From 62f9819a6697e40543310b7671fbc414ff72577a Mon Sep 17 00:00:00 2001 From: a2nr Date: Tue, 19 Nov 2019 14:02:10 +0700 Subject: [PATCH] Tambah sub bab kendali robot --- BAB4/bab4.tex | 324 ++++++++++++++++++++++++++++++++++++-------- thesis (copy 1).pdf | Bin 0 -> 671741 bytes thesis.bbl | 29 ++++ thesis.blg | 68 +++++----- thesis.run.xml | 4 +- 5 files changed, 331 insertions(+), 94 deletions(-) create mode 100644 thesis (copy 1).pdf diff --git a/BAB4/bab4.tex b/BAB4/bab4.tex index e2c0d7f..e82a38f 100644 --- a/BAB4/bab4.tex +++ b/BAB4/bab4.tex @@ -1,70 +1,280 @@ -%-----------------------------------------------------------------------------% \chapter{\babEmpat} -%-----------------------------------------------------------------------------% -\todo{tambahkan kata-kata pengantar bab 1 disini} -%-----------------------------------------------------------------------------% -\section{thesis.tex} -%-----------------------------------------------------------------------------% -Berkas ini berisi seluruh berkas Latex yang dibaca, jadi bisa dikatakan sebagai -berkas utama. Dari berkas ini kita dapat mengatur bab apa saja yang ingin -kita tampilkan dalam dokumen. +\section{Strategi Kendali Multi Robot} + +Analisa akan dilakukan dalam beberapa bagian agar mudah dipahami dan diterapkan. +Analisa tersebut adalah mengenai kendali dari model dinamika robot dan kendali formasi, +dan mengenai metode percobaan akan dibahas secara matematis, simulasi, dan HIL.\ + +\subsection{Kendali Robot} +Pada kendali robot akan dibahas mengenai analisis kendali robot menggunakan +state-space feedback. Kendali robot ini adalah kendali tahap akhir dari kendali ke- +seluruhan. Dapat diperhatikan pada persamaan~\eqref{eq:ss-formasi}, sebagai kendali tahap awal, +bahwa state yang digunakan adalah koordinat. Maka koordinat tersebut akan men- +jadi set point bagi robot. Variable yang dikendalikan pada kendali robot adalah +koordinat robot dari kondisi inisial. Koordinat disini adalah koordinat state pada +persamaan~\eqref{eq:ss1}. Pada sub bab ini akan didefinisi mengenai kriteria pencapaian +set point dan membahas parameter kendali state-space feedback agar mencapai kri- +teria yang diinginkan. + +\subsubsection{State Feedback} +\todo{Gambar grafik state space feedback} +Pada persamaan~\eqref{eq:ss1} diketahui bahwa state memiliki dimensi $6 \times 1$. Dimensi +tersebut tidak menunjukan sistem memiliki orde 6. Apabila diperhatikan orde +dari sistem adalah orde 2. Dengan membaginya kedalam 3 persamaan state-space +akan lebih mudah dalam analisis parameter kendalinya. Berikut adalah persamaan + +\begin{align} + \begin{bmatrix}\dot{x}_p \\ \ddot{x}_r \end{bmatrix} &= + \begin{bmatrix}0 & A_{14} \\ 0 & A_{44} \end{bmatrix} + \begin{bmatrix}{x}_p \\ \dot{x}_r\end{bmatrix} + + \begin{bmatrix}0 & 0& 0 \\ B_{11} & B_{12} & B_{13} \end{bmatrix} + \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + K_{44}sgn(\dot{x}_r) \\ + \begin{bmatrix}\dot{y}_p \\ \ddot{y}_r \end{bmatrix} &= + \begin{bmatrix}0 & A_{25} \\ 0 & A_{55} \end{bmatrix} + \begin{bmatrix}{y}_p \\ \dot{y}_r\end{bmatrix} + + \begin{bmatrix}0 & 0& 0 \\ B_{21} & B_{22} & B_{23} \end{bmatrix} + \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + K_{55}sgn(\dot{y}_r) \\ + \begin{bmatrix}\dot{\theta}_p \\ \ddot{\theta}_r\end{bmatrix} &= + \begin{bmatrix}0 & A_{34} \\ 0 & A_{66} \end{bmatrix} + \begin{bmatrix}{\theta}_p \\ \dot{\theta}_r\end{bmatrix} + + \begin{bmatrix}0 & 0& 0 \\ B_{31} & B_{32} & B_{33} \end{bmatrix} + \begin{bmatrix}u_1 \\ u_2 \\ u_3 \end{bmatrix} + K_{66}sgn(\dot{\theta}_r) +\end{align} + +State feedback membutuhkan kembalian nilai state dari sistem dan mengka- +likanya dengan besaran tertentu agar nilai karakteristik sistem tetap dalam keadaan +stabil atau sesuai ketentuan. Secara umum, state tidak dapat diperoleh langsung dari +sistem. Kemampuan untuk memperoleh state dari sistem langsung disebut dengan +kemampuan Observablity. Apabila sebuh sistem tidak Observable, maka dalam +kendalinya dibutuhkan Observer. Dimana tugasnya adalah mengestimasi state pada +sistem dengan membandingkan keluaran dan masukan. Syarat untuk dapat diterap- +kan state feedback, sistem harus observable dan controlable. Berikut adalah rumus +untuk menguji apakah sistem bersifat controlable atau tidak (Dorf, dkk (2010)). + +\begin{align*} P_c = \begin{bmatrix} B & AB & A^2B & \dots & A^{n-1}B \end{bmatrix}\end{align*} +\begin{align} rank[P_c] = n \end{align} + +Apabila hasil dari $rank(P_c ) \neq n$ maka sistem tidak controlable. Sedangkan untuk +menguji observable dapat menggunakan rumus berikut. + +\begin{align*} P_o = \begin{bmatrix} C \\ CA & CA^2 & \vdots & CA^{n-1} \end{bmatrix}\end{align*} +\begin{align} rank[P_o] = n \end{align} + +Apabila sistem observable determinan dari matriks Observablity $P_o$ tidak nol. + +Menggunakan parameter robot oleh~\kutip{CORREIA20127} yang diterapkan pada per- +samaan~\eqref{eq:ss1}-\eqref{eq:ss2}, hasil pengujian controlable $rank[P_c] = 6$, maka dapat disim- +pulkan sistem robot controlable. Hasil pengujian observable $rank[P o] = 6$, maka +sistem robot juga observable. Karena sistem robot observable, maka dalam desain +kendali tidak diperlukan observer. + +\subsubsection{Desain Kendali} +\todo{Kriteria} +Kriteria didefinisi menggunakan analisis sistem orde dua pada domain waktu. +Berikut adalah transfer fungsi tertutup dari sistem orde dua (\kutip{Richard2010}). +\begin{align} + Y(s) = \frac{\omega_n^2}{s^2+2\zeta\omega_n+\omega_n^2} R(s) +\end{align} +Dengan input $R(s) = 1/s$ sebagai unit impulse, maka didapat persamaan keluaran sistem dalam domain waktu. +\begin{align} + y(t) = 1 - \frac{1}{\beta}\exp^{-\zeta\omega_n t}sin(\omega_n\beta t + \theta) +\end{align} +dimana $\beta = \sqrt{1-\zeta^2}$, $\theta = \cos^{-1}\zeta$, dan $0<\zeta<1$. +Dari persamaan domain waktu tersebut didapat 4 kriteria dalam sistem, +yaitu \textit{satling time} ($T_s$), Prosentase \textit{overshoot} ($P.O$), +\textit{peek time} ($T_p$), dan \textit{Transient Time} ($T_{r1}$). -%-----------------------------------------------------------------------------% -\section{laporan\_setting.tex} -%-----------------------------------------------------------------------------% -Berkas ini berguna untuk mempermudah pembuatan beberapa template standar. -Anda diminta untuk menuliskan judul laporan, nama, npm, dan hal-hal lain yang -dibutuhkan untuk pembuatan template. + +\todo{Parameter K} +\todo{Parameter N} -%-----------------------------------------------------------------------------% -\section{istilah.tex} -%-----------------------------------------------------------------------------% -Berkas istilah digunakan untuk mencatat istilah-istilah yang digunakan. -Fungsinya hanya untuk memudahkan penulisan. -Pada beberapa kasus, ada kata-kata yang harus selalu muncul dengan tercetak -miring atau tercetak tebal. -Dengan menjadikan kata-kata tersebut sebagai sebuah perintah \latex~tentu akan -mempercepat dan mempermudah pengerjaan laporan. +\subsection{Kendali Formasi Multi Robot} +Pada sub bab~\ref{subbab:KendaliFormasi} dijabarkan bagaimana kendali formasi menggunakan +kendali-PI dan menghasilkan persamaan~\eqref{eq:ss-formasi}. +Persamaan tersebut adalah persamaan \textit{state-space} kendali formasi. +Apabila diperhatikan \textit{state} yang digunakan adalah koordinat relatif dari robot. +Akan tetapi dalam batasanya, robot hanya bisa mengetahui nilai jarak dari robot lain. +Dengan kata lain, yang dibutuhkan dalam metode kendali formasi adalah jarak dalam bentuk koordinat, +$x \in \mathbb{R}^2$. Sedangkan dalam kenyataanya yang diketahui adalah jarak, $r \in \mathbb{R}$. +Apabila hanya variable jarak tersebut sebagai acuan kendali, maka robot tidak mengerti kearah mana +harusnya robot itu bergerak untuk meminimalisasi error jaraknya. +\subsubsection{Strategi Penentuan Koordinat Tetangga} +Penentuan koordinat tentangga dapat ditemukan dengang mengubah koordinat polar menjadi koordinat kartesian. +Koordinat polar membutuhkan panjang, $d_a$, dan sudut, $\alpha$. +Variable $d_a$ dapat diperoleh dari sensor, akan tetapi sudu $\alpha$ tidak bisa dideteksi secara langsung oleh sensor. +Dengan menggunakan \textit{cosinus} pada segitiga dimungkinkan untuk mendapatkan sudut tersebut. -%-----------------------------------------------------------------------------% -\section{hype.indonesia.tex} -%-----------------------------------------------------------------------------% -Berkas ini berisi cara pemenggalan beberapa kata dalam bahasa Indonesia. -\latex~memiliki algoritma untuk memenggal kata-kata sendiri, namun untuk -beberapa kasus algoritma ini memenggal dengan cara yang salah. -Untuk memperbaiki pemenggalan yang salah inilah cara pemenggalan yang benar -ditulis dalam berkas hype.indonesia.tex. +\begin{figure} + \centering + \includegraphics[scale=.5]{BAB3/img/estimate_coordinate.png} + \caption{Strategi Penentuan Koordinat} + \label{fig:strategiPenentuanKoordinat} +\end{figure} +Dapat diperhatikan pada gambar~\ref{fig:strategiPenentuanKoordinat} untuk gambaran strateginya. +Robot $B \in \tetangga_A$, adalah tetangga dari robot $A$. +Pertama-tama, sebelum robot $A$ bergerak, disimpan terlebih dahulu nilai $d_a$, +atau dinotasikan dengan $d_a[k]$ sebagai jarak sebelum bergerak. +Lalu robot $A$ berjalan secara random kesegala arah dengan jarak $l_a$. +Disimpan kembali nilai jara $d_a$, atau dinotasikan dengan $d_a[k+1]$. +Setalah itu dapat ditentukan sudut $\alpha[k+1]$ +\begin{align} + \alpha[k+1] = cos^{-1}\Bigg[ \frac{l_a^2 + d[k+1]^2 -d_a[k]^2}{2d_a[k+1]l_a} \Bigg]. +\end{align} +Sebelum $\alpha[k+1]$ digunakan, jarak $d_a[k+1]$ dan $d_a[k]$ berpengaruh dalam penentuan koordinat. +Sehingga diperlukan sedikit algoritma +\begin{align} + \alpha_i= + \begin{cases} + \alpha[k+1] & ,d_a[k+1] > d_a[k] \\ + 180-\alpha[k+1] & ,d_a[k+1] < d_a[k] + \end{cases}.\label{eq:init_relatif_koordinat} +\end{align} -%-----------------------------------------------------------------------------% -\section{pustaka.tex} -%-----------------------------------------------------------------------------% -Berkas pustaka.tex berisi seluruh daftar referensi yang digunakan dalam -laporan. -Anda bisa membuat model daftar referensi lain dengan menggunakan bibtex. -Untuk mempelajari bibtex lebih lanjut, silahkan buka -\url{http://www.bibtex.org/Format}. -Untuk merujuk pada salah satu referensi yang ada, gunakan perintah \bslash -cite, e.g. \bslash cite\{latex.intro\} yang akan akan memunculkan -\cite{latex.intro} +Strategi pada gambar~\ref{fig:strategiPenentuanKoordinat} hanya berlaku apabila target ukur berhenti. Apabila dinotasikan koordinat $x_B^A$ adalah koordinat relatif robot $B$ terhadap $A$, +maka $\dot{x}_B^A$ adalah notasi kecepatan koordinat dari robot B. +Dengan menggunakan persamaan~\eqref{eq:kinematika_robot} untuk menyelesaikan koordinat dalam +keadaan robot $B$ bergerak, yaitu mengirimkan informasi kecepatan koordinatnya +ke robot $A$. Lalu robot $A$ dapat mengkalkulasi koordinat relatif dengan persamaan berikut +\begin{align} + \alpha[k+1] & = \alpha[k]+tan^{-1} \Big[ \frac{\dot{x}_B^A}{\dot{y}_B^A} \Big] +\end{align} +dimana kondisi inisial adalah $\alpha[k] = \alpha_i$ diperoleh dari hasil strategi pada persamaan~\eqref{eq:init_relatif_koordinat}. +Dengan memanfaatkan kedua strategi tersebut dapat digunakan untuk +mengkalkulasi koordinat robot $B$ relatif terhadap robot $A$ +\begin{align} + x_B^A = \begin{bmatrix} + x_B = d_a[k].\cos \alpha[k] \\ + y_B = d_a[k].\sin \alpha[k] + \end{bmatrix} +\end{align} +Dalam strategi ini akan terjadi ketidak akuratan dalam pengukuran apabila target ukur +berada pada sudut $90^\circ$. +Akan tetapi, \kutip{Cao2007} sudah menjelaskan mengenai kriteria posisi agent ketika dalam kondisi inisial. +Yaitu semua agent tidak berada pada kondisi sejajar secara koordinat global pada kondisi inisial. +\section{Kestabilan Perangkat Percobaan} +Sub bab ini akan dibahas mengenai prangkat penunjang sebagai pembatu dalam menyelesaikan penelitian. +Sebagai langkah awal pengembangan, metode yang digunakan adalah \textit{Hardware-In Loop}. +\begin{figure} + \centering + \begin{subfigure}[t]{.4\textwidth} + \includegraphics[scale=.5]{BAB3/img/hil_graph.png} + \caption{} + \label{fig:hil_graph} + \end{subfigure} + \begin{subfigure}[t]{.4\textwidth} + \includegraphics[scale=.5]{BAB3/img/hil_graph_1.png} + \caption{} + \label{fig:hil_graph_1} + \end{subfigure} + \caption{(a)Grafik Hardware-in-the-loop (\kutip{Jim1999}). (b) HIL Kendali Multi-Robot. } +\end{figure} +\textit{Hardware-in-the-loop} (HIL) adalah metode untuk pengembangan prangkat kendali dengan memanfaatkan model sebagai objek kendalinya. Seperti pada gambar~\ref{fig:hil_graph}, +bahwa HIL terdiri dari dua prangkat, yaitu prangkat untuk menjalankan objek kendali atau dapat +disebut sebagai model/plant dan prangkat sistem kontrolnya, dalam kasus ini sistem kontrol menggunakan sistem tertanam (\textit{embedded system}). +Metode HIL, banyak digunakan oleh peneliti dalam proses pengembangan dengan pertimbangan efisiensi terhadap berbagai hal. +Seperti yang digunakan oleh~\kutip{Irwanto2018}, mengembangkan kendali UAV menggunakan HIL; +dan \kutip{QUESADA2019275}, mengembangkan prangkat pankreas buatan yang digunakan untuk mengendalikan kadar gula pada pengidap diabetes. -%-----------------------------------------------------------------------------% -\section{bab[1 - 6].tex} -%-----------------------------------------------------------------------------% -Berkas ini berisi isi laporan yang Anda tulis. -Setiap nama berkas e.g. bab1.tex merepresentasikan bab dimana tulisan tersebut -akan muncul. -Sebagai contoh, kode dimana tulisan ini dibaut berada dalam berkas dengan nama -bab4.tex. -Ada enam buah berkas yang telah disiapkan untuk mengakomodir enam bab dari -laporan Anda, diluar bab kesimpulan dan saran. -Jika Anda tidak membutuhkan sebanyak itu, silahkan hapus kode dalam berkas -thesis.tex yang memasukan berkas \latex~yang tidak dibutuhkan; contohnya -perintah \bslash include\{bab6.tex\} merupakan kode untuk memasukan berkas -bab6.tex kedalam laporan. +Pada penelitian ini akan digunakan \textit{microcontroller}(MCU) STM32F466 sebagai prangkat kendalinya. +MCU tersebut ber-arsitektur ARM Cortex-M4 dengan clock 180MHz, menampung ukuran program sampai 256K didalam memori Flash, serta fitur komunikasi standart MCU dengan lengkap. +\textit{Platform Library} yang digunakan dalam pembuatan aplikasi didalamnya adalah \textit{Mbed}, +yang menyediakan berbagai banyak fungsi yang lengkap dan mudah untuk berinteraksi dengan fitur-fitur MCU. \textit{Mbed} juga menyediakan fungsi untuk mengaplikasikan RTOS (Real-time Operating System) dengan mudah dan terdokumentasi secara jelas didalam lamannya. +Pada prangkat PC akan dikembangkan program berbasis \textit{Python} yang akan +menjalankan simulasi model dan berkomunikasi dengan MCU secara \textit{real-time}. +Program \textit{Python} akan menjalankan model pada persamaan~\eqref{eq:ss1}-\eqref{eq:ss2} +dengan metode yang dijabarkan pada sub bab~\ref{bab:solusi_ODE}. +Dapat diperhatikan pada gambar~\ref{fig:hil_graph_1}, pada HIL untuk kendali multi robot akan +menggunakan tiga kendali untuk mempresentasikan tiga robot. +Setiap prangkat pengendali akan saling terhubung satu sama lain dan semua prangkat pengendali terhubung dengan prangkat PC. +Komunikasi antar prangkat pengendali akan digunakan untuk pertukaran informasi. +Sedangkan komunikasi dengan PC akan mempresentasikan aktuator dan sensor untuk setiap prangkat +kendali. PC akan merekam setiap keluaran dari model dan masukan dari setiap prangkat kendali +sebagai tampilan pergerakan robotnya. +\subsection{Kestabilan Model} +Pada persamaan~\eqref{eq:disstab} apabila model dikalkulasi akan bergantung dengan besarnya \textit{step size}, $h$. +Oleh karena itu, setelah persamaan~\eqref{eq:ss1}-\eqref{eq:ss2} dilakukan parameterisasi harus dilakukan penentuan \textit{step size} agar model tersebut stabil dalam mensimulasikan modelnya. +Penentuan \textit{step size} harus berdasarkan kriteria kestabilan pada gamabar~\ref{fig:explicit_euler}. + +Apabila didefinisi ulang \textit{state} pada persamaan~\eqref{eq:ss1}-\eqref{eq:ss2} dengan +$x(t) = \begin{bmatrix}\dot{x}_r & \dot{y}_r & \dot{\theta}_r \end{bmatrix}^T$, +maka akan lebih mudah untuk menghitung kestabilan dari matriks $A \in \mathbb{R}^{3 \times 3}$. +Dengan menggunakan parameter dari penelitian oleh \kutip{CORREIA20127}, maka akan diperoleh matriks $A, B, K,$ dan $C$. +\begin{align*} + A & = \begin{bmatrix} + -6.69666 & 0.00000 & 0.00000 \\ + 0.00000 & -6.71000 & 0.00000 \\ + 0.00000 & 0.00000 & -4.04200 \\ + \end{bmatrix} ; \quad + B = \begin{bmatrix} + 0.00000 & 0.57735 & -0.57735 \\ + -0.66667 & 0.33333 & 0.33333 \\ + 4.00000 & 4.00000 & 4.00000 \\ + \end{bmatrix} ; \\ + K & = \begin{bmatrix} + -1.46667 & 0.00000 & 0.00000 \\ + 0.00000 & -1.00000 & 0.00000 \\ + 0.00000 & 0.00000 & -0.06600 \\ + \end{bmatrix}; \quad + C = \begin{bmatrix} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 + \end{bmatrix}. +\end{align*} + +Dengan menggunakan pendekatan pada persamaan~\eqref{eq:desdotode1} untuk persamaan~\eqref{eq:ss1} maka diperoleh bentuk diskretnya +\begin{align} + x[k+1] & = (I + A.h).x[k] + B.h.u[k] + K.h.sgn(x[k]). \\ +\end{align} +Pengali $sgn(.)$ bersifat penambah dari sistem, maka dalam penentuan kestabilan ini akan dianggap penambah dari matriks sistem. +\begin{align} + x[k+1] & = (I + (A+K).h).x[k] + B.h.u[k]. \\ +\end{align} +Kriteria kestabilan akan bergantung dari hasil penentuan $h$ pada $I+(A+K)h~=~\Lambda$. +Untuk semua nilai $\lambda$ pada matriks $\Lambda$ harus memenuhi kriteris $\lambda \leq 1$. +Dimungkinkan akan mengalami kebingungan ketika menentukan besar $h$, +akan tetapi nantinya persamaan ini akan diterapkan dan diselesaikan oleh komputer. +Alangkah baiknya apabila diidentifikasi terlebih dahulu konsumsi waktu yang dibutuhkan untuk menyelesaikan +satu iterasi dari persamaan tersebut. +Setelah dilakukan identifikasi, waktu yang dibutuhkan untuk satu kali iterasi berkisar $0.001$ ms (Pembulatan). +Sehingga penentuan \textit{step size} sebesar $0.1$ ms sangat dimungkinkan, dengan pertimbangan +sisa dari waktu yang digunakan kalkulasi dapat digunakan untuk waktu \textit{idle} dan menjalankan program yang lain. Berikut adalah matriks $\Lambda$ setelah dikalkulasi menggunakan $h=0.1$ +\begin{align*} + \Lambda = \begin{bmatrix} + 0.18367 & 0.00000 & 0.00000 \\ + 0.00000 & 0.22900 & 0.00000 \\ + 0.00000 & 0.00000 & 0.58920 \\ + \end{bmatrix}. +\end{align*} +Terbukti bahwa semua nilai item didalam matriks kurang dari sama dengan satu. +Sehingga menggunakan algoritma \textit{Expilicit Euler} sudah cukup untuk menjalankan model robot \textit{omni 3-wheel} sebagai model \textit{holonomic} yang akan digunakan untuk percobaan kendali multi robot. +Hasil plot dari simulasi model dapat dilihat pada gambar~\ref{fig:sim_model}. +\begin{figure} + \centering + \begin{subfigure}[t]{.6\textwidth} + \includegraphics[scale=.4]{BAB3/img/speedRobot_-6_3_3.png} + \caption{} + \end{subfigure} + \begin{subfigure}[t]{.6\textwidth} + \includegraphics[scale=.4]{BAB3/img/speedRobot_0_6_-6.png} + \caption{} + \end{subfigure} + \begin{subfigure}[t]{.6\textwidth} + \includegraphics[scale=.4]{BAB3/img/speedRobot_6_6_6.png} + \caption{} + \end{subfigure} + \caption{(a)$w_1=-6; w_2=3; w_3=3$. (b) $w_1=0; w_2=6; w_3=-6$ (c) $w_1=6; w_2=6; w_3=6$} + \label{fig:sim_model} +\end{figure} +\subsection{Rencana Hardware-in-Loop} +\todo{kutip hasil HIL yang sudah ada lalu gabungkan model dan kendali jadi satu secara sederhana} + +\subsection{Rencana Uji Lapangan} +\todo{Membahas mengenai cara pengambilan data penerapan pada robot aslinya} diff --git a/thesis (copy 1).pdf b/thesis (copy 1).pdf new file mode 100644 index 0000000000000000000000000000000000000000..b141b486acce1afcaacb649b21f82aa7f24829be GIT binary patch literal 671741 zcmagFWl&vRuq}!N_uv-XZQ~9>f+}+(>0tA=f8r?w$V3-=i`HCQgX93B4+xmZ?5QQ z1q1g)M(LF@FcjwbU`u%DB%sku<^Teas5A4G<1^4Cr znlK1MJd_j!uZk{=`P0_$0>4SfP?9D(iujxGGwc`YC-6r7izs$)aDwyeE+^RB2wyPy z;P)FA5Ll}3mJpS|WXOI&->Fl?`clw@TY5y6L|}eyYO@0c#W;< z4Dkt72^RD$UNGXK><^Vvt+x$#v@~TNlj&p4ZY8rrX8@|Y|68HT+os2LfY zW5|)a>B=Hw{U|^4fubpGv|y;vW>@Hju0z;>t+cwm?_qQczeDT#2eZxG_%rG1lgs^f 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+cite$ -- 24 +duplicate$ -- 7545 +empty$ -- 6627 +format.name$ -- 836 +if$ -- 14218 int.to.chr$ -- 0 -int.to.str$ -- 24 +int.to.str$ -- 26 missing$ -- 0 -newline$ -- 466 -num.names$ -- 477 -pop$ -- 3797 +newline$ -- 495 +num.names$ -- 518 +pop$ -- 4138 preamble$ -- 1 -purify$ -- 201 +purify$ -- 216 quote$ -- 0 -skip$ -- 1908 +skip$ -- 2079 stack$ -- 0 -substring$ -- 6887 -swap$ -- 1991 -text.length$ -- 334 -text.prefix$ -- 11 +substring$ -- 7256 +swap$ -- 2143 +text.length$ -- 366 +text.prefix$ -- 12 top$ -- 1 -type$ -- 395 +type$ -- 429 warning$ -- 0 -while$ -- 834 +while$ -- 895 width$ -- 0 -write$ -- 455 -(There were 2 warnings) +write$ -- 483 +(There was 1 warning) diff --git a/thesis.run.xml b/thesis.run.xml index 08eee91..e9a7229 100644 --- a/thesis.run.xml +++ b/thesis.run.xml @@ -41,7 +41,7 @@ > ]> - + latex thesis.aux @@ -63,7 +63,7 @@ english.lbx - + bibtex bibtex

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