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! Undefined control sequence.
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l.2 ...40\000p\000h\000y\000s\000i\000q\000u\000e}{}
                                                  % 2
The control sequence at the end of the top line
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spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
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l.3 ...e\000n\000\040\001\123\000u\000v\000r\000e}{}
                                                  % 3
The control sequence at the end of the top line
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spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.4 ...40\000d\000'\000e\000r\000r\000e\000u\000r}{}
                                                  % 4
The control sequence at the end of the top line
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and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.5 ...i\000t\000i\000a\000l\000e\000s}{section.0.4}
                                                  % 5
The control sequence at the end of the top line
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spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.6 ...00\351\000r\000i\000q\000u\000e}{section.0.4}
                                                  % 6
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
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spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.7 ...1\000r\000a\000t\000i\000o\000n}{section.0.4}
                                                  % 7
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.8 ...000\040\000e\000r\000r\000e\000u\000r\000s}{}
                                                  % 8
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.9 ...i\000s\000a\000t\000i\000o\000n}{section.0.5}
                                                  % 9
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.10 ...\000l\000a\000t\000e\000u\000r}{section.0.5}
                                                  % 10
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.11 ...\000p\000t\000e\000u\000r\000s}{section.0.5}
                                                  % 11
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@OutlineName #1#2#3#4->\pdfoutline 
                                      goto name{#2}count#3{#4}
l.12 ...o\000n\000c\000l\000u\000s\000i\000o\000n}{}
                                                  % 12
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

)
\@outlinefile=\write11

\openout11 = naviguer-presque-sans-gps-grace-a-la-navigation-inertielle.out

! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.11 \begin{document}
                   
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> \pdfpagewidth 
              
l.11 \begin{document}
                   
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing number, treated as zero.
<to be read again> 
\protect 
l.11 \begin{document}
                   
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
\protect 
l.11 \begin{document}
                   
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

! Undefined control sequence.
<argument> \pdfpageheight 
               
l.11 \begin{document}
                   
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Missing number, treated as zero.
<to be read again> 
\let 
l.11 \begin{document}
                   
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
\let 
l.11 \begin{document}
                   
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

Package hyperref Info: Link coloring ON on input line 11.
(./naviguer-presque-sans-gps-grace-a-la-navigation-inertielle.glsdefs)
\GPT@outputbox=\box122

Overfull \hbox (203.51234pt too wide) in paragraph at lines 1--12
[]\TU/lmr/m/n/12 goto na-me-sec-tion.0.1count0\376\377\0001\000\040\000I\000n\0
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 []


Overfull \hbox (341.06015pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sec-tion.0.2count0\376\377\0002\000\040\000P\000r\000i\000
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 []


Overfull \hbox (265.77255pt too wide) in paragraph at lines 1--12
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 []


Overfull \hbox (309.42572pt too wide) in paragraph at lines 1--12
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Overfull \hbox (974.61903pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sub-sec-tion.0.4.1count0\376\377\0004\000.\0001\000\040\00
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40\000c\000o\000n\000d\000i\000t\000i\000o\000n\000s\000\040\000i\000n\000i\000
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Overfull \hbox (847.35214pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sub-sec-tion.0.4.2count0\376\377\0004\000.\0002\000\040\00
0E\000r\000r\000e\000u\000r\000\040\000d\000’\000i\000n\000t\000\351\000g\000
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00egoto
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Overfull \hbox (892.13426pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sub-sec-tion.0.4.3count0\376\377\0004\000.\0003\000\040\00
0E\000r\000r\000e\000u\000r\000\040\000d\000e\000\040\000m\000e\000s\000u\000r\
000e\000\040\000d\000’\000a\000c\000c\000\351\000l\000\351\000r\000a\000t\000
i\000o\000ngoto
 []


Overfull \hbox (497.57414pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sec-tion.0.5count3\376\377\0005\000\040\000M\000i\000t\000
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00e\000u\000r\000sgoto
 []


Overfull \hbox (1612.85834pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sub-sec-tion.0.5.1count0\376\377\0005\000.\0001\000\040\00
0E\000r\000r\000e\000u\000r\000\040\000d\000e\000\040\000p\000o\000s\000i\000t\
000i\000o\000n\000\040\000e\000t\000\040\000v\000i\000t\000e\000s\000s\000e\000
\040\000i\000n\000i\000t\000i\000a\000l\000e\000\040\000e\000t\000\040\000i\000
n\000i\000t\000i\000a\000l\000i\000s\000a\000t\000i\000o\000ngoto
 []


Overfull \hbox (1387.91281pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sub-sec-tion.0.5.2count0\376\377\0005\000.\0002\000\040\00
0E\000r\000r\000e\000u\000r\000\040\000d\000’\000i\000n\000t\000\351\000g\000
r\000a\000t\000i\000o\000n\000\040\000e\000t\000\040\000p\000u\000i\000s\000s\0
00a\000n\000c\000e\000\040\000d\000u\000\040\000c\000a\000l\000c\000u\000l\000a
\000t\000e\000u\000rgoto
 []


Overfull \hbox (1405.36462pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sub-sec-tion.0.5.3count0\376\377\0005\000.\0003\000\040\00
0E\000r\000r\000e\000u\000r\000\040\000d\000e\000\040\000m\000e\000s\000u\000r\
000e\000,\000\040\000f\000i\000l\000t\000r\000a\000g\000e\000\040\000e\000t\000
\040\000f\000u\000s\000i\000o\000n\000\040\000d\000e\000\040\000c\000a\000p\000
t\000e\000u\000r\000sgoto
 []


Overfull \hbox (151.84421pt too wide) in paragraph at lines 1--12
\TU/lmr/m/n/12 na-me-sec-tion.0.6count0\376\377\0006\000\040\000C\000o\000n\000
c\000l\000u\000s\000i\000o\000nnameDoc-
 []

(compiling luc: /var/lib/texmf/luatex-cache/generic/fonts/otl/lmromanslant12-reg
ular.luc)(load luc: /opt/zds/.texlive2016/texmf-var/luatex-cache/generic/fonts/o
tl/lmromanslant12-regular.luc)(compiling luc: /var/lib/texmf/luatex-cache/generi
c/fonts/otl/lmroman8-regular.luc)(load luc: /opt/zds/.texlive2016/texmf-var/luat
ex-cache/generic/fonts/otl/lmroman8-regular.luc)(compiling luc: /var/lib/texmf/l
uatex-cache/generic/fonts/otl/lmroman6-regular.luc)(load luc: /opt/zds/.texlive2
016/texmf-var/luatex-cache/generic/fonts/otl/lmroman6-regular.luc)
LaTeX Font Info:    Try loading font information for U+msa on input line 12.

(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
)
(/usr/share/texlive/texmf-dist/tex/latex/microtype/mt-msa.cfg
File: mt-msa.cfg 2006/02/04 v1.1 microtype config. file: AMS symbols (a) (RS)
)
LaTeX Font Info:    Try loading font information for U+msb on input line 12.

(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
)
(/usr/share/texlive/texmf-dist/tex/latex/microtype/mt-msb.cfg
File: mt-msb.cfg 2005/06/01 v1.0 microtype config. file: AMS symbols (b) (RS)
)
(/usr/share/texlive/texmf-dist/tex/latex/microtype/mt-cmr.cfg
File: mt-cmr.cfg 2013/05/19 v2.2 microtype config. file: Computer Modern Roman (
RS)
)
! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.12 \maketitle
             
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
<argument> ...MaketitleInfos \HyInfo@GenerateAddons \pdfinfo 
                                                  {/Author(\@pdfauthor )/Tit...
l.12 \maketitle
             
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

[0{/opt/zds/.texlive2016/texmf-var/fonts/map/pdftex/updmap/pdftex.map}


\openout9 = naviguer-presque-sans-gps-grace-a-la-navigation-inertielle.ist
]
*geometry* verbose mode - [ newgeometry ] result:
* driver: luatex
* paper: <default>
* layout: <same size as paper>
* layoutoffset:(h,v)=(0.0pt,0.0pt)
* modes: 
* h-part:(L,W,R)=(42.67912pt, 512.14969pt, 42.67912pt)
* v-part:(T,H,B)=(28.45274pt, 788.14145pt, 28.45274pt)
* \paperwidth=597.50793pt
* \paperheight=845.04694pt
* \textwidth=512.14969pt
* \textheight=788.14145pt
* \oddsidemargin=-29.59087pt
* \evensidemargin=-29.59087pt
* \topmargin=-83.69225pt
* \headheight=18.125pt
* \headsep=21.75pt
* \topskip=12.0pt
* \footskip=50.75pt
* \marginparwidth=49.79233pt
* \marginparsep=12.8401pt
* \columnsep=10.0pt
* \skip\footins=10.8pt plus 4.0pt minus 2.0pt
* \hoffset=0.0pt
* \voffset=0.0pt
* \mag=1000
* \@twocolumnfalse
* \@twosidefalse
* \@mparswitchfalse
* \@reversemarginfalse
* (1in=72.27pt=25.4mm, 1cm=28.453pt)


! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.12 \maketitle
             
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! You can't use `\prevdepth' in horizontal mode.
\caption@startanchor ...n@hypcapspace }\prevdepth 
                                                  \@tempdima \endgroup \else...
l.12 \maketitle
             
Sorry, but I'm not programmed to handle this case;
I'll just pretend that you didn''t ask for it.
If you're in the wrong mode, you might be able to
return to the right one by typing `I}' or `I$' or `I\par'.

! Missing number, treated as zero.
<to be read again> 
\endgroup 
l.12 \maketitle
             
A number should have been here; I inserted `0'.
(If you can't figure out why I needed to see a number,
look up `weird error' in the index to The TeXbook.)

! Illegal unit of measure (pt inserted).
<to be read again> 
\endgroup 
l.12 \maketitle
             
Dimensions can be in units of em, ex, in, pt, pc,
cm, mm, dd, cc, nd, nc, bp, or sp; but yours is a new one!
I'll assume that you meant to say pt, for printer's points.
To recover gracefully from this error, it's best to
delete the erroneous units; e.g., type `2' to delete
two letters. (See Chapter 27 of The TeXbook.)

[\directlua]:1: attempt to call global 'formatAuthors' (a nil value)
stack traceback:
	[\directlua]:1: in main chunk.
\maketitle ...hor }, \luastringO {\@authorlink })}
                                                   \end {minipage} & \vspace...
l.12 \maketitle
             
The lua interpreter ran into a problem, so the
remainder of this lua chunk will be ignored.


Package hyperref Warning: Suppressing link with empty target on input line 12.

Package epstopdf Info: Source file: </opt/zds/app/dist/licenses/copyright.svg>
(epstopdf)                    date: 2018-06-18 11:41:01
(epstopdf)                    size: 265 bytes
(epstopdf)             Output file: </opt/zds/app/dist/licenses/copyright-svg-co
nverted-to.pdf>
(epstopdf)             Command: <rsvg-convert -f pdf /opt/zds/app/dist/licenses/
copyright.svg -o /opt/zds/app/dist/licenses/copyright-svg-converted-to.pdf>
(epstopdf)             \includegraphics on input line 12.
system(rsvg-convert -f pdf /opt/zds/app/dist/licenses/copyright.svg -o /opt/zds/
app/dist/licenses/copyright-svg-converted-to.pdf) executed.
Package epstopdf Info: Result file: </opt/zds/app/dist/licenses/copyright-svg-co
nverted-to.pdf>.

! Package luatex.def Error: File `/opt/zds/app/dist/licenses/copyright-svg-conv
erted-to.pdf' not found.

See the luatex.def package documentation for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.12 \maketitle
             
Using draft setting for this image.
Try typing  <return>  to proceed.
If that doesn't work, type  X <return>  to quit.

Package microtype Info: Loading generic settings for font family
(microtype)             `SourceCodePro' (encoding: TU).
(microtype)             For optimal results, create family-specific settings.
(microtype)             See the microtype manual for details.

Overfull \hbox (22.76183pt too wide) in paragraph at lines 12--12
 [] 
 []

<default_logo.png, id=36, 240.9pt x 60.225pt>
File: default_logo.png Graphic file (type png)
 <use default_logo.png>
Package luatex.def Info: default_logo.png used on input line 12.
(luatex.def)             Requested size: 240.8994pt x 60.22484pt.
(compiling luc: /var/lib/texmf/luatex-cache/generic/fonts/otl/lmroman17-regular.
luc)(load luc: /opt/zds/.texlive2016/texmf-var/luatex-cache/generic/fonts/otl/lm
roman17-regular.luc)
! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.12 \maketitle
             
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

[1


<./default_logo.png>]
! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.12 \maketitle
             
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

[2

][\directlua]:1: attempt to call global 'makeAuthorsPage' (a nil value)
stack traceback:
	[\directlua]:1: in main chunk.
\maketitle ...e}{1} \directlua {makeAuthorsPage()}
                                                   \newpage 
l.12 \maketitle
             
The lua interpreter ran into a problem, so the
remainder of this lua chunk will be ignored.

! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.13 \tableofcontents
                   
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

Package microtype Info: Loading generic settings for font family
(microtype)             `SourceSansPro' (encoding: TU).
(microtype)             For optimal results, create family-specific settings.
(microtype)             See the microtype manual for details.
Package tocbasic Info: character protrusion at toc deactivated on input line 13.

(./naviguer-presque-sans-gps-grace-a-la-navigation-inertielle.toc)
\tf@toc=\write12

\openout12 = naviguer-presque-sans-gps-grace-a-la-navigation-inertielle.toc

! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.15 \levelOneTitle{Introduction}
                               
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

\@gls@deffile=\write13

\openout13 = naviguer-presque-sans-gps-grace-a-la-navigation-inertielle.glsdefs
! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.30 \levelOneTitle{Principe physique}
                                    
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
                            name{#1}#2\relax 
l.34 
   
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

[1


]
! Undefined control sequence.
l.50 a(t) = \deriv
                {v}{t}(t) $$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.


! LaTeX Error: Bad math environment delimiter.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.52 ...pression de la vitesse par intégration : \]
                                                  
Your command was ignored.
Type  I <command> <return>  to replace it with another command,
or  <return>  to continue without it.

! Undefined control sequence.
l.60 $$v(t) = \deriv
                  {p(t)}{t}$$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.68 $$p(t) = p(0) + \integ
                         {0}{t}{v(t)}{t}$$
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.76 $$p(t) = p(0) + \integ
                         {0}{t}{\left( v(0) + \integ{0}{t}{a(t)}{t} \right)}...
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

! Undefined control sequence.
l.76 ...) = p(0) + \integ{0}{t}{\left( v(0) + \integ
                                                  {0}{t}{a(t)}{t} \right)}{t}$$
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l.89 \item u
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l.90 \item u
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l.91 \item u
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\Hy@DestName #1#2->\pdfdest 
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spelling (e.g., `I\hbox'). Otherwise just continue,
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spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.


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[4]
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spelling (e.g., `I\hbox'). Otherwise just continue,
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\Hy@DestName #1#2->\pdfdest 
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and I'll forget about whatever was undefined.

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and I'll forget about whatever was undefined.

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and I'll forget about whatever was undefined.

[7</opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigation
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\Hy@DestName #1#2->\pdfdest 
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l.247 
    
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misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.

[8</opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigation
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! Undefined control sequence.
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[9]
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[10</opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigatio
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\Hy@DestName #1#2->\pdfdest 
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[11


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! Undefined control sequence.
\Hy@DestName #1#2->\pdfdest 
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[12


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latex.tex

\documentclass[small]{zmdocument}

\usepackage{blindtext}
\title{Naviguer (presque) sans GPS grâce à la navigation inertielle}
\author{Aabu}
\licence[/opt/zds/app/dist/licenses/copyright.svg]{Tous droits réservés}{}

\smileysPath{/opt/zds/app/dist/smileys}
\makeglossaries

\begin{document}
\maketitle
\tableofcontents

\levelOneTitle{Introduction}


De nos jours, il est facile de savoir où l’on est. Grâce au \abbr{GPS}{Global Positioning System} ou à des systèmes concurrents, connaître précisément sa position est devenu un jeu d’enfant. Les récepteurs \abbr{GPS}{Global Positioning System} sont d’ailleurs partout, que ce soit dans les téléphones, les voitures, les avions, ou encore les bateaux.



Pourtant, le \abbr{GPS}{Global Positioning System} ne répond pas à tous les besoins. Pour certaines applications, il est trop cher, trop peu précis, ou tout simplement impossible à mettre en œuvre. Par exemple, pour les sous-marins en plongée, il n’est pas possible de capter les signaux du \abbr{GPS}{Global Positioning System}. Sur un champ de bataille, le positionnement doit rester précis même en cas de brouillage. Sur les systèmes grand public, les contraintes de coût limitent encore la précision.



Quand le \abbr{GPS}{Global Positioning System} seul n’est pas une option viable, on recourt souvent à la \textbf{navigation inertielle}. Cette technique permet d'estimer le déplacement d'un objet avec seulement un capteur d'accélération et un chronomètre. Cet article en présente le principe de fonctionnement.



\levelOneTitle{Principe physique}


La navigation inertielle est rendue possible grâce aux les lois physiques régissant le mouvement des corps, car elles permettent de déterminer la position d'un objet en connaissant seulement :



\begin{itemize}
\item le point de départ,
\item la vitesse initiale,
\item l'accélération à chaque instant,
\item le temps écoulé depuis le départ.
\end{itemize}


Pour ce faire, on part de la définition de l'accélération. Il s'agit de la dérivée de la vitesse par rapport au temps :



\[ \newcommand{\integ}[4]{\int_{#1}^{#2} {#3} \mathrm{d}{#4}}
a(t) = \deriv{v}{t}(t) $$

On en déduit une expression de la vitesse par intégration : \]



La vitesse est, par définition, la dérivée de la position par rapport au temps :



$$v(t) = \deriv{p(t)}{t}$$



On procède alors à une deuxième intégration pour en déduire la position en fonction de la vitesse :



$$p(t) = p(0) + \integ{0}{t}{v(t)}{t}$$



Si on réunit tous les morceaux, on obtient :



$$p(t) = p(0) + \integ{0}{t}{\left( v(0) + \integ{0}{t}{a(t)}{t} \right)}{t}$$



En fin de compte, on obtient la position exprimée en fonction d'éléments connus ou calculables : la position et la vitesse initiale, l'accélération, le temps et des intégrales.



Illustrons ce principe physique par un exemple. Imaginez une voiture de sport sur une piste rectiligne, et qui enchaîne les cinq phases suivantes :



\begin{enumerate}
\item une phase sans accélération pendant 15 secondes,
\item une phase d'accélération pendant les 25 secondes suivantes,
\item une nouvelle phase sans accélération pendant les 10 secondes suivantes,
\item une phase de freinage pendant 12,5 secondes,
\item une nouvelle phase sans accélération à la fin.
\end{enumerate}


Avec les expressions précédentes, on peut déduire la vitesse, puis la position à partir de l'accélération. La figure ci-dessous montre le résultat qu'on obtient pour notre exemple.



\image{/opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigation-inertielle__building/extra_contents/images/B1qLYMr-X/S1xqLYGrZX.png}



\levelOneTitle{Mise en œuvre}


Les lois physiques vues ci-avant peuvent être simulées en temps réel par un calculateur, en utilisant des techniques d'intégration numérique.



Nous utilisons ici la \externalLink{méthode d'Euler}{https://zestedesavoir.com/tutoriels/528/simulez-des-systemes-physiques-avec-la-methode-deuler/}, qui est certainement la technique la plus simple. D'autres approches plus raffinées existent, mais elles ne sont pas utiles pour notre article.



La méthode d'Euler consiste à approximer les dérivées par le taux de variation entre deux instants suffisamment proches.



Todo : image approximation.



En appliquant cette approximation dans notre cas, on déduit deux récurrences : une pour mettre à jour la vitesse, et une pour mettre à jour la position.



Le calcul se déroule de la manière suivante. On part d'une estimation de position initiale $p_0$ et de vitesse initiale $v_0$. Ensuite, à chaque instant $t_k$ où l'on souhaite estimer la position, on commence par mettre à jour l'estimation de vitesse :



$$v_k = v_{k-1} + (t_k - t_{k-1}) v_{k-1}$$



Après, on met à jour l'estimation de position :



$$p_k = p_{k-1} + (t_k - t_{k-1}) v_k$$



Et on recommence à un nouvel instant en partant de l'estimation précédente.



Pour que la méthode d'Euler estime suffisamment bien la véritable intégrale, il faut que les instants $t_k$ soient suffisamment proches les uns des autres, c'est-à-dire que la fréquence de mise à jour soit suffisamment élevée.



Reprenons notre voiture de sport de la section précédente et estimons sa position  avec les calculs vus ci-avant. Le résultat est montré dans la figure ci-dessous. La réalité physique et l'estimation sont presque indiscernable !



\image{/opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigation-inertielle__building/extra_contents/images/B1qLYMr-X/BkW98KzrZX.png}



\levelOneTitle{Sources d'erreur}


La section précédente fait une démonstration de la navigation inertielle dans un cas idéalisé. En pratique, différents phénomènes introduisent des erreurs dans le processus d'estimation.



\levelTwoTitle{Erreur sur les conditions initiales}


Les conditions initiales, position et vitesse, sont souvent issues de mesures, entachées d'erreur plus ou moins importantes.



Par exemple, en mer, un bateau pourrait se repérer en utilisant les étoiles, mais les instruments de mesure n'étant pas parfaitement précis, la position ne serait connue que de manière approximative.



De même, toutes sortes d'instruments permettent de mesurer la vitesse avec une erreur variable.



L'erreur de position initiale créée un décalage entre l'estimation et la réalité, qui va perdurer indéfiniment. L'erreur de vitesse initiale, quant à elle, fait dériver l'estimation, en introduisant un décalage qui croît proportionnellement avec la temps, rendant à terme la l'estimation inutile.



\image{/opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigation-inertielle__building/extra_contents/images/B1qLYMr-X/BJzcUFzB-7.png}



\levelTwoTitle{Erreur d'intégration numérique}


La méthode d'intégration numérique introduit des erreurs qui grandissent à mesure que la fréquence de mise à jour diminue. Ce phénomène s'explique par le fait que l'approximation de la dérivée devient de moins en moins bonne quand la fréquence de la mise à jour diminue.



En pratique, les calculateurs modernes permettent une précision respectables. Cependant, si on attend suffisamment longtemps, cette erreur finira toujours par se manifester sensiblement.



L'exemple ci-dessous reprend le cas de la voiture de sport. Cette fois-ci, la fréquence de mise à jour est beaucoup plus faible, et crée des erreurs qui s'accumulent au cours du temps. Pour que le phénomène soit bien visible, la fréquence de mise à jour est choisie volontairement très basse.



\image{/opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigation-inertielle__building/extra_contents/images/B1qLYMr-X/rJQ9IYMH-7.png}



\levelTwoTitle{Erreur de mesure d'accélération}


L'erreur la plus gênante en pratique est sans doute celle sur la mesure d'accélération.



Les capteurs d'accélération présentent différents types d'erreurs liés à la constitution et à la chaîne de traitement du signal. On peut notamment citer le biais, le bruit de mesure, ou les non-linéarités.



Bien qu'on puisse les limiter, ces erreurs sont inévitables et entraînent une dérive à mesure que l'estimation prend en compte le bruit comme s'il s'agissait d'une véritable mesure.



L'exemple ci-dessous montre comment la dérive affecte l'estimateur de position de notre voiture. La voiture est immobile, mais un bruit entraîne la dérive de l'estimateur,que j'aime appeler le « voyage immobile ». :-)



\image{/opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigation-inertielle__building/extra_contents/images/B1qLYMr-X/rkE5UKfr-X.png}



Il existe encore d'autres erreurs spécifiques à certaines applications, qui ne seront pas traitées dans cet article.



\levelOneTitle{Mitigation des erreurs}


Il existe différentes techniques pour mitiger les erreurs.



\levelTwoTitle{Erreur de position et vitesse initiale et initialisation}


Pour mitiger l'erreur de position et de vitesse initiales, il est possible d'effectuer l'initialisation dans des conditions précises connues. Par exemple, pour un véhicule terrestre, il s'agirait de faire une mesure à l'arrêt (vitesse nulle) en un point précisément mesuré (erreur de position faible).



Ce n'est pas toujours possible, si on ne peut pas se mettre à l'arrêt ni savoir précisément où l'on est. C'est le cas notamment pour un bateau en pleine mer : le courant et le vent empêchent de mesurer la véritable vitesse, c'est-à-dire la vitesse par rapport au fond et la précision de la position initiale est limitée par les moyens du bord.



\levelTwoTitle{Erreur d'intégration et puissance du calculateur}


Nous avons vu dans la partie précédente que si on mettait à jour l'estimation trop lentement, l'erreur de position s'accumulait. Pour limiter ce type d'erreur, il suffit alors de procéder plus vite.



La performance du calculateur peut devenir bloquante. D'une part, les contraintes de coûts limitent la puissance des calculateurs dans les applications grands publics (dans un téléphone par exemple) ; d'autres part, plus les accélérations maximales du véhicule sont importantes et le positionnement nécessaire précis, plus la mise à jour doit se faire vite (dans un avion de chasse par exemple), ce qui peut devenir limitant pour du calcul en temps réel.



\levelTwoTitle{Erreur de mesure, filtrage et fusion de capteurs}


Erreur de mesure très gênante et cause de « voyage immobile ».



Possibilité de filtrer le bruit ou le biais. Différentes approches dont Kalman.



Possibilité de fusionner l'information de différents capteurs pour obtenir un résultat meilleur que chacun pris séparément



Exemple : mélanger \abbr{GPS}{Global Positioning System} et navigation inertielle pour se recaler.



\image{/opt/zds/data/contents-public/naviguer-presque-sans-gps-grace-a-la-navigation-inertielle__building/extra_contents/images/B1qLYMr-X/SyBqUtzHZm.png}



\levelOneTitle{Conclusion}


\externalLink{https://fr.wikipedia.org/wiki/Navigation\_inertielle}{https://fr.wikipedia.org/wiki/Navigation\_inertielle}



\externalLink{https://fr.wikipedia.org/wiki/Centrale\textit{à}inertie}{https://fr.wikipedia.org/wiki/Centrale\_à\_inertie}



\externalLink{https://fr.wikipedia.org/wiki/Calcul\textit{numérique}d'une\_intégrale}{https://fr.wikipedia.org/wiki/Calcul\_numérique\_d'une\_intégrale}



\externalLink{https://en.wikipedia.org/wiki/Kalman\_filter}{https://en.wikipedia.org/wiki/Kalman\_filter}



\textit{Illustration de la miniature : Carte du monde faite à partir des données de Natural Earth (projection: planisphère Hölzels centrée 10° est), par \externalLink{Ktrinko}{https://commons.wikimedia.org/wiki/User:Ktrinko}, sous licence \externalLink{CC BY-SA 3.0}{https://creativecommons.org/licenses/by-sa/3.0}.}


\end{document}