensi i e
axes may be al e na i ely highe ).
While3sensi i eaxesa e equi ed,ins eado employing
1 iaxial o 1 mul iaxial accele ome e , i may be conside ed
o apply 3 uniaxial o 2 biaxial accele ome e s, o 1 uniaxial
and 1 biaxial accele ome e . Such app oach is mo e labo ious
and expensi e; howe e i p o ides an oppo uni y o build
a mo e accu a e il senso . This concep is accep ed by
such companies as Mic oS ain Inc. o Sen e a Technol-
ogy Co p., whose senso s ha e he sensi i e axes o he
cons i uen accele ome e s p ecisely calib a ed and aligned
(o ha e he ele an misalignmen s compensa ed o )
[39].
Howe e , e en hough some ime ago iaxial MEMS
accele ome e s we e no comme cially a ailable, nowadays
i is qui e he opposi e: some manu ac u e s o e only
iaxial de ices, ha ing abandoned p oduc ion o biaxial o
uniaxial e sions. Ye , some imes i is wo hwhile o apply
uniaxial o biaxial MEMS accele ome e s. Fo ins ance, he
alignmen e o o he sensi i e axes o biaxial accele ome e s
is some imes e y small, ha is, o 0.01∘[28],unlikein hecase
o iaxial accele ome e s, whose z-axis may be poo ly aligned
and addi ionally ea u e wo se accu acy due o highe noise,
wha esul s om hei mechanical s uc u e de e mined by
he ab ica ion p ocess ( his sho coming is e iden especially
in he case o su ace mic omachining), which in ac is semi-
h ee-dimensional [40].