Selection of MEMS Accelerometers for Tilt Measurements

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].